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.

Arithmetick.

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.

CHAP. II. The Explication, and use of the Ta∣bles of LOGARITHMS.

SECT. I. The Explication of the Tables of the Lo∣garithms, and of parts proportional.

THE Logarithms, were first inven∣ted,* 1.13 found out and framed, by that never to be forgotten and thrice Honourable Lord, the Lord Nepeir: which Numbers, so found out and framed by his diligent industry he was pleased to call Logarithms; which in the Greek signifies the Speech of Numbers, I shall not here trouble you with the manner or the Construction of those Tables of Logarithms but shall first lay down some brief and ge∣neral Rules, that thereby the better you ma•• Understand those Tables, and then I shall e••∣plain their manifold uses, in sundry Exam∣ples Arithmetical, &c.

PROP. I. Any Number given under 10000, or 100000, to find the Logarithm corresponding thereunto.

1. If the number propounded consist of one place whose Logarithm is required to be found, as suppose (5,) look for 5, in the top of the left hand Column under the Letter* 1.14 N, and right against 5, and in the next Column under LOG.* 1.15 you will find this number or rank of figures, 0698970, which is the Logarithm of the number 5 required.

2. If the number consisteth of two places as if it were 57, look 57 under N, and opposite to it and under LOG. you will find this num∣ber 1. 755875, which is the Logarithm of 57, the number propounded.

3. If the number propounded consist of three places as 972, look for 972, under N, and opposite to 972, and under ••o) the Column, you shall find this number 2. 987666, which is the Logarithm of 972, the number which was propounded.

4. But if the number consist of four places as 685, look the three first figures 168, under the Column N, and opposite to that, and un∣••er 5 at the top of the page, you will find this number 3. 226599, which is the Logarithm of 1685, the number propounded.

5. But if the number* 1.16 given be above 10000, and under 100000, you may find its Logarithm by the Table of parts proportional, printed at the latter end of this Book. Thus if the Lo∣garithm of 35786, be sought, first seek the Log. of 3578, which will be 553649, and the com∣mon Difference under D is 121; with this dif∣ference 121, Enter the Table of parts propor∣tional, and finding 121 in the first Column un∣der D, you may then lineally under 6, find the number 72, which add to the Log. of 3578, that is 553649, it produceth, 553712, which is the Log. of 35786 the number propounded: now be∣cause the number propounded 35786, ariseth to the place of X. M. therefore there must be the figure 4 prefixed before its Logarithm, and then it will be thus 4, 553712, which 4, is cal∣led the Index, as shall be hereafter shewed.

Now before we proceed to find numbers cor∣responding to Logarithms, it will be necessary to explain the meaning of the first figure to the left hand of any Logarithm placed, Mr. Briggs calleth it a Cha∣racteristick* 1.17 or Index, which doth represent the distance of any the first figure of any whole number from Unity, whose Index is 0, a Cypher; so the Index o•• 10 is 1, and so to 100 whose Index is 2, and s•• to 1000 whose Index is 3, and so to 10000 whose Index is 4, and so if you persist furthe•• the Characteristick is always one less in dignity

than the places or figures os the number pro∣pounded.

PROP. II.

To find the Logarithm belonging to a Vulgar Fraction, and a Mixt number.

First as is before shewed if it be a Vulgar Fraction, find the Log. of the Numerator, and the Log: of the Denominator, then substract the Log: of the Numerator, from the Log: of the Denominator, the remainder* 1.18 is the Log: of the Fraction propounded: Now if you would find the Logarithm of 5/7, do as is prescribed whose Log. I find to be 0. 146121, Now to find the Log. of a Mixt Number, reduce it into an Improper Fraction, and then do as before, so the Log of 15 ⅖, Improper 77/5;, is 1, 187, 52, and so do for any other Mixt number.

PROP III. A Logarithm propounded to find the whole, or Mixt number, corresponding thereunto.

For the more speedy finding the number, answering unto the Logarithm propounded, ob∣serve that if the Index be 0, then the Number sought may be found between 1 and 10; If 1,

between 10 and 100; if 2, between 100, and 1000; if 3 between 1000 and 10000, and so on still observing the Rules of the Characteris∣tick, or Index, therefore loo, in the Table un∣till you find the Logarithm proposed, and a∣gainst it in the Margent according to the afore∣going directions under N, you shall find the number belonging thereunto. This Rule holds in force in Mixt Numbers also.

Thus.

0. 845098

1. 556302

2. 130334

3. 980276

Are the Lo∣garithms of,

7

36

135

9556

NOTE.

But if you cannot find the Logarithm exact∣ly in the Table, as in many operations it so hapneth, you must then take the nearest Lo∣garithm Number to the Logarithm propoun∣ded, and so take the number belonging thereto ••or the desired number.

SECT. II. Of the Admirable use of the Logarithms in Arithmetick.

PROP. I. To Multiply one number by another.

Admit 90, be to be multiplied by 42, what is the product? To find which first find the Log. of the Multiplicand 90, whose Log. is 1. 95424: Then find the Log. of* 1.19 the Multiplier 42, whose Log: is 1 62324, then add these two Log: together, viz. the Log: of the Multiplicand, and Mul∣tiplier, their sum is 3, 57748, which is the Log: of 3780, the product of 90; and 42, Multipli∣ed together.

PROP. II. To Divide one number by another.

Admit the Dividend (or number to be divi∣ded) be 648, and the Divisor 72, what is the number that the Quotient shall consist off? To find which, first write down the Logarithm of the Dividend 648, which is 2. 81157 and al∣so

write down the Logarithm of the Divisor 72, which is 1.* 1.20 85733. Now substract the Log: of the Divisor, out of the Log: of the Dividend, the remainder is 0. 95424, which is the Loga∣rithm of 9, so I conclude that the Divisor 72, is contained in the Dividend 648, 9 times, and so do for any other.

PROP. III. To find the Square-Root of a Number.

Admit it be required to Extract the Square∣Root of the Number 144, to perform which first write down the Log: of 144 which is 2. 15836. Then take the half thereof which is 1. 07918 which number 1. 07918, is the Log: of 12, the Root of 144 propounded, and so do for any other.

NOTE.

Now on the Contrary by doubling the Log. of any number, you have the Geometrical Square thereof.

PROP. IV. To find the Cube Root of any Number.

Admit it be required to Extract the Cube Root of 1728, to perform which, First write down the Log of 1728 which is 3. 23754, then take the third part thereof which is 1. 07918, which is the Log. of 12; which is the Cube∣root of the Number propounded 1728, and so for any other. Note on the contrary if you multiply the Log. of any Number propounded by 3, it produceth the Log. of the Cube thereof.

PROP. V. A Summ of Money being forborn for any number of years, to find how much it will amount unto, reckoning Interest on Interest, according to any Rate propounded.

Admit 300 pounds Sterling, be put out for 4 years, for Compound Interest at 6 l. per Cent. what will it amount to when the four years are expired? To find which substract the Log of ••00 l. the principal, whose Log. is 2. 477121, out of the Log. of 318 l. Principal and Interest for a year whose Log. is 2. 502427, the re∣mainder is 0. 025306, which being multiplyed by 4, the number of years of its continuance, produceth 0. 101224, which added to the Log. of the principal 300l. to wit, to 2. 477121, makes ••, 578345, which is the Log. of 378 l. 14 s. 10d. 2q. very near, and so much will 300 l. amount to.

PROP. VI. A Summ of Money being to be paid hereafter, to find what it is worth in ••eady Money.

Admit 100 pounds Sterling, to be paid at 30 years end; I demand how much it is worth in ready Money? after the rate of Interest of 6 l. per Cent. To find which substract the Logarithm of 100 the principal, whose Log. is 2. 000000 from the Log. of 106 Principal and Interest, whose Log. is 2. 025306, the remainder is 0 025306, which Multiplyed by 30 the num∣ber of years to succeed, produceth 0. 759180, which substracted out of 2. 000000, leaveth 1. 240820, which is the Log. of 17 411/1000, which sheweth the said 100 l. is worth but 17 l. 8s. 2d 3q. fere.

PROP. VII. A yearly rent, or Annuity to continue any number of years, to find what it is worth in ready Money, at any Rate of Interest propounded.

What is 100 pound per annum to continue 30 years, worth in ready money at 6 l. per Cent. To find which first substract the Log of 100 l. the principal, which is 2. 000000 from the Log. of 106 l principal and interest for a year, whose Log. is 2. 025306 the remainder is 0. 025306: Then Multiply 0. 025306, by 30 the number of years of its continuance, it produceth the number 0. 759180; Then Divide 100 l. by 6

the rate of interest and the Quotient is 16 6667/10000, &c. which: 16 6667/10000, is the proportional parts of 100 l. the principal, then add the Log. there∣of which is 1. 221829 to the former Log. 0. 759180 it produceth 1. 981009, which is the Log. of 95 7215/10000 parts the Arrearages with the said some for that Time, then from those Arrea∣rages 95 7215/10000, substract the parts proportional of 100, to wit 16 6667/10000, the remainder is 79 ••••48/10000, which is the bare Arrearages for that proportio∣nal part; Then take the Log. of 79 0548/10000, which is 1. 897929, out of the which take the Log. found by Multiplication of years, to wit 0. 759180, there remains 1. 138749, which is the Log. of the value of the Arrearages in ready money, Then to the Log. 1. 138749, add the Log. of 100 l. principal, 2. 000000, it produ∣ceth this number 3. 138749; the Log. of 137 6 48/100, reduced is 1376 l. 9. sh. 7d. 80/100 or ⅘ fere: and so much is the said Annuity worth in ready money.

CHAP. III. The Explication of the SINES, TANGENTS, and SE∣CANTS.

SECT. I. Of Right Signs, Tangents, Secants, Co∣sines, Tangents, and Secants: Of any Arch, or Angle of a Triangle.

PROP. I. To find the right Sine, or Tangent of any Arch or Angle of a Triangle containing any number of Degrees and Minutes.

IF the Angle or* 1.21 Arch of the Tri∣angle propounded beless than 45 Deg. the Sine,

or Tangent belonging thereunto, is found in the Column under the Title SINE, or TANGENT, at the top of the Table; and if there be any Mi∣nutes annexēd unto the Degrees, you must find them out in the first Co∣lumn under M. signify∣ing Minutes, and oppo∣site to those Minutes, and under the title aforesaid, you shall have the Loga∣rithm of the Sine or Tan∣gent, of the Arch or An∣gle required.

But if the Arch or Angle of a Triangle exceed 45 Degrees, you must then look for the Sine or Tangent belonging thereunto, in the bottom of the said Table, and if thereunto are Minutes annexed, you must look for them in the first Column to the Right hand under M. and so opposite to those Minutes in the Column above the Title, Sine, or Tang; there have you the Log. of the Sine, or Tangent, of the Arch or Angle, of the Triangle propoun∣ded.

Examp. Suppose it were required to find the Log-Sine or Log-Tangent; of an Angle of 25 D. 37 M. whose Log Sine, whereof according to the former directions I find to be 9. 635833. and Tangent thereof to be 9. 680768. and so for any other under 45 degrees.

Again, suppose it were required to find the Log-Sine or Log-Tangent, of an Angle of 64D. 23M the Sine whereof, I find to be this num∣ber'9, 955065, and the Tangent thereof, 10. 319231, and so for any other Arch, or Angle of a Triangle, above 45 degrees.

PROP. II. To find the Co-Sine or Co-Tangent of any Arch, or Angle propounded.

* 1.22 The Co-sine or Co tangent, of an Angle or Arch, is the remaining part of the Angle propounded, to a Quadrent or 90 De∣grees; and is by some called the Complement of an Angle, thus the Arch or Angle of 64D. 23M. taken out of 90D. leaves 25D. 27M. for its Complement, on the contrary if 25D. 37M. were taken out of 90 Degrees, there would remain 64D. 23M. for its Complement. So you see that these two Angles, are the Complements of each other, because they two are equal to a Quadrent or 90 Degrees.

Now the Logarithm of the Complement, may be exactly found with ease, for the Sines and Tangents of every degree, and Minute of the Quadrent in one Column is joyned with his Complement in the next Column, so that

without substracting the Angle from 90D. you may readily find the Complement thereof ei∣ther the Arch in Degrees and Minutes, or the Log. Sine, or Tangent thereof, as you have oc∣casion: Thus the Log. of the Sines Comple∣ment before mentioned, to wit, 64D. 23M. Comp. is 25D. 37M. is 9. 635833, Tang. is 9. 680768; so 64D. 23M. is the others Compl. whose Sine is 9. 955065, and his Tang. is 10, 319231; so for any other.

PROP. III. To find the Secant of any Arch or Angle propounded.

In this little Book I have not room to set down the Tables of Artificial Secants at large, as I have done with the Sines and Tangents: Nevertheless I will not here omit to shew how they may be easily found out, by the Tables of Sines. The method is thus, substract the Lo∣garithm Sine, of the Sines compl of an Angle, from the double Radius of the Tables, and the remainder shall be the Secant required: As if I desire the Secant of 25D. 37M. I find the Loga∣rithm-sine of his complement to be 9. 955065, which substracted from the double Radius, that is 20. 000000: there remains 10, 044935 which is the Secant of it, and so the Secant of 64D. 23M. is 9. 955065; which is the Complement of the former, because they both are Equal to 20. 000000, the double Radius; and so may a∣ny other be found out.

CHAP. IV. Of GEOMETRY.

* 1.23THE End and Scope of Geometry is to measure well: for as Number or Multitude, is the Subject of Arithmetick: so is Magnitude that of Geometry: to measure well is therefore to consider the Nature of e∣very thing that is to be measured; to compare such like things one with another: and to un∣derstand their Reason and proportion, and also their similitude: And this is the End and Scope of Geometry* 1.24.

I shall not trouble you with the Definitions of Geometry, they being largely handled by many, and herein every one meanly conver∣sant in the study of the Mathematicks is ac∣quainted,

but shall immediately fall in hand with the principal Propositions, which chiefly concern the other following parts of this treatise.

SECT. I. The Explication of some Geometrical Pro∣positions.

PROP. I. To erect a perpendicular on any part of a line assigned.

LET the Line be A, B, and on the point D,* 1.25 'tis required to raise a perpendicular to A, B, To operate which first open your Compasses to any convenient distance, and placing one foot thereof in D, with the other make the two marks C, and E, equidistant from D; then open the Compasses to some other convenient distance, and set one foot in E, and describe the Arch FF; then likewise in C, describe the Arch GG, then through the Intersections of these two Arches, and to the point D, draw H D, per∣pendicular to A B; as was required.

PROP. II. To Erect a Perpendicular, on the End of a Line.

Let the given line be A B, and on the End* 1.26 thereof at B, 'tis required to raise a Perpen∣dicular

line: To perform which open your Compasses to the distance B D, then on B as a Center, describe the Arch D, E, F, then from D, to E, place BD; then placing one foot in E, describe the Arch CF, then remove your* 1.27 Compasses to F, and draw the Arch CE; Lastly through their Intersection draw C B, which is a Perpendicular to AB, on the end B; as required.

PROP. III. From a Point above to let fall a Perpendicular on a Line.

Let the line given be B A, and 'tis required* 1.28 from the point above at C; to let fall a Per∣pendicular to the said Line: To perform which place one foot of your Compasses in C, and o∣pen them beyond the given line A B, and de∣scribe the Arch EF; divide EF, into two parts in D; Lastly draw CD, which shall be per∣pendicular unto AB, falling from the point above at C, as was so required.

PROP. IV. To draw a right line Parallel to a right line, at a∣ny distance assigned.

Let the distance assigned be O E, and the* 1.29 Lime given be A B, and 'tis required to draw C D, Parallel to A B; at the distance O E: To perform which, take in your Com∣passes the distance O E, and on A, describe

the Arch H, and on B, the Arch K; then draw C D, so as it may justly touch the two Arches, but cut them not, so shall C D; be pa∣rallel to A B, at the assigned distance O E, as was required.

PROP. V. To Protract an Angle of any Quantity of Degrees propounded.

Let it be required to Protract, or lay down an Angle, of 40 degrees: To perform which first draw a right line as A B, then open y••••r Compasses to 60 degrees, in your line of Chords: and with that Distance on A, de∣scribe* 1.30 the Arch E F, then take 40 degrees in your Compasses out of your line of Chords, and place it on the Arch, from F, to H; Lastly through the point H, and from A draw A C; so shall the Angle CAB contain 40 degrees as required.

PROP. VI. To measure an Angle already protracted.

Let the Angle given be C A B, and 'tis ••equired to find the Quantity thereof: To ••erform which take in your Compasses 60 de••rees from your line of Chords; and on A,* 1.31 ••escribe the Arch EF; then take in your Com∣••asses the Distance FH, and apply it to your 〈…〉〈…〉ne of Chords; and you will find the Angle, 〈…〉〈…〉 AB to contain 40 degrees.

PROP. VII. To divide an Angle into two Equal parts.

Let the Angle given be BAC, and 'tis required to divide it into two equal parts: To perform which do thus: first take in your Com∣passes any convenient distance, and placing one foot in A, describe the Arch FKHE,* 1.32 then on H, describe the Arch KK, and on K, the Arch HH; lastly through the Intersections of these two Arches, draw the line AD, to the Angular point A; so shall the Angle BAC, be divided into two equal parts, viz. BA••, and DAC; as required.

PROP. VIII. To divide a right line into any Number of Equal or Unequal parts; or like to any divided line propounded.

Let the line A B, be given to be divided into 5 equal parts; as the line CD. To per∣form which do thus: first on the point C, draw out a line making an Angle with CD at plea∣sure: then make CF, equal to AB; and joyn their Extremities FD, then draw Parallel lines* 1.33 to FD, through all the 5 points of CD, (by the 4 prop. aforegoing) which shall divide AB, into 5 equal parts; as required: This way is to be observed, when the line given to be divided, is greater than the divided line propounded.

CASE II.

But if AB, be shorter than the given di∣vided line CD; take the line AB, in your Compasses, and on D strike the Arch F, then* 1.34 draw the Tangent CF, then take the nearest distance from the first division of CD, to the Tangent-line CF, which distance shall divide AB* 1.35 into 5 equal parts, as the given divided line CD; as required.

PROP. IX. How to Protract or lay down any of the Regular Figures, called Polygons.

To perform which divide 360 degrees, (the number of degrees in a Circle) by the number of the Poligon his sides: as if it be a Pentagon by 5, if a Hexagon by 6, &c. the Quotient is the Angle of the Center; its Complement to 180D. (or a Semi circle) is the Angle at the Figure, half whereof is the Angle of the Tri∣angle at the Figure: Now I will shew how to delineate any Poligon three ways, viz. 1 by the Angle at the Center, 2. by the Angle at the Figure, 3. by the Angle of the Triangle at the Figure: I have hereunto annexed a Table, which gives at the first sight, (without the trouble of Division) 1. the quantity of the Angle at the Center; 2. the quantity of the Angle at the

Figure; and 3 the Quantity of the Angle at the Triangle of the Figure, from a Triangle to a Decigon.

Names of the Poli∣gons.	Sides	Angles at the Center	Angles at the Figure	Angles at the Trian.
D	M	D	M	D	M
Triangle	3	120	00	60	00	30	00
Square	4	90	00	90	00	45	00
Pentagon	5	72	00	108	00	54	00
Hexagon	6	60	00	120	00	60	00
Heptagon	7	51	43½	128	34½	64	17¼
Octogon	8	45	00	135	00	67	30
Nonigon	9	40	00	140	00	70	00
Decigon	10	36	00	144	00	72	00

CONSTRUCTION I.

First by the Angle at the Center, to delineate a Hexagon, whose Angle at the Center is 60 degrees, first lay down an Angle of 60 deg.* 1.36 (by prop. the 5. aforegoing) making its sides of a convenient length at pleasure, then take such a distance from O the Center of the figure, equally on both sides, as may make the third side equal to the side of the Poligon gi∣ven; which here is 100 parts: * 1.37 Then divide the third side equally into two equal parts, and draw a line through it, from ☉ the Center:

set each half of the side of the Poligon 100, to wit 50, on each from the middle of the third line.† 1.38 thus having placed the side of the Hexagon PP, 100 parts, in order; de∣scribe the whole Hexa∣gon PPPPPP, as was re∣quired.

CONSTRUCTION II.

Now by the Angle of the Figure, to de∣lineate any regular Poligon, Let it be required to protract a Hexagon, whose side as afore is 100 parts; first I draw a line and make it 100 of those parts, then I sind in the precedent Table the Angle of a Hexagon at the figure to be 120 degrees: Then on each side of the drawn line, I lay down an Angle of 120 deg. (according to the 5 precedent propositions) and so work 6 times, (or as many times as your Poligon hath sides) making each side 100 parts,* 1.39 and each Angle 120 degrees; so shall you have enclosed the Poligon PPPPPP, as required.

CONSTRUCTION. III.

To Protract or lay down a Hexagon, or any other regular Poligon, by the Angle of the Triangle, do thus; First draw the side of the* 1.40 Hexagon P P, make it 100 parts. I find in the precedent Table that the Angle of the Triangle is 60 deg; then at each end of the line P P, I

lay down an Angle of 60 * 1.41 deg. (by prop. 5. prece∣dent) and continue the two lines PO, and PO; untill they intersect each* 1.42 other in O: then on O, as a Center (OP: be∣ing Radius) describe a Circle, and within it de∣scribe the Hexagon PPPPPP, as you see in the figure: and so may you delineate any other Poligon: whose Angels from a Triangle, to a Decigon, are all specified in the precedent Table.

PROP. X. To divide a line according to any assigned pro∣portion.

Admit the right line given to be AB, and 'tis required to divide the same into two parts, bearing proportion the one to the other as the lines E, and F doth: To perform which, first draw the line CD, equal to the given line AB: Then draw the line HC, from C, to contain* 1.43 an Angle at pleasure. Then from C to G, place the line F, and from G, to H, place the line E: Then draw the line HD. And lastly, draw GK parallel to HD, (by the 4 prop. precedent) so is the line DC, equal to AB, and divided into two parts, bearing such proporti∣on to each other, as the two given lines E, and F, as was required.

PROP. XI. To two lines given, to find a third proportional to each of them.

Admit the two given lines be A and B, and* 1.44 'tis required to find a third proportional to A, as A, to B: First make an Angle at pleasure; as HIK. Then place the line B, from I, unto P; and the line A, from I, unto L; and draw PL. then also place the line A, from I unto M, and draw QM, parallel unto LP, (by 4 prop.) so shall the line IQ, be a third proportional unto the two given lines A, and B, as was required. For as B, is to A, so is A, unto the proportional found IQ.

PROP. XII. To three lines given to find out a fourth proportio∣nal unto them.

Admit the three given lines to be A, B, and C; and 'tis required to find a third proportional to them, which shall have such proportion unto A, as B, hath unto C. To perform which, first make an Angle at pleasure as DKG, now see∣ing the line C, hath such proportion to B, as the line A, unto the line sought: Therefore place* 1.45 the line C, from K, unto H and B, from K, to F, and draw FH. Again, place the line A, from K, to I, and draw IE, parallel unto FH, (by 4 prop.) until it cutteth DK, in E; so have you the line KE, a fourth proportional, as was required. For as C, is unto B, so is A, unto the found line KE.

PROP. XIII. To find a mean proportional Line between any two right lines given.

Let the two given lines be A, and B, be∣tween which it is required to find a mean pro∣portional line. To perform which, first joyn the two lines A, and B together, so as they make the right line CED: Then describe thereon a* 1.46 Semicircle CFD. Then on the point E, erect the perpendicular EF, (by 1 prop.) to cut the limb of the Semi-circle in F, so shall EF, be a mean proportional line, between the two given lines A, and B, as required.

PROP. XIV. To find two mean proportional Lines between any two right Lines given.

Let the two given lines be A, and B; be∣tween* 1.47 which 'tis required to find two mean proportionals. To per∣form which, first make an* 1.48 Angle containing 90 deg. making the sides CD, and CE of a convenient length: then from C, place the line B, unto F, and the line A, from C, unto G; and draw FG, which divide equally in H, and describe the Se∣mi-circle F K G. Then take the line B in your Compasses, and place∣ing one soot in G, with the other make a mark

in the limb of the Semi-circle in K, then draw ST, in such sort that it may justly touch the Semi-circle in K, and may cut through the two sides of the Angle, equidistant from the Cen∣ter of the Semi-circle H; so shall SF, and TG, be two mean proportionals, betwixt the two gi∣ven lines A, and B, as required.

PROP. XV. To make a Geometrical square equal to divers Geo∣metrical squares.

Let there be given the 5 sides of five Geome∣trical Squares, viz. A, B, C, D, E; and 'tis requi∣red to make one Geometrical Square, equal to the said five Sqares: To perform which first make a Right Angle as ABC, making its contain∣ed* 1.49 sides of a convenient length. Then from B, place A, to D, and from B, place B, to E, and draw Ed. Then place Ed, from B, to F, and C, from B, to G; and draw GF. Then place GF, from B, to H, and D, from B, to I; and draw ••H. Lastly from B, unto K, place IH, and from B, unto L, place the line E; and draw LK. So shall LK, be the side of a Square, equal to the five Squares propounded.

PROP. XVI. To make a Circle equal to divers Circles propounded.

Let the two Circles propounded be A,* 1.50 and B, and 'tis required to make a third Circle, ••e••ual to the said Circles propounded. To per∣form

which, first take the Diameter, of the les∣ser Circle A, and place it as a Tangent, on the Diameter of the greater Circle B, at right An∣gles;* 1.51 as ECD. Then draw the Diagonal ED, which divide equally in F, on which as a Cen∣ter describe the Circle K, making E D, the Diameter of which Circle K shall be equal unto the two given Circles A, and B, as required* 1.52

SECT. II. Of Planometry, or the way to measure any plain Superfice.

PLanometry is that part of the Mathematicks, derived from that Noble Science Geometry, by which the Superficies or Planes of things are measured, and by which their Superficial Content is found, which is done most com∣monly by the Squares of such Measures, Viz. a Square Inch, Square Foot, Square Yard, Square Pace, Square Perch, &c. That is whose side is an Inch, Foot, Yard, Pace, or Pearch Square. So that the Content of any Figure is said to be found, when you know how many such Inches, Feet, Yards, Paces, &c. are contained therein: Thus the End and Scope of Geometry is to measure well.

PROP. I. To find the superficial Content of a Geometrical square.

Let the side of the Square AA be 4 Perch,* 1.53 what is the Area, or superficial content thereof? To find which multiply its side 4, by its self, it produceth 16, which is the content of that Square AAAA, propounded.

PROP. II. To find the superficial content of a Parallelogram, or long Square.

Multiply the length in parts, by the breadth in parts; the product is the content thereof. So in the Parallelogram, or long Square ABCD, the length of the side AB, or CD is 20 Paces,* 1.54 and the breadth AC, or BD is 10 paces, and his superficial content is required. I say there∣fore if according unto the Rule, you multi∣ply the length 20, by the breadth 10, it produ∣ceth 200 Paces; which is the content of the Pa∣rallelogram or long Square ABCD.

PROP. III. To find the superficial Content of any Right-lined Triangle.

Although right-lined Triangles are of seve∣ral kinds, and forms; as first in respect unto their Angles, they are either Right-angled, or

Oblique-angled, i. e. Acute-angled, or Obtuse∣angled. Secondly in respect of their sides, they are either an Equilateral, Isosceles, or Scaleni∣um Triangle: But now seeing they are all measur'd by one and the same manner, I shall therefore add but one Example for all; which take for a general Rule: which is,

Multiply the length of the Base, by the* 1.55 length of the Perpendicular, half their product is the Area or superficial content thereof. So if the content of the Triangle ABC, be required. To find which first from the Angle B, let fall the Perpendicular DB, on the Base AC, (by prop. 3. §. 1.) let therefore the length of the* 1.56 Perpendicular BD be 24, and the Base AC 44 parts. Now if the Base AC 44, were multiply∣ed by BD 24, the product is 1056, half where∣of is 528, the Content of the Triangle ABC, pro∣pounded.

PROP IV. To find the superficial Content of a Rhombus.

First let fall a Perpendicular from one of the Obtuse-angles, unto its opposite side, (by prop. 3. §. 1.) and then Multiply the length of the side thereof, by the length of the Perpendicu∣lar, their product is the Content thereof.

So in the Rhombus ABCD, the side AC, or BD is 16 Inches, and the Perpendicular KC is* 1.57 14 Inches, which multiplyed into the side 16, produceth 224 Inches; which is the Area, or superficial Content, of the Rhombus ABCD, pro∣pounded.

PROP. V. To find the superficial content of a Rhomboides.

Frst let fall a Perpendicular, as in the former proposition, then the length thereof multiply by the length of the Perpendicular; the pro∣duct is the Area, or superficial content thereof. For in the Rhomboides EDAH, whose length AH, or ED is 32 Feet, and the length of the* 1.58 Perpendicular HK is 16 Feet, which multiply∣ed together produceth 512 Feet, which is the Area or superficial content of the Rhomboides AHED, propounded.

PROP. VI. Te find the superficial Content of any Poligon, or many equal sided Superficies.

First from the Center unto the middle of either of the sides of the Poligon, let fall a Per∣pendicular, (by 3. prop §. 1.) Then multiply the length of half the Perifery, by the Perpen∣dicular, the product shall be the Superficial Con∣tent of the Poligon.

Admit the Poligon to be an Hexagon AAAA* 1.59 AA, whose side AA is 22 Feet, and the Per∣••end••cular BE 19 Feet; now, if 66 half the Pe∣rifery, be multiplyed by 19 it produceth 1254 Feet; which is the Content of the Poligon AA, &c. as required.

PROP. VII. To find the superficial Content of a Circle.

Multiply half the Circumference, by one half of the Diameter, their product is the superficial Content thereof.

Admit the Circumference of a Circle ACBD, be 44 Inches, what is the Area or Content thereof. (by the 9. prop. §. 2.) I find the Dia∣meter* 1.60 to be 14 Inches, therefore I say if 22, half the Circumference, be multiplied by 7, half the Diameter, it shall produce 154 Inches; which is the superficial Content of the Circle ACDB, as required.

PROP. VIII. By the Diameter of a Circle given, to find the Cir∣cumference.

Suppose the Diameter be 14, what is the Circumference? The Analogy or Proportion holds thus, as 7, to 22, so is 14, unto 44, the Circumference required.

PROP. IX. By the Circumference of a Circle given, to find the Diameter.

Suppose the Circumference of a Circle be 44 what is the Diameter? the Analogy or Proportion is, as 22, to 7, so is 44, unto 14, the Diameter re∣quired.

Now the proportion of the Diameter, unto the Circumference is as 7, unto-22; or as 113, to 355; or as 1, unto 3, 1415926, &c. so is the Diameter to the Circumference.

PROP. X. By the Content of a Circle given, to find the Cir∣cumference.

Suppose the Content of a Circle be 154, what is the Circumference, the Analogy or Proportion?

As 7, unto 4 times 22, which is 88, so is 154 the Content of the given Circle; to the square of the Circumference 1936, whose root being Extracted, as is taught (in prop. 8. §. 1. chap. 1.) gives the Circumference 44, as required.

PROP. XI. By the Content of a Circle given, to find the Dia∣meter.

Suppose the Superficial Content of a Circle be 154 parts, what is the Diameter thereof? to find which this is the Analogy or Proportion.

As 22,

To 4 times 7, which is 28,

So is 154, the given Content,

To the Square of the Diameter 196, whose Root being Extracted (by 8 prop chap. 1. §. 1.) ••iveth the Diameter 14, as required.

PROP. XII. By the Diameter of a Circle given to find the side of a square equal thereto.

To find which this is the Analogy or Propor∣tion.

As 1, 000000,

To 0, 886227.

So is the Diameter of the Circle propounded.

To the side of a Square, whose superficial Content, is equal unto the superficial Content, of the Circle propounded.

PROP. XIII. By the Circumference of a Circle given, to find the side of a square equal to it.

This is the Analogy or Proportion.

As 1. 000000,

To 0. 282093.

So is the Circumference of the Circle pro∣pounded, to the side of a Square equal to the Circle.

PROP. XIV. By the Content of a Circle given to find the side a square equal to it.

To do which, Extract the Square-Root o•• the Content propounded, (by prop. 8 chap. •••• §. 1.) so is the Root, the side of a Geometrica•• Square, equal thereunto.

PROP. XV. By the Diameter of a Circle given, to find the side of an Inscribed square.

This is the Analogy or Proportion.

As 1. 000000,

To 0. 707107,

So is the Diameter of the Circle propounded, To the side of the inscribed Square.

PROP. XVI. By the Circumference of a Circle given, to find the side of an Inscribed Square.

This is the Analogy, or Proportion.

As 1. 000000,

To 0. 225079.

So is the Circumference of the Circle pro∣pounded, To the side of the inscribed Square.

PROP. XVII. ••o find the Superficial Content of an Oval, or El∣leipsis.

Let the Oval given be ABCD, and 'tis re∣••uired to find the Area or Superficial Content 〈…〉〈…〉ereof? To do which multiply the length A 〈…〉〈…〉 40 Inches, by the breadth CD 30 Inches, the* 1.61 •••• is 1200. Which divide by 1. 27324; 〈…〉〈…〉e Quotient is 942 48/100 parts. Which is the 〈…〉〈…〉ea or Superficial Content of the Oval ABCD 〈…〉〈…〉opounded

PROP. XVIII. To find the Superficial Content of any Section, or Portion of a Circle.

Multiply half the Circute of the Section, by the Semidiameter of the whole Circle, and the product thence arising is the Area or superfi∣cial Content thereof.

Suppose there be a Circle whose Diameter is* 1.62 14 parts, and the Circute of the Quadrent ABC is 11 parts, and the Content of the said Qua∣drent is desired? To find which multiply 5 ½ or, 5., 5 half the Circute of the Quadrent, by 7 the Semidiameter, the product is 38 5/10, which is the Content of the Quadrent ABC pro∣pounded.

SECT. III. Of STEREOMETRY, or the way how to measure any Regular Solid.

STereometry is that part of the Mathema∣ticks, springing from Geometry, by which the Content of all Solid Bodies are discovered by two Multiplications, or three Dimention and is valued by the Cube of some famous Mea sure; as an Inch-Cube, a Foot-Cube, a Yard Cube, or a Perch-Cube, &c.

PROP. I. To find the solid Content of a Cube.

Multiply the side into its self, and that pro∣duct by its side again; their product is the so∣lid Content thereof.

Suppose there be a Cube A, whose side is* 1.63 2 Feet; and his solid Content is required? I say if his side 2, be multiplyed by its self, it pro∣duceth 4, which again multiplyed by 2, it pro∣duceth 8 Feet, which is the solid Content of the Cube propounded.

PROP. II. To find the solid Content of a Parallelepipedon.

First get the Superficial Content of the End, (by prop. 1, or 2, §. 2.) which multiply into the length, the product is the solid Content.

Suppose there be a Parallelepipedon B, whose* 1.64 sides of the Base is 40, and 30 Inches, and length 120 Inches, and his Solid Content is demanded? I say if you multiply 30, by 40, the product is 1, 200, which is the superficial Content at the Base. Which multiplyed by the length 120 In∣ches produceth 144000 Inches, which is the so∣lid Content of the Parallelepipedon B, propounded.

PROP. III. To find the solid Content of a Cylinder.

First get the superficial Content of the Circle at the Base, (by prop. 7. §. 2.) and by it multi∣ply its length, their product is the solid Content thereof.

Suppose there be a Cylinder as D, whose* 1.65 Diameter of the Circle at the Base is 7 parts, and the length of the Cylinder is 14 parts, and 'tis required to find the solid Content thereof? First I find the superficial Content of the Base to be 38. 5, which multiplied into 14 the length, giveth 539 parts, which is the solid Content of the Cylinder propounded.

PROP. IV. To find the solid Content of a Pyramid.

First get the superficial Content of the Base of the Pyramid, (by some of the aforegoing pro∣positions in Planometria) and then multiply that into ⅓ of his Altitude, the product is the solid Content thereof.

Suppose there be a Pyramid H, whose side of the Base is 4 ½ parts, or 4 5/10, and his Altitude 12 parts, and his solid Content is required? First I find, (by prop. 1. §. 2.) the superficial Content* 1.66 of the Base to be 20 25/100 or 20 ¼, which multiply∣ed by 4, (which is ⅓ of the Altitude 12) produ∣ceth 81 parts, for the solid Content of the Py∣ramid propounded.

PROP. V. To find the solid content of a Cone.

First find the superficial Content of the Circle at the Base, (by prop. 7. §. 2.) then multiply it by ⅓ of its Altitude or Heighth, the product is the solid Content thereof.

Suppose there be a Cone as B, whose Dia∣meter* 1.67 of the Base is 7, and his Altitude or Heighth is 15 parts, and his solid Content is re∣quired? First I find the superficial Content of the Base to be 38½ or 38. 5; which multiply∣ed into 5, ⅓ of its Altitude or Heighth) produ∣ceth 192. 5, or ½, which is the solid Content of the Cone propounded.

PROP. VI. By the Diameter of a Globe to find his solid Content.

This is the Analogy or Proportion.

As 6 times 7, which is 42.

Is to 22,

So is the Cube of the Diameter of the Sphere, or Globe propounded.

To the solid Content thereof.

Suppose there be a Sphere or Globe, whose Diameter is 12 Inches; what is the solid Con∣tent thereof? say, (see the Globe R.)

As 42,

Is to 22,* 1.68

So is 1728, the Cube of the Diameter,

To the solid Content 905 6/42 or 1/7 of the Globe, or Sphere propounded: This and all other such* 1.69 like Propositions, are performed by the help of the first Proposition, of the first Chapter of this Book.

PROP. VII. By the Circumference of a Sphere, or Globe, to find his solid Content.

This is the Analogy or Proportion.

As 1. 000000,

To 0. 016887,

So is the Cube of the Circumference of the Globe or Sphere propounded

To the solid Content thereof.

PROP. VIII. By the Axis of a Globe, to make a Cube equal there∣unto.

This is the Analogy or Proportion.

As 1. 00000,

To 0. 80604,

So is the Axis of the Sphere propounded,

To the ••u••••-Root, which shall be equal to it.

PROP. IX. By the Circumference of a Globe, to make a Cube equal thereunto.

This is the Analogy or Proportion.

As 1. 000000,

To 0 256556.

So is the Circumference of the Globe propoun∣ded,

To the Cube-Root, which shall be equal to the Sphere, or Globe, propounded.

PROP. X. By the solid Content of a Sphere or Globe, to make a Cube equal thereunto.

Extract the Cube-root of the solid Content of the Sphere or Globe, (by prop. 9. § 1. chap. 1.) so shall the Root, so found, be the side of a Cube, equal unto the Globe or Sphere propounded.

PROP. XI. A Segment of a Sphere being given to find the solid Content thereof.

To find which first say, As the Altitude of the other Segment, is to the Altitude of the Seg∣ment

given: so is that Altitude of the other Segment increased by half the Axis, unto a fourth: Then say, As 1, to 1, 0472, so is the product of the Quadrant of half the Chord of the Circumfe∣rence of that Segment, multiplyed by that fourth, To the solid Content of the Segment propounded.

CHAP. V. Of TRIGONOMETRY. Or the Doctrine of Triangles.

SECT. I. Some general Maxims, belonging to plain or Right-lined Triangles.

TRIGONOMETRY is necessary in most parts of the Mathematicks, and herein indeed consisteth the most fre∣quent use of the Logarithms, Sines, Tangents, and Secants: It is conversant in the measuring of Triangles, Plain or Spherical, comparing their Sides, and Angles together; according unto their known Analogies, or Proportions: So that any three parts of a Triangle being given, the other parts may be found out, and known: Now in the Doctrine of Right-lined Triangles, it will be necessary to know these Maxims fol∣lowing.

1. That a Right-lined Triangle, is a Figure constituted, by the Conjunction, or Intersection, of the three Right, or Streight-lines thereof; in their Angles or Meeting-places. So that e∣very Triangle hath six distinct parts, Viz. Three Sides, and three Angles.

2. That all Right-lined Triangles, are either* 1.70 Right-angled, That is, which hath one Right∣Angle, as ABC Fig. 34. Or Oblique-angled, whose three Angles are all Acute; that is, less than a Quadrant, or 90 deg; or else they have One Angle Obtuse, or greater than a Quadrent: So all Triangles, that have not one Right-angle,* 1.71 are called Oblique-Triangles; as Fig. 36. to wit, the Triangle ABC.

3. That the three Angles, of any Right-lined Triangle, are equal unto two Right-angles; or 180 Degrees. So that any two of their Angles being known, the third Angle is also found, being the Complement of the other two; unto 180 Degrees: But this is more readily found in a Rectangled Triangle, for the Rectangle being a Quadrent, or 90 degrees, one of the acute Angles therefore being given, the other is rea∣dily known, being the Complement thereof unto 90 Degrees.

4 That the three sides, comprehending the* 1.72 Triangle, some call Leggs, others Sides, but in Rectangled Triangles, as in the Triangle ABC, I call AB, the Base, BC the Cathetus or Perpendi∣cular; and AC the Hypothenuse.

5. That the Sines, of the Angles are proporti∣onal unto their opposite Sides; and their Sides, to their opposite Angles. So that if the Side of a Triangle were desired, put the Sine of the oppo∣site

Angle in the first place. Also if an Angle be required, put the Logarithm of his opposite side in the first place.

6. That the sides of any Rectangled Tri∣angle may be measured by any Scale of equal parts, as Inches, Feet, Yards, Poles, Miles, Leagues, &c.

7. That if an Angle propounded, be greater than 90 deg. and so not to be found in the Ta∣bles, take the Complement thereof, unto 180 deg. and work by the Sine, or Tangent there∣of, and the work will be the same.

And here for the more short, and speedy performance of these conclusions in Trigono∣metry; I have annexed, and used, these fol∣lowing Symbols; which I would have you take notice of.

• = Equal, or Equal to.
• + More.
• - Less.
• × Multiplyed by.
• ° Degrees as 15°.
• ' Minutes as. 40'.
• cr. A Side.
• cr^s, Sides.
• V An Angle.
• VV Angles.
• Z Sum.
• X Difference.
• S Sine.
• Sc Co-sine.
• T Tangent.
• Tc Co-tangent.
• Se Secant.
• Sec Co-secant.
• Co. Ar. Compl. Arithmetic.
• R A Right-angle.
• 2R Two Right-angles.
• Q Square.

SECT. II. Of Plain Rectangled Triangles.

PROP. I. Two Angles and the Base of a Rectangled Trian∣gle given, to find the other parts.

ADmit the Triangle given be ABC: Now* 1.73 the Angle at B, is an Angle of 90°, or a Right-angle; And the Angle at C is 57° 35', and the Base AB; is 736 parts.

Now first I find the Angle at A, to be 32° 25': it being the Complement of the Angle at C, unto 90°: Secondly, to find the Cathe∣tus, or Perpendicular, this is the analogy or proportion.* 1.74

Add the Log. of the third and second Terms together, and from their Sum, deduct the Log. of* 1.75 the first number, so is the Remainder, the Log. of

the fourth Term, or Number sought, as you see in the aforegoing Example.

Thirdly to find the Hypothenuse AC, the Analogy or Proportion hold thus.

As S. V, C 57° 35',

To Log. Base AB 736 ••arts.* 1.76

So Radius or S. 90°,

To Log. Hypothenuse AC 871 8/10 parts re∣quired: Thus are the three required parts, of the given Triangle ABC found, viz. the Angle A to be 32° 25', the Cathetus BC to be 467 4/10 parts, and the Hypothenuse AC to be 871 8/10 parts, as was so required to be found.

PROP. II. The Hypothenuse, Base, and one of the Angles Of a Rectangled Triangle given, to find the o∣ther parts thereof.

In the Triangle ABC, the Hypothenuse AC is 871 8/10 parts, the Base AB is 736 parts, and the Angle at B, is known to be a Right-angle; or 90°: First to find the Angle at the Cathetus C, the analogy or proportion holds thus.

As Log. Hypothen: AC 871 8/10 parts* 1.77

To Radius or S. 90°.

So Log. Base AB 736 parts,

To the S. V. at Cathetus C 57° 35'.

Secondly, now having found the Angle at the Cathetus C, to be 57° 35'; I say the Angle of the Base A is 32° 25', being the Compl. of the Angle C, unto 90°.

Thirdly to find the Cathetus BC, this is the ••nalogy, or proportion.

As Radius or S. 90°,

To Log. Hypothen. AC 871 8/10 parts,

So S. V. at Base A 32° 25'.* 1.78

To Log. Cathetus BC, 467 4/10 parts required. It may also be found, as in the former Proposi∣tion.

PROP. III. In a Rectangled Triangle, the Base, and Cathetus given to find the other parts thereof.

In the Triangle ABC, the Base AB is 736 parts, and the Cathetus BC is 467 4/10 parts, and the Angle B, between them is a right angle o•• 90°: And here you may make either side of the Triangle, Radius, but I shall make BC the Cathetus Radius, and then to find the Angle at the Cathetus C, this is the Analogy or ••••••∣portion.

As Log. Cathet, BC 467 4/10 parts,* 1.79

To Radius or S 90°.

So Log. Base AB 736 parts,

To T. V. Cathe C 57° 35', as required.

Secondly, I find the other Angle, at A to be* 1.80 32° 25', it being the Complement, to C 57 35', unto 90°.

Thirdly, To find out the Hypothenuse AC this is the analogy or proportion.

As S. V. Cathe C. 57° 35',

To Log. Base AB 736 parts,

So Radius or S. 90°,

To Log. Hypothenuse AC 871 8/10 parts, 〈…〉〈…〉∣quired. But making the Base AB Radius, yo•• may find the Hypothenuse AC, by this anal〈…〉〈…〉 or proportion.

[illustration]

Plate 1 Page 65

As Radius or S. 90°,

To Log. Base AB 736 parts.

So Sc. V. Base A 32° 25',

To Log. Hypothenuse AE 871 8/10 parts required,* 1.81 and thus you have all the parts of the Trian∣gle propounded.

PROP. IV. The Base, and Hypothenuse, with the Angle be∣tween them given, to find the other parts of a Rect-angled Triangle.

In the Triangle ABC, the Base AB is 736 parts, and the Hypothenuse AC is 871 8/10 parts, and the Angle A included between them is 32° 25'. First to find the Angles, and first remem∣ber that the Angle B is a right Angle; or 90°. Secondly, that the Angle at C, is the Comple∣ment to the Angle at A 32° 25' unto 90°:* 1.82 and therefore is 57° 35': Now these being known, you may find the Cathetus, by this analogy or proportion.

As S. V. Cathe. C. 57° 35',

To Log. Base AB 736 parts.

So S. V. Base A 32° 25',

To Log. Cathe. BC 467 4/10 parts required. Thus I have sufficiently explained all the Cases of Plain Rect-angle Triangles, for to these rules they may be all reduced.

SECT. III. Of Oblique-Angled Plain Triangles.

PROP. I. Two Angles, and a side opposite, in an Oblique-Ang∣led Triangle given, to find the other parts there∣of.

IN the Triangle ABC, the Angleat A is 50°, and at C is 37°, and the side AB is 30 parts, and opposite to the Angle C

First, to find the Angle B, remember that (as 'tis said, in the third Maxim aforegoing) 'tis the Complement, to the Angles A 50°, and C 37°, to 180°, and therefore is the Angle at B 93°.

Secondly, having thus found the Angles, the* 1.83 two unknown sides, may be found by the pro∣portion they bear to their opposite Angles, for that proportion holds also in these; thus to find the side BC, this is the analogy or proportion.

As S. V. C 37° 00',

To Log. side AB 30 parts.

So S. V. A. 50° 00',

To Log. side BC 38 19/100 parts required to be found.

But it may be more readily found, and per∣formed in such case as this, where you have a Sine, or Tangent, in the first place, by the A∣rithmetical Complement thereof, and so save the Substraction.

Now the readiest way to find the Arithmetical* 1.84 Complement is that of Mr. Norwood, in his Doctrine of Triangles; which is thus: begin with the first Figure towards the left hand of any Number and write down the Complement, or the re∣mainder thereof, unto 9:* 1.85 And so do with all the* 1.86 rest of the Figures, as you see here done. Saying 9, wants of 9, 0: and again 9, wants 0: 6, wants 3; 2, wants 7: 3, wants 6; 9, wants 0: only when you come to the last Figure to the right hand, take it out of 10, so 8, wants 2; of 10: Thus you may readily find the Co-Ar. of any Sine, al∣most as soon as the Sine it self.

But if you want the Complement Arithmetical of any Tangent, you may take the Co-tang. which is exactly the Co-Arith. of the double Radius, so that the Tangent, and Co-tangent, of an Arch makes exactly 20. 000000.

Now if the Radius be in the first place, then there is no need of taking the Co-Arith. of the first Number, only you must cut off, the first I, to the left hand thus X, and you will have the Logarithm of the Number desired.

Thirdly, now to find the side AC, by the opposite Angle B; which is 93° 00': And see∣〈…〉〈…〉ng the Angle B, exceeds 90°, you must work 〈…〉〈…〉y the Complement to 180°) as in the seventh 〈…〉〈…〉ork in page 61 is taught.

Thus having found all the parts of the Tri∣angle * 1.87 * 1.88 propounded, Viz. The Angle B, to be 93° 00', the side AC to be 49 78/100 parts, and the side BC to be 38 19/100 parts, as was required to be found.

PROP. II. Two sides, and an Angle opposite to one of them in an Oblique-angled Triangle given, to find the other parts thereof.

In the Triangle ABC, the side AB is 30 parts, and the side AC, is 49 78/100 parts, and the opposite Angle C, is 37° 00'.

First, To find the Angle at B, this is the A∣nalogy* 1.89 or Proportion.

As Log. cr. AB 30 parts,

To S. V. at C 37° 00'.

So Log. cr. AC 49 78/100 parts,

To Sc. V. B 93° 00', as was required to be found.

Now seeing that the Angle C, is 37° 00', and the Angle B, is 93° 00', which makes 120° 00', therefore must the Angle A be 50° 00'; the

Complement to 180°: so having found all the three Angles, you may find the other side CB,* 1.90 38 19/100 parts, as afore in the first proposition, by his opposite Angle.

PROP. III. Two Sides of an Oblique-angled Triangle, with the Angle included between them given, to find the other parts thereof.

In the Triangle ABC, the side AC is 49 78/100 parts, the side DB is 30 parts, and the Angle A between them is 50° 00'; and 'tis required to find the other parts of the Triangle propoun∣ded. To resolve this Conclusion, let fall a Perpendicular DB, from the Angle B, on the side AC; (by prop. 3. §. 1. chap. 4) and then pro∣ceed thus.

First, Seeing the Oblique-angled Triangle,* 1.91 ABC is divided into two Rectangled Triangles, Viz. ADB, and BDC: Now I will begin with the Triangle ADB, in which is given the An∣gle A 50° 00', and the Angle D is a right An∣gle, or 90°, and the side AB 30 parts, and the sides AD, and DB, and the Angle at B, are required.

First to find the Angle at B, remember that it is the Complement unto the Angle A 50° 00', unto 90° 00', and therefore must the Angle B be 40° 00'; Now for to find the Cathetus BD, (as in prop. 1. and 2 §. 2. chap. 5.) by the Rule of opposition, the Analogy or Proportion holds thus.

As Radius or S. 90°,

To Log. Hypoth. AB 30 parts.

So S. V. at A 50° 00',

To Log. Cath. BD 22 98/100 parts sought.

And AGAIN, say.

As Radius or S. 90°,

To Log. Hypoth. AB 30 parts.

So S. V. at B 40° 00',

To Log. Base AD 19 28/100 parts sought.

Thus in the Triangle ADB, you have found the Angle B, to be 40° 00', the Cathetus BD, to be 22 98/100 parts; and the Base AD to be 19 28/100 parts, as was so required.

Now for the other Triangle which is BDC, in which there is given the side BD, 22 98/100 parts, and the Angle at D, is a Right-angle, or 90°, and the sides DC, and CB, and the Angles B, and C, are required.

First to find the side DC, substract AD, 19 28/100* 1.92 parts, out of AC, 49 78/100 parts; there remains the Base DC; 30 50/100 parts: Thus have you the two sides of the Triangle, to wit the Base DC, 30 50/100 parts, and the Cathetus BD, 22 98/100 parts, and the Angle D between them is a Right-angle or 90°. Now you may find the Angle at B, by the Tangent (as in prop. 3. §. 2. chap. 5.) thus.

As Log. Cath. BD, 22 98/100 parts,

To Radius or S. 90°.

So Log. Base CD 30 50/100 parts.

To T. V. B. 53° 00'.

Secondly, For the Angle C, remember 'tis the Complement of the Angle B, 53°, to 90°; and therefore is the Angle C, 37° 00', required.

Thirdly, To find the Hypoth. BC, this is the Analogy or Proportion.

As S. V. B. 53° 00',

To Log. Base DC 30 50/100 parts.

So Radius or S. 90°,

To Log. Hypoth. BC 38 19/100 parts: Thus have you found all the required parts of the Tri∣angle ABC propounded, viz. the Angle C to be 37° 00', the Angle B, to be 93° 00', * 1.93 and the Side BC, 38 19/10•• parts, as required to be found.

Another way to perform the same.

Take the Sum of the* 1.94 two sides, and the diffe∣rence of the two sides; and work as followeth.

Now to find the two* 1.95 Angles B, and C, this is the Manner, and by this Analogy or Proportion, they are found out and known.

As Log. Z. cr^s. AB, and CA, 79 78/100 parts,

To Log. X. cr^s. AB, and CA; 19 78/100 parts,

So T. of ½ VV unknown, 65° 00',

To T. ½X. of VV, 28° 00'.

This difference of Angles 28° 00', add unto 65° 00', (half the difference of the unknown Angles) and it shall produce 93° 00', which is the greater Angle, and substracted from it, leaves 37° 00', which is the lesser Angle C: so have you the required Angles.

PROP. IV. The three sides of an Oblique-angled Triangle gi∣ven, to find the Angles.

In the Triangle ABC, the side AC, is 49 78/100 parts, the side AB, is 30 parts, and the side BC, is 38 19/100 parts; and the three Angles of the Tri∣angle are required.

The resolution of* 1.96 this Conclusion is thus. Take the Summ and Dif∣fer.* 1.97 of the two sides AB, and BC; And then work as follows: To find a Segment of the Base AC, to wit CE; say:

As Log. Base AC, 49 78/100 parts,

To Z. cr^s. AB, and BC; 68 19/100 parts,

So X. cr^s. AB, and BC; 8 19/100 parts,

To Log of a Segment of the Base AC, to wit C E 11 22/100 parts.

This Segment of the Base CE, 11 22/100 parts, being substracted from the whole Base AC, 49 78/100 parts, the remainder is EA 38 56/100 parts, in the middle of which as at D, the Perpendicu∣lar DB, will fall from the Angle B; and so di∣vide* 1.98 it into two Rectangled Triangles, to wit, ADB, and CDB, whose Base DA is 19 28/100 parts, which taken from AC 49 78/100 parts, leaves the Base of the greater Triangle CD 30 50/100 parts.

Now having the two Bases of these two Tri∣angles, and their Hypothenuses; to wit CD 30 50/100 parts, DA 19 28/100 parts, CB 38 19/100 parts, and BA 30

parts; you may find all their Angles, by the Rule of Opposite sides, to their Angles as afore.

I. In the Triangle CDB.

To find the Angles, this is the Analogy or Proportion.

As Log. BC 38 19/100 parts,

To Radius or S. 90°.

So Log. DC 30 50/100 parts,

To S. V. B 53° 00': whose Complement is the* 1.99 Angle at C 37° 00' unto 90: or a Quadrant.

II. In the Triangle ADB.

To find the Angles, this is the Analogy or Proportion.

As Log. AB, 30 parts,

To Radius or S. 90°.

So Log. AD 19 28/100 parts,

To S. V. B, 40° 00'.

The Complement whereof, unto 90° 00', is the Angle at A 50° 00'.

Now in the first Triangle CDB, there is found the Angle C, to be 37° 00', and the Angle B, to be 53° 00'.

In the second Triangle ADB, there is found the Angle A; to be 50° 00', and the Angle B, to be 40° 00'.

Now the two Angles at B, to wit 53° 00'; and 40° 00'; makes 93° 00', which is the Angle of the Oblique-angled Triangle ABC, at B: Thus the three Angles of the said given Triangle ABC, are found as was required, viz. the An∣gle A to be 50° 00', the Angle B to be 93° 00', and the Angle C to be 37° 00', as sought.

Thus I have sufficiently, fully and plainly explained all the Cases of Plain Right-lined Tri∣angles, both Right and Oblique-angled: I shall now fall in hand with Spherical Triangles, both Right and Oblique-angled.

SECT. IV. Of Spherical Rectangled Triangles.

And here first it will be necessary also to understand those few general Maxims or Rules, that are of special Moment, in the Doctrine of Spherical Triangles.

1. THat a Spherical Triangle is comprehen∣ded and formed, by the Conjunction and Intersection of three Arches of a Circle, described on the Surface of the Sphere or Globe.

2. That those Spherical Triangles, consisteth of six distinct parts, viz. three Sides and three Angles, any of which being known, the other is also found out and known.

3. That the three Sides of a Spherical Trian∣gle, are parts or Arches of three great Circles of a Sphere, mutually intersection each other: and as plain or Right-lined Triangles, are mea∣sured by a Measure, or Scale of equal parts: So these are measured, by a Scale or Arch of equal Deg••ees.

4. That a Great Circle is such a Circle that doth bessect the Sphere, dividing it into two equal parts; as the Equinoctial, the Ecliptick, the Meridians, the Horizon, &c.

5. That in a Right-angled Spherical Triangle, the Side subtending the Right-angle we call the Hypothenuse, the other two containing the Right∣angle we may simply call the Sides, and for distinction either of them may be called the Base or Perpendicular.

6. That the Summ of the Sides of a Spherical Triangle are less than two Semicircles or 360°.

7. That if two Sides of a Spherical Triangle be equal to a Semicircle; then the two Angles at the Base shall be equal to two Right-angles; but if they be less, then the two Angles shall be less; but if greater, then shall the two Angles be greater than a Semicircle.

8. That the Summ of the Angles of a Spheri∣cal Triangle, is greater than two Right-angles.

9. That every spherical Triangle is either a Right, or Oblique-angled Triangle.

10. That the Sines of the Angles, are in pro∣portion, unto the Sines of their opposite Sides; and the Sines of their opposite Sides, are in proportion unto the Sines of their opposite Angles.

11. That in a Right-angled Spherical Triangle, either of the Oblique-angles, is greater than the Complement of the other, but less than the Diffe∣rence of the same Complement unto a Semicircle.

12. That a Perpendicular is part of the Arch of a great Circle, which, being let fall from any Angle of a spherical Triangle, cutteth the oppo∣site Side of the Triangle at Right-angles, and so

divideth the Triangle into two Right-angled Tri∣angles, and these two parts (either of the Sides or Angles) so divided must be sometimes added together, and sometimes substracted from each other, according as the Perpendicular falls with∣in or without the Triangle.

PROP. I. Case 1. A Side and an Angle adjacent thereunto being gi∣ven, to find the other Side.

In the Triangle ABC, there is given the Side AB 27° 54'; and the Angle A 23° 30', and the Side BC is required, to find which this is the Analogy or Proportion.* 1.100* 1.101

PROP. II. Case 2. A Side and an Angle adjacent thereunto being gi∣ven, to find the other Oblique-angle.

In the Triangle ABC, there is given the Side AB 27° 54', and the Angle A 23° 30',* 1.102 and the Angle at C is required, to find which say by this Analogy or Proportion.

As the Radius or S 90° 00',

To Sc. of cr. AB 27, 54.

So is S. V. at A 23, 30,

To Sc. V. at c 69, 22 required.

PROP. III. Case 3. A Side and an Angle adjacent thereunto being given, to find the Hypothenuse.

In the Triangle ABC, there is given the Side* 1.103 AB 27° 54', and the Angle at A 23° 30', and the Hypothenuse AC, is required; which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. of V. at A 23, 30.

So is Tc cr. AB, 27, 54.

To Tc. Hypothenuse AC, 30, 00 required.

PROP. IV. Case 4. A Side and an Angle opposite thereunto being given, to find the other Oblique-angle.

In the Triangle ABC, there is given the Side* 1.104 BC 11° 30', and the Angle A 23° 30', and the Angle C is required, to find which, say by this Analogy or Proportion.

As Sc. cr. BC, 11° 30',

To Radius or S. 90, 00.

So is Sc. V. at A, 23, 30,

To S. V. at C. 69, 22, as required.

PROP. V. Case 5. A Side and the opposite Angle given, to find the Hypothenuse.

In the Triangle ABC, there is given the* 1.105 side BC 11° 30', and the Angle at A 23° 30', and the Hypothenuse AC, is required, which may be found by this Analogy or Proportion.

As S. V. at A 23° 30',

To Radius or S. 90, 00.

So is Ser. BC 11. 30,

To S. Hypothenuse AC 30, 00. as required.

PROP. VI. Case 6. A side and the opposite Angle given, to find the other side.

In the Triangle ABC, there is given the side BC 11° 30', and the Angle at A 23° 30', and the side AB is required, to find which this is the Analogy or Proportion.

As Radius or S 90° 00',* 1.106

To Tc. of V. at A. 23. 30,

So is T. cr. BC 11, 30,

To S. of cr. AB 27. 54 as was required.

PROP. VII. Case 7. The Hypothenuse, and an Oblique Angle given, to find the side adjacent thereunto.

In the Triangle ABC, there is given the

Hypothenuse AC, 30° 00', and the Angle A 23° 30', and the side AB, is required, which is found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. V. at A, 23, 30.* 1.107

So is T. Hypoth. AC, 30, 00,

To T. cr. AB, 27, 54, as was required.

PROP. VIII. Case 8. The Hypothenuse, and an Oblique-angle given, to find the opposite Side.

In the Triangle ABC, there is given the Hypothenuse AC, 30° 00', and the Angle at A 23° 30', and the Side BC, is required, which is found by this Analogy or Proportion.* 1.108

As the Radius or S. 90° 00',

To S. Hypoth. AC, 30, 00.

So is S. V. at A, 23, 30,

To the S. cr. BC, 11, 30. which was required.

PROP. IX. Case 9. The Hypothenuse, and an Oblique-angle given, to find the other Oblique-angle.

In the Triangle ABC, there is given the Hy∣pothenuse AC 30° 00', and the Angle A, 23° 30', now the Angle at C, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. Hypoth. AC, 30, 00.

So is T. of V. at A, 23, 30,* 1.109

To Tc. of V. at C. 69, 22, as was required.

PROP. X. Case 10. The sides given, to find the Hypothenuse.

In the Triangle ABC, there is given the side AB 27° 54', and the side BC 11° 30', and the Hypothenuse AC is required, to find which say by this Analogy or Proportion.

As the Radius or S. 90° 00',* 1.110

To Sc. cr. BC. 11, 30.

So is Sc. cr. AB 27, 54,

To Sc. Hypothenuse AC 30, 00. required.

PROP. XI. Case 11. The sides given, to find an Angle.

In the Triangle ABC, there is given, the side AB 27° 54', and the side BC 11° 30', and the Angle at A, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',* 1.111

To S. cr. AB. 27, 54.

So is Tc. cr. BC. 11, 30,

To Tc. of V. at A. 23. 30. as required.

PROP. XII. Case 12. The Hypothenuse, and a side given, to find the o∣ther side.

In the Triangle ABC, there is given, the Hy∣pothenuse* 1.112 AC 30° 00', and the side AB 27° 54' and the side BC is required, which may be

found by this Analogy or Proportion.

As Sc. cr. AB. 27° 54',

To Radius or S. 90 00.

So is Sc. Hypothenuse AC. 30° 00',

To Sc. cr. BC. 11° 30' as required.

PROP. XIII. Case 13. The Hypothenuse, and a Side given, to find the contained Angle.

In the Triangle ABC, there is given the Hy∣pothenuse* 1.113 AC 30° 00', and the side AB 27° 54', and the Angle at A is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To T. cr. AB. 27° 54'

So is Tc. Hypoth. AC 30° 00',

To Sc. of V. at A, 23° 30', as required.

PROP. XIV. Case 14. The Hypothenuse, and a Side given, to find the oppo∣site Angle.

In the Triangle ABC, there is given the Hy∣pothenuse AC 30° 00', and the side AB 27° 54', ••ow the Angle C, is required, which may be* 1.114 〈…〉〈…〉ound by this Analogy or Proportion.

As the S. Hypoth. C, 30° 00',

To Radius or S. 90° 00'.

So is S. of cr. AB, 27° 54',

To S of V. at C. 69 22, as required.

PROP. XV. Case 15. The Oblique Angles given, to find either Side.

In the Triangle ABC, there is given the An∣gle A 23° 30', and the Angle at C 69° 22', and the side BC, is required, which may be found by this Analogy or Proportion.

As the S. of V. at C, 69° 22',* 1.115

To the Radius or S. 90° 00'.

So is the Sc. of V. at A, 23° 30',

To the Sc. of cr. BC, 11° 30', as required.

PROP. XVI. Case 16. The Oblique-angles given, to find the Hypothenuse.

In the Triangle ABC, there is given the An∣gle A 23° 30', the Angle C, 69° 22', and the Hypothenuse AC, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',* 1.116

To Tc. of V. at C. 69°, 22',

So is Tc. of V. at A, 23 30,

To Sc. Hypoth. AC, 30 00, as required.

SECT. V. Of Oblique-angled Spherical Triangles.

PROP. I. Case 1. Two Sides, and an Angle opposite to one of them given, to find the other opposite Angle.

IN the Triangle ADE, there is given the Side* 1.117 AE, 70° 00', the Side ED, 38° 30', and the Angle A, 30° 28', now the Angle at D, is re∣quired, to find which this is the Analogy or Proportion.

As S. cr. DE, 38° 30',

To S. V. at A, 30 28.

So is S. cr. AE, 70 00,

To S. V. at D, 130 03,* 1.118 required.

PROP. II. Case 2. Two Angles and a Side opposite to one of them gi∣ven, to find the Side opposite to the other.

In the Triangle ADE, there is given the An∣gle* 1.119 at D, 130° 03', the Angle E, 31° 34', and the Side AE, 70° 00', now the Side AD, is re∣quired, which may be found by this Analogy or Proportion.

As S. V. at D, 130° 03',

To S. cr. AE, 70 00.

So is S. V. at E, 31 34,

To S. cr. AD. 40 00, required.

PROP. III. Case 3. Two Sides and an Angle included between them being known, to find the other Angles.

In the Triangle ADE, there is given the Side* 1.120 AE, 70° 00', the Side AD, 40° 00', and the An∣gle A 30° 28', Now the Angles D, and E, are re∣quired, which is thus found: take the Sum and Difference of the two Sides, and work as follow∣eth, saying.* 1.121

As S. ½ Z. cr^s. AE and AD, 55° co',

To S. ½ X. cr^s. AE and AD, 15 00.

So is Tc. ½ V. at A, 15 14,* 1.122

To T. ½ X. VV. D and E. 49 1430".

AGAIN.

As Sc. ½ Z. cr^s. AE and AD, 55° 00',

To Sc. ½ X. cr^s. AE and AD, 15° 00'.

So is Tc. ½ V. at A, 15° 14',

To T. ½ Z. VV. D and E, 80 48 30".

This difference of the Angles unknown D and E, 49° 14' 30", being added unto the half Sum of the Angles 80° 48' 30", (unknown) produceth the Greater Angle D 130° 03', and substracted from it, leaves the Lesser Angle E, to wit 31° 34'.

PROP. IV. Case 4. Two Angles, and their Interjacent side being known, to find the other sides.

In the Triangle ADE, there is given the Angl••* 1.123 at A 30° 28', and the Angle at D 130° 03', and their Interjacent-side AD 40° 00', and the Sides DE and EA, are required: Which is thus found.

Take the Sum and Dif∣fference of the two An∣gles,* 1.124 and work as fol∣loweth, saying.

As S. ½ Z. of VV. A and D, 80° 15' 30", * 1.125 * 1.126

To S. ½ X. of VV. A and D, 49 47 30.

So is T. ½ cr. AD, 20 00 00,

To T. ½ X. cr^s. DE and EA, 15 45 00.

AGAIN Say.

As Sc. ½ Z. of VV. A and D, 80° 15' 30",

To Sc. ½ X. of VV. A and D, 49 47 30.

So is T. ½ cr. AD, 20 00 00,

To T. ½ cr^s. Z. DE and AE. 54 15 00.

Add the half Difference of the Sides DE and AE, 15° 45', unto half the Sum of the Sides DE and AE, 54° 15'. It produceth the greater Side, the Side AE 70° 00', but if deducted from it, leaves the lesser Side ED, which is 38° 30', as was required.

PROP. V. Case 5. Two Sides and an Angle opposite to one of them gi∣ven, to find the third side.

In the Triangle ADE, there is given the Side* 1.127 AE 70° 00', the Side DE 38° 30', and the Angle A 30° 28', the Side AD is required.

First by Case 1. Prop. 1. I find the Angle at D to be 130° 03', and then proceed thus

First take the Sum and Difference of the two Angles; then also find the Difference of the two Sides given, and then work as followeth.* 1.128 * 1.129

Now say,

As S. ½ X. VV. D and A, 49° 47' 30",

To S. ½ Z. VV. D and 〈…〉〈…〉, 80 15 30.

So is T. ½ X. cr^s. AE and ED, 15 45 00,

To T. ½ cr. AD. 20° 00' 00": which doubled giveth the Side AD, 40 00 00, as was required.

PROP. VI. Case 6. Two Angles and a Side opposite to one of them gi∣ven, to find the third Angle.

In the Triangle ADE, there is given the An∣gle* 1.130 A 30° 28', the Angle D 130° 03', and his opposite Side AE 70° 00', and 'tis required to find the Angle at E.

First by Prop. 2. Case 2. I find the Side DE, opposed to the Angle A; to be 38° 30', then proceed thus.

Fi••st find the Sum and Difference of the Sides. Then find the Difference of the Angles.* 1.131 * 1.132

Now say,

As S. ½ X. cr^s. DE and AE 15° 45',

To S. ½ Z. cr^s. EA and DE 54 15.

So is T. ½ X. VV. D and A 49 47 30",

To Tc. ½ V. at E 15° 47' 00". which doubled giveth the Angle at E 31 34, as required.

PROP. VII. Case 7. Two Sides and an Angle opposite to one of them gi∣ven, to find the Included Angle.

In the Triangle ADE, there is given the Side* 1.133 AE 70° 00', the Side ED 38° 30', and the An∣gle opposite thereunto at A 30° 28', and the Angle E is required.

First by Prop. 1. Case 1. I find the Angle D, opposite to AE, to be 130° 03', then proceed thus.

First find the Difference of the Angles, then find the Sum and Difference of the Sides.* 1.134 * 1.135

Now say,

As S. ½ X. cr^s. AE and ED 15° 45',

To S ½ Z. cr^s. AE and ED 54 15.

So is T. ½ X. of VV. D and A 49 47 30",* 1.136

To Tc. ½ V. at E 15° 47'. Which doubled is the Angle at E 31° 34', as was required.

PROP. VIII. Case 8. Two Angles and a Side opposite to one of them be∣ing known, to find the Interjacent Side.

In the Triangle ADE, there is given the An∣gle E 31° 34', the Angle D 130° 03', and his opposite Side AE 70° 00', Now the Side ED is required.

First by Prop. 2. Case 2. I find AD opposed to E, to be 40° 00', and then work thus.

Take the Sum and Difference of the Angles, then also find the Difference of the two Sides:* 1.137 * 1.138 * 1.139

Now say,

As S. ½ X. VV D and E 49° 14' 30",

To S. ½ Z VV D and E 80 48 30.

So is T ½ X cr^s. AD and AE 15 00 00",

To T. ½ cr^s. ED, 19° 15' 00", which being doubled is the Side ED 38° 30', as required.

PROP. IX. Case 9. Two Sides and their Included Angle being known, to find the third Side.

In the Triangle APZ, there is given the Side* 1.140 ZP 38° 30', the Side PA 70°, and the Angle P, let be 31° 34, and the Side AZ is required.

The Resolution of this Case depends on the Catholike proposition of the Lord of Marchiston, by supposing the Oblique-Triangle to be divided (by a supposed Perpendicular falling either within or without the Triangle) into two Rect∣angulars.

Now in the Triangle AZP, let fall the Per∣perpendicular ZR; so is the Triangle AZP divi∣ded into two Rectangulars ARZ and ZRP. Now the Side AZ may be found at two Opera∣tions thus: say,

As the Radius or S. of 90° 00'

To Sc. of the included V, P. 31 34.

So is T. of the lesser Side PZ. 38 30,

To T. of a fourth Arch. 34 08.

If the contained Angle be less than 90°, take this fourth Arch from the greater Side; but if it be greater than 90°, from its Complement unto 180°, the Remainder is the Residual Arch: Now again say,

As Sc. of the fourth Arch. 34° 08'

To Sc. Residual Arch. 35 52

So Sc. of the lesser Side PZ. 38 30

To Sc. AZ the Side required. 40 00

☞ 1.141 But note that many times the Perpendicu∣lar will fall without the Triangle, as it doth* 1.142 now within; in such case the Sides of the Tri∣angle must be continued, so will there be two Rectangulars, the one included within the o∣ther: as in the Triangle HIK, the Perpendicular let fall is KM, falling on the Side HE, and so the two Rectangulars found thereby will be IM K, and KMH, and so by the directions in the former proposition find out the Side IK, if re∣quired to be found.

PROP. X. Case 10. Two Angles and their Interjacent Side known, to find the third Angle.

In the Triangle AZP, there is given the Side ZP 38° 30', the Angle P 31° 34', and the An∣gle Z 130° 03', and the Angle at A is requi∣red.

First the Oblique-Triangle AZP, being redu∣ced into two Rectangulars ARZ, and ZRP, by* 1.143 Case 9 aforegoing, I find the Angle RZP, to be 64° 19', (in the Triangle ZRP.) which ta∣ken out of Angle AZP 130° 03', leaves the Angle AZR 65° 44': Now the Angle A is found by this Analogy or Proportion.

As S. V. PZR, 64° 19',

To S. V. AZR 65 44,

So is Sc. V. at P 31 34,

To Sc. V. at A 30 28: which was required to be found out and known.

PROP. XI. Case 11. Three Sides given, to find an Angle.

In the Triangle APZ, the Side AZ is 40° 00', the Side ZP is 38° 30', the Side AP is 70° 00', and the Angle Z is required. To find which do thus.

Add the three Sides: together, and from half* 1.144 their Sum, deduct the Side opposite, to the re∣quired Angle: and then proceed as you see in the Operation following.* 1.145* 1.146

½Sum is 65° 07' 30", the Sc. ½ V. at Z which doubled is 130° 03' 12"; the Angle at Z required.

PROP. XII. Case 12. Three Angles given, to find a Side.

In the Triangle AZP, the Angle A is 30° 28'* 1.147 11", the Angle Z 130° 03' 12", the Angle P is 31° 34' 26", and the Side AZ, opposite to P, is required.

This Case is likewise performed as the former Case or Proposition, the Angles being conver∣ted into Sides, and the Sides into Angles, by taking the Complement of the greatest Angle unto 180°: see the work.* 1.148* 1.149 which being doubled, gives the Side AZ 40° 00 required to be found out and known

☞ But if the greater Side AP were required the Operation would produce the Complem〈…〉〈…〉 thereof unto a Semicircle or 180°; therfo〈…〉〈…〉

substract it from 180°, it leaves the remaining required Side sought.

Thus I have laid down all the Cases of Tri∣angles, both Right-lined and Spherical; either Right, or Oblique-angled; I might hereunto have annexed many Varieties unto each Case, and some fundamental Axioms, which somewhat more would have Illustrated and Demonstrated those Cases, and Proportions; but because of the smallness of this Treatise, which is intended more for Practice than Theory, I have for brevi∣ty sake omitted them, and refer you for those things to larger Authors, who have largely discoursed thereon to good purpose.

CHAP. VI. Of ASTRONOMY.

* 1.150 ASTRONOMY is an Art Ma∣thematical, which measureth the distinct course of Times, Days, Years, &c. It sheweth the Distance, Magnitude, Natural Motions, Appearances and Passions, pro∣per unto the Planets, and fixed Stars, for any time past, present and to come; by this we are certified of the Distance of the starry Sky, and of each Pla∣net, from the Center of the Earth, and the Mag∣nitude of any fixed Star or Planet, in respect of the Earth's Magnitude.

SECT. I. Of Astronomical Definitions.

• * 1.151 1. ASphere or Globe is a solid Body, containing onely one Superficies, in whose middle there is a point (called the

• Center,) from which all right or streight lines drawn unto the Circumference or Superficies, are Equal.
• 2. The Poles of the World, are two fixed points* 1.152 in the Heavens Diametrically opposite the one to the other, the one called the Artick or North∣Pole; noted in the Scheme by P. The other is called the Antartick, or South-Pole; as S. and is not to be seen of us, being in the lower Hemisphere.
• 3. The Axis of the World, is an imaginary* 1.153 line drawn from the North-Pole, through the Center of the Earth, unto the South-Pole, about which the Diurnal motion is performed, from the East to the West; as the line PS.
• 4. The Meridians are great Circles, concur∣ring* 1.154 and intersecting one another, in the Poles of the World, as PES, and Pc S.
• 5. The Equinoctial, or Equator, is a great Cir∣cle, 90 deg. distant from the Poles of the World, cutting the Meridians at Right-angles, and di∣videth the World into two Equal parts, called* 1.155 the Northern, and Southern Hemispheres, as E ♎ Q. in Scheme 42.
• 6. The Ecliptick is a great Circle, crossing the Equinoctial, in the two opposite points Aries and Libra, and maketh an Angle therewith (called, its Obliquity) of 23° 30', represented by ♋ ♎ ♑. This Circle is divided into 12 Sines, each containing 30° 00': As A∣ries* 1.156 ♈, Taurus ♉, Gemini ♊, Cancer ♋, Leo ♌, Virgo ♍, (which are called Northern Sines) Libra ♎; Scorpio ♏, Sagitarius ♐; Capricornus ♑, Aquarius ♒, and Pisces ♓; these are called Sou∣thern Sines.

• 7. The Zodiack is a Zone or Girdle, having 8 deg. of Latitude on either side the Ecliptick, in which space the Planets make their revolu∣tion. This Circle is a Circle which regulates the Years, Months, and Seasons, * 1.157 and is distin∣guished* 1.158 in the Scheme by the 12 Sines.
• 8. The Colures are two Meridians, dividing the Ecliptick, and the Equinoctial, into four e∣qual parts; one of which passeth by the Equi∣noctial* 1.159 points Aries, and Libra, and is called the Equinoctial Colure, as P ♎ S. The other by the beginning of Cancer, and Capricorn, and is cal∣led the Solstitial Colure, as P ♋, S ♑.
• 9. The Poles of the Ecliptick are two points, 23° 30' distant from the Poles of the World, as I and K.* 1.160
• 10. The Tropicks are two small Circles, Pa∣rallel unto the Equinoctial, and distant there∣from 23° 30', limiting the Sun's greatest declina∣tion. The Northern Tropick passeth by the be∣ginning of Cancer, and is therefore called the Tropick of Cancer, as ♋ a D. The Southern Tro∣pick* 1.161 passeth by the beginning of Capricorn, and is therefore called the Tropick of Capricorn; as B b ♑.
• 11. The Polar Circles, are two small Circles parrallel to the Equinoctial, and distant there∣from 66° 30'; and from the Poles of the World

• 23°. 30'. That which is adjacent unto the North Pole, is called the Artick Circle, as G d I. and the other the Antartick Circle, as Kd M.* 1.162
• 12. The Zenith, and the Nadir, are two points, Diametrically opposite the one to the o∣ther: the Zenith is the Vertical point, or the point over our heads, as Z, The Nadir, is oppo∣site thereto as the point N.
• 13. The Azimuths or Vertical Circles are great Circles of the Sphere, concurring and in∣tersecting each other, in the Zenith, and Nadir, as Z f N.
• 14 The Horizon, is a great Circle, 90 deg.* 1.163 distant from the Zenith, and Nadir; cutting all the Azimuths, at Rightangles, and dividing the World into two equal parts, the upper and visible Hemisphere, and the lower and invisible Hemisphere, represented by H ♎ R.
• 15. The Meridian of a Place, is that Meridi∣an, which passeth by the Zenith, and Nadir, of* 1.164 the place as P Z S N.
• 16. The Alinicanthars, or Parallels of Alti∣tude, are small Circles, parrallel unto the Hori∣zon, (imagined to ••pass through every degree and minute of the Meridian, between the Ze∣nith, and Horizon, B a F.
• 17. Parallels of Latitude, or Declination, are small Circles parallel unto the Equinoctial; they* 1.165 are called Parallels of Latitude, in respect to any place on the Earth, and Parallels of Declination, in respect of the Sun, or Stars, in the Heavens.
• 18. The Latitude of a place, is the height of the Pole above the, Horizon; or the distance be∣tween the Zenith and the Equinoctial.

• 19. The Latitude of a Star, is the Arch of a Circle, contained betwixt the Center of a Star, and the Ecliptick line: this Circle making Right∣angles, with the Ecliptick, is accounted either Northward or Southward; according to the Sci∣tuation of the Star.
• 20. Longitude on Earth is measured by an Arch of the Equinoctial, contained between the Primary Meridian, (or Meridian of that place where Longitude is assigned to begin) and the Meridian of any other place, counted always Easterly.
• 21. The Longitude of a Star, is that part of the Ecliptick, which is contained between the Star's place in the Ecliptick, and the beginning of Aries, counting them according unto the suc∣cession of Sines.
• 22. The Altitude of the Sun or Stars, is the Arch of an Azimuth, contained betwixt the Center of the Sun, or Star, and the Horizon.
• 23. Ascension is the rising of any Star, or part of the Equinoctial, to any degree above the Horizon; and Descension is the setting of it.
• 24. Right Ascension, is the number of Degrees and Minutes of the Equinoctial; (i. e. from the beginning of Aries) which cometh unto the Meridian, with the Sun or Stars; or with any portion of the Ecliptick.
• 25. Oblique-Ascension, is an Arch of the Equi∣noctial, between the beginning of Aries, and that part of the Equinoctial which riseth with the Center of a Star; or with any portion of the Ecliptick in an Oblique Sphere: and Oblique De∣scention, is that part of the Equinoctial, tha•• setteth therewith.

• 26. The Ascentional difference, is an Arch of the Equinoctial, being the difference betwixt the Right and Oblique-Ascension.
• 27. The Amplitude, of the Sun or Stars, is* 1.166 the distance of the rising or setting thereof, from the East or West point of the Horizon.
• 28. The Parallax, is the difference between the true and apparent place of the Sun or Star; so the true place in respect of Altitude, is in the line ACE, or ADG, the Sun or Star being at C, or D, and the apparent place in the Line BCF, and BDH, so likewise the Angles of the Parallax are ACB, or ECF; and ADB, or GDB: also in the said Scheme, ABK representeth a* 1.167 Quadrent (of the Globe or Earth,) on the Earth's Superficies: A the Center of the Earth, and B any point of the Earth's Surface.
• 29. The Refraction of a Star, is caused by the Atmosphere, or Vapourous thickness of the Air near the Earth's Superficies, whereby the Sun and Stars seem always to rise sooner, and and set la∣ter than really they do.

SECT. II. Of Astronomical Propositions.

PROP. I. The Distance of the Sun from the next Equinoctial point (either Aries or Libra) being known, to find his Declination.

THE Analogy or Proportion.

As Radius or S. 90°,

To S. of the Sun's distance from the next Equi∣noctial point,

So it S. of the Sun's greatest Declination,

To the S. of the Sun's present Declination sought.

PROP. II. The Sun's place given, to find his Right-Ascen∣sion.

This is the Analogy or Proportion.

As Radius or S. 90°,

To T. of the Sun's Longitude from the next Equi∣noctial point,

So is the Sc. of his greatest Declination,

To T. of his Right-Ascension from the next Equi∣noctial point.

PROP. III. To find the Sun's place or longitude from Aries, his Declination being given.

This is the Analogy or Proportion.

As S. of the Suns greatest Declination,

To Radius or S. 90° 00',

So is S. of his present Declination,

To S. of the Suns Place or Longitude from Aries* 1.168

PROP. IV. By knowing the Suns Declination, to find his Right Ascension.

This is the Analogy or Proportion.

As Radius or S. 90°,

To Tc. of the Suns greatest Declination,

So is T. of the Declination given,

To S. of the Suns right Ascension required† 1.169.

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PROP. V. By knowing the Latitude of a Place, and the Suns Declination, to find the Ascensional Difference.

This is the Analogy or Proportion.

As Radius or S. 90°,

To Tc. of the Latitude given,

So is T. of the Suns Declination given,

To the S. of the Ascensional difference required.

☞ Note that if you reduce the degrees, &c. of the Ascensional difference into hours, it will shew you how much the Sun riseth, or setteth before, or after six a Clock, in that Latitude.

PROP. VI. To find the Suns Oblique Ascension or Descension.

First find the Ascensional Difference by the 5th Proposition, and his Right-ascension by the fourth: Now if the Suns Declination be Northerly, de∣duct the Ascentional Difference out of his Right Ascension, from the beginning of ♈, (for the six Northern Signs ♈ ♉ ♊ ♋ ♌ ♍) it leaves the Oblique Ascension; and added unto the Right∣ascension, giveth the Oblique-descension.

But if the Suns Declination be Southerly, the Ascentional Difference, added to the Right-ascensi∣on, (for the six Southern Signs ♎ ♏ ♐ ♑ ♒ ♓) giveth the Right-ascension, and substracted there from leaves the Oblique-descension.

[illustration]

Plate 11 Page 105

PROP. VII. By knowing the Suns Declination, and the Latitude of a Place, to find the Suns Amplitude.

This is the Analogy or Proportion.

As Sc. of the Latitude,

To the Radius or S. 90°.

So is the S. of the Suns Declination,

To the S. of the Amplitude from the East or West Points of the Horizon.

PROP. VIII. By knowing the Suns Declination and Amplitude, from the North part of the Horizon, to find the Latitude.

This is the Analogy or Proportion.

As Sc. of the Amplitude from the North,

To Radius or S. 90° 00'

So is S. of his Declination given,

To Sc. of the required Latitude.

PROP. IX. By knowing the Latitude of a place, and the Sun's Declination, to find at what time the Sun will be on the true East or West Points.

The Analogy or Proportion is.

As T. of the given Latitude,

To T. of the Sun's Declination propounded,

So is Radius or S. 90° 00',

To, Sc. of the Hour from Noon.

〈1 page duplicate〉〈1 page duplicate〉

PROP. X. By knowing the Sun's Declination, and Latitude of a place, to find his Altitude at six a Clock.

This is the Analogy or Proportion.

As Radius or S. 90° 00',

To S. of the Sun's Declination,

So is S. of the Latitude of the place,

To S. of the Sun's Altitude at six a Clock.

PROP. XI. By knowing the Latitude of a place, and the Sun's Declination, to find the Azimuth at six.

This is the Analogy or Proportion.

As Radius or S. 90° 00',

To the T. of the Sun's Declination,

So is Sc. of the Latitude of the place,

To the T. of the Azimuth sought.

PROP. XII. By knowing the Latitude of a place, and the Sun's Declination, to find the Sun's Altitude when he i•• on the true East or West points.

This is the Analogy or Proportion.

As S. of the Latitude,

To the Radius or S. 90° 00',

So is the S. of the Declination,

To the S. of the Sun's Altitude being due Ea•••• or West.

PROP. XIII. To find the Sun's Altitude at any time of the day.

The Analogy or Proportion is.

As Radius or S. 90° 00',

To Tc. of the Poles height,

So is S. of the Sun's Distance,

From the Hour of Six,

To the T. of an Arch: which being substract∣ed from the Sun's Distance from the Pole; say,

As Sc. of the Arch found,

To Sc. of the remaining Arch of the Sun's Di∣stance from the Pole,

So is S. of the Poles height,

To the S. of the Sun's Altitude at the Hour required.

PROP. XIV. By knowing the Latitude of a Place, with the Sun's Declination, and Altitude, to find the Hour of the Day.

To solve this Conclusion do thus: get the Sum of the Complements of the Latitude, Declina∣tion and Altitude given* 1.170, Then find the Difference betwixt their half Sum, and the Complement of the Altitude; then say,

As Radius or S. 90° 00',

To Sc. of the Sun's Altitude,

So is Sc. of the Latitude of the Place,

To a fourth Sine: then again say,

As the fourth S.

To the S. of ½ Z. of the Lat. Declin. and Alt.

So is the S. of X. of the Altitude to the ½ Z,

To a fifth S. unto which Sine, if you add the Radius or 90° 00', half that Sum shall be the Sine of an Arch, whose double Complement is the Hour from the Meridian.

PROP. XV. To find the Time of the Sun's Rising or Setting, and consequently the Length of the Day or Night.

To resolve this Conclusion, first by prop. the 5. find the Ascensional Difference, which redu∣ced into Hours, and Minutes of Time, by allow∣ing for every 15 Deg. one Hour, and for every Deg. less than 15°, 4', of Time, and for every 15 Min. one Minute of Time.

Secondly, If the Sun's Declination be Norther∣ly, the Ascentional Difference added unto 6 Hours, gives the Time of Sun-setting, and sub∣stracted therefrom, leaves the Time of Sun∣Rising: On the contrary, if the Sun's Declinati∣on be Southerly, the Ascentional Difference added unto 6 Hours, gives the Time of Sun-Rising, and deducted therefrom, the Time of Sun∣setting.

Thirdly, If you double the Time of Sun∣Rising, it gives you the length of the Night; and the Time of Sun-setting, the length of the Day.

PROP. XVI. The Sun's Declination, Altitude and Azimuth known, to find the Hour of the Day.

The Analogy or Proportion is.

As the Sc. of the Sun's Declination,

To the S. of the Azimuth,

So is the Sc. of the Altitude,

To the S. of the Hour from Noon: which con∣verted into Time, will shew the Hour of the Day.

PROP. XVII. By knowing the Sun's Declination, Altitude, and Hour from Noon, to find the Azimuth.

The Analogy or Proportion is.

As Sc. of the Sun's Altitude,

To S. of the Hour from Noon,

So is Sc. of the Sun's Declination,

To the S. of the Azimuth, required.

PROP. XVIII. By knowing the Latitude of a place, the Altitude of the Sun, and the Hour from Noon, to find the Angle of the Sun's Position.

This is the Analogy or Proportion.

As Sc. of the Sun's Altitude,

To S. of the Hour from Noon,

So is Sc. of the Latitude,

To S. of the V. of the Sun's Position, at the time of the Question.

PROP. XIX. By knowing the Sun's Altitude, Declination, and Azimuth; to find the Latitude.

The Analogy or Proportion is.

As S. of the Sun's Azimuth,

To S. of his Distance from the North-pole,

So is S. of V of the Sun's Position,

To Sc. of the Latitude required.

PROP. XX. To find the length of the Crepusculum, or Twilight

The Crepusculum or Twilight, is nothing else but the Refraction of the Sun's Beams in the Density of the Air. Which the Learned Pet. Nonnius found the length of the Crepusculum (by his many strict obser∣vations * 1.171) to continue from the time of the S〈…〉〈…〉 passing below the Hori∣zon of a place, untill the Sun had run below the said Horizon 18° 00', and then followed the shut∣ting in of the Twilight, and untill the Sun was

departed so low the Twi∣light continued. — To find which observe this Ana∣logy or Proportion.

As Radius or S. 90°,

To Sc. of the Sun's Declination,

So is Sc. of the Poles-height,

To a fourth Sine: which keep.

Then out of the Sun's Distance from the South-Pole, subduct the Complement of the Pole; and of that remains and the degrees 62, being added to it, their Sum and Difference found, say again.

As the fourth Sine found,

To S. ½ Z of the remainder and 62° 00',

So is S. ½ X. of the remainder and 62° 00',

To a Number, which being multiplyed by the Radius is equal unto the Quadrat of the Sine of the ½ Angle of the Sun's Distance at the Ending of the Twilight, from Noon next ensuing.

Then from the Sun of the whole Angle con∣verted into Hours, substract the Hour of the Sun's setting * 1.172, it gives you the length of the Crepusculum, or Twilight.

But the Sun being in the Winter Tropick, makes the Twilight longest of a∣ny* 1.173 Twilight, the whole Winter half year: Now in a certain Parallel, be∣twixt that Tropick, and the Equinoctial is the shortest Crepusculum: the Declination of which Parallel, is thus found.

As the Tc. of the Pole,

To the S of the Pole,

So is the T. of 99° 00',

To S. of the Declination of the Parallel, in which the Sun maketh the shortest Crepusculum of the Year.

But before the Crepusculum come to be short∣est, there is another Parallel, in which the Crepusculum is equal to that of the Equinoctial: the Declination of which is found thus.

As the Radius or S. 90° 00',

To S. of the Poles Elevation or Altitude,

So i•• S. of 18° 00',

To S. of the Declination of the Parallel, in which the Sun maketh the Crepusculum equal to that in the Equinoctial.

PROP. XXI. To find the Quantity of the Angles, which the Cir∣cles of the 12 Houses make with the Meri∣dian.

This is the Analogy or Proportion.

As the Radius or S. 90°,

To T. of 60°: for the 11th, 9th, 5th and 3d House, Or to the T. 30° for the 12th, 8th, 6th, and 2d House,

So is the Sc. of the Pole,

To the Tc. of any House with the Meridian.

PROP. XXII. To find the Right Ascension of the Point in the Equinoctial: and also the Point in the Ecliptick; called Medium Coeli or Cor Coeli.

First, To find the Right Ascension of the Point of the Equinoctial; called Medium Coeli, vel Cor Coeli, find out the Sun's Right Ascension, by prop. 2. Then reduce the whole Time from Noon last past into degrees, which add unto the right Ascension of the Sun, so shall their A∣gragat, be the right Ascension of the point, which in the Equinoctial, is called Medium Coeli, vel Cor Caeli, required to be found.

Secondly, By the 2 propositions aforegoing, you may find the right Ascension of the point in the Ecliptick Culminant in the Meridian, cal∣led Medium Coeli vel Cor Coeli, which is the Cuspis of the tenth House: and his Declination by prop. the first.

PROP. XXIII. To find the Angle of the Ecliptick with the Meri∣dian.

The Analogy or Proportion is.

As the Radius or S. 90°,

To S. of the Sun's Greatest Declination,

So is Sc. of the Sun's right Ascension, from the next Equinoctial point,

To Sc. of the V. of the Ecliptick, with the Me∣ridian.

PROP. XXIV. To find the Angle of the Ecliptick with the Ho∣rizon.

The Analogy or Proportion is.

As Radius or S. 90°,

To Sc. of the Altitude of Cor Coeli,

So is S. of the V. Ecliptick with the Meridian,

To Sc. of the V. of the Ecliptick and Horizon sought.

PROP. XXV. To find the Amplitude Ortive of the Ascendent, or Horoscopus.

This is the Analogy or Proportion.

As Radius or S. 90°,

To S. of Altitude of Med. Coeli,

So is T. of V. Ecliptick with the Meridian,

To Tc. of the Amplitude Ortive of the Ascen∣dent, or the distance of the Azimuth from the Meridian.

PROP. XXVI. To find the Ascendent degree of the Ecliptick, or the Cuspis of the first House.

The Amplitude Ortive of the Ascendent, is equal to the Distance of the Azimuth of 90°, from the Meridian, wherefore the Cuspis of the

first House, or Ascendent Degree of the Ecliptick, may be found thus.

As Radius or S. 90°,

To Sc. of the V. Ecliptick with the Meridian,

So is Tc. of the Altitude of Med. Coeli,

To T. of the Distance of Med. Coeli, from the Ascendent Degrees.

PROP. XXVII. To find the Distance of the Cuspis of any House, from Med. Coeli.

This is the Analogy or Proportion.

As Sc. of the remaining part of V. of the E∣cliptick with the Meridian, (found by prop. 28.)

To Sc of the former part of the V,

So is T. of the Altitude of Med. Coeli,

To T. of the Distance of the Cuspis of that House sought, from Med. Coeli.

PROP. XXVIII. To find the parts of the Angle of the Ecliptick with the Meridian, cut with an Arch perpendicular to the Circle of any of the Houses.

The Analogy or Proportion is:

As Radius or S. 90°,

To Sc. Altitude of Med. Coeli,

So is T. of the Circle of any House with the Meridian,

To Tc. of that part of that Angle which is next the Meridian:

Then substract that part found, out of the whole Angle, for the remaining or latter part

PROP. XXIX. To find the Pole's Altitude, above any of the Circles of the Houses.

The Analogy or Proportion is.

As the Radius or S 90°,

To S. of V. of the Circle of the House with the Meridian: (found by the 21 prop.)

So is the S of the Poles Elevation, above the Horizon of the Place,

To S. of the Altitude of the Pole, above the Circle of Position.

PROP. XXX. By knowing the Latitude and Longitude of any fixed Star, to find his Right Ascension and De∣clination.

The Analogy or Proportion is.

1. As Radius or S. 90°,

To S. of the Stars Longitude from the next Equinoctial point,

So is Tc. of the Stars Latitude,

To T. of a fourth Arch.

Which compared with the Arch of Distance betwixt the Poles of the World and the Ecliptick 23°, 30'; And if the Latitude and Longitude of the Star be both of one Dignity, i e. when the Star hath North Latitude in the six Northern Sines, ♈, ♉, ♊, ♋, ♌, ♍, or South Latitude in the six Southern Sines, ♎, ♏, ♐, ♑, ♒, ♓: Then shall the difference between this found Arch, and the

[illustration]

[illustration]

[illustration]

[illustration]

Page 216 Plate III

Distance of the Poles be your fifth Arch: But if the Latitude and Longitude of the Star be of contrary qualities, i. e the one Northern, and the other Southern, then add this fourth Arch to the Distance of the Poles 23° 30', and the Sum thereof shall be your fifth Arch; with which,

AGAIN, say.

2. As S. of the fourth Arch,

To S. of the fifth Arch,

So is T. of the Stars Longitude,

To T. of the Stars Right-ascension from the next Equinoctial point.

3. As Sc of the fourth Arch,

To Sc. of the fifth Arch,

So is S of the Stars Latitude,

To S. of the Stars Declination.

I might also shew how by having the La∣titude and Longitude of any two fixed Stars, to find their Distance: but because 'tis the very same with finding the Distance of any two Places on Earth, I refer you to the Directions of Prop 1, 2 and 3. of Chap. 7, ensuing, where you will see the plain Demonstration thereof.

PROP. XXXI. By knowing the Pole's Altitude, to find when any fixed Star shall be due East or West.

This is the Analogy or Proportion.

As Radius or S. 90°,

To T. of the Stars Declination,

So is Tc. of the Pole,

To Sc. of the Stars Horary Distance from the Meridian.

PROP. XXXII. By knowing the Poles Altitude, to find the Eleva∣tion of any fixed Star above the Horizon, being due East or West.

This is the Analogy or Proportion.

As S. of the Poles Altitude,

To Radius or S. 90°,

So is S. of the Stars Declination,

To S. of the Stars Elevation, above the Hori∣zon, at due East or West.

PROP. XXXIII. To find out the Horizontal Parallax of the Moon.

The Analogy or Proportion.

As the Moons Distance from the Center of the Earth,

To the Earth's Semidiameter,

So is Radius or S. 90°,

So S. of the Moon's Horizontal Parallax in that Distance.

PROP. XXXIV. The Horizontal Parallax of the Moon being known, to find her Parallax in any apparent Latitude.

This is the Analogy or Proportion.

As Radius or S. 90°,

To S. of the Moon's Altitude,

So is S. of the Moon's Horizontal Parallax,

To S. of the Parallax in that Altitude.

PROP. XXXV. By knowing the Moon's Place in the Ecliptick, (having little or no Latitude) and her Paral∣lax of Altitude, to find the Parallaxes of her Longitude and Latitude.

First, If the Moon be in the 90° of the Eclip∣tick, she hath then no Parallax of Longitude, and the Parallax of the Latitude, is the very Parallax in that Altitude.

Secondly, But if the Moon be not in the 90th. Degree of the Ecliptick, to find the Parallaxes of the Latitude and Longitude, the Analogy or Proportion is,

1. As Radius or S. 90°,

To T. of the V. of the Ecliptick and Horizon,

So is Sc. of the Moon's Distance from the As∣cendent, or Descendent deg. of the Ecliptick,

To Tc. of the Ecliptick's V, with the Azimuth of the Moon.

AGAIN say,

2. As the Radius or S. 90°,

To S. of that V. found,

So is the Parallax of the Moon's Altitude,

To the Parallax of her Latitude sought.

LASTLY say,

3. As the Radius or S. 90°, 00'

To Sc. of the former V. found,

So is the Parallax of the Moon's Altitude,

To the Parallax of his Longitude sought, which being added to the true Motion of the Moon, if she be on the East part of the 90° of the Eclip∣tick. Or from it to be deducted if she be on the West part of the 90° of the Ecliptick.

PROP. XXXVI. How by knowing the Refraction of a Star, to find his true Altitude.

For the speedy performance of which I have annexed this Table of Refractions of the Stars observed by Tycho Brabe a Nobleman of Den∣mark, and a most famous Astronomer.

A Table of the Refraction of the Stars observed by Tycho Brabe.

Altitude.	Refraction.
o°	30'	00"
1	21	30
2	15	30
3	12	30
4	11	00
5	10	00
6	9	00
7	8	15
8	6	45
9	6	00
10	5	30
11	5	00
12	4	30
13	4	00
14	3	30
15	3	00
16	2	30
17	2	00
18	1	15
19	0	30
20	0	00

The USE of which Table is thus.

EXAMPLE.

Suppose the Altitude of a Star were found by Observation to be 13°; the correspondent Re∣fraction is 4' 00", which substracted from 13° leaves 12°, 56', which is the true Altitude

CHAP. VII. Of GEOGRAPHY.

* 1.174 GEOGRAPHY is an art Ma∣thematical, which sheweth how the Situations of Kingdoms, Pro∣vinces, Cities, Towns, Villages, Forts, Castles, Mountains, Woods, Havens, Rivers, Creeks, &c. being on the Surface of the Terrestrial Globe, may be described, and designed, in commensuration A∣nalogical to Nature, and Verity: and most aptly to our view may be represented.

Ptolomy saith of Geography, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. That it is a description of all the known Earth, imitated by writing and delineation: with all other things belonging thereunto. Of all which I shall say somewhat, as to its Situation, Commodity, Cus∣toms, &c. concerning which Ovid saith, Met. lib. 2.

Terra, viros, Urbesque gerit, frugesque, ferasque, Fluminaque; haec super est Caeli fulgentis imago.

In English Thus.

The Earth, Men, Towers, Fruits, Beasts, and Rivers bears, And over these are place'd the Heavenly Spheres.

SECT. I. Of GEOGRAPHICAL Definitions.

• * 1.175 1. THE Globe of the Earth is a Sphe∣rical Body composed of Earth, and Water, and is divided into Continents, Islands and Seas.
• 2. A Continent is a great Quantity of Land not separated, interlaced or divided by the Sea, wherein are Kingdoms, Principalities and Nations, as EUROPE, ASIA and AFRICA, are one Continent: and AMERICA is another.
• 3. An Island is such a part of the Earth that is environed round with Water on every Side, as the Isle of Great Britain, Java, Wight, &c.
• 4. A Peninsula is such a Tract of Land which being almost cut off from the Main Land, and encompassed round with Water, yet neverthe∣less is joyned unto the firm Land, by some little Isthmus, as Peloponesus, Peruviana, Taurica, Cym∣tryca and Morea in the Levant.

• 5. An Isthmus is a little narrow Neck of Land which joyneth the Peninsula unto the Continent.
• 6. A Promontory is some high Mountain, which shooteth it self into the Sea, the utmost end of which is called a Cape: as Cape-boon, Esperance, Cape d'Verde, and Cape d'Coquibocao.
• 7. The Ocean is a general Collection of Wa∣ters, which environeth the World on every side, and produceth Seas, Straits, Bays, Lakes, and Rivers: Of which and other Waters Ovid thus speaks in his Metamorphosis.
  Tum Freta diffudīt, rapidisque tumescere ventis Jussit, & ambitae circundare littora terrae. He spread the Seas, which then he did command To swell with Winds, and compass round the Land.
• 8. The Sea is part of the Ocean, to which we cannot come but through some Strait, as the Mediterranean, or Baltick Sea.
• 9. A Strait is a part of the Ocean restrained within narrow bounds, yet openeth a way to the Sea, as the Straits of Gibralter, Helespont, &c
• 10. A Creek is a crooked Shoar thrusting, as. it were, two Armes forth to hold the Sea; as the Adriatick, Persian, and Corinthian Creeks: from whence are produced Rivers, Brooks and Fountains: which are engendred of Congealed Air in the Earths Concavity, and seconded by Sea-water creeping through the hidden Cranies of the Earth.
• 11. A Bay is a great Inlet of Land, as the Bay of Mexico, and Biscay.
• 12. A Gulph is a greater Inlet of Land and

• deeper than a Bay, as the Gulph of Venice, and Florida.
• 13. A Climate is a certain space of Earth and Sea, included within the space of two Parallels; and there have been anciently accounted these seven: viz. 1. Dia Meros, 2. Dia Syenes, 3. Dia Alexandria, 4. Dia Rhodes, 5. Dia Rhomes, 6. Dia Boristhenes, and 7. Dia Ripheos.
• 14. A Zone is a certain space of Earth con∣tained betwixt certain Circles of the Sphere, of which there are five: viz. The Torrid or Burning Zone, two Temperate, and two Frigid or Frozen Zones.

The Torrid Zone is that which lieth on each side the Equinoctial, whose bounds are the two Tropicks of ♋ and ♑.

The two Temperate Zones are those which lieth betwixt the two Tropicks of ♋ and ♑, and the Palar Circles.

The two Frigid Zones lieth between the Ar∣tick and Antartick Circles, and their respec∣tive Poles: Of which Ovid thus speaks.

Metam. 1.

Utque duae dextrâ Coelum, totidemque sinistrâ Parte secant Zonae, quinta est ardientior illis: Sic onus inclusum munero distinxit eodem Cura Dei, totidemque plaga tellure premuntur: Quarum quae media est, non est habitalis aestu; Nix tegit alta duas: totidem inter utram{que} locavit Temperiemque dedit mistâ cum Frigore Flammâ.

SECT. II. Geographical Descriptions of the Earth.

THE whole Earth is divided into four parts.

VIZ.

• EUROPE,
• ASIA,
• AFRICA and
• AMERICA.

EUROPE, the first part of the World, is* 1.176 divided from ASIA by the Mediterrane∣an Sea; bounded on the West with the Western Ocean; East with the River Tanais. It is lesser than ASIA, or AFRICA, yet doth ex∣cell all the other parts, in Worthiness, Fame, Power, multitudes of well builded Cities, strong Fortifications, full of a Wity and Learned People, Courageous Wariours, and the knowledge of God, better than all the Riches of the World. It once had the dominion of ASIA and A∣FRICA, and in it were fourteen Mother Tongues, and doth contain these Provinces: Viz. Italy, Spain, Alps, France, Britain, Belgia, Germany, Denmark, Sweden, Russia, Poland, Hungary, S••lavonia, Dacia, and Greece, with its several Islands, which shall be mentioned in their due places.

Italy* 1.177 the Mother of Latine learning, is boun∣ded* 1.178 East with the Adria∣tick and Tuscan Seas, West with France, North with Germany, and South with the River Varus, and the Alps. It hath had seven kinds of Govern∣ments: First Kings, Dic∣tators, Consuls, Decimivires, Tribunes, Emperours, and lastly Popes. It far excel∣leth all the other Lands in EUROPE in fruit∣fulness and pleasantness. The Inhabitants are wit∣ty and frugal, yet hot and lascivious, and very jealous of their Wives; they are of the Popish Re∣ligion, and its chief Com∣modities are Rice, Silk, Velvets, Sattins, Taffeties, Grogerams, Arras, Gold and Silver, Threed, Venetian Glasses, &c.

Italy at this day contains the Kingdom of Na∣ples, Sicily, Sardina, the Lands of the Pope, now Innocent the XI. the Dukedom of Tuscany, Urbin, the Republick of Venice, Genoa, and Luca. The Estates of Lumbardy, being the Dukedom of Millain, Mantua, Modena, Parma, Mount∣ferrat, and the Principality of Piemont, of all which we shall treat in their order.

The Kingdom of Naples is environed with* 1.179 the Adriatick, Ionian, and Tuscan Seas; except where it is joyned to the Lands of the Church, from which 'tis separated by a line drawn from the mouth of the River Tronto, falling into the Adriatick, and to the spring-head of Axofenus, taking in all the East of Italy, 1468 miles. It is very fertile, abounding with all things neces∣sary for the life of Man, delight, and Physick: from hence come the Neopolitan Horses. It hath had 13 Princes, 24 Dukes, 25 Marquesses, 19 Earls, and 900 Baronets, and 26 Kings of several Countries of the Norman and Spanish race, whom 'tis now under: here the Disease called the French-Pox derived its Original: the Arms are Azure Seme of Flower-d'-lices, or a File of three Lables Gules; its revenues are 2500000 Crowns, 20000 of which belongs to the Pope, and the rest are imployed to maintain the Garisons against the Turks; so that scarcely 60000 Crowns falls to the King of Spain s share; it hath 20 Archbishops, and 124 Bishops Sees.

Sicily is situated under the fourth Climate, it shoots forth into the Seas with three Promonto∣ries; the Inhabitants are Eloquent, Ingenious, and Pleasant, but very unconstant, and Talka∣tive; the first Inventors of Oratory. It's a fruit∣full Soil, it yields Wine, Grain, Oyl, Hony, Gold, and Silver, Agats, Emeralds, Allom, Salt, Sugar, and Silks. Here is the Hill Aetna, supposed to be Hell, and by the Papist Purgatory, because of its vomiting Smoak and Fire: it hath many Cities, Rivers, Lakes, whose descriptions must here be omitted; it hath had eight Kings; the first were of the Arragon Family, and began

Rule Anno 1281. But it's now united to the Crown of Spain; its Revenues are 800000, or a 1000000 of Ducats, which is disburst on the* 1.180 Account of the Vice-Roy, and Defence of the Countrey; the Arms are four Pallets Gules, Sable for Aragon, between two Flanches Argent, charged with as many Eagles Sable beaked Gules. It hath had seven Princes, four Dukes, thirteen Marquesses, fourteen Earls, one Viscount and forty∣eight Barons, they are of the Romish Religion, and have three Archbishops, and nine Bishops.

The Kingdom and Isle of Sardina, lieth West from Sicily and Cap Bara, whose length is 180, and breadth 90 Miles; the People are low of Stature, and of a swarthy Complection, rude, slothfull, and rebellious, their diet mean, yet rich in their Apparel, they are of the Romish Religion; but have an ignorant and illiterate Clergy. It belongs to the King of Spain, and governed by a Vice-Roy, under whom are two Deputy Spaniards: but other inferiour Officers may be Natives. It hath neither Wolf, nor Serpent, nor venemous Beast, but the Fox only, and a little Spider, which cannot endure the light of the Sun; they are destitute of Water, and are therefore forced to keep the Rain that falls in Summer for their Use in Winter, the Air is unhealthfull and Pestilential; the Soyl Fertile, but ill manured; it hath plenty of Cattel, their Horses will last very long, the Natives ride on their Bullocks as we on our Horses, here is also a Beast called Mufrones, resembling a Stagg, whose Hide is used as Armour, and an Herb which eaten produces Death with excessive 〈…〉〈…〉aughter, it yields to the King of Spain but a

small Revenue. The Arms are Or, a Cross Gules betwixt four Sarazens Heads, Sabled Curled* 1.181 Argent, it hath several Isles belonging thereun∣to, it hath three Archbishops, and fifteen Bi∣shops.

The Lands of the Church or the Pope's Domi∣nion in Italy, lieth West of Naples, extended North and South from the Adriatick to the Tus∣can Seas, bounded on the North with the Ri∣ver Trontus; on the South-east with Axofenes: hy the Rivers Poe and Frore, separated from the Republique of Venice, on the South-west by the River Piscio; by which 'tis divided by the Mo∣dern Tuscany, it is in the middle of Italy, its breadth is 115, and length 300 Miles, it's most exceeding fruitfull, very Populous, there have been 15 Exarches of Ravena in Romandiola, 17 Dukes and Marquesses of Ferata; the Revenue thereof to the Pope is 250000 Crowns, there hath also been 6 Dukes of Urbin, its Revenue are 100000 Crowns, but the most splendid Glo∣ry of Italy is the City of Rome, sometimes the Empress of the World, and was the Seat of the past Popes, and the now present Pope Inno∣cent the XI. the inferior spiritual Governours, are these, Viz. Cardinals, Friers of the Order of St. Basil, Austin, Jerome, Carmelites, Crouchedfriers, Dominicans, Benedictines, Franciscans, Jesuits and Oratorians; and of Nuns, the Order of St. Clear and Bridget, which to name wholly doth de∣serve a particular Treatise, here are 44 Arch∣bishops, and 57 Bishops.

The Republique of Venice, lieth Northward of the Popes Dominions, from Romandiola to the Alps, limited on the South with the Territories

of Ferrata, and Romandiola; on the West with the Dukedom of Millain; on the North with the Alps; and on the East with the Adriatiok Sea,* 1.182 and the River Arsia. It is a very fruitfull Coun∣trey, well peopled, their Government Aristo∣cratical, and popular, their Religion Popish, they baptize the Sea yearly; they have had a hundred Dukes, they have two Principal Orders of Knighthood of St. Mark the Patron of the famous City of which the Poet speaks.

Viderat Adriacis venetiam Neptunus inundis Stare Urbem, & toto ponere jurae mari: Nunc mihi Tarpeias, quantumvis Jupiter, arces Objice, & illa tui moenia martis ait. Sic Pelago Tibrim praefers, urbem aspice utram{que} Illam bomines dices, hanc posuisse Deos.

Instituted 1330, and renewed 1562, they are to be all of Noble Blood: their Motto is Pax tibi Marce. The other is of the Glorious Virgin, iri∣stituted 1222, their Duty is to be a refuge to Widows and Orphans, and to procure the peace of Italy; their Habit a White Surcoat over a Russet Cloak, representing Religion, as well as Belliar∣city, there are two Patriarchs, and sixteen Bi∣shops.

The Dukedom of Florence, being the Seat of the Great Duke of Tuscany, is bounded on the East by the River Pisca; on the West by the River Macra, and the Fort Sarzana; on the North by the Appenuine Hills; and on the South with the Tuscan Seas. Its length is 261 Miles, and breadth not known; the Order of Knight∣hood is that of St. Stephen, instituted 1567, they

are to be Nobly born, and in lawfull Wedlock, without Insamy: their Robe is of White Cham∣blet, with a Red-Cross on the left Side of* 1.183 their Midway Garment, their Number I can∣not certainly know, the Grand Duke is their Sovereign; the Revenue of this Countrey is great, their Duke is also a Merchant, and re∣ceiveth Excise of all Commodities, the Arms is Or, five Tortecax Gules, two, two, and one, one, on chief Azure charged with three Flower∣de-Luces, of the first. They are of the Popish Reli∣gion, and they have three Archbishops, and twen∣ty six Bishops.

The Estate of Luca, lieth betwixt the Estate of the Grand Duke, and the Republique of Genoa: The Government is Aristocratical, and Democra∣cy, their Principal Magistrate is called, Gon Fa∣tinere, and is changed every second Month: being assisted by a certain Number of Citizens, which are changed every six Months, during which time they lie together in the Common Hall; their Protector is Elective from some Neighbouring Prince: they are a very generous People, good Merchants, they sell rich Cloths of Gold and Silver; the Revenues yearly are 80000 Crowns, it can raise for War 15000 Foot, and 3000 Horse, they are of the Popish Religion, and have two Bishops, and acknowledge the Bi∣shop of Florence for their Metropolitan.

The Republique of Genoa, lieth West of Tus∣cany, from whence 'tis divided by the River Macra, it was formerly a large State but have now only Liguria, and the Isle of Corsica; the Inhabitants are good Warriours, Merchants, and subtle Userers; here the Women have the most

••iberty of any in all Italy, so that they may convense with whom they will, either publiquely, or privately; from hence ariseth a* 1.184 Proverb, That Genoa is a Country of Mountains without Woods; Seas without Fish; Men without Faith; and Women without shame. They have a Duke, with Eight Assistants, all subject to the General Council of 400 Men; which hold but two years, they are of the Popish Religion, and have one Archbishop, and fourteen Bishops.

The Estates of Lumbardy is bounded on the East with Romandiola, and Ferrata; on the West and North with the Alps; and on the South with the Apenuine hills: Now as Italy is the Gar∣den of EUROPE, so is Lumbardy of Italy, for its exceeding Fruitfulness.

The Dukedom of Millain hath on the East the State of Mantua, and Parma; on the West Piemont, and part of Switzerland; on the North Marca Trevigana; and on the South the Ape∣nuine, parting it from Liguria: it was once the chief Dukedom in Christendom, and is now in the Spanish Territories; its Revenues are 8000 Ducat's, their Arms are Argent, a Serpent A∣zure Crown'd, Or, in his George an Infant Gules, their Religion is Popish, they have one Archbi∣shop, and six Bishops.

The Dukedom of Mantua, is bounded West with Millain; East with Romandiola; North with Marca Triugiana; and South with the Dukedom of Parma. The Countrey yields good store of Corn, Fruit and Wine, the Inhabitants are rustick, foolish in their Apparel, it is a free state and hath had many Dukes, the Order of Knighthood is that of the Blood of Christ: in∣stituted

1608, it consisteth of twenty Knights, the Mantuan Duke is their Sovereign; the Col∣ler hath threads of Gold, layed on with Fire,* 1.185 with this Motto Domine probasti, to the Collar are pendent two Angels, supporting three drops of Blood, and circumscribed with this Motto, Nihil ista triste receptô. Its Revenue is 500000 Ducats; Its Arms are Argent, a Cross Patere Gules, between four Eagles, Sable membred of the second, under an Eschucheon in Fife, charged quarterly with Gules, a Lion Or, and Or, three barrs Sable: Their Religion is Popish, here is one Archbishop, with four Bishops.

I shall pass by the Dukedoms of Modena, Par∣ma and Mountferrat, they being but small E∣states of Italy, having but four Bishops: they are of the Popish Religion, the Arms of Modena and Parma are as Ferrata; and the Arms of Mount∣ferrat, a chief Argent.

And here we should describe Piemont the last part of Italy, but being but part in Italy, and the Alps belonging to the Duke of Savoy, I shall de∣fer it to the Alpian Descriptions.

Now Italy hath these most famous Cities, viz. Genoa, Milain, Venice, Florence, Rome, Bo∣logne and Naples, the Rivers most famous are Arnus, Po and Tiber, and so much for Italy.

The Alps begin about the Ligustick Seas, and crosseth all along the Borders of France and Germany, and extend as far as the Gulph of Cor∣nero; It hath these Provinces, viz. the Duke∣dom of Savoy (to which Piemont belongs) Geneva, Wallisland, Switzerland, and the Countrey of the Grizons, of all which I shall give a short and plain Description.

Piemont is part of the Alps, situated at the Foot of the Mountains, bounded North with* 1.186 the Switzes; East with Millain and Mountferrat; West with Savoy; and on the South it runneth into a Narrow Vally to the Mediterranean, ha∣ving Mountferrat on one side, and Province and part of the Alps on the other: it's very fruitfull compar'd with Savoy, but yet inferiour to any part of Italy: The Arms are Gules, a Cross Argent, charged with a Lebel of three points Azure.

Savoy is bounded East with Wallisland and Pie∣mont; West and South with Daulphin, and La∣Bress; and North with Switzerland, and the Lake of Geneva: this is a Mountainous Count∣rey, very healthfull, but not very fruitfull. The Inhabitants are dull and slothfull, it hath had thirty Dukes and Earls, it is a place of Natural strength; its Revenues is yearly 1000000 of Crowns. The order of Knighthood is that of Anunciado, instituted 1480, their Coller hath 50 links, (to shew the Mystery of the Virgin) ap∣pendent to it is her Effigies, and instead of a Motto these Letters F. E. R. T. i. e. Fortitudo ejus Rhodum tenuit, which is engraven on each link of the Chain, interwoven like a True-lovers∣knot. The Number fourteen, besides the Duke Soveraign of the Order, their Arms are G. a Cross A.

Geneva was a City of the Dukedom of Savoy, but now a free State: having both cast off the Duke and his Holiness the Pope, with all the Clergy. They are now Calvanist Protestants: their Government Presbyterial; their Language the worst of French, they are an industrious People, and good Merchants.

Wallisland reacheth from the Mount De Burken, to the Town of St. Maurice, where the Hills do shut up the Valleys, so that a Bridge* 1.187 is lain from one Hill to 'tother, under which passeth the River Rosue, which Bridge is defen∣ded by a Castle and two strong Gates; on the other side 'tis surrounded with steep and hor∣rid Mountains; covered with a Crust of Ice not passable by Armies; the Inhabitants are courteous to Strangers, but unnatural to each other: they are of the Romish Religion, and sub∣ject to the Bishop of Sion; the Deputies of the seven Resorts, have voices in his Election, and joyn with him in Diets, for chusing Magistrates; desiding Grievancies, and determining matters of State. The Valleys of this Countrey is very fruitfull in Saffron, Corn, Wine and Delicate Fruits, they have a Fountain of Salt, many hot Baths, and Spaw-Waters, they have plenty of Cattle, with a wild Stag footed as a Goat, and horned as a fallow Deer: who in Summer is blind with heat.

Switzerland is bounded East with the Grisons; West with Mount-Jove and the Lake of Gene∣va; North with Suevia; and South with Wallisland, and part of the Alps; this Land is a very Mountainous Countrey, but yet hath some rich Meddows. It is 240 in length and 180 Miles in breadth, the Inhabitants are rich, but rugged like their Soyl: good Souldiers: they are some Papists and some Protestants, o∣thers Zwinglians, yet have they toleration under a Popular Government.

The Countrey of the Grisons is bounded East with Tyrol, North with Switzerland, South

with Suevia, Switzerland, and Lumbardy; it is a very Mountainous and Barren Land, their Religion Protestant, their Government* 1.188 Popular; there are in this Alpin Provinces two Archbishops, and thirteen Bishops. Its chief Cities are Turin, Geneva, Basil and Zurich, in all of which are Universities.

France is bounded East with Germany; and South and East with the Mediterranean Seas and Alps; North with the British Seas; It hath been esteemed the worthiest Kingdom in Chris∣tendom, it yields plenty of Grains and Wines, wherewith it supporteth other Lands, it con∣sisteth of many great Dukedoms and Provinces. It hath great and mighty Cities, the People are Ingenious and good Warriers, the Govern∣ment is Monarchial, their Religion Popish, but intermixt with Protestants, which of late hath endured grievous Persecutions. Their Orders of Knighthood are that of St. Michael insti∣tuted 1409, consisting of 300 Persons, their Habit is a long Cloak of White Damask down to the Ground, with a Border interwoven with Cockleshels of Gold, interlaced and furred with Ermins, with a Hood of Crimson-velvet, and a long Tippit about their Necks, and a Coller woven with Cockleshells, with this Mot∣to, Immensitremor Oceani, to it hangs appendent the Effigies of St. Michael conquering the Dragon. Their Seat is St. Michael's Mount in Normandy. 2dly the Order of the Holy Ghost instituted 1579, so that whosoever was admitted to the Order of St. Michael, must and was first digni∣fied with this; proving their Nobility by three Descents; and be bound by Oath to maintain

the Romish Religion; and persecute all Dissenters thereunto. Their Robe is a Black Velvet Mantle, portrayed with Lillies and Flames of Gold: the* 1.189 Coller of Flower-de-Luces, and Flowers of Gold, with a Dove and Cross appendent to it. The Arms of France, are Azure three Flower-de∣Luces, Or; It hath seventeen Archbishops, 107 Bishops, 132000 Parishes, and hath these Magni∣sicent Cities, viz. Amiens, Rouen, Paris, Troys, Nants, Orleans, Diion, Lyons, Burde∣aux, Toulose, Marsailles, Grenoble and Anverse; the Rivers of most Note are the Loyre, Garone, Rhone and the Seyne.

The Pirenean-hills lyeth betwixt France and Spain, and are two Potent Kingdoms, esteem∣ed 240 Miles long, the People are barbarous and scarce of no Religion at all.

Spain is separated from France by the Pirene∣an-hills; on all other sides environed with the Sea; this Land yieldeth all sorts of Wine, Oyl, Sugars, Grains, Metals, as Gold and Silver, and it is Fertile; the Inhabitants are Ambitious, Proud, Superstitious, Hypocrites and Lascivious, yet good Souldiers; by enduring Hunger, Thirst, Labour, &c. It containeth divers Kingdoms. 1. Goths. 2. Navars, it hath had 41 Kings. Their Arms are Gules, a Carbuncle Nowed Or, their Order of Knighthood was of the Lilly, their Blazon a Pot of Lillys, with the Effigies of the Virgin on it, their Duty is to defend the Faith, and daily to repeat a certain Number of Ave-Maries. 3dly Biscay and Empascon, hath had nineteen Lords, their Arms Argent, two Wolves Sable, each in his Mouth a Lamb of the second. 4ly Leon and Oviedo hath had thirty

Kings, the Arms Argent, a Lion Passant crown∣ed Or. 5ly. Galicia hath had ten Kings, the Arms Azure Sema of Cressels siched a Chalice* 1.190 crowned Or. 6ly. Corduva hath had twenty Kings, the Arms Or, a Lion Gules armed and crowned of the first, a Border Azure charged with eight Towers Argent. 7ly. Granado hath had twenty Kings, the Arms Or, a Pomgranet slipped Vert. 8ly. Marcia. 9ly. Tolledo hath had eleven Morish Kings. Ioly. Castile hath had twenty Kings, the Order of Mercia is his chief Order, here the Armes is a Cross Argent, and four Beads, Gules in a Field Or, their Habit white, the Rule of their Order that of St. Augustine, they are to redeem Captives from Turky. 11ly. Portugal (the Native soyl of the most serene Catharine Queen Dowager) hath had 21 Kings, the Orders of Knight here is first Avis, wearing a Green Cross, 2dly of Christ instituted 1321, their Robe is a black Cassock under a white Surcoat with a Red Cross hanging in the midst a white Line, and their Duty is to expell Mores out of Boetica, the Arms are Argent, on five Escucheons Azure, as many Befants in saltire of the first pointed, Sa∣ble within a powder Gules, charged with seven Towers Or. 12ly. Majorica hath had four Kings. 13ly. Arragon hath had twenty Kings, their Order of Knighthood is of Mintesa, their Robe a red Cross on their Breast, the Arms Or, four Pallets Gules, all which Kingdoms are now united into one Monarchy, under the King of Spain, their Religion Popish: the King is not rich by reason of his great Expences to keep his Dominions, in which are eleven Archbishops,

and 52 Bishops, and hath these most notable Cities, viz. Toledo, Madrid, Leon, Fax, Sivil∣le, Grenado, Mursy, Saragosa, Bracelon, Pam∣phelune,* 1.191 Bilbo, Priede, St. James of Compostella, and Lesbone, and Rivers famous are the Dower, Tagus, Gadian and Guadalguinr.

Great Britain consisteth of England and Scot∣land, and is the Biggest Isle in EUROPE, and the Glory thereof; it is a temperate Soyl, a sound Air, and yieldeth all manner of good things, 'tis environed all round with the Seas; I shall begin first with England.

England hath many pleasant Rivers well stored with Fish, excellent Havens, commo∣dious Mines of Silver, Lead, Iron and Tinn, abundance of Woods, good Timber, plentifull in Cattle, good Wool of which is made fine Cloath, which serves not only themselves but vended into other Countreys, the chief City is London, in which are two of the Wonders of the World, viz. the Monument and Bridge over the Thames, the People are brave Warriers, both by Sea and Land, as Europe has felt and can testifie to their Grief, they are learned in all manner of noble Sciences; the Order of Knighthood is that of St. George, or the Garter, there are 26 Knights of it, whereof the King is the Soveraign, their Ensign is a blue Garter buckled on the left Leg, with this Motto — Hony Soit Qui Mall 〈…〉〈…〉 Pense, and about their Necks they do wear a blue Ribbon, at the End of which hangs the Image of St. George, upon which day this Or∣der is Celebrated: secondly of the Bath, institu∣ted 1009, they use to be Created at the Coro∣nation of Kings and Queens, and at the Enstalling

of the Prince of Wales. The Knights thereof distinguished by a red Ribbon, which they wear about their Necks, their Duty is to defend Re∣ligion,* 1.192 Widows, Maids and Orphans, with the Kings right. Thirdly of Barronets and Heredi∣tary Honour, the Arms are Mars, three Lions Passant, Gardant Sol, their Religion is the Pro∣testant; they have two Archbishops and twenty Bishops. The length of England is 320, and breadth 250 Miles, it hath 857 famous Bridges, 325 Rivers, it's defended and invironed with Turbulent Seas; guarded by unaccessible Cleves and Rocks; and defended by a strong and Puissant Navy; so that of it may well be said,

Insula praedives, que toto vix eget Orbe; Et cujus totus indiget Orbis ope. Insula praedives, cujus miretur & optet Delicias Solomon octavianus opes.

Its chief Cities are London, York, Bristol, and Rivers are the Thames, Severn, Humber, and the Ouze.

Wales is bounded on all sides with the Sea, except towards England on the East; it is a barren and mountainous Countrey: Its chief Commodities are their Freeze, and Cottons. The Inhabitants are faithfull in their promises to all men, but yet much enclined to Choler, and subject to Passion, which Aristotle calleth 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. It contains 14 Shires, 13 Forests, 36 Parks, 230 Rivers, and 1016 Parishes. They are so resolute and valiant (saith Henry III. writing to Emanuel then Emperour of Constanti∣nople)

That they dare encounter Naked with

armed Men, being ready to spend their Blood for their Countrey, and pawn their Life for Praise.

They are Protestants, and have four Bishops* 1.193 but no Towns of Note.

Scotland is the Northern part of Britain, en∣vironed all round with the Sea, unless where it is joyned to England. Polydore saith it is 480 in length, and 60 miles in breadth, divided in∣to Highlands and Lowlands: the Highlands are Irish-Scots, and the Lowlands English-Scots. It is not so fruitfull as England; the chief City is Edenbrough. Its Commodities is course Wool, and Cloth, Malt, Hides, and Fish. The Order of Knighthood is that of St. Andrew, the Knights did wear about their Necks a Coller interlaced with Thistles, with the Picture of St. Andrew appendant thereunto, having this Motto — Nemo me impune lacessit. 2. Of Nova Scotia in∣stituted by King James, Anno 1622, hereditary the Knight hereof distinguished by a Ribbond of Orange-tawny. The Arms of Scotland is Sol, a Lion Rampant, Mars within a double Tressure counterflowered; they are Protestants, and have 2 Archbishops and 12 Bishops. The Cities most Famous are Edenbrough, Sterlin, Aberdeen and St. Andrews: and they have the Famous Rivers Tay and Tweed.

Ireland is on all sides environed round with the Irish Seas, and St. Georges Channel. In length is 300 and breadth 120 miles. The Natives are strong and nimble, haughty, careless, hardy, bearing cold and hunger with patience and in a word, if they are bad you shall neve•• find worse; but if good scarcely find better. The Wild Irish have a custom to kneel down to

the New Moon, praying it to leave them in as good health as it found them. They received the Christian Faith 435. The Soil is fruitfull* 1.194 and it hath good Pasture, yet full of Boggs and Woods, and multitude of Fowls, and in it will dwell no venomous Creature. The Revenues yearly have been 40000 li. The Air is Tempe∣rate, cooler in Summer, and hotter in the Win∣ter than in England. Their Arms are Azure an Harp or stringed Argent, they are some Prote∣stants and Papists mixt: they have 4 Archbishops, and 19 Bishops; the chief City is Dublin.

The Islands belonging to Great Britain, are 1. Wight (the place where I first drew my Breath, and the Land of my Nativity) 2. Sur∣lings, 3. Garnsey, 4. Jersey, 5. Anglesey, 6. Man, 7. Hebrides, 8. Orcades, 9. Portland, 10. Sunder∣land, 11. Holy Island. And thus I have done with the British Empire; all these Parts descri∣bed belong to it, and are under the Royal Sceptre of his Sacred Majesty JAMES the Second (whom God long preserve.) Thus I have finished the description of Great Britain having this only to say — Quae Deus conjunxit nemo separet.

Belgia, or the Low Countreys, consisteth of several wealthy Provinces: viz. The Duke∣dom of Brabant, Guelderland, Lymburge, Flan∣ders, Artois, Henault, Holland, Zeland, Mamen, Zukfen, the Marquisate of the Holy Empire, Freezeland, Michlen, Ouserisen, and Graving. All which Lands are very fertile and populous, having 208 Cities, and 6300 Villages, with Pa∣rish-Churches, Castles, and Forts; and is watered

with the Rhine and the Mose, the Mara and the Sheld. It hath commodious Havens, the Inhabitants are brave Warriours, good Mecha∣nicks,* 1.195 their chief Commodity is Rhenish Wine, Linnen and Woollen Cloth, Camericks, Lace of Gold and Silver, Silk, Taffatys, Velvet, Groge∣rams and Sayes, all manner of Twined threds, re∣fined Sugars, Buff, Ox-hides, Spanish-leather, Pictures, Books, Cables, Ropes and Herrings. Now Belgia is bounded East with Westphalen, Gulick, Cleve, and the Isle of Triers; West with the Main Ocean; North with the River Ems; and South with Picardy and Champagne. The People are of the reformed Religion all except Flaunders and Artoise, and they have the Po∣pish Tenents, here are three Archbishops, and fifteen Bishops. The Order of Knighthood is that of the Golden Fleece, instituted 1439. their Habit is a Coller of Gold, interlaced with Iron, Or. Ex ferre Flammam, at the end thereof hangs a Golden Fleece. Their chief Cities are Mentz, Antwerp, Amsterdam, Roterdam, and Rivers are the Sheild and Mosa.

Germany is the greatest Province in all EU∣ROPE, and is bounded East with Russia, Poland and Hungary; West with France, Swit∣zerland and Belgia; North with the Baltick Seas, and part of Denmark; and South with the Alps and parted from Italy: it contains Bo∣hemia and Pragu, it is adorned with Magnifi∣cent Towers, strong Fortifications, Castles and Villages, very Popular; the Soyl is Fertile; many Navigable Rivers do to it belong, Good Spaws, Hot Baths, Mines of Gold and Silver, Tinn, Copper, Lead and Iron; they are some Pa∣pists,

others Protestants, Zwinglians, Calvinists, and Lutherans. The Arms is Sol, an Eagle dis∣played with two Heads, Saturn armed, and* 1.196 Crowned Mars. There are six Archbishops, and 34 Bishops. They are a People much given to drinking; which made the Poet say —

Germani possunt cunctos tolerare labores, O utinam possent tam bene ferre sitim.

The chief Cities of Germany are these, viz. Strasborough, Cologn, Munster, Norimbergh, Ausburg, Numick, Vienna, Prague, Dresda, Berlin, Stetin, Lubeck; Its chief Rivers the Rhine, Weser, Elbe, Oder, and the Danow, and Cities of Bohemia, are Cutenberg, and Budrozu.

Denmark and Norway, are two great Regions and bounded South with Germany; they have North Latitude 71° 30', toward the East they border on Sweden; and elsewhere environed with the Sea. Their Commodities are Oxen, Grain, Fish, Tallow, Sand, Nuts, Oyl, Hides, Goat∣skins, Fir-trees for Masts, Boards, &c. Pitch, Tarr and Brimstone: they are Lutherans. The Order of Knighthood is that of the Elephant, their Badge a Coller powdered with Elephants Towred, supporting the Kings Arms; having appendent the Effigies of the Virgin Mary; the Arms of the Land are Quarterly. 1. Or, three Lions passant Vert, Crowned of the first, for the Kingdom of Denmark. 2dly, Gules a Lion Ram∣pant, Or, Crowned and Armed of the first, in his Paws a Dansk hatchet; Argent, for the Kingdom of Norway. There are two Archbi∣shops, and 13 Bishops; its chief City is Coppenhagne.

Sweden is a mighty Kingdom, is bounded East with Muscovia, West with the Dorfirin hills, North with the Frozen Ocean, and South with* 1.197 Denmark, Liesland and Mare Balticum; the Com∣modities are Copper, Iron, Lead, Furr, Buff, &c. They are brave Warriers, their Religion is Lu∣therans. The Arms Azure three Crowns Or, it hath two Archbishops, and eight Bishops.

Russia is bounded East with Tartaria, West with Livonia and Finland, North with the Frozen Ocean, and South with Lituania, and Mare Caspium, This Countrey is extreme cold: but yet Nature hath counterpoized it by sup∣plying the Land plentifully with the best of Furrs, viz. Sable, White-fox, Martin, &c. It's subject to the Emperour of Russia; a vast Tract, and as wild a Government. The Inhabitants are Base and Ignorant, Contentious and Foolish, they deny the proceeding of the Holy-Ghost, they bury their Dead upright, with many o∣ther foolish Ceremonies; Muscovia is the Seat of the Empire. Its Commodities are Furrs, Flax, Ropes, Hides, Fish, and Whales-grease. The Arms are Sable, a Portal open of two Leaves, and as many degrees Or, they are of a mixt Romish Religion, not observing Learning as a∣ny thing: They have one Patriarch, two Arch∣bishops, and eighteen Bishops. Its chief Cities are Mucon, Wolodimax, St. Michael, Cazan, and Astracan, it's chief Rivers are the Dwine, Vola∣ga, and the Tana.

Poland is bounded South of Moldavia and Hungary; East with Moscovia, and Tartaria; West with Germany; and North with the Baltick Seas. The Commodities are Spruce-Beer, Amber,

Wheat, Rye, Hony, Wax, Hemp, Flax, Pitch, Tarr; it hath Mines of Tinn and Copper; their Religion is partly Romish, and partly of the Greek-Church,* 1.198 and so there are of the Greek Church, two Arch∣bishops, and six Bishops, and of the Romish Church three Archbishops, and nineteen Bishops: The Arms are Quarterly. 1. Gules an Eagle Argent Crowned and Armed Or, for Poland, and two Gules a Chevalier armed Cap-a-peid, advancing his Sword Argent, Mounted on a Barbed Course of the Second, for the Dukedom of Latuania. Its chief Cities are Cracovia, Warsovia, Damzerk, Vil∣na, Kion, Cameneca, and Smolensco; and Rivers are Vistula, Niemen, Dunae and the Boristhenes.

Hungary is bounded East with Transilvania, and Walachia, West with Stiria, Austria and Moravia, North with the Carpathian Moun∣tains, and South with Sclavonia and part of Dacia. The People are valiant, and shew their Antiquity to be Scythians by their barbarous Manners, and neglect of Learning Their Sons equally inherit without Priviledge of Birthright, and their Daughters portion is only a New Attire. Its Commodities are Colours, Wheat, Beef, Salt, Wine and Fish, the German Empe∣rour and Turk hath it between them. The Arms is eight Barrs Gules, and Argent, they are some of the Romish, and others Mahometans. There are two Archbishops, and thirteen Bishops, and its chief Cities are Transilvania, Valastia, Moldavia, Buda, Presbrough, Hermonstada, Ter∣goguis, Czuchan, Craffa and Bargos. Its Ri∣vers are the Drin, Oxfeus, Peneus, Vardax, Ma∣rize and the Danubus.

Sclavonia is bounded South with the Adria∣tick Seas, East with Greece, North with Hunga∣ry, and West with Carniola. It is fruitfull of all* 1.199 those Commodities found in Italy, and is under several Governments, viz. Turks, Venetians, Hungarians and Austrians. The Arms are Ar∣gent, a Cardinals Hat, the strings Pendant, and Pleated in a True-lovers-knot, meeting in the Base Gules. They are some Christians, and some Mahometans. There are four Archbishops, and twenty six Bishops. Its chiefest Cities are Nova, Zara, Nonigrad, Tinu, Sebenico, S. Nicolo, Trau, Spalato, Salona, Almisse, Starigrad, Ve∣sicchio, Catara, and Doleigne.

Dacia is bounded East with the Euxine Seas, on the West with Hungary and Sclavonia, North with Podolia, and South with Thrace, and Mace∣donia. The Soyl is fruitfull in Corn and Wine. It yieldeth medicinal Plants, they have plenty of Fowls, both wild and tame, very Populous and of Nature like the Hungarians; they are all Mahometans; Its most famous Cities are Triste and Pedena.

Greece is bounded East with Propontick Helles∣pont, and Aegean Seas, West with the Adria∣tick, North with Mount Haemus, and South with the Ionian Seas. It was once the Mother 〈…〉〈…〉 Arts and Sciences, but now the very Den 〈…〉〈…〉 the Turkish Empire. The Soyl is very fruitfull 〈…〉〈…〉 well manur'd, which made the Poet say —

Impius haec tam culta novalia miles habebit? Barbarus has segetes? en queis consevimus arva

Its Commodities are Gold, Silver, Copper, 〈…〉〈…〉

Wines, Velvets, Damask; here is the Mount of Parnassus: Here was the Temple of Delphos, consecrated to Apollo; where the De∣vil* 1.200 through the Oracle did deceive the People, but after the Crucifixion of Christ the Oracle ceased. Augustus (saith Suidas, in whose time Christ was born) consulting with the Oracle, received this Answer —

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉

An Hebrew Child, whom the blest Gods adore, Hath bid me leave these Shrines, and pack to Hell, So that of Oracle I can no more: In silence leave our Altar, and farewell.

Their Religion is mixt but they are chiefly Mahometans. The Arms of this Empire were Mars a Cross, Sol, between four Greek Beta's of the second; Bodin saith the four Beta's signified 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

The most famous Cities in Greece are Buda, Sa∣lonique, Adrianopolis, Scutary, Durazzo, La Va∣lone, L'Armiro, Brevezza, Larta, Lepanto, Setines or Athens, Thebes, Corinth, Patras, Mi∣sira or Lacaedomia.

I shall pass over the Islands of Sicily, Sardinia, Candia and Corsica: and thus we have finished the description of the first part of the World, called by the Name of EUROPE.

ASIA. * 1.201

ASIA the second part of the World is bounded on the North with the Northern Ocean; South, with the Red Sea; East, with the East Indian Ocean; and on the West with the Flood Tunais. It is bigger than EUROPE, or AFRICA, and is far more rich, Viz. in Pre∣tious Stones, and Spices, and hath been renown∣ed by the first and second Monarchs of the World. Here Man was Created, placed in Eden, seduced by Satan, and redeemed by our Blessed Saviour. In it was done most of the History mentioned in the Old Testament. It hath been Ruled by the Kings of China, of Persia, the Great Turk, and the Emperour of Rushia, and contains these Provinces. Viz. Anatolia, Cyprus, Syria, Palestine, Arabia, Chal∣dea, Assyria, Mesopotamia, Turcomania, Media, Persia, Tartaria, China, India and the Oriental Isles.

Anatolia is bounded West with the Thracian∣Bosphorus, Helespont and the Aegean Seas; East with Euphrates; North with the Black Sea; and South with the Rhodian, Lydian and Pam∣phylian Seas. Its length is 630, and breadth 210 miles; the Air is sound, the Soil fruitfull, but in some places desolate: it is inhabited by Greeks and Turks. It hath these Cities of note. Viz. Anatolia, Bruce, Chiontai, Augoure, Trebi∣sond, Sattalie: and Rivers are Alie, Jordan, Eu∣phrates, and Tygris.

Syprus is bounded all round with the Syrian and Sicilian Seas; whose length is 200, and compass 550 miles. It is stored with plenty of* 1.202 all things, so that it wanteth no help of other Nations. Its Commodity is Wine, Oyl, Corn, Sugar, Cotton, Honey, &c. for which plenty of all things 'twas Consecrated to Venus, as Ovid saith:

Festa dies Veneris, tota celeberrima Cypro, Venerat; Ipsa suis aderat Venus aurea festis.

The People are Warlike, Strong and Nimble, and very hospitable to Strangers Their Arms are Quarterly. 1. Argent, a Cross, Patent, be∣twixt 4 Crosses Or. 2. Cross-wise of 8 pieces Argent, and Azure, supporting a Lion Passant, Azure Crown'd Or. 3. A Lion Gules. 4 Argent a Lion Gules: they are of the Popish Religion, and have 2 Archbishops and 6 Bishops.

Syria is bounded East with Euphrates; West with the Mediterranean Seas; North with Cili∣cia; and South with Palestine and Arabia. Its length is 525 miles, and breadth 470. They are inhabited by Mahometans, Christians, and Pagans. They are a stout and warlike people. In this Countrey there are said to be Sheep whose Tails weigh some 30, and some 40 pounds; the People are also gluttenous; it is almost overrun by the Turks: Its most famous Cities are Aleppo, Te, Tripoly and Damal.

Palestine is bounded West with the Mediter∣ranean Seas; East with the Arabian Desarts; North with the Anti-Libianus; and South with Arabia. The Inhabitants are of a middle stature,

strong constitution, yet a stiffe necked and mur∣muring People and Idolaters. In this is the Land of Canaan, and the famous City Hierusalem, tho'* 1.203 now a Den of Idolatrous Mahometans. It abounds with all good things.

Arabia is bounded East with Chaldea and the Gulph of Persia; West with Palestine, Aegypt, and the Red Sea; North with Euphrates; and South with the Southern Ocean. The Inhabitants are Mahometans. Job's Habitation was here. It yields Frankincense, Pretious Stones, &c. It is now under the Great Turk's Sceptre. The most famous Cities are Herac, Ava, Medina, and Mectar: and it hath the famous River Cayban.

Chaldea is bounded East with Susiana; West with Arabia Deserta; North with Mesopotamia; and South with the Persian Bay. The Country is exceeding fruitfull; in it is supposed to have been the Garden of Eden; they were great Southsayers, and therefore flouted by the Sa∣tyrist. —

Chaldeis sed major erit fiducia, quicquid Dixerit Astrologus, credent à fonte relatum Ammonis, &c.

The Inhabitants are stout and valiant; they are Mahometans. Here Julian the Apostate breathed his Soul out to Satan, in these dying words, — Vicisti tandem Galileae: the chief Cities are Babylon, Bagdad, Balfora, and Sippa∣rum, with the famous River Fazze.

Assyria is bounded East with Media; West with Mesopotamia; South with Susiana; and North with Turcomania and Chaldea. This is a

very plain and level Countrey, and very fruit∣full, having good Rivers: the Natives are brave stout Warriours, formal in their Habit.* 1.204 It is under the Turk's command, and governed by one of his Bassa's; who is able to bring into the Field at any time 100000 Souldiers: here are also a Sect of the Nestorians, and fifteen Christian Churches: its most famous Cities are Calach, Cittace, and Arbela.

Mesopotamia is bounded East with the River Tigris; West with Euphrates; North with Mount Taurus; and South with Chaldea, and Arabia Desertae. It aboundeth with all good things necessary for the life of Man; they are Mahometans, and a people unable to defend themselves but by the assistance of their Neigh∣bours: It belongs to the Mahometan Empire. Its chief Cities are Edessa, and Cologenbar. I shall not describe Mount Taurus, because it is of no moment.

Turcomana is bounded East with Media and Mare Caspium; West with the Euxine Seas, Cap∣padocia, and Armenia major; North with Tar∣taria; and South with Mesopotamia and Assyria. It is a very mountainous Countrey; the people are handsome, stout and brave Warriours: the Women are good Archers. It hath Gold and Silver Mines: It yields Grain, Fruit and Wine; and in Colchis (a part thereof, and in Assyria) they sell their Children: The Arms are the Half Moon Or. It is inhabited by Mahometans, and under the Turkish Empire. Its chief Cities are Musol, Bagded, Batfora, Sanatopdy, and Derbent; with the famous River Arais.

Media is bounded East with Parthia; West with Aremenia; North with Mare Caspium; and South with Persia. The Countrey is of a* 1.205 large extent and very different, even to a Mi∣racle, for in the North part it is cold and bar∣ren, their Bread is dryed Almonds, and Drink the Juice of Herbs and Fruits. Their Food is Venison, and other Wild Beasts, which they catch by hunting. And in the South side the Country is of a rich Soil, plentifull in Corn, Wine, &c. They have been brave Warriours, and it was a custom with them to poyson their Arrows, in an Oyl called Oleum Mediacum: they are Mahometans.

Persia is bounded East with India; West with Media, Assyria, and Chaldea; North with Tar∣taria; and South with the Southern Ocean. This is a mighty rich Countrey governed by the Sophy, the people are strong and valiant, and though Mahometans, yet they War with the Turks for the Mahometan Religion in ex∣pounding the Alcoran. From hence comes Be∣zoars and other pretious Stones, Pearls, and Silk Works. It hath these famous Cities with Media: Viz Taurus, Gorgia, Cogsolama, Hysphan, Erat, Sus, Schiras, and Ormutz: and these Ri∣vers Tiriditiri, and Bendimuz.

Tartary bounded East with China, the Orien∣tal Ocean, and the Straits of Anian; West with Russia and Podolia; North with the Frozen Oce∣an; and South with China. Now the Tartari∣ans are divided into certain Collonies, and differ in manners and Trade of Living, and are Men of a Square Stature, broad Faces, and look a Sq••int; they are hardy and valiant; they will

eat either Horse-flesh or Man's Flesh. They drink Blood and Mares-milk; their Habit is very homely; they are some Mahometans, and* 1.206 some Pagans; their chief Commodity is rich Furrs, and they are governed by the great Cham of Tartary, and hath these famous Cities, viz. Zahasp, Samarcanda, Thibet, Cambalu and Tatur; and Rivers famous are Joniscoy, Oby, Chezel and Albiamu.

China is bounded East with the Oriental Oce∣an; West with India and Cathay; North with Altay and the Eastern Tartaries; and South with Canchin-China. It hath 591 Provinces, 1593 Walled Towns, 1154 Castles, 4200 unwalled Towns, and such an infinite Number of Villa∣ges, that the whole Countrey seems as one Town. It is reported that the Prince can bring into the Field 300000 Foot and 200000 Horse. The Land is fruitfull in Grain, full of wild and tame Beasts, it yields Silk, Pretious Stones, Gold, Copper, &c. The People are ingenious and great Artists, Witness their Wagon made to sail over the Land driven by the Wind: and Historians tells us, that the Art of Printing and of making Guns, is more Ancient with them than with us. They are Idolaters and worship the Sun, Moon and Stars, also they worship the Devil himself, that he may not hurt them. And it hath these most famous Cities, viz. Pa∣guin, Quinjay, Caneun, Macao, Mancian and Magaia, with the great River Quinam.

India is bounded East with the Oriental Ocean, and part of China; West with the Persi∣an Empire; North with Mount Taurus; and South with the Indian Ocean. This Countrey hath an

Exact temperature of Air; two Summers and a double encrease, blest with all things neces∣sary for the Life of Man. It hath Mines of* 1.207 Gold and Silver, Pretious Stones, Spices, and Me∣dicinal Druggs, abundance of Cattle, and Cam∣mels, Apes, Dragons, Serpents, also multitude of Elephants; a Creature of a vast Bigness, some of which are said to be nine Cubits high, and as many long, and five Cubits thick. It is a Creature of wonderfull Sence: for 'tis repor∣ted of the Elephant on which King Phorus sate in the Warrs of Alexander, finding his Master strong and lusty, rushed boldly into the thick∣est of the Enemies Army: But when he once perceived him to be faint and weary, he with∣drew himself out of the Battel, kneel'd down, and into his own Trunck received all the Ar∣rows, directed at his Master. It also is of a most prodigious strength, for it is reported to carry a Wooden Tower on his Back, with thirty fight∣ing men besides the Indian that Rules him. The Sea yields variety of Pearls and Fish; here is also the Leviathan or Whale, of which Pliny says there are some of 960 Foot long; here is the Rhinoceros also found: (such as hath of late been publickly shewed at the Bell∣savage Inn on Ludgate-hill in London) a deadly and cruel Enemy to the Elephant, for though he be less, yet he will whet his horn against the Rocks, and then therewith strive to rip up the Elephants Belly, and is by many Natu∣ralists supposed to be the Unicorn, for all the parts of his Body, especially his Horn, is a soveraign Antidote against Poyson. This Coun∣trey is inhabited by Indians, Moors, Arabians,

Jews, Tartars and Portugeze. The Natives are Tawny, tall and strong, and very punctual to their word. They eat no Fish nor Flesh, but* 1.208 live on things without life; being Pythagoreans. It is also reported that when the Husband dies, and is burning on the Funeral Pile, that then the Wife leaps into the Fire, and so the li∣ving and the dead burn together, which made the Poet say —

Et certamen habent lethi, quae viva sequatur, Conjugium; pudor est non licuisse mori. Ardent victrices, & praebent pectora flammae, Imponuntque suis ora perusta viris.

In India these are the chief Cities, viz. A∣medabur, Cambaia, Gouro, Diu, Bengala, Pangab, Agra, Goa, Calicut, Visnagor, Pegu, Arracan, Malaca, Camboge, and Faefo. The fairest Ri∣vers are Indus, Ganges and Mecon.

The Oriental Islands are these, viz. 1. Japan, 2. The Phillepinae Isles, 3. The Maluccose, 4. Bantam, 5. The Selebes, 6. Borneo, 7. The Isles of Java, 8. Sumatra, 9. Zeiland, and other lesser Isles of which we shall not treat.

1. Japan is a rich Island abounding with Gold: So that Paulus Ventius saith, that in his time the King's Palace was covered therewith. It is a Mountainous Countrey, a healthfull Air; here the Wheat is ripe in May. It's full of Woods of tall Cedars, abundance of Beasts, Wild and Tame; and also Fowls. The Inhabitants are strong, and witty, and have but one Language. They are Christians, and Idolaters, and the chief Cities are these, viz. Bungo, Meaco and Sacay.

The Phillipine Isles are in Number 40, called so in honour to Philip II. King of Spain, and are now inhabited by the Natives, and Spani∣ards,* 1.209 they are in a good Air and stored with rich Commodities; and in them are these Ci∣ties, Lusor, Manille, and Mindanao.

The Moluccoes Islands are many in Number, their Commodity is Cinnamon, (which grows in whole Woods; it is the Bark of a Tree, stript and laid in the Sun till it looks red; and in three years time the Tree receives his Bark again.) Ginger, Nutmegs, Mastick, Aloes, Pepper and Cloves: now the Clove groweth on a Tree like a Bay Tree; yielding blossoms first white, then green, and at last red and hard, and then are Cloves. In it is also found the Bird of Paradice, and no where else, which for the strangeness and fairness of Feathers exceeds all the Birds in the World. The People are Pagans. Here is a Mountain of a prodigious height, above the Clouds, and agreeing to the Element of Fire, which it seems to mount unto, through Flames, wherewith, a dreadfull Thunder, and a dark Smoak it sends forth continually.

The Isles of Bantam are in Number seven, one of which is continually burning, the Inha∣bi••ants are Barbarous, Weak of Bodies, Slothfull, Dull, and lying most confusedly together, with∣out Rule, and are Mahometans. Its Commodi∣ties are Nutmegs, and both the yellow and white Saunderses. Now the Nutmeg grows on a Tree like a Peach Tree, the innermost part of whose Fruit is the Nutmeg, and is covered over with a Coat which ripe is called Mace; they yield their Fruit thrice in the Year, to wit, at April, August and December.

The Selebes are a Number of Isles full of Bar∣barous People, and Man-Eaters, they have a∣bundance of strange Birds: It yields Sugar,* 1.210 Cocanuts, Cloves, Oranges, &c. In some of these Isles they make Bread of the Pith, and Drink of the Juice of the Tree called Sagu: It hath these chief Towns, viz. Senderem and Macassar.

Borneo lieth West of the Celebes, and is in compass 2200 Miles, the Countrey yields As∣ses, Oxen, Herbs of Cattle and Horses. It yields Camphire, Agarick, and Mines of Adamants: They think the Sun and Moon to be Husband and Wife, and the Stars their Children, they re∣verently salute the Sun at his first rising. Their Affairs of State they Treat of in the Night, at which time the Councellor of State meets, and ascends some Tree, viewing the Heavens till the Moon ariseth, and then they go to their House of State. In it are these Towns, viz. Borneo, Taiopura, Tamaoratas, Malno and Saga∣dana. It is under the Government of the Kings of Borneo and Laus; the People are Idolaters.

Java Major, and Java Minor, are two I∣slands opposite to Borneo. They have plenty of Fruits, Grains, Beasts, Fish and Fowls, Gold and Pretious Stones. The Natives are of a middle Stature, broad faced and tawny, their Religion Mahometans, and they will eat their nearest of kin: the chief Town is Panarucan near a burning Hill, which in 1586 broke forth, and cast huge Stones into the City for three Days together, and destroyed much People. From the top of this vast high Mountain the Devil environed with a white and shining Cloud, doth sometimes shew himself unto his Worship∣pers, which live about those Hills.

Sumatra lieth North of Java Major, betwixt it and the straits of Sincapura, its length is 900 Miles, and breadth 200; it is full of Fenns* 1.211 and Rivers, with thick Woods, and hath a very hot Air; it is not fruitfull in Grain. Its Com∣modities are Ginger, Pepper, Agarick, Cassia, Wax, Honey, Silk, Cotten, Iron, Tinn and Sul∣phur. It hath also Mines of Gold, and is sup∣posed to be Solomon's Ophyr. The King's Furni∣ture of his House, and Trapping for his Ele∣phants was beaten Gold, and he intituleth himself King of the Golden Mountains. Here is the notable Mountain Balalvanus, said to burn continually; out of which or not far off do a∣rise two Fountains, the one is said to run pure Oyle, and the other Balsamum Sumatra; the People are Mahometans. The chief City and Seat of the King is Achen, beautifyed with the Royal Pallace, to which you pass through se∣ven Gates one after another, with green Courts betwixt the two outermost; which are guard∣ed with Women, that are expert at their Weapons, and use both Sword and Guns with great dexterity, and are the only Guard the King hath for his Person. The Government is Absolute and Arbitrary, merely at the King's pleasure.

Zeiland lies West of Sumatra, it is a good Soyl, and yields these Commodities, viz. Cin∣namon, Oranges, Lemmons, most delicate fruit, Gold, Silver and Pretious Stones, it's full of wild and tame Beasts, Fish and Fowls, yet destitute of the Vine: the People are strong and tall, given to Ease and Pleasure, and are in general Maho∣metans. The chief Towns are Candia, Ventane,

and Janasipata. They have Fish-shells passing currant for money, there are other lesser Isles which we do for brevity sake omit, and thus* 1.212 we have done with the description of the second part of the World called ASIA.

AFRICA.* 1.213

AFRICA the third part of the World is bounded East with the Red-Sea; West with the Atlantick-Ocean; South with the Sou∣thern-Ocean; and North with the Mediterranean Sea; and contains these Provinces, viz. Egypt, Barbary, Numidia, Lybia, Terra-Nigritarum, Aethiopia-superior and Aethiopia-inferior, with the Islands thereto belonging. Its Commodities are Balm, Ivory, Ebony, Sugar, Ginger, Dates, Myrrh, Feathers, &c.

Egypt is bounded East with Idumea, and the Bay of Arabia; on the West with Barbary, Nu∣midia, and Lybia; North with the Mediterranean Sea; and South with Aethiopia-superior. Its length is 562 Miles, and breadth 160. The Natives are of a Tawny Complection, their Wives are the Merchants, whilst the Husband attends the Houshold Affairs. They were the Inventers of Mathematical Sciences; they were also Magicians, and are still endued with a spe∣cial Dexterity of Wit: They worship in every Town a particular God, but the God by them most adored was Apis. This Land is very fruitfull

in all manner of Cattle, Cammels, and abundance of Goats; they have plenty of Fowls both wild and tame: It hath Metals and Pretious Stones,* 1.214 Good Wines and rare Fruits, as Oranges, Lemmons, Cittons, Pomgranets, Figgs, Cherries, &c. Here also groweth the Palm-Tree, which grow the Male and Female together; both put out Cods of Seeds, but the Female is not fruitfull unless she grow by the Male, and have her Seed mixt with his. The Pith of this Tree is good for Sallads, of the Wood they make Bedsteads, of the Leaves Baskets, Mats, and Fanns, of the outward husk of the fruit Cordage, of the in∣ward brushes. Its fruit is the Dates, good for food, and finally 'tis said to produce all things necessary for the Life of Man, and its Branches are worn in token of Victory, as saith Horace.

— Palmaque nobilis, Terrarum Dominos evebit ad Deos.

It hath many other Rarities which I am forc'd to omit. In it are these famous Cities, viz. Sabod, Cairo, Alexandria, Rascha, Damietta, Cosir and Surs, with the famous River Nilus, which by its overflowing makes the Land fertile, according to that of Lucia. —

Terra suis contenta bonis, non indiga mercis, Aut Jovis; in solo tanta est fiducia Nilo.

Barbary is bounded East with Cyrenaica; West with the Atlantick-Ocean; North with the Mediterranean, the Straits of Gibralter, and part of the Atlantick-Ocean; and on the South by

Mount Atlas. It is full of Hills and Woods, sto∣red with Wild Beasts: as Lyons, Bears, &c. Large Herds of Cattle; it hath Dragons, Leopards,* 1.215 and Elephants; beautifull, swift, and strong Horses; it is the fruitfullest Countrey •••• the World in some parts of it; for ••liny saith that not far from the City Tacape, you shall see a great Palm-Tree overshadowing an Olive; under that a Figg-Tree; under that a Pomgranat; under that a Vine; and under all Pease, Wheat and Herbs; all growing and flourishing at one time, which the Earth produceth of it self: Its length is 1500 Miles, and breadth 300 Miles, the Na∣tives are of a Tawnyish Colour, rare Horsemen, crafty and unfaithfull, and above measure Jea∣lous of their Wives. It contains these Kingdoms, Viz. Tunis, Algiers, Morocco and Feze, and it hath these Isles, Viz. Pantalaria, Carchana, Zerby, Gaulos and Malta, the two latter of which Isles are inhabited by Christians, and are of the Ro∣mish Religion; but for the other parts of Barba∣ry, they are either Mahometans or Pagans. The most famous Cities are Morocco, Feze, Tangier, (which formerly was a Principal City of Barbary; but is now demolisht and lain level with the Ground, by the Command of His late Majesty Carolus II. of blessed Memory, and performed by the indefatigable skill and industry of the right Honourable George Lord Darmouth Anno, 1683.) Teleusin, Oran, Algi••r, Constantine, Tunis, Tripoly and Barca, with these famous Rivers, Ommiraby and Magrida.

Mount Atlas is a ridge of Hills of no small length, but of an exceeding heighth, above the Clouds, and is always covered with Snow,

Summer and Winter, full of thick Woods, and a∣gainst ASIA so fruitfull, that it affords ex∣cellent Fruit of it's natural growth; it received* 1.216 it's Name from Atlas a King of Mauritania, fain. ed by the Poets to be turned into that Hill, by the Head of Medusa; he was seigned to be so high that his Head touched Heaven: The ground of this Fiction I suppose was from his extraordinary knowledge in Astronomy, which Virgil seems to intimate —

Jamque volans apicem & latera ardua cernit Atlantis duri, Coelum qui vertice foluit.

N••vidia is bounded East with Egypt; West with the Atlantick-Ocean; North with Mount Atlas; and South with Libia Deserta. The Natives are a wandring and unstable People, for they spend their Lives in Hunting, and continue not above four or five Days in one place, but so long as it will graze their Camels. Here grow abun∣dance of Dates, with which they feed them∣selves, and with the Stones fat their Goats. The Air here is so sound that it will cure the Fr••nch-Pox without any Course of Physick. They are Mahometans: its chief Provinces are Dara, Pescara, Fighig, Tegorarin and Biledulge∣rid; and its chief Cities are Taradath, Dara and Zev; they belong to the Scepter of M••∣rocco.

Lybia is bounded North with Numidia; East with Nuba; South with Terra-Nig〈…〉〈…〉∣tarum; and West with Gualata. This is well termed a Desart, for in it may a man travel eight or ten Days and not see any Water, no〈…〉〈…〉

Trees, nor Grass. So that Merchants are for∣ced to carry their Provision with them on Ca∣mels, which if it fails they kill their Camels,* 1.217 and drink the Juice of their Entrails It con∣tains these Provinces, Viz. Zahaga, Zv••nz••ga, Targa, Lembta and Bordea. They are governed by the chief of the Clans, and are a People only differing from Brute-Beasts, by their Shape and their Speech.

Terra Nigritarum is bounded East with Ethio∣pia-superior; West with the Atlantick-Ocean; North with Lybia; and South with the Ethiopick-Ocean. The Countrey is under the Torrid-Zone, full of People, and most excessive hot; the soyl is exceeding fruitfull, brave Woods, Multitudes of Elephants and other Beasts: they have Mines both of Silver and Gold, very fine and pure; the Natives are Cole∣Black, or very Tawny, and are now some of them Mahometans, but most of them Pagans. It hath now these Provinces, Viz. Ora, Ante∣rosa, Gualata, Agadez, Cano, Ca••••na, Sanaga, Gambra, Tombrutum, Melli, Gheneoa, G••ber, Gialofi, Guinea, Benin, Guangara, Bornum and Goaga, (in which groweth a Poyson, which if any eateth but the tenth part of a Grain it will end his Days) Bito, Temiano, Zegzeg, Zanfara, Gothan, Medra and Daum. And in it are these most remarkab'e Cities: Gue, Eata, Gueneha, Tomta, Agad••s, Cu〈…〉〈…〉a, Tuta, Waver and Sanfara. The Rivers here that are most famous are Sernoga, Cambua and Ri••-Degrand.

Aethiopia-superior is bounded East with the Red-Sea, and Sinus Barbaricus; West with Lybia-inferior, Nubia, and Congo; North

with Egypt, and Lybia Marmarica; and South with Monta-Luna. Now its length is said to be* 1.218 1500 Miles, and Circute 4300. It is under the Command of the Abassine Emperour: here the Air and Earth is so hot and pieircing, that if the Inhabitants go out of their Doors without Shoes they lose their Feet; here they also roast their Meat with the Sun: they have some grain, their Rivers are almost choaked up with Fish, their Woods stuffed with Deer, yet they will not trouble themselves to catch them. The Inhabitants are Lazy and destitute of all Lear∣ning, ••hey are of an Olive Tawny: here is also a Fountain, that if a man drinks thereof he either falleth mad, or else for a long time is troubled with a continual Drowsiness, of which Ovid thus speaks —

Aethiopesque Lacus; quos si quis faucibus hausit, Aut furit, aut patitur mirum gravitate saporem.

And it contains these Provinces, Viz. Gua∣gere, Tigremaon, Angote, Amara, Damut, Goja∣my, Bagamedrum, Barnagasse, Dancali, Dobas, Adel, Adea, Fatigar, Xoa and Barus. Now as for the Government of these Empires'tis merely Regal: here is the Order of St. Anthony, to which every Father that is a Gentleman, is to give one of his Sons: out of which they raise about 12000 Horse, which are to be a standing Guard of the Emperour's Person: their Oath is to de∣fend the Frontiers of their Kingdom, to preserve Religion, and to root out the Enemies of their Faith; the Principals of their Religion are these. First, they circumcise their Children both Males

and Females. Secondly, they Baptize the Males at 40 and Females at 80 Days after Circumcisi∣on. Thirdly, after the Eucharist they are not* 1.219 to spit till Sun set. Fourthly, they profess but one Nature and one Will in Christ. Fifthly, they accept but of the three first general Coun∣cils. Sixthly, the Priests live by the own la∣bour of their hands, and are not to beg. Se∣venthly, they baptize themselves every Epipha∣ny in Lakes or Ponds, because that Day they say Christ was baptized by John in Jordan. Eighthly, they eat not of those Beasts which Moses pronounced unclean, keeping the Jews Sabbath, with the Lord's Day. Tenthly, they administer the Lord's Supper to Infants presently after Baptism. Eleventhly, they teach the Reasonable Soul of Man comes by Seminal Propagation. Twelfthly, that Infants dying unbaptized are saved, being sanctifyed by the Eucharist in the Womb, and finally they produce a Book of Eight Volumes, writ as they say by the Apostles at Jerulalem for that pur∣pose, the Contents whereof they observe most solemnly, and thus they differ from the Papists.

Now the chief Cities in this Empire are these, Viz Barone, Caxumo, Amarar, Damont, ••••••••∣tes, Narre, Goyame, and Adeghena with the fa∣mous Rivers Zaire and Quilm••nei.

Aethiopia-inferior is bounded East with the Red-Sea; West with the Aethiolick Ocean; North with Terra-Negritarum, and the higher aethipy; and South with the Main Ocean And it con∣tains these Provinces, Viz. Zanzibar, M••no∣motapa, Cafravia, and Manigongo. The Na∣tives are Black, with curled Hair, and are Pagans.

In it are great Herds of Cattle, abundance of Deere, Antelopes, Baboons, Foxes, Hares, Ostri∣ches, Pelicans and Herons, and in a Word what* 1.220 else is necessary for the Life of Man. In it are these most famous Cities, Viz. Banza, Loanga, S. Salvador, Cabazze, Sabula, Simbaos, Butua, and Tete. The Rivers are Cuama, Spiritus Sancto, and Dos Infante.

The Islands in AFRICA are these, Viz. the Aethiopick-Isles, Madagascar, Socofara, Mohelia, Mauritius, St. Helens, the Isles of Ascention, St. Thomas-Isles, the Princes-Isles, the Isles of Annibon, the Isles of Cape d'Verd, the Canaries, Madera, Holyport and the Hesperides. The Description of all which I am forced to omit because I have been so very large in the Description of the third part of the World cal∣led AFRICA.

AMERICA.

* 1.221

AMERICA, the fourth part of the World, was first discovered by Christopher Columbus, Anno 1492, but it hath received its Name from Americus Vesputius, who in the year of Christ 1597 did fail about it. Now this fourth part of the World is bounded East with the Atlantick Ocean; West with the West-Indian Ocean; South with the Magellanick Sea; and on the North with the Northern Ocean.

When first the Spaniards had entred on A∣merica they found the people without Apparel, and their Bread was made of the Jucca-Root, whose Juice is a strong poyson: but it being squeezed out and dried it makes Bread. They worshipped Devillish Spirits, which they call Zema; in remembrance of which they keep Images made of Cotton Wool, to which they did great reverence, supposing the Spirits of their Gods were there; and to blind them the more, the Devil would cause these Puppets to seem to move and to make a noise, so that they feared them so greatly that they durst not of∣fend them; which if they did, then the Devil would come and destroy their Children. They were so ignorant that they thought the Spani∣ards to be immortal; but the doubt continued not long, for having taken some of them Priso∣ners, they put them under Water untill they were dead, and then they knew them to be mortal

like other Men. They were quite destitute of all good Learning, reckoning their Time by a confused knowledge of the course of the Moon;* 1.222 they were honest and kind in their Entertain∣ments, encouraged thereunto by an Opinion that there was a certain place to which the Souls of those that so lived, and dyed for the defence of their Countrey, should go to, and there be for ever happy. So natural is the knowledge of the Soul's Immortality, and of some Ubi, for its future reception, that we find some tract of it in the most Barbarous Nations. The Americans were of a fair and clear Com∣plexion. This Countrey is very plentifull in Spices, and Fruits; and such Creatures which the other parts never knew: So fu l of Cows and Bulls, that the Spaniards kill thousands of them yearly only for the Hides and Tallow. Blest with abundance of Gold, that in some Mines they have found more Gold than Earth. They have Grey Lyons, their Dogs snowted like Foxes, neither can they bark; their Swine hath Talons sharp as Rozors, and their Navel on the ridges of their Backs; the Stags and Deer without Horns; their Sheep are so strong that they make them carry burthens of 150 pound weight; they have a Creature with the forepart as a Fox, and hinder as an Ape, ex∣cept the Feet which are like a Man's; beneath their Belly is placed a Receptacle like a Purse, in which their young remains till they can shift for themselves, never coming thence but when they suck and then go in again. The Arma∣dilla is like a barbed Horse, armed all over with Scales that seem to shut and open. The Vieugue

resembling a Goat, but bigger, in whose Belly is found the Bezoar, good against Poyson. A Hare having a Tail like a Cat, under whose* 1.223 Skin nature hath placed a Bagg, which she useth as a Store-house: for having filled her self she putteth the residue of her provision therein. Pigritia, a little Beast that can go no further in fourteen days than a Man will cast a Stone. For their Birds they are of such variety of Colours and Notes, which are so rare and charming, that they surpass all other Birds in any other parts.

Now America is divided into two parts, viz. Mexicana, whose compass is said to be 13000 miles, and that other part called Peruana, whose Circumnavigation is esteemed 17000 miles. The Provinces of Mexicana are these: Viz Estotilant, Canada, Virginia, Florida, Cali∣formia, Nova-Gallicia, Nova-Hispania, and Guatimala. Peruana contains these Provinces: viz. Castella-Aurea, Nova-Granado, Peru, Chile, paragnay, Brasila, Guiana, and Paria. To Pe∣ruana belongs these principal Isles: viz Hispa∣niola, Cuba, famaica, Porto-Rict, Barbadoes, the Charibe-Isles, Insula-Margaretta, Molaque-Isles, Remora, Insula Solamnis, and some other small Isles. But first of Mexicana.

Estotilant hath on the East the Main Ocean; South Canada; West Terra Incognita; and North Hudson's Bay. It comprehends Estotilant, so principally called, Terra Corterialis, New-found∣land, and the Isles of Baccala••s. It is well stockt with all things necessary for the life of Man: the Natives are barbarous, fair, swift of Foot, and good Archers. They are Pagans.

Canada is bounded North with Cortelialas; South with New England; East with the Main Ocean; and West with Terra Incognita. It* 1.224 contains these several Regions: viz. Nova Francia, Nova Scotia, Norumbegne, and four small Isles adjoyning thereto. The people when first discovered were very rude and barbarous, going Naked only a piece of Fishes Skin to cover their private parts, and had two or three Wives a piece, which never Marry after the death of their Husbands The Soil is fruitfull, and yields all manner of good things Here groweth the Sea Horse whose Teeth is an Anti∣dote against Poyson It hath these principal Cities: viz. Hochelaga and Quebeque.

Virginia hath North Canada; South Florida; East Mare-del-Noo〈…〉〈…〉; West with Terra Incognita. And it is now divided into New England, New Belgium, and Virginia strictly so called. It is in some parts (yea most parts) Mountainous, Wooddy and Barren, and full of Wild Beasts. It yields plenty of Cattle, wild and tame Fowls. Its Commodities are Furrs, Amber, Iron, Rop••s, Tobaco, Sturgecn, &c. The Natives are but few in number, and those very different both in Speech and Size, to a Miracle: those whom they call Sasques Honoxi, are to the English as Giants clad in Bears Skins; those whom they call Wig••ocomici, are as Dwarfs; for the most part without Beards; they hide their nakedness with a Skin, the rest of their Body they paint over in the figures of horrid Creatures The chief Towns are `fames's, and Plimouth, and Isle of Bermoodus, which I here omit.

Florida is bounded North-East with Virginia; East with Mare-del-Noort; South with the Gulph of Mexico. It was first discovered by the Eng∣lish,* 1.225 Anno 1497. The Soil is very fertile in Grain and Fruit, Beast wild and tame, and so also for Fowls: It yields lofty Cedars, and Sassafras: It hath Gold and Silver Mines, and also Pearls.

The Natives are of an Olive-Colour, strong and fierce, and are clad like the former Natives of America. The Women when their Husbands are dead cut off their Hair, and cannot Marry till their Hair is grown out again. To it be∣longs these Islands: viz. the Isles of Tortugas, Martyres, and Lucaios: there are also about 24 small ones more which are insignificant. The Women here are most extreamly beautifull; the Natives are Pagans. Its chief Towns are St. H••elens, Ax Carolina, and Port-Royall.

Califormia is an Island having on the West New Spain, and New Gallicia; and so unto those undiscovered parts which lie furthest North, to the Straits of Anian; and 'tis divided into these four parts: viz. Quivira, Cibola, Ca∣liformia, specially so called, and Nova Albion. All which Countreys are indifferent fruitfull, full of Woods, and both wild and tame Beasts; plenty both of Fish and Fowl wild and tame: They worship the Sun as their chief God: They go naked both Men and Women in some parts, others are half way cloathed; and so very various that I cannot in this small Tract describe them. Its chief Town is Chichilticala. And here I cannot chuse but remark that in Quivira their Beasts are of strange forms, and are to them both Meat, Drink and Cloathes.

For the Hides yields them Houses; their Bones and Hair, Bodkins and Threed; their Sinews, Cords; their Horns, Guts and Bladders, Vessels;* 1.226 their Dung, Fire; their Calveskins, Buckets to draw and keep Water in; their Blood, Drink; and their Flesh, Meat; and so much for Cali∣formia.

Nova Gallicia is bounded East and South with Nova Hispania; West with the River Buena, Guia, and the Gulph Califormia; and North with Terra Incognita. It comprehendeth these Provinces: viz. Chialoa, Contiacan, Xa∣lisco, Guadalajara, Zacatecas, New Biscay, and Nova Mexicana. In which Provinces the Air is indifferently temperate, yet sometimes given to Thunder, Storms, and Rain. It is full of Mountains, yields Brass, Iron, &c. They have plenty of Fish, Beast, Fowls, Fruit, and abundance of Honey. The Natives are wavering, crafty and lazy, given to singing and dancing. They go not naked: they are subject to the King of Spain. Its chief Cities are Guadalajara, and St. Johns.

Nova-Hispania is bounded East with the Gulph of Mexico, and the Bay of New-Spain; West with Nova Gallicia, and Mare-del-Zur; on the North with part of Nova Gallicia, and part of Florida; and on the South with the South Sea. It comprehendeth these Provinces: Viz. Mexicana, Mechoacan, Panuco, Trascala, Guaxata, Chiapa and Jucutan. In all which the Air is healthfull and temperate, rich in Mines of Gold and Silver, Cassia, Coccineel, which grows on a shrub called Tuna, yields grain, and delicate Fruit, Birds and Beasts both Wild and

Tame: their Harvest is in October and in May.

The Natives are witty and hardy, yet so igno∣rant that they thought the Spanish-horse and Man* 1.227 to have been but one Creature, and thought when the Horses Neighed they had spoken. The Spaniards whose Cruelties will never be forgot∣ten, did in less than 17 years kill of the Natives 6000000; here is a Tree called Meto, it bears 40 kinds of Leaves, of which they make Con∣serves, Paper, Flax, Mantles, Matts, Shoes, Girdles; it yields a Juce like Syrup, which boyled becomes Hony, if purified Sugar; the Bark roasted is a good Emplaisture for Punctures or Con∣tusions; and it yields a Gum Sovereign against Poyson: here is also a Burning Mountain called Propaeampeche, which sends forth two streams the one of Red and the other of Black Pitch: the Inhabitants are Pagans.

Guatimala is bounded North with Jacuta, and the Gulph Honduras; South with Mare-del∣Zur; East with Castella-Aurea; and West with New Spain. The Soyl and People are as in Nova Hispania: it contains these Provinces, Viz. Chiapa, Verapaz, Guatimala, Hondarus, Nicerag∣na and Teragna. And Towns of most Note are Cutrinidao and St. Michael's, the People are Pagans. And so much for Mexicana.

Peruana the Second Part of AMERICA, so called from Peru a Place of Note therein, and it doth contain these Provinces, Viz. Castella∣Aurea, Nova-Granada, Peru, Chile, Paragnay, Brasile, Guyana, and Paria and its Isles. But such Isles that fall not properly under some of these must be referred to the general Heads of the American Islands.

Castella-del Oro, is bounded East and North with Mare-del-Noort; West with Mare-del-Zur; and South with Granada. And it containeth* 1.228 these Provinces, Viz. Panama, Darien, Nova∣Andaluzia, St. Martha and De-la-Hacha. In all which Provinces the Air is very hot and un∣healthfull: the Soyl either Mountainous and Barren, or low and Miry: plenty of Beast and Fowls. Here is said to be a Tree which if one touch he is poysoned to death: the old Natives are now almost quite rooted out. Its chief City is Carthagena, which Sir Francis Drake in 1585 took by Assault. This Land hath abundance of Gold.

Nova-Granada is bounded North with Castella Aurea; West with Mare-del-Zur; East with Venez••••la; and South with Terra Incognita. Its length is 390 Miles, and as much in breadth. It doth consist of these two parts, Viz. Granada, specially so called, and Popayan, both which hath a temperate Air, brave Woods, well stored with Cattle, and Fowls both wild and tame, plen∣ty of Emeralds and Guacum: the People tall and strong; the Women handsome and better drest than their Neighbours: The chief Towns are S. Toy d'Bagota and Popayan.

Peru is bounded East with the Andes; West with Mare-del Zur; North with Popayan; and South with Chile. It is 2100 Miles in length, and its breadth is 300 Miles: it is a Mountai∣nous Country: And here 'tis to be noted that in the Plains it never raineth; and that on the Hills it continually raineth from September to April, and then breaks up. In the Hilly Coun∣treys the Summer begins in April, and endeth

in September. In the Plains the Summer begin∣neth in October and endeth in April, So that a man may travel from Summer to Winter both* 1.229 in one Day; be frozen in the Morning when he setteth out, and scorched with heat before the dawning of the Day. It is not very plenti∣full of Corn nor Fruits, but they have a kind of Sheep which they call Pacos as bigg as an Ass, profitable both for fleece and burthen, but in tast as pleasant as our Mutten: So subtile that if it be overladen it will not for blows move a foot till the burthen be lessened, and it is a very hardy Creature. Here is a Figg-tree, the North part of which looketh towards the Mountains, and yieldeth its Fruit in Summer only, and the Part facing the Sea in Winter only. They have another Plant, that if put into the hands of the Sick and the Patient looks merry, they will recover; but if sad, die. It yieldeth also Multitudes of Rarities more. It's chief Commodities are Gold, Silver, Tobacco, Sarsaparilla and Balsamum d'Peru, and many other rich Drugs. The Natives are almost now rooted out of the Country. They are fierce and Barbarous. Now it contains these Provinces: viz. Quito, Los Quinxos, Lima, Cus∣co, Charcos and Colla••.

Chile is bounded North with Deserta Alaca∣ma; West with Mare del Zuz; South with the Straits of Magellan; and East with Rio de la Plata. Its length is 1500 miles, and breadth uncertain. The Soil hereof in the Mid-land is mountainous and unfruitfull; towards the Sea∣side level and fertile; with products of Maize and Wheat, plenty of Gold and Silver, Cattle

and Wine. The Natives are very tall and war∣like, some of them affirmed to be eleven foot high; their Garments of the Skins of Beasts;* 1.230 they are of a white Complexion; their Armes Bows and Arrows. It is divided into Chile (especially so called) and Magellanica. Here Sir Walter Rawleigh planted two Collenies, who for want of timely Succors were either starved at home, or eaten by the Salvages, as they ranged the Countrey for food.

Paraguay is bounded South with Magellanica; East with the main Atlantick; North with Brazila; and West with Terra Incognita. It is said to be of a fruitfull Soyl, well stored with Sugar-Canes, Fraught with Mines of Gold, Brass, and Iron: great plenty of Amathyses, and Mon∣keys, Lyons and Tigers, the People are as the other Salvages, and it contains these Provinces, viz. Rio de la Plata, Tucaman and La Crux de Sierra, and it's chief Towns are Puenas Agrees, and Chividad.

Brazila is bounded East with Mare del Noort; West with Terra Incognita; North with Guiana; and South with Paraguay. Its said to be 1500 Mi'es long and 500 broad. The Countrey is full of Mountains, Rivers and Forests, the Air sound and healthfull; the Soyl is indifferent fruitfull: Its chief Commodities are Sugar and Brazele-wood. There is a Plant called Copiba which yields Balsam, soveraign for Poyson. An Herb called Viva, which if touched will shut up and not open till the Toucher is out of fight. A Creature which hath the Head of an Ape, the Foot of a Lyon, and the rest of a Man. The Ox-Fish with Arms, Fingers and Duggs, the

rest as a Cow. So that it may be said of Bra∣sila — Semper aliquid apportat novi.

The people are witty as appears by their* 1.231 sayings to the Christians (holding up a Wedge of Gold) say'd they, Behold your God oh ye Christians! on their Festival-days they go Na∣ked, both Men and Women; and are able Swimmers, staying under water an hour and half: the Women are delivered without great pain: some of the Natives are all over Hairy, like Beasts: it containeth not Provinces, but these Captain-Ships: viz. Vincent, Rio de Juneiro, Holy Ghost, Porto-Seguro, Des Ilheos, Todos los Santos, Fernambuck, Tamaraca, Paraiba, Rio Grande, Saiara, Maragnon, and Para. Its chief Cities are, Meranhan, Tamaracai, and Olinda, and the great River Zoyal.

Guiana is bounded East with the Atlantick; West with Mount-Peru; North with the Flood Orenoque; and South with the Amazones. The Air here is indifferently good: near the Sea it is plain and level, up in the Countrey Mountai∣nous; here the Trees keep their leaves all the year, with their fruit always ripe, and grow∣ing. The Inhabitants are under no settled Government: they punish only Murder, Theft, and Adultery; their Wives are their Slaves, and they may have as many as they please; they are without Religion or Notion of a Deity. It doth contain these Provinces: viz. Rio de las Amazones, Wiapoce, Orenoque, and the Isles of Guiana. Its Comodities are Sugar, and Cotton: in it are plenty of Beast, Fish, and Fowles; they are Swarthy in Complection, and great Idola∣ters; as for Cities it hath none of note.

Paria is bounded on the East with Guiana; West with the Bay of Venezuela; North with the Atlantick Ocean; and South with Terra In∣cognita:* 1.232 and contains these Provinces: Viz. Cumana, Venezuela, S. Margarita, Cutagana, and its Isles. All which are not very fruitfull; it is well stored with Pearls; the People paint their Teeth and Bodies with Colour: The Women are trained up to ride, run, leap and swim: and also to Till the Land. In it are these most noted Cities: Viz. St. Jago, St. Michael de Ne∣very, and Mahanao.

As for the Descriptions of the American Isles I must beg the favour to omit: I shall there∣fore only name them having been so very large already; and they are these: Viz. Los. Ladro∣nes, Fernandes, the Caribes; as Granada, S. Vin∣cent, Barbados, Matinino, Dominica, Desrada; Guadalupe, Antego, S. Christopher, Nieves, Sancta Crux, and some lesser Isles belonging to them: As also Portorico, Monico Hispaniola, Cuba and Jamaica. Thus I have finished the Description of the known Earth.

Now the Names of the Seas are these: Viz. the Ocean Sea, Narrow Sea, Mediterranean Sea, Mare Major, Mare Pacificum, Mare Caspium, the East-Indian Sea, Perfian Sea, Red Sea, and Mare-del-Zuz, which are all the Principal-Seas.

Thus through the Blessing of God I have given you a brief, tho'true Description of all the known Earth and Seas, and have thus fi∣nished my Geographical Descriptions of the Divi∣sion of the Earthly Globe.

The Author on the Difficulties in the Collecti∣on of his 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, or little Description of the great World.

Oh thou Urania! Thou that hast now brought Our Ship to Harbour sound, and richly fraught. Tho'Aeolus his blustring Gales did send, And foaming Billows high, the Skies did rend: Tho'Blustring storms, and Thunder loud did roar, And darkness Grim, opprest our Souls all'ore; So that we could not view the Stars, nor Sky, Nor Sun, nor Moon, nay Earth, could not espy. Yet by thy Art, such safety we did find, Safely to pass both raging Seas, and Wind. And at the last a Harbour, safe did gain: Rejecting fears; we quite cast off our pain. When Seas are calm, and Winds more serene be, Then we again will put our Ship to Sea; That when refresht we farther may descry, And search into this Noble Treasury.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉

SECT. III. Of Geographical Propositions.

PROP. I. How to find the Distance of any two Cities or Places, which differ onely in Latitude.

IN this Proposition there are two Varieties which are these.

• 1. If both the Places lie under one and the same Meridian, and on one and the same side the Equinoctial, either on the North or South side thereof; then substract the lesser Latitude from the greater, and convert their difference into Miles (by allowing 60 Miles for a Degree) so have you the distance of the two Places propounded.
• 2. But if the two Places lie under one and the same Meridian, but the one on the South side of the Equinoctial, and the other on the North side, then add both their Latitudes toge∣ther their Sum is their Distance.

PROP. II. To find the Distance of any two Places which differ only in Longitude.

There are also in this Proposition two Va∣rieties.

• 1. The two Places may both lie under the Eqinoctial, and so have no Latitude: and if so,

• substract the lesser Longitude out of the greater, and convert the remainder into Miles; so have you the distance of any two Places so posuited.
• 2. But if the two Places differ only in Longi∣tude, and lieth not under the Equinoctial, but under some other Intermediate Parallel of Lati∣tude, between the Equinoctial, and one of the Poles: Then to find their distance, this is the Analogy or Proportion.

  As Radius or S. 90°,

  To Sc. of the Latitude:

  So is S. of ½ X. of Longitude,

  To S. of ½ their distance, which being doubled, and converted into Miles, giveth the required distance.

PROP. III To find the Distance of any two Places, which dif∣fer both in Latitude, and in Longitude.

In this Proposition three Varieties do present themselves to our View.

• 1. One of the Places may lie under the Equi∣noctial and have no Latitude, and the other under some Parallel of Latitude between the Equinoctial and one of the Poles. In such case observe this Analogy or Proportion.

  As Radius or S. of 90°,

  To Sc. of their X. of Longitude:

  So is Sc. of their Latitude,

  To Sc. of their Distance required.

• 2. But if both the Places proposed shall be without the Equinoctial, but on the one side either both towards the North, or both towards the South, then their Distance may be found, by this Analogy or Proportion.

  As Radius or S. 90° 00',

  To Sc. of their X. of Longitude:

  So is To. of the greater Latitude,

  To the T. of a fourth Arch, which substracted from the Complement of the lesser Latitude, the remainder must be the fifth Arch; Then say,

  As Sc. of the fourth Arch,

  To Sc. of the fifth Arch:

  So is S. of the greater Latitudes,

  To Sc. of the Distance of the two proposed Places.

• 3. The two Places propounded may be so situated, that one of them may lie on the North, and the other on the South side the Equinoctial: the Distance of Places so situated may be obtai∣ned, by this Analogy or Proportion.

  As Radius or S. 90°,

  To Sc. of X. of Longitude:

  So is Tc. of the greater Latitude,

  To T. of a fourth Arch, which being substrac∣ted out of the Summ of the other Latitude, and the Radius or 90° Deg. the remainder is a fifth Arch; Then say,

  As Sc. of the fourth Arch,

  To Sc. of the fifth Arch:

  So is S. of the Latitude first taken,

  To Sc. of the Distance required.

These are all the Varieties of the Positions of Places on the Terrestrial Globe: For if the Di∣stance of any two Places be required, they must fall under one or other of these Varieties, and may be obtained by one or other of the Proportions, mentioned in the three afore∣going Propositions.

Also if you know the Latitude and Longitude of any two fixed Stars, or their Right Ascension and Declination, then by these Rules their Distance may be found, which is of good use to Astronomy. It may also be applyed to Circular Sailing; of all other ways the most perfect: which is treated of in its due Place.

CHAP. VIII. of NAVIGATION.

NAVIGATION So called from Navis a Ship, is an Art Mathematical, which sheweth how by the shortest good Way, by the aptest Direction, and in the shortest Time, to conduct a Ship from any one place unto any other place assigned: it hath been highly esteem∣ed by the Ancients; it is the Glory, Beauty, Bull∣wark, Wall and Wealth of Britain, and the Bridge that joyns it to the Universe. Navigation is commonly divided into three sorts of Sailing: viz. Plain sailing, Mercator's sailing, and Circular sailing: Of all which three Parts I shall treat in their Order.

SECT. I. Of Plain sailing, or sailing by the Plain Chart.

PLain Sailing, or sailing by the Plain Chart, is the most plainest way, and the Founda∣tion of all the Rest: and although the Ground and Projection of the Plain Chart is erroneous, yet seeing it is more facile to the Learner, and may serve indifferently near the Equinoctial, because there the Degrees of Longitude, as well as the Degrees of Latitude, are Equal: Each Degree being divided into 60 Minutes, or Milles, though they are somewhat more than English Miles, Each Minute or Mile containing about 6000 Feet.

PROPI. The Rumb, and Distance sailed thereon being gi∣ven,* 1.233 to find the Difference of Latitude, and the Departure from the Meridian.

Admit a Ship sails N. W. by N. 372' or Miles, or 124 Leagues, I demand her Difference of Latitude and departure from the Meridian?

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In the Triangle ABC, the Hypothenuse AC representeth the distance sailed, or Rumb-line,* 1.234 BC the departure from the Meridian, and AB the difference of Latitude.

1. To find which say,

As Radius or S. 90°,

To Log. distance sailed 372'.

So is Sc. V. A of the Course 56° 15',

To Log. cr. AB 309 3/10 Minutes, which being divided by 60' giveth 5°. 9'. 18" for the Diffe∣rence of Latitude.

2. To find the Departure from the Meridian* 1.235 say,

As Radius or S 90°,

To Log. Rumb-line AC 372'.

So is S. of V. of the Course A 33° 45',

To Log. cr. BC 206 6/10 Minutes, the departure from the Meridian, which divided by 60 giveth 3°. 26'. 36" for the difference of Longitude.

Note that by this Proposition you may keep an Account how much you have sailed either East or West, North or South.

PROP. II. By the Rumb and Difference of Latitude given, To find the Distance, and the Departure from the Meridian.

Admit a Ship sail N. W. by W. untill her* 1.236 difference of Latitude be 309 3/10 Minutes, I de∣mand her distance sailed, and her departure from the Meridian?

1. To find the distance, say,

As Sc. of V. of the Course A 56° 15',

To Log. cr. AB the X. of Lat. 309 3/10 Minutes.

So is the Radius or S. 90°,

To Log. AC 372 the distance sailed.

2. For the Departure, say,* 1.237

As Sc. of A V. of the Course 56° 15",

To Log. cr. AB. X of Lat. 309 3/10 Minutes.

So is S. of V. of the Course A 33° 45',

To Log. cr. AB 206 6/10 Minutes, the Departure required.

By the help of this Proposition, when your Latitude by Observation doth not agree with your dead reckoning, (kept by the former Proposition) Then according to this Rule, you may make your way saild agree with your Observed Latitude, and so correct your Ac∣count or dead Reckoning.

PROP. III. By knowing the Distance of the Meridians of two Places, and their Difference of Latitude, to find the Rumb, and Distance.

Admit A, to represent the Lizard, AB the Parallel thereof, C. St. Mary's Islands, being one of the Azores, and CB the Meridian thereof.

In the Triangle ABC, there is given the side AB 816 Minutes, the Distance of the Lizard, from the Meridian of St. Marys, and the side CB their difference of Latitude 768 Minutes, I* 1.238 demand the Rumb: i.e. the Angle at C, and the Distance of the Lizard from St. Marys?

1. For the Rumb or Angle at C, say,

As Log. cr. CB 768',

To radius •••• S 90°.

So is Log. cr. AB 816',

To T. of the B〈…〉〈…〉, or Angle at C 46° 44', and is from the Lizard unto St. Marys to the fourth Rumb of the Meridian, and 1° 44' more, Viz. S. W. and •• 44'. Westerly, or from St. Marys, to the Lizard, N. E. and 1° 44' Easterly: and thus it shall be by the Plain Chart.

2. For their Distance AC, say,

As S. Rumb. or V. at C. 46° 44',

To Log. cr. AB 816' Minutes.

So is Radius or S. 90°,

To Log. Hypoth. AC 1120' 1/10 which is the Di∣stance* 1.239 of the Lizard, unto St. Marys Istand, and such should be the distance by the Plain Chart.

PROP. IV. Admit two Ships to set sail from one Port, one Ship sails W. S. W. 40', the other W. by N. so far untill she finds the first Ship to bear from her S. E. by E. I demand the second Ships distance from the Port, and their Distance asunder?

In the Triangle ADE, let A represent the Port, AD the W. S. W. course, and AE the* 1.240 Course W. by North.

1. To find the second Ships distance from the Port, say,

As S. of V. at E. 22° 30',

To Log. cr. AD 40' Minutes.

So is S. of V. at D 123° 45',

To Log. cr. AE 86 98/100 Minutes, which is the* 1.241 distance required.

2. To find the two Ships their distance Asun∣der, say,

As S. of V. at E 22° 30',

To Log. cr. AD 40 Minutes.

So is S. of V. at A 33° 45',

To Log. cr. DE 58 12/100 Minutes, which is the Distance required.

PROP. V. Two Ships sets sail from two Ports, which lie N. and South of each other, the one sails from the Northermost Port 72 29'/100, and then meets she other Ship, which came from the Southermost Port, on a N. W. Course, and had sailed from thence 56 80'/100 I demand the Rumb on which the first ship made her way, and also the Distance be∣tween the two Ports?

In the Triangle ADE, let A be the Souther∣most* 1.242 Port, AD the Course and way of the second Ship N. W. 56 80'/100, let E be the Norther∣most Port, ED the Course and Way of the other Ship 72 29'/100, and D the Place where they both meet.

1. To find the Rumb on which the first Ship sailed, say,

As Log. cr. DE 72 29/100 Minutes,

To S. of V. at A 45° 00'.

So is Log. cr. DA 56 80/100 Minutes,

To S. of V. at E 33° 45', which sheweth the Course of the first Ship to be S. W. by South.

2. To find the Distance between the two* 1.243 Ports A and E, say,

As S. of V. at A 45° 00',

To Log. cr. DE 72 29/100 Minutes,

So is S. of V. at D 101° 15',

To Log. cr. EA 100', which is the required Distance.

PROP. VI. Admit a Ship coming off the Main Ocean and I had sight of a Promontory or Cape, by which it is my desire to sail, I find it to bear from me S. S. E. and distant by Estimation 33', or Miles: But keeping still on my Course S. untill the Evening, having sailed 36' or Miles, I would then know how the Cape bears, and its distance from the Ship?

In the Triangle ADE, admit that at A, I do observe the Cape D, to bear from me S. S. E. 33', and having sail'd from A, to E 36' South, I desire to know its Distance, and bearing. In the Triangle, there is therefore given, AD 33',* 1.244 AE. 36', and the Angle at A 22° 30'.

1. To find the Angle at E, say,

As Z. cr^s. AE, and AD 69',

To X. cr^s. AD, and AE 03'.

So is T. ½ VV unknown D and E 78° 45',

To the T. of 12° 20', which taken from 78° 45', leaves the Angle at E 66° 25', so that the

Cape D then bears from me E. N. E. and 01° 05' Northerly.

2. To find the Distance of the Cape ED from the Ship, say,

As S. V. at E 66° 25',

To Log. cr. AD 33'.

So is S. V. at A 22° 30',

To Log. cr. ED 13 78/100 Miles distant, so that the Cape is then distant from the Ship 13 78/100 Miles.

PROP. VII. Two Ports both lying in one Latitude, distant 64' or Miles, the Westermost of those Ports lieth op∣posite to an Island, more Northerly distant there∣from 47' or Miles, which Island is also distant from the Eastermost Port, 34' or Miles, I de∣mand the Course from the Westermost Port to that Island?

In the Triangle ADE, let A be the Westermost Port, and E, the Eastermost Port, distant Asun∣der 64'; and let D be the Island, distant from A 47', and from E 34': Then is the Angle at* 1.245 A required, which is the Course or Rumb, from the Westermost Port, unto the Island: To find which, say,

As Log. cr. AE 64',

To Log. Z. cr^s. AD, and ED 81'.

So is Log. X. cr^s. AD, and ED 13',

To Log. os a certain line AO 16 454/〈…〉〈…〉.

Which added to AE 64, is 80 454/1000,

The ½ whereof is AB, 40 227/1000

Then again say,

As Log. cr. AD 47',

To Radius or S. 90°.

So is Log. AB 40 227/1000,

To Sc. V. at A, 58° 51', that is N. E. by E. 2° 36' Easterly, which is the Course from the Westermost Port A, unto the Island D.

SECT. II. Of sailing by the true Sea Chart, commonly called MERCATOR'S Chart.

THE true Sea Chart, commonly called MERCATOR'S Chart* 1.246, performs the same Conclusions as the Plain Chart, and almost as speedily, but far more exactly: Because all Pla∣ces may be laid down hereon, with the same truth as on the Globe it self: both to their Lati∣tudes, Longitudes, Bearing and Distance from each other.

And here it will be necessary to have a Table of Meridional Ports, which I have extracted out of Mr. Wright's Tables, to every tenth Minute of Latitude; accounting it in single Miles, or Minutes of the Equinoctial, and have hereunto annexed the said Table.

A Table of Meridional Miles

The Deg. of Lat.	The Minutes of each Degree.	The Difference.
0	10	20	30	40	50
The Meridional Miles.
0	0	10	20	30	40	50	10
1	60	70	80	90	100	110	10
2	120	130	140	150	160	170	10
3	180	190	200	210	220	230	10
4	240	250	260	270	280	290	10
5	300	310	320	330	340	350	10
6	360	370	380	390	400	410	10
7	421	431	441	451	461	471	10
8	481	491	501	511	521	532	10
9	542	552	562	572	582	592	10
10	603	613	623	633	643	653	10
11	664	674	684	694	704	715	10
12	725	735	745	755	766	776	10
13	786	797	807	817	827	838	10
14	848	858	869	879	889	900	10
15	910	920	931	941	951	962	10
16	972	983	993	1004	1014	1024	10
17	1035	1045	1056	1066	1077	1087	10
18	1098	1108	1119	1129	1140	1150	10
19	1161	1172	1182	1193	1203	1214	10
20	1225	1235	1246	1257	1267	1278	11
21	1289	1299	1310	1321	1332	1342	11
22	1353	1364	1375	1386	1396	1407	11
23	1418	1429	1440	1451	1462	1473	11
24	1484	1499	1505	1516	1527	1538	11
25	1549	1561	1572	1583	1594	1605	11
26	1616	1627	1638	1649	1661	1672	11
27	1683	1694	1705	1717	1728	1738	11
28	1751	1761	1773	1785	1796	1808	11
29	1819	1830	1842	1853	1865	1867	11

The Deg. of Lat.	The Minutes of each Degree.	The Difference.
0	10	20	30	40	50
The Meridional Miles.
30	1888	1899	1911	1923	1934	1946	12
31	1958	1969	1981	1993	2004	2016	12
32	2028	2040	2052	2063	2075	2087	12
33	2099	2111	2123	2135	2147	2159	12
34	2171	2183	2195	2207	2219	2231	12
35	2244	2256	2268	2281	2293	2305	12
36	2318	2330	2342	2355	2367	2380	12
37	2392	2405	2417	2430	2442	2455	12
38	2468	2481	2493	2506	2519	2532	13
39	2544	2557	2570	2583	2596	2609	13
40	2622	2635	2648	2662	2675	2668	13
41	2701	2714	2718	2741	2754	2768	13
42	2781	2795	2808	2822	2835	2849	13
43	2863	2876	2890	2904	2918	2931	14
44	2945	2959	2973	2987	3001	3015	14
45	3030	3044	3050	3072	3086	3101	14
46	3115	3130	3144	3159	3173	3188	14
47	3202	3217	3232	3247	3261	3276	15
48	3291	3306	3321	3336	3351	3366	15
49	3382	3397	3412	3428	3443	3459	15
50	3474	3490	3505	3521	3537	3553	16
51	3568	3584	3600	3616	3632	3649	16
52	3665	3681	3697	3714	3730	3747	16
53	3763	3780	3797	3814	3830	3847	17
54	3864	3881	3899	3616	3933	3950	17
55	3968	3985	4003	4020	4038	4056	18
56	4074	4092	4110	4128	4146	4164	19
57	4182	4201	4219	4238	4257	4275	19
58	4294	4313	4331	4351	4370	4390	20
59	4409	4428	4448	4468	4487	4507	20

The Deg. of Lat.	The Minutes of each Degree.	The Difference.
0	10	20	30	40	50
The Meridional Miles.
60	4527	4547	4567	4588	4608	4629	20
61	4643	4670	4691	4711	4733	4754	21
62	4775	4796	4818	4839	4861	4883	22
63	4905	4927	4949	4972	4994	5017	23
64	5039	5062	5085	5018	5132	5155	23
65	5179	5203	5226	5250	5275	5299	24
66	5324	5348	5373	5390	5423	5449	25
67	5474	5500	5520	5552	5678	5704	26
68	5631	5658	5685	5712	5739	5767	27
69	5795	5823	6021	5879	5908	5937	28
70	5966	5996	6125	6055	6085	6115	30
71	〈…〉〈…〉	6177	6208	6239	6271	6303	31
72	6335	6368	6401	6431	6468	6501	33
73	6535	6570	6605	6640	6675	6718	35
74	6747	6783	6620	6857	6895	6933	37
75	6972	7010	7050	〈…〉〈…〉	7130	7170	40
76	7211	7253	7295	7338	7381	7424	43
77	7469	7513	7559	7605	7651	7651	7698	46
78	7746	7795	7844	7894	7944	7996	50
79	8048	8100	8154	8209	8264	8320	55
80	8377	8435	8495	8555	8616	8678	60
81	8742	8806	8872	8939	9007	9077	68
82	9148	9221	9295	9371	9449	9523	77
83	9609	9692	9778	9865	9954	〈…〉〈…〉	88
84	10141	10238	〈…〉〈…〉	10441	10547	10656	105
85	10770	10887	11007	11133	〈…〉〈…〉	〈…〉〈…〉	128
86	11539	11686	11839	11999	12168	12344	165
87	12521	12718	〈…〉〈…〉	13150	13388	〈…〉〈…〉	〈◊〉〈◊〉
88	13920	14221	14550	14914	15321	15783	386
89	16318	16950	17726	18729	20152	22623

PROP. I. To find by the Table, what Meridional parts are contained in any Difference of Latitude.

The Use of the Table is demonstrated by the several Examples following, after this Manner.

In this Proposition three Varieties present themselves unto our View.

• 1. When one Place is under the Equinoctial, the other having North, or South Latitude, his Meridional parts corresponding, is to be esteem∣ed for the Meridional Difference of Latitude.
• 2. When both Places are towards one of the Poles, then the Meridional parts of the lesser, taken from the Meridional parts of the greater Latitude, the remainder is the Meridional dif∣ference required.
• 3. When one Place hath North, and the o∣ther South Latitude, their corresponding Me∣ridional parts added together gives the Meridio∣nal difference of Latitude sought: thus having sound them out they may thus be applyed.

PROP. II By knowing the Latitudes, and the difference of Longitude of any two Places, to find the Rumb, and Distance.

Admit there be a Port in the Latitude of 50° 00' North, and another in the Latitude of 13° 12' North, and their Difference of Longitude is 52° 57' West, I demand the Rumb and Di∣stance?

In the Triangle A b c, let A b represent the proper difference of Latitude, bc the Departure, Ac the distance sailed, A the Angle of the* 1.247 Course, c the Complement of the Course.

In the Triangle ABC, AB is the Meridional difference of Latitude, BC the Difference of Lon∣gitude, A the Angle of the Rumb, C the Compl. of the Angle of the Rumb: These things being understood the work evidently appears to be the same as in Rightangled Plain Triangles.

There is then required first the Difference of Latitude, and this falls under the second Va∣riety.〈 math 〉〈 math 〉

1. To find the Rumb or Course say,

As Merid. X. Lat. 2676',

To Radius or S. 90°.

So is X. of Longitude 3177',

To T. of the Rumb 49° 53', the Course there fore is S. W. ½ W, &c.

2. To find the Distance,* 1.248 * 1.249

As Sc. Course 40° 07',

To proper X. of Lat. 2208'.

So is Radius or S. 90°,

To the Distance 3426 Minutes as required.

PROP. III. By knowing the Latitudes, and distance of two Places, to find the Rumb, and Difference of Lon∣gitude.

1. To find the Rumb or Course say,

As the Distance sailed,

To Radius or S. 90°.

So is the X. of Latitude,

To Sc. of the Rumb required.

2. To find the Difference of Longitude say,

As Radius or S. 90°,

To the X. of Latitude in Merid. Parts.

So is T. of the Rumb,

To the X. of Longitude required.

PROP. IV. By knowing the Latitudes, and Rumb of two Pla∣ces, to find their Distance, and Difference of Longitude.

1. To find the Distance say,

As Sc of the Rumb,

To the X of Latitude.

So is Radius or S. 90°,

To the Distance required.

2. To find the Difference of Longitude say,

As Radius or S. 90°,

To the X. of Latitude in M. Parts.

So is T. of the Rumb,

To the X. of Longitude required.

PROP. V. By knowing the Rumb, Difference of Longitude, and one Latitude, to find the other Latitude, and the Distance.

1. To find the other Latitude say,

As T. of the Rumb,

To the X of Longitude in parts.

So is Radius or S. 90°,

To the Merid. X. of Latitude required.

2. To find the Distance say,

As Sc. of the Rumb,

To the X. of Latitude.

So is Radius or S. 90°,

To the required Distance,

PROP. VI. By knowing the Distance, one Latitude, and Rumb, to find the other Latitude, and Difference of Longitude.

1. To find the Difference of Latitude say,

As Radius or S. 90°,

To the Distance.

So is Sc. of the Rumb,

To the X. of Latitude required.

2. To find the Difference of Longitude say,

As Radius or S. 90°,

To the Merid. X. of Latitude.

So is T. of the Rumb,

To the X. of the Longitude required.

SECT. III. Of Circular Sailing, or Sailing by the Arch of a Great Circle.

THIS is of all other the most exact way of sailing, and above all other most perfect, shewing the nearest way, and distance between any two Places: and although it is hardly pos∣sible to keep close unto the Arch of a great Cir∣cle, yet it is of great advantage to keep conve∣niently near it, especially in an East or West

Course: In the former Propositions of sailing, we used Meridians, Parallels and Rumbs, as the Sides of every Triangle, whether by the Plain or Mercator's Chart: but in Circular sailing the Rumbs are not used so, because they are Helis∣pherical-lines, and not Circles; nor the Parallels, because they are not great Circles: Whereas the sides comprehending every Spherical Triangle are Arches of great Circles: Therefore here we use Arches of the Meridians, of the Equinoctial, and of other great Circles described, or so ima∣gined to be described, from one Place unto ano∣ther, on the Spherical Superficies of the Earth and Sea.

Therefore here ariseth two things observa∣ble: and,

• 1. If the two places lie under the Equinoctial, then is their Position East and West, and their distance is their Difference of Longitude, con∣verted into Miles: or,
• 2. If the two Places proposed be in one and the same Meridian, then is their Position North and South, and their Distance is their Difference of Latitude converted into Miles.

And thus far doth Circular sailing agree with the former; their difference will evidently ap∣pear by these following Propositions.

PROP. I. Two Places, the one under the Equinoctial, the other in any Latitude given; also their difference of Longitude given, to find.

• 1. Their Distance in the Arch of a great Circle.
• 2. The direct Position of the first place from the second.
• 3. And of the second Place from the first.

Here we call the Angles which the Arch makes with the Meridians of the places pro∣pounded, the Angles of the Direct Positions of those places one from the other: because the Arch of a great Circle. drawn between two places is the nearest distance; and the most di∣rect way of the one, to the other Place. Now I shall not here demonstrate it by Schemes, as I have done in the other two Sections, but shall only lay down the proportions, whereby the re∣quired parts may be found; and so leave the in∣genious Seamen to practice it with Schemes at his leasure: and,

1. To find the nearest distance from Place to place, in the Arch of a Great Circle: Say ac∣cording to the 10 Case of Rectangled Spherical Triangles.

As the Radius,

To Sc. of X. of Longitude.

So is Sc. of X. of Latitude;

To Sc. of the Distance in the Arch required.

2. For the Direct Position, say by the 11 Case thus,

As the Radius,

To S. of X. of Latitude.

So is Tc. of X. of Longitude,

To Tc. of V. of Position required.

3. For the Direct Position of the second Place from the first, say by the 11 Case thus,

As the Radius,

To S. of the X of Longitude.

So is Tc. of X. of Latitude,

To Tc. of V. of Position required.

PROP. II. Two Places proposed, the one lying under the Equi∣noctial, the other in any Latitude given; with their distance in a great Circle of the same Places being also known, to find.

• 1. Their Difference of Longitude.
• 2. The direct Position from the first to the second Place.
• 3. And from the second to the first Place.

1. For their Difference of Longitude, say by Case 12,

As Sc. of the Latitude,

To Radius.

So is Sc. of their Distance in the Arch,

To Sc. of their Difference of Longitude requi∣red.

2. Now to find the Direct Position from the first place to the second, say by the 13 Case; and thirdly, for the Direct Position from the second place to the first, say by the 14 Case of Rectan∣gulars.

PROP. III. Two Places lying in one Latitude given, their dif∣ference of Longitude being also known, to find.

• 1. The nearest distance of those two Places.
• 2. The direct Position of one Place from the other.

The Resolution of this Proposition depends on the 9 Case of Oblique Spherical Triangles: by sup∣posing the Oblique Triangle, to be transfigured or converted into two Rectangulars, by a supposed Perpendicular: and then,

1. To find the nearest distance in the Arch, say by the 8 Case of Rectangulars.

As the Radius,

To Sc. of the Latitude.

So is S. of half X. of Longitude,

To S. of half the required distance, which doubled giveth the distance of the two places in the Arches, as sought.

2. For the Direct Position, say by the 9 Case.

As the Radius,

To S. of the Latitude.

So is T. of half X. of Longitude,

To Tc. of V. of Position required.

PROP. IV. Two Places lying both in one Latitude given, and the nearest distance being also known, to find.

• 1. Their Difference of Longitude.
• 2. The direct Position of the one Place from the other.

The Resolution of this Proposition falls un∣der the 11 Case of Oblique Spherical Triangles: for here you have the three sides of the Triangle given, viz. the Arch of Distance, and the other two sides (are both equal) being the Comple∣ments of the places Latitude: and here seeing the two sides are equal, therefore the two Angles of Position are also equal: now there is required the three Angles,

1. To find their Difference of Longitude, add the double of the Complement of Latitude to the Arch of Distance; then from half this Sum, de∣duct the Arch of Distance, and then proceed in all points as you see in Case the 11th. So shall their Difference of Longitude be obtained.

2. To find their direct Position:

First, to the double Complement of Latitude, add the Arch of Distance, then from half that agra∣gate, deduct the Complement of Latitude, and then work as before, so shall the direct Position be attained.

PROP. V. Two Places proposed lying in one Latitude, and the distance of those Places in their Parallel given; to find.

• 1. Their Difference of Longitude,
• 2. Their distance in the Arch of a great Circle,
• 3. The direct Position of the one from the other.

Now you must understand, that as the Semi∣diamiter of a Parallel, is in proportion to the Semidiamiter of the Equinoctial: so is any num∣ber of Miles in that Parallel, to the Minutes of Longitude answering to those Miles: fo that if we suppose the Semidiameter of the Equinoctial to be Radius, then the Semidiameter of any Pa∣rallel is the Sine of that Parallel's distance from the Pole, that is the Sc. of the Latitude of that Parallel: Therefore,

1. To find the Diff. of Longitude say,

As Sc. of the Latitude,

To the Radius,

So is the Distance in that Parallel,

To the Diff. of Longitude required.

2. Now the Difference of Latitude being ob∣tained, the nearest distance may be found, as in the third proposition aforegoing: 3. so likewise may the Angles of Position also.

PROP. VI. By knowing the nearest Distance of two Places, their Difference of Longitude, and one of their Latitudes; to find the Direct Position thereof from the other.

This Proposition falls under the first Case of Oblique Spherical Triangles, and is thus resolved: therefore,

As S. of the Distance of the two Places,

To S. of their X. of Longitude.

So is Sc. of the Latitude of the one Place gi∣ven,

To S. of the Direct Position from the other as was so required.

PROP. VII. By knowing the Latitudes of two places, and like∣wise their Difference of Longitude; to find,

• 1. The distance in the Arch.
• 2. The direct Position from the first to the second place.
• 3. The direct Position from the second to the first place.
• 4. The Latitudes and Longitudes by which the Arch passeth.
• 5. The Course and Distance from Place to Place through those Latitudes and Longitudes according to Mercator.

I shall here make use of M. Norwood's example of a Voyage from the Summer-Islands, unto the

Lizard: now because the work is various I have therefore illustrated it with a Scheme, and shall be as brief and facile as possible. Therefore,

In the Triangle ADE, let A be the Summer∣Islands, whose Latitude is 32° 25', AD the Com∣plement* 1.250 thereof 57° 35', let E represent the Lizard whose Latitude is 50° 00', and ED the Complement thereof 40° 00', and let their Dif∣ference of Longitude, namely the Angle ADE be 70° 00', now Drepresenteth the North-Pole, and AE an Arch of a great Circle passing by these two Places: now see the operation.

1. By having the Complements of the Latitudes of the two Places, viz. AD 57° 35', and ED 40° 00', and their Difference of Longi∣tude, namely the Angle EDA 70° 00': you may find the nearest distance EA to be 53° 24'; by Case the 9. § 5. chap. 5.

2. Then having found the nearest distance in* 1.251 the Arch EA to be 53° 24', (or 3204 Miles) the Angle of Position from the Summer Islands to the Lizard, namely the Angle DAE, may be found by Case the 1. § 5. chap. 5. to be 48° 48', that is N. E. and 03° 48' Easterly.

3. And also by the same Case, may the Direct Position from the Lizard, to the Summer-Islands, namely the Angle AED befound to be 81° 10', that is W. by N. and 2° 25' Westerly.

4. In order to the finding the Latitudes and Longitudes by which the Arch passeth, first let fall the Perpendicular DB, so is the Oblique* 1.252 Triangle ADE converted into two Rectangulars, viz. ABD, and DBE: secondly, by Case the 8. § 4. chap. 5. you may find the length of the

Perpendicular DB to be 39° 26', whose Comple∣ment is 50° 34', which is the greatest Latitude by which the Arch ABE passeth, so the greatest Ob∣liquity* 1.253

BDc	48°	31
BDf	38	31
BDg	28	31
BDh	18	31
BDj	08	31

of the Equinoctial from that Circle is 50° 34'. — Thirdly, by Case the 9. § 4 chap. 5. you must find the vertical Angles, viz. ADB, and BDE, which will appear, the Angle ADB to be 58° 31', and EDB to be 11° 29': now these things being obtained, the Lati∣tudes by which the Arch passeth at every tenth degree of Longitude from A, may be found by resolving the several Right-Angled Triangles, viz. BDc, BDf, &c. substracting 10° from ADB 58° 31', there remains BDc 48° 31', and so for the rest as in the Table. Now by knowing these Angles last found, and the Perpendicular BD before found to be 39° 26', you may by Case the 3. §. 4. chap. 5. find the Latitudes of the several points A. c. f. g. h. i. B. and E. to be as in the subsequent Table.

5. Thus having* 1.254

Latitude.	Longitude.
A.	32°	25'	00	00'
c.	38	51	10	00
f.	43	34	20	00
g.	46	54	30	00
h.	49	04	40	00
i.	50	15	50	00
B.	50	34	60	00
E.	50	00	70	00

found the Latitudes and Longitudes of the Arch, and the other required parts afore∣mentioned, we now come to shew how the Course, and the Distance from place to place according to Mercator may be found. So to find, first the Course and Distance

Ac. now there is given the Latitude of A 32° 25', and of c 38° 51', and their Difference of Longitude is 10° 00', now the Proper Difference* 1.255 of Latitude is 6° 26', or 386', and Meridional Difference of Latitude is 475'. Now knowing these things by proposition 2. § 2. chap. 8. yon may find the Course from A to c, to be N. E. 51° 38'; and the Distance Ac to be 622', and so those Rules prosecuted will shew the course and distance from c to f; from f to g; from g to h, &c. So of the rest, which for brevity sake I shall omit, and leave the Ingenious Seaman to Calculate at his Pleasure.

I might hereunto annex many more proposi∣tions of Circular Sailing, but because of the smallness of this Treatise, and that those Pro∣positions already handled, being by the Inge∣nious Seaman well understood, will be sufficient to enable him to perform any other Conclusion in Circular Sailing whatsoever, I therefore here omit, and hasten forwards unto the other parts of this Mathematical Treasury.

A Table of Angles, which every Rumb makethwith the Meridian.

These on this side the W. incline to∣wards the N. end of the Meridian	Angles of In∣clination with the Meridian.	These on this side the E. incline to the N. end of the Meridian.
Rumbs.	North	Rumbs.
N. by W.	11°	15'	N. by E.
N. N. W.	22	30	N. N. E.
N. W. by N	33	45	N. E. by N.
North West	45	00	North East
N. W. by W.	56	15	N. E. by E.
W. N. W.	67	30	E. N. E.
W. by N.	78	45	E. by N.
West	90	00	East
W. by S.	78	45	E. by S.
W. S. W.	67	30	E. S. E.
S. W. by W.	56	15	S. E. by E.
South West	45	00	South East
S. W. by S.	33	45	S. E. by S.
S. S. W.	22	30	S. S. E.
S. and by W.	11	15	S. and by E.
Rumbs	South	Rumbs
These on this side the W. in∣cline unto the S. end of the Meridian.	These on this side the E. incline to∣wards the S. end of the Meridian.
Note that if you account in quarter of Points, add for one quarter 2° 48', for one half 5° 37', for three quarters 8° 26', (not regarding the Seconds in Navigation.)

CHAP. IX. Of SURVEYING.

IT hath been a custom among Modern Au∣thors, that have treated on this Subject, that before they entred on the Work it self, to give the Description of the In∣struments, used in; and chiefly appertaining to the Art of Surveying: viz. the Circumferentor, the Theodolite, the Plain-Table, and the Semi∣circle: concerning the descriptions of which In∣struments I shall not here treat, but refer you unto those Authors that have largely and am∣ply described them. I shall in this place onely demonstrate the Use of the Semicircle in taking the Plots of Enclosures, Champain-Plains, Woods and Mountains divers ways* 1.256; and also in ta∣king of Accessible, and Inaccessible Heights and Di∣stances; and also I shall shew the use of a little Instrument called a Protractor, in the delinea∣ting on Paper the Plot of a Field, &c. which Instrument being so commonly known, and so generally used makes me omit the description thereof as superfluous.

As for your Chain, I would have you, have it made of good round Wyre; to contain in length four Poles, or Perch, to be divided into an hundred equal parts called Links.

And here before we enter on the Work it self, it will be necessary to understand how by the Protractor to lay down an Angle of any quantity of degrees propounded, or to find the quantity of an Angle given.

SECT. I. Of the use of the Protractor.

PROP. I. By the Protractor, to Protract an Angle of any quantity of degrees propounded.

AN Angle may be laid down easily accord∣ing to the directions of Prop. 5. §. 1. Ch. 4.* 1.257 but because this is more usefull in Surveying, Know that if it be required to protract an Angle of 50 deg. having drawn the line A B at plea∣sure, place the Centre of your Protractor on C, and moving it by your Protracting Pinn, untill the Meridional line thereof be directly on the line A B, then make a Mark by the division of 50° on the limb of the Protractor as at D, and 〈1 page duplicate〉〈1 page duplicate〉

〈1 page duplicate〉〈1 page duplicate〉

draw the line CD, so shall the Angle DCB, be an Angle of 50 degrees.

PROP. II. By the Protractor given, to measure an Angle given.

This is performed by the line of Chords also, according to prop. 6. §. 1. chap 4. and by the Protractor is found thus: Suppose DCB were an Angle whose Quantity were desired, to find which, first the Center of the Protractor applyed* 1.258 unto the Angular point C, and its Meridional line lying justly with CB; you shall perceive the Point D, to touch the limb of the Circle at 50 deg. Therefore I conclude the Measure of the Angle DCB, to be 50 degrees.

SECT. II. Of the Manifold Use of the Semicircle, in taking the Plots of small Enclosures, Plains, Woods, or Mountains divers Ways.

PROP. I. How to take the Plot of a Field, by the Semicircle at one Station taken in any part thereof, from whence all the Angles may be seen, and measu∣ring from the Station unto every Angle thereof.

SUppose ABCDEF were a Field, and 'tis re∣quired to take the Plot thereof: Having placed marks at all the Angles thereof, and made choice of your Station, which let be K; at which, place your Instrument, and turn∣ing* 1.259 it about untill the Needle hang over the Meridian Line of the Chart, there screw it fast: Then directing your sight to A, you'l find the Degree out by the Index to be 40° 15': Then measuring KA with your Chain it appears to be 5 Chains and 20 Links, which note down in your Field-book: and so do by all the rest untill you have found all the Angles and Distances from your Station K, to each respective Angle, which finished your work will stand thus.

Angles.	D.	M.	C.	L.
A.	40	15	5	20
B.	88	00	6	10
C.	130	00	5	50
D.	200	00	7	00
E.	250	00	5	00
F.	310	00	5	20

PROP. II. How to delineate on Paper any Observation taken according to the Doctrine of the last Proposition.

Upon your Paper draw a Line to represent the Meridian line as M, H, then Placing the Center of your Protractor on the point K, lay∣ing the Meridian line of the Protractor on the Meridian line M, H, then seeing the Angle at A was 40° 15', make a Mark against 40° 15' of the Protractor, as at A, and so do with all the* 1.260 other Angles, as you find them in your Table: Then remove your Protractor, and draw the Lines KA, KB, &c. This done lay down on each line his respective Measure, as it appear∣eth in the Table. Lastly draw the Lines AB, BC, &c. So have you on the Paper the exact Figure of the Field.

PROP. III. How by the Semicircle to take the Plot of a Field at one Station in any Angle thereof, from whence you may view all the other Angles, by measuring from the Stationary-Angle, unto all the other Angles.

Admit A, B, C, D, E, F, G, to be a Field, whose Plot is required: Place your Semicircle at G, and turning it about untill the Needle hang over the Meridian line of the Chart, and there screw it fast: Then direct your sights to the several Angles, viz. B, C, D, &c. in order one after the other, and so shall eace respective* 1.261 Angle be found, as in the subsequent Table: Then with your Chain, measure from your Sta∣tionary-Angle G, to all the other respective Angles, which done you have finished, and the work standeth thus.

Angles.	D.	M.	C.	L.
B.	40	00	5	00
C.	88	00	6	00
D.	120	15	6	40
E.	165	00	6	30
F.	193	00	3	40
A.	348	07	4	00

PROP. IV. How to delineate any Observation taken according to the Doctrine of the last Proposition.

Upon your Paper draw a streight line as M, N, then take a point therein as G, to represent the Stationary-Angle, to which point apply the Center of your Protractor, (in all respects as is* 1.262 before taught) then according to the Notes in the Table, prick off all the Angles, viz. B, C, &c. according to their due quantity, then draw all the lines, viz. GB, GC, GD, &c. and on them place their respective measure (as appeareth in your Notes) lastly draw the lines AB, BC, CD, &c. So is there on the Paper the exact Figure of the Field, as was required.

PROP. V. How by the Semicircle to take the Plot of a Field at two Stations, by measuring from each Station to the visible Angles: the Field being so Irregu∣lar that from no one Place thereof, all the Angles can be seen.

Admit A, B, C, D, E, F, G, H, I, K, to be the Figure of a Field, whose Plot is required: having made choice of your two Stations, viz. Q, and P, and placed Marks in all the Angles: Then place your Semicircle at Q, and there* 1.263 six it with the Needle hanging over the Meri∣dian of the Chart, represented by R, Q, X, and direct your sights unto all the visible Angles, viz.

A, B, C, D, E, and F, and note down the Quantity of each Angle in your Field-book: Then measure with your Chain from your Station Q, to the Angles A, B, C, D, E, and F, and their length so found, note down in your Field-book also.* 1.264

This done direct your sight unto your second Station P, and note down in your Field-book the degree of Declination, of your second-station P, from the Meridian. Then measure the Stationary Distance PQ with your Chain, and note it down in your Field-book also.

Then remove the Instrument unto P, your second-station, and there fix it with the Needle hanging over the Meridian line of the Chart re∣presented by TPB, then direct your sights to the several visible Angles at this second Station, viz. F, G, H, I, and K, in order one after ano∣ther, and note down the Quantity of each An∣gle in your Field-book: Then with your Chain measure from your Station P, to these several Angles G, H, I, and K, (in all respects as at the first station Q.) and their length so found note down in your Field-book likewise: So have you finished your Observation, and your work stan∣deth thus.* 1.265

The Observation taken at the first Station Q.

Angles.	D	M	C.	L
A	50	00	6	60
B	80	00	7	65* 1.266
C	140	12	12	00
D	220	07	11	10
E	270	05	12	60
F	330	00	6	00

The Declination of the Station P, from the Meridian R Q X, is 30° 00', and the Stationary distance Q P is 9 Chains.

The Observation taken at the second Station P.

Angles.	D	M	C	L
F	227	11	00	00
G	297	00	12	00
H	347	16	9	90
I	60	00	6	00
K	90	00	6	26* 1.267

☞ Note that the manner of taking the Plot of a large Champain Field, at many Stations, is almost the same with this Proposition; for he that can do the one, can also perform the o∣ther: therefore for brevity sake I here omit it as superfluous.

PROP. VI. How to delineate any Observation taken according to the Doctrine of the last Proposition.

Upon your Paper draw the Meridian-line R Q X, then place the Center of your Protrac∣tor on Q, (representing your first Station) and its Meridional-line lay equal to R Q X, then prick off the Angles visible at your first Station Q, viz. A, B, C, D, E, and F, Of their due quantity, then draw Q A, Q B, &c. laying on them their corresponding measure, noted in your Field-book. Now because your second Station P, doth decline 30° 00', from the Me∣ridian RQX, prick off 30° 00', and draw PQ,* 1.268 making it 9 Chains as in your Field-book ap∣peareth, so doth P represent your second Stati∣on. Then in all respects as before, place your Protractor at P your second Station, and draw the Meridian T P B parallel to R Q X, then prick off the several Angles, viz. F, G, H, I, and K, Of their due quantity, and then draw PF, PH, PI, &c. of their due length. Lastly draw the lines AB, BC, CD, &c. and so shall you have on your Paper the exact Figure of the Field as required.

PROP. VII. How by the Semicircle, to take the Plot of a Field at t〈…〉〈…〉 Stations, which lieth remote from you, when either by opposition of Enemies you may not, or by some other Impediment you cannot come into the same.

Admit the Figure A, B, C, D, E, F, to be a Field into which by no means you can possibly enter, and yet of necessity the Plot thereof must be had, for the obtaining of which chuse any two Stations, it mattereth not whether near at hand or far off, so that all the Angles may be seen. Let your two Stations be H and L, (the full length of the Field if possible) then place your Instrument at H, and fixing it as is* 1.269 afore shewed, direct your sights to the several Angles of the Field, viz. A, B, C, &c. orderly one after another, observing their degrees as is afore taught, noting it down in your Field-book: then take up your Instrument, leaving a mark in its room at H, And measure with your Chain from Hunto L, your second Station, which note down in your Field-book; Then placing your In∣strument at L, your second Station, and as is be∣fore taught, fixing it there, make the like Obser∣vation to the several Angles, viz. A, B, C, D, &c. as at the first Station H, and note it down in your Field-book also, And having so done you have finished, and your Work standeth thus.

Observations at the first Sta∣tion H, are

The Angle from H the first Station, unto L the se∣cond Station, is 180° 00', the* 1.270 Stationary distance HL, is 60 Chains.

Observations at the second Sta∣tion L, are

1 Angles	D	M
A	104	00
B	88	07
C	59	00
D	48	00
E	26	00
F	21	30
2Angles	D	M
A	16	00
B	39	00
C	50	09
D	74	00
E	100	00
F	29	15

PROP. VIII. How to delineate any Observation taken according to the Doctrine of the last Proposition.

Upon your Paper draw a Line as HL, which make equal to 60 Chains, then placing the Center of your Protractor on H, your first Sta∣tion, prick off all the Angles A, B, C, &c. as* 1.271 you find them in your Field-book, and draw HA, HB, HC, &c. at pleasure: then remove your Protractor unto your second Station L, pla∣cing it as before, and prick off all the Angles

A, B, C, D, &c. as you find them in your Field notes; and draw the lines LA, LB, LC, &c. at length untill they intersect the former lines, HA, HB, &c. in the Points A, B, C, &c.* 1.272 which Points of Intersection are the Angles of the Field. Lastly draw AB, BC, CD, &c. So shall you have on your Paper the Figure of your Field, required.

PROP. IX. How by the Semicircle, to take the Plot of a great Champain-Plain, Wood, or other overgrown Ground, by measuring round about the same, and making Observation at every Angle thereof.

Admit A, B, C, D, be the figure of a Large overgrown Champain-Field; whose Plot is requi∣red. First Place your Instrument at A, laying the Index on the Diameter; and turn it about, untill you espy the Angle at D, and there fix it fast: and direct your sights to B, and note the Degree cut by your Index, in your Field-book, (as afore is taught) then remove your Instru∣ment to B, and there make the like observation, and so to C, and D, noting it down in your Field-book, as asore. Then with your Chain, measure the Sides AB, BC, CD, and DA, whose* 1.273 length note down in your Field book, and so you have finished and your work standeth thus.

Angles.	D	M	C	L
DAB	100	00	12	20
ABC	117	15	10	00
BCD	71	30	19	20
CDA	71	15	12	20

PROP. X. How to delineate any Observation taken according unto the Doctrine of the last Proposition.

Upon your Paper draw the line AB, at Plea∣sure,* 1.274 and placing the Center of your Protractor on the Point A, prick off an Angle of 100°, and draw AD, setting on it, and also on AB, their corresponding measure, in your notes: Then on B, protract an Angle of 117° 15', draw BC of its due length: Then draw the line CD, so have you the exact figure of the Field, on your Paper.

PROP. XI. How to take the Plot of any Field, by the help of the Chain only.

Admit the Figure A, B, C, D, E, to represent a Field whose Plot is required. To obtain the* 1.275 which, first measure the sides CD, CB, and BD, and note their due length down in your Field∣book,

and then measure the Sides CA, and BA, and then note* 1.276 down their Length in your Field∣book. Then measure the sides BE, and ED, for the sides BC, and* 1.277 BD, were be∣fore known) which note down in your Field∣book. So is your Field A, B, C, D, E, reduced into three Triangles, viz. CBD, CAB, and BED, the length of whose sides are all known, thus you have finished, and the works stands as you see.

PROP. XII. How to delineate any Observation, taken according to the Doctrine of the last Proposition.

Upon your Paper, draw a streight line, as CD, make it 5 Chains, 97/100, take CB in your Compasses, and strike an Obscure Arch; then take BD, and with that extent in D, cross the for∣mer* 1.278 Arch in B, and draw BC, and BD. Then take in your Compasses BE, and on B, strike an Obscure Arch, then take DE, and also cross

the former Arch in E, and draw BE, and ED. Lastly take the line CA, and on C strike an Obscure Arch, then take AB, and on B, inter∣sect* 1.279 the former Arch in A, then draw CA, and AB, so have you on your Paper the exact figure of the Field A, B, C, D, E, as was re∣quired.

SECT. III. Of finding the Area or superficial Content of any Field, lying in any Regular or Ir∣regular Form: by reducing the Irregular Fields into Regular Forms.

HAving already shewed how to take the Plot of any Field divers ways, by the Semicircle and Chain, and also by the Protractor how to delineate the Draught thereof on Pa∣per, &c. I now come to shew how the Area or superficial Content of a Field may be attained, i. e. how many Acres, Roods and Perches are there∣in contained. To which end know; That a Statute Pole or Perch contains 16½ Feet; that 40 of those Perches in length, and 4 in breadth makes an Acre. So that an Acre contains 160 Perches, and a Rood 40 Perches; according to the Statute 33, of Edward the First.

Now the Original of the Mensuration of Land, and all other Superficies, depends on the Mensuration of certain Geometrical Figures; as a Triangle, Square, &c. which may be measured according to the directions of §. 2. chap. 4 of Geometry: It would therefore here be superflu∣ous to make a repetition of things already handled: I shall therefore omit it, and come to shew how any Field lying in any Irregular Form, may be measured by converting it into Regular Figures; for it seldom happeneth, but that the Plot of a Field, is either a Trape∣zium* 1.280* 1.281, or a many-sided Irregular Figure: there∣fore I shall first shew how to find the Content of a Trapezium. Secondly, of any many sided Irregular Figure; and thirdly, how to reduce any num∣ber of Perches into Acres, &c. and on the con∣trary any number of Acres, into Roods and Perches.

PROP. I. How to find the Area or superficial Content of a Trapezium.

Trapeziums are Quadrangles of sundry forms: yet take this as a general Rule, whereby their Content may be found. Admit it be required to find the Area or superficial Content of the Tra∣pezium ABCD, to find which, first by drawing

the Diagonal AD, you reduceth it into two Tri∣angles, ABD, and ADC. Then by prop. 3. §. 1. of Chap. 4 let fall the two Perpendiculars on AD, from B, and C, Then by prop 3. §. 2 Ch.* 1.282 4. find the superficial Content of the two Trian∣angles ABD, and ADC, which added together, is the Content os the Trapezium; by which Rule the Content of the Trapezium, A, B, C, D, is found to be 630 Perches.

PROP. II. To find the Area or superficial Content of a many∣sided Irregular Figure.

Admit A, B, C, D, E, F, G, to be an Irregu∣lar many-sided Figure, representing a Field whose Content is required: now in regard the Field is Irregular, therefore reduce it into Triangles, viz. ABC, ACG, EDG, DEG, and DFG, and* 1.283 then find the Content of all the said Triangles, by prop. 3. §. 2. Chap. 4 and add their Con∣tents together; so shall that Sum be the Con∣tent of the said Figure; and so do for any other.

PROP. III. How to reduce any Number of Perches into Acres, and on the contrary, Acres into Perches.

To find how many Acres are contained in a∣ny Number of Perches given, you must consi∣der that 160 Perches do make a Statute Acre, therefore if you divide the Number of Perches

propounded, by 160, the Quotient is the number of Acres contained therein; and if there be a remainder which exceed 40, then divide it by 40, the Quotient shall be Roods, and the remainder Perches.

But on the contrary, if it were required to find how many Perches are contained in a cer∣tain Number of Acres propounded. You must multiply the Number of Acres, by 160: the product shall be the Perches contained there∣in.

It may be here expected, that I should shew how to reduce customary Measure to statute Mea∣sure; and also that I should treat of the Division and Separation of Land. But because Mr. Rath∣borne, and of late Mr. Holwell, hath sufficiently explained the same, by many varieties, I shall for brevity sake omit it, and leave you to consult those Authors.

SECT. IV. Of the Use of the Semicircle in taking Al∣titudes, Distances, &c.

PROP. I. How by the Semicircle to take an Accessible Alti∣tude.

ADmit AB, be the Height of a Tower, which is required to be known. First placing your Semicircle at D, (with the Arch downwards* 1.284 and the two sights fixed) place it Horizontal* 1.285 and screw it fast; Then move your Index, till through the sights thereof, you espy the top of the Tower at B, and observe what degree the lower part of the Index cutteth and that will be equal unto the Angle at D 50 deg Then measure the di∣stance DA, which let be 299 Feet. Now the heighth of the Tower AB, is found, according to prop. 1. §. 2. Chap. 5. thus,

As Sc. V. at A 50° 00',

To Log. cr. DA 299 Feet.

So is S. V. at A 50 00,

To Log. AB 356 3/10 Feet the height of the Tower AB required.

PROP. II. How by the Semicircle to take an Inaccessible Al∣titude, at two Stations.

Let AB be a Tower whose height is required; having placed your Instrument at E, as before direct your sights unto the Top of the Tower at B, and finding the Degree cut by the Index,* 1.286 to be 23° 43', I say it is the Quantity of the Angle at E: Now by reason of Water, or such like Impediment, you can approach no nearer the Base of the Tower, than D, Therefore measure ED, which is found to be 512 Feet, then at D, make the like Observation, and the Angle at D, appeareth to be 50° 00', whose Complement is the Angle DBA, 40° 00', and the Complement of the Angle E 23° 43', is the An∣gle EBA 66° 17': Now if the lesser Angle at B, be taken out of the greater, the remainder is 26° 17', the Angle EBD: Now first to find the side BD, of the Trangle EBD, say according to prop. 1. §. 3. chap. 5. thus.

As S. of V. EBD, 26° 17',

To Log. cr. ED 512 Feet.

So is S. of V. at E 23° 43',

To Log. cr. BD 465 2/10 Feet required.

Now to find the Height of the Tower AB, say according to prop. 2. §. 2. chap. 5. thus.

As Radius or S. 90°,

To Log. cr. DB 465 2/10 Feet found.

So is S. of V. BDA 50° 00',

To Log. cr. BA 356 3/10 Feet, which is the height* 1.287 of the Tower required.

☞ Note that in taking any manner of Alti∣tude the height of your Instrument must be added unto the height found, and that will give you the True Altitude required.

PROP. III. How by the Semicircle to take an Inaccessible Dis∣tance at two Stations.

Admit A, and B, be the two Stations, from either of which it is required to find the distance unto the Church at C; placing your Instrument at B, the Index lying on the Diameter, and di∣rect your sights unto the Church at C, fasten your Instrument, and turn your sights about un∣till you see through your sights, your second Station at A, so will you find your Index to cut* 1.288 30° 00', which is the Quantity of the Angle ABC. Then measure the distance AB, which is found to be 250 Yards, then with your In∣strument at A, make the like Observation as before, and you will find the Angle BAC to contain 50° 00'. Now by the third Maxim of Plain Triangles §. 1. Chap. 5 you find also the Angle ACB, to be 100° 00': now to find the distance AC, and BC, you may by their oppo∣site proportion according to prop. 1. §. 3. chap. 5. find the distance of AC, thus.

As S. of V. at C 100° 00',

To Log. cr. AB 250 yards.

So is S. of V. B 30° 00',

To Log. cr. AC 127 yards. Which is the di∣stance of the Church from A.

Now to find the distance BC, say,

As S. of V. at A 100° 00',

To Log. cr. AB 250 yards.

So S. is of V. at A 50° 00',

To Log. cr. BC 194 4/10 yards, which is the di∣stance of the Station B, from the Church at C.

PROP. IV. How to find the Horizontal line of any Hill or Mountain, by the Semicircle.

Let Figure 63 be a Mountain, whose Hori∣zontal-line AB is required to be found: to find which, place your Instrument at A, and ha∣ving caused a Mark to be placed on the Top of the Mountain at C; (of the just height of your Instrument) then move your Index, untill through the sights thereof you espy the Mark at C, so will you find the Quantity of the An∣gle CAD, to be 50° 00', and by consequence* 1.289 the Angle ACD to be 40° 00', then measure up the Hill AC, which is 346 yards. Now ha∣ving obtained these several things, 'tis required to find the length of AD part of AB; to find which say,

As Radius or S. 90°,

To Log. cr. AC 346 Feet.

So is Sc. of V. at A 50° 00',

To Log. cr. AD 222 4/10 Feet.

Now seeing the Hill or Mountain descendeth on the other side, you must place your Instru∣ment at C, and direct your sights unto the Bottom at B, and the Angle DCB will be found 50° 00', and the Angle CBD 40° 00'. Then measuring down the Mountain as CB, it appea∣reth

[illustration]

Plate IV Page 237

* 1.290* 1.291* 1.292

to be 415 Feet; then have you the An∣gles DCB, and CBD.

To find DB, part of AB, say,

As Radius or S. 90°,

To Log. cr. CB 415 Feet.

So is S. of V. BCD 50° 00',

To Log. cr. DB, 318 Feet: Now AD 222 4/10* 1.293 Feet added thereunto produceth AB 540 4/10 Feet, which is the Horizontal line required of the Mountain ACBD.

☞ Note that when you come to delineate a Field wherein are Hills, you must protract the line AB, instead of the Hypothenusal Lines AC, and CB, and 'twill be necessary to distinguish those kind of Fields, by shadowing them off with Hills and Dales.

SECT. V. How to find whether Water may be conveyed from a Spring-Head unto any appointed Place.

THE Art of conveying of Water from a Spring-Head unto any appointed Place, hath a special respect unto measuring, and therefore I think it not amiss to assert it in this place, and enroll it under the Title of Sur∣veying.

In the performance of which we make use of a Water-level, the Construction and ma∣king whereof is sufficiently known to those who make Mathematical Instruments: Now if it were required to find whether Water may be conveyed in Pipes, &c. to any Place assigned: to perform which observe these Rules.

First at some 10, 20, 30, 40, 60, or 100 yards distant from the Spring-head in a right∣line towards the Place unto which your Water is to be conveyed. Place your Water-level, be∣ing prepared of two Station Staves with move∣able Vanes on each of them, graduated also af∣ter the usual Manner: Cause your first Assis∣tant to set up one of them at the Spring-Head; Perpendicular unto the Horizon, and your second Assistant to erect another, as far from your Water-level towards the Place to which the Wa∣ter is to be conveyed, as your Water-level is distant from the Spring-head: Now the Station∣staves in this order erected, and your Water∣level placed precisely Horizontal, go unto the end of the Level, and looking through the sights, cause your first Assistant to move a Leaf of Paper, up or down your Station staff, untill through the sights you espy the very edge there∣of, and then by some known sign or sound, inti∣mate to your Assistant that the Paper is then in its true position, then let the first Assistant note against what Number of Feet, Inches, and parts of an Inch the edge of the Paper resteth; which he must note down in a Paper. Then your Water-level remaining immoveable, go to the other end thereof, and looking through the sights towards your other Station-staff, cause

your second Assistant to move a Leaf of Paper along the Staff, till you see the very edge thereof through the sights, and then cause him by some known sign or sound, to take notice what number of Feet, &c. are cut by the said Paper, which let him keep, as your first Assis∣tant did.

This done let your first Assistant bring his Station-staff from the Spring-head, and cause your second Assistant to take that Staff, and carry it forwards towards the Place, unto which the Water is to be conveyed; some 30, 40, 60, or 100 yards, and there to erect it Perpendicular as before, letting your second Assistant's staff stand immoveable, and your first Assistant to stand by it: Then in the Midway between your two Assistants, place your Water-level exactly Horizontal, and looking through the sights thereof, cause your first Assistant, and after that your second, to make their several observations in all respects as before.

In this manner you must go along from the Spring-head, to the place unto which you would have the Water conveyed, and if there be ne∣ver so many several Stations, you must in all of them observe this manner of work precisely; so that by comparing the notes of your two Assistants together, you may easily know whe∣ther the Water may be conveyed from the Spring-head, or not, by calling your two Assis∣tants together, and causing them to give in their notes of observation at each Station, which add together severally: Then if the Notes of the second Assistant, exceed the Notes of the first Assistant, take the lesser out of the greater, and

the remainder will shew you how much the appointed Place, to which the Water is to be conveyed, is lower than the Spring-head.

The first Assistant's Note.

Station.	Feet.	Inch.	Parts.
1	15	3	. 50
2	2	1	. 25
3	1	6	. 00
Sum	18	10	. 75

The second Assistant's Note.

Station.	Feet.	Inch.	Parts.
1	3	2	. 75
2	14	0	. 25
3	3	11	. 00
Sum	21	2	. 00

By these two Tables you may perceive that the Notes of the first Assistant collected at his several Stations, being added together, amounts unto 18 Feet, 10 Inches, and 75/100 or ¾ of an Inch: and the Notes of your second Assistant collected at his several Stations, amounts unto 21 Feet, 2 Inches: So that the number of the first Assi∣stant's Observations, being taken from the se∣cond's, there will remain 2 Feet, 3 Inches, and 25/100 or ¼ of an Inch. And so much is the place unto which the Water is to be brought, lower than the Spring-Head, according to the sleight Water-Level, and therefore the Water may easily be conveyed thither. And here observe these Notes.

1. In your Passage between the Spring head, and the appointed Place, from Station to Station, you must observe this order, that your first Assistant at every Station must stand between the Spring-head, and your Water-level: other∣wise great Errours will ensue.

2. That if the Notes of your first Assistant, exceed the Notes of the second Assistant, then 'tis impossible to bring the Water from that Spring-head unto the appointed place, but if their Notes are equal, it may be done, if the distance be but short.

3. That the most approved Authors con∣cerning this particular do aver, that at every Mile's end there ought to be allowed 4½ Inches more than the Streight-level, for the current of the Water.

4. That if there be any Mountains lying in the way betwixt the Spring-head and the Place to which the Water is to be conveyed, you must then cut a Trench by the side of the Mountain, in which you must lay your Pipes equal with the streight Water-level, with the former allow∣ance: and in case there be a Valley, you must then make a Trunk of strong wood, well un∣der-propped with strong pieces of Timber, well Pitched, or Leaded, as is done in divers places between Ware and London.

5. That when the Spring will have too vio∣lent a Current, you must then convey your Wa∣ter to the place assigned, by a Crooked or Winding line, and you also ought to lay the Pipes, the one up, and the other down, that thereby the Violence of the Current may be stopped.

CHAP. X. Of MEASURING, Of Board, Glass, Tiling, Paving, Timber, Stone, and Irregular Solids, such as Geo∣metry can give no Rule for the Measuring thereof.

SECT. I. Of the Measuring of Board, Glass, Paving, Tiling, &c.

I Have already in the fourth Chapter of this Book, and the second Section thereof, ap∣plyed Geometry to the finding out of the Su∣perficial Content of all Regular Superficies. I have also in the ninth Chapter, and the third Section thereof, shewed how the Superficial Content of any Irregular Superficies may be found, by redu∣cing

them into Regular Forms: which I have explained amply in that Section, I shall there∣fore here be as plain and brief as is possible.

PROP. I. To Measure a Piece of Board, Plank, Glass, &c.

In Measuring of Board, Glass, &c. Carpen∣ters and other Mechanicks measure by the Foot, 12 Inches unto the Foot; so that a Foot of Board, or Glass, contains 144 Square Inches.

Now if a Piece of Board, Plank, or Glass, be required to be measured, let it be either a Parallelogram, or Tapering Piece: first by the Rules aforegoing find the Content thereof in Inches, and that Product divide by 144, the Quotient is the Content of that Superficies in Feet.

PROP. II. To measure Tiling, Flooring, Roofing, and Parti∣tioning-works.

In Tiling, Flooring, Roofing, and Partitioning∣work, Carpenters, and other Workmen, reckon by the Square, which is 10 Feet every way; so that a Square containeth 100 Feet: Exam∣ple.

There is a Roof 14 Feet broad, what length thereof shall make a Square? Divide 100 by 14, it yields 7 1/7 Feet.

Now if you have any Number of Feet gi∣ven, and the Number of Squares therein contai∣ned are required, divide that Number by 100, the product is Squares.

PROP. III. To measure Paving, Plaistering, Wainscotting, and Painting-work.

In Paving, Plaistering, Wainscotting, and Painting-work, Mechanicks reckon by the Yard Square, so each Yard is equal unto 9 Square Feet.

By the Rules aforegoing find the Superficial Content of the Court, Alley, &c. in Feet: which divide by 9, the Quotient is the Number of Yards in that work contained.

SECT. II. Of the Measuring of Timber, Stone, and Irregular Solids.

IN Superficial Measure a Superficial Foot con∣tains 144 Square Inches; but in Solid Measure a Foot contains 1728 Cubick Inches. Now ha∣ving already in the fourth Chapter of this Book, and the third Section thereof, largely applyed Geometry unto the Measuring of all

Regular Solids, I shall therefore in this Place be as brief as possible, only I shall be somewhat larger in the Mensuration of Irregular Solids, which is of special Moment in sundry parts of the Mathematical Practices.

PROP. I. How to Measure any kind of Timber, or Stone, whether Three-square, Four-square, Many-square, Round, or of any other fashion, provided it be streight and equal all along.

To perform which first by the Rules afore∣going in Chap. 4. §. 2. get the Superficial Content at the End, and then say,

As 144, the Inches of the Superficial Content of the End of a Cubick Foot,

To a Cubick Foot containing 1000 parts;

So is the Superficial Content of the End of any piece of Timber,

To the Solid Content of one Foot length of the said piece of Timber.

According to which Mr. Phillips calculated the ensuing Table, which I have thought fit hereunto to annex.

Case 2 Or the solid Content in Feet, &c. may be found otherwise thus.

By the Rules aforegoing find the Content of the End of the piece of Timber in Inches, which Content multiply by the length of the said piece of Timber, or Stone in Inches, and that Product divide by 1728, it produceth the Solid

Content of that Piece of Timber, or Stone, in Feet, and parts of a Foot.

A Table shewing the Solid Content of one Foot-length of any Piece of Timber, according to the Superfi∣cial Content at the End thereof.

Feet.	Parts.	Feet.	Parts.
The Inches of the Content at the End.	1	0	007	The Inches of the Content at the End.	200	1	398
2	0	014	300	2	083
3	0	021	400	2	778
4	0	028	500	3	472
5	0	035	600	4	167
6	0	042	700	4	861
7	0	049	800	5	556
8	0	056	900	6	250
9	0	062	1000	6	944
10	0	069	2000	13	888
20	0	139	3000	20	833
30	0	208	4000	27	778
40	0	278	5000	34	722
50	0	347	6000	41	666
60	0	417	7000	48	711
70	0	485	8000	55	555
80	0	556	9000	62	500
90	0	625	10000	69	444
100	0	694	20000	138	888

PROP. II. To measure Round Timber which is Hollow: or a∣ny other Hollow Body.

If Hollow Timber be to be measured, first measure the Stick as though it were not Hollow, then find the Solidity of the Concavity, as though it were Massie Timber, then substract this last found Content, out of the whole Content be∣fore found, the remainder is the Content of that Hollow Body.

PROP. III. To Measure Tapering Timber, or Stone.

Those Tapering Bodies are either Segments of Cones, or Pyramids: now the way to measure such bodies, is demonstrated in Prop. the 4. and 5. §. 3. Chap. 4: But now to find the Content of these Segments do thus: measure the Solidity of the whole Cone, or Pyramid, and then find the Content of the Top part thereof cut off, (as if it were a Cone, or Pyramid of it self) and the Content thereof, deduct from the Content of the whole Cone, or Pyramid: so shall the re∣mainder be the Content of the Segment requi∣red: which reduced into Feet gives the Solid Content of that Piece of Timber in Feet. Now to find the length of the Top part cut off, from the Cone, or Pyramid, say,

As the Difference of the breadth of the two Ends, To the length between them:

So is the breadth of the greater End, To the whole length of the Cone, or Pyramid.

PROP. IV. How to find the Solid Content of any Solid Body, in any strange form, such as Geometry can given: no Rule for the measuring thereof.

These strange forms are either Branches in Metal, Crowns, Cups, Bowles, Pots, Screws, or Twisted Ballisters,* 1.294 or any other Irregular-Solid, that keep not in thick∣ness one Quantity, but are thicker in one place, than in another, so that no man by Geometry, is possible to measure their Solidity.

Now for the finding the Content of any such like Irregular Body in Inches or Feet, do thus: Cause to be made a Hollow Cube, or Parallelepipe∣don, so that you may measure it with an Inch∣Rule without Difficulty, and so to know the true Content of the whole, or any part thereof at pleasure within the Concavity: Then take some other convenient Vessel, and put pure Spring∣water therein; then having filled the Vessel to a known Measure, make a Mark precisely round the very edge of the Water, then take the solid body and put it therein, then take out as much of the Water (as by means of the body put therein) is arisen above the Mark, untill the Water do justly touch at the Mark again: then put the Water taken forth into the

Hollow Cube, and find the solid Content thereof (being transformed into a Cubick Body) in Feet, Inches, and parts of an Inch: Which Con∣tent is the just solidity of the Body put into the Water. (Archimedes by this Proposition found the deceit of the Crown of Gold which Gelo the Son of Hiero had vowed unto his Gods: now the Workmen had mixed Silver with the Gold, which Theft was disco∣vered by the great skill of Archimedes)* 1.295 And herein you must be very curious not to spill any of the Water, or take out of the Vessel, or put into the Hollow Cube, any more than the just quantity arisen above the Mark, for if you do it will produce infinite Errours, and thus may the Solidity of any Irregular Body be found.

CHAP. XI. Of GAUGING.

IN GAUGING there are two things chiefly necessary to be noted, yet both controver∣ted. First, that seeing all manner of Casks, made to hold Liquor in, are for the most part the Trunk of a Sphereroid, cut off with two Circles, at Rightangles with the Base, and there∣fore Irregular, Therefore they must, first be re∣duced into a Regular Proportion. — And the second thing necessary to be noted, is to find the true quantity of an Ale, or Wine-Gallon in Cubick-Inches or parts of a Foot, that thereby the Content of the Vessel or Cask in Gallons may be known.

SECT. I. Of Gauging any Beer, Ale, or Wine-Cask, also any manner of Brewers Tuns.

PROP. I. To find the Solid Content in Inches of any Cask.

I Shall follow Mr. Oughthred's method, which is, Take the Diameter of the Cask both at Head and Bung, by which find the Area's of their Circles, which done, then take two thirds of the Area of the Bung, and one third of the Area at the Head, which added together, shall be the Mean Area of the Cask; which multi∣plyed into the length of the Vessel, it will shew how many solid Inches are contained therein.

Example: Suppose the Diameter at the Head of a Vessel be 18, and at the Bung 32, and length is 40 Inches.

Now I find the Aggregate of the two Circles to be 620, and 989, Cubick Inches: which mul∣tiplyed by 40, the length, produceth 24839, 56/100 Cubick Inches, for the whole Content of that Cask in Cubick Inches.

PROP. II. To find the Content of a Vessel in Wine, or Ale Gallons.

The Wine Gallon is established by the Con∣sent of Artists, in these and other Nations, to contain 231 Cubick Inches* 1.296. Yet Dr. Wybard affirms it to be somewhat less, to wit 225, at most: The Ale Gallon contains 282 Cubick Inches, accor∣ding to the Establish∣ment of Excise. Here∣in Artists differ somewhat in their Experi∣ments.

Now having already shewed how to find the Content in Inches of any Cask, I now come to shew how to find the Content in Gallons, of any Beer, Ale, or Wine Cask, which is thus: Divide the Number of Inches given by 231, for Wine Measure, and 282, for Ale Measure. In the former Example I find the said Cask to con∣tain 107, 53 Wine Gallons, and 88, 8, &c. Gal∣lons in Ale Measure.

PROP. III. How to Gauge or Measure Brewers Tuns, &c.

Those Tuns are most commonly Segments of Cones or Pyramid, whose Basis is either a Square Parallelogram, Circle, or Oval; to measure which,

let their form be what it will you must do thus. By the former Rules of Measuring such Seg∣ments or Bodies, you must find their Solid Con∣tent in Cubick Inches, (as in prop. 3. §. 2. chap. 10.) which Content divide by 282 Inches, (the Inches in one Gallon) it sheweth the Content in Gallons, and dividing the Gallons by 36, (the number of Gallons in a Barrel) it shews the Content in Barrels.

SECT. II. Of Gauging or Measuring, and the Moulding of Ships.

PROP. I. To Gauge a Ship, thereby to find how many Tuns her burthen is.

IN the Gauging or Measuring of Ships, Naupegers, or Ship-Wrights, observe these three Particular Rules: First, that if you measure the Ship within, you shall find the Content, or the Burthen the Ship will hold or take in. Secondly, if the Ship be measured on the outside, to her light mark as she swims be∣ing unladen, you shall have the Content of the Empty Ship. Thirdly, but if you measure from the light mark, to her full draught of Water be∣ing laden, you shall have the true Burthen of the Ship.

Now to find the Content of the King's Royal Ships: Measure the length of the Keel, the breadth of the Mid ship Beam, and the depth of the Hold: which three multiply into one ano∣ther, and divide their Product by 100; so shall you find how many Tuns her Burthen is.

But for Merchant's Ships, which give no al∣lowance for Ordnance, Masts, Sails, Cables, Anchors, &c. which are all a Burthen, but no Tonnage, you must divide the product by 95, so shall their true Burthen be found.

PROP. II.

By knowing the Measure of a Ship, of one Burthen, to make another Ship, of the same Mould, which shall be double, or triple, or in any proportion, either more or less than the said Ship.

First you shall multiply the Keel Cubically; and in like manner every Beam; the Mid ship Beams multiply them Cubically; and also the Reaking of the Ship, both at Stem, and Stem∣Post, multiply them Cubically; likewise the principal Timbers, that doth mould the Ship, multiply them Cubically; and the depth of the Hold, multiply it Cubically; and so conse∣quently every Place, or Places, which doth lead any work, multiply them Cubically; then if it be required to have a Ship as big again, or thrice as big; double, or triple each respec∣tive Cubical number; then by prop. 9. §. 1. chap. 1: Or by prop. 4. §. 2. chap. 2. find the Cube∣roots thereunto belonging; then according unto these respective Numbers, make your Keel, your Timbers, Beams, &c. which being done, you shall make a Ship of the Mould and Pro∣portion desired.

CHAP. XII. Of DIALLING.

HOROLOGIOGRAPHIA, or the Art of DIALLING, is an Art Mathematical, which demonstrateth the precise Distinction of Times, by the Sun, Moon and Stars, whereby the Time of the Day, or Night, may be known* 1.297. Now the Demonstrative de∣lineation of Dials, con∣sisteth chiefly in the find∣ing out the Hour-lines, and their true distance one from the other: which lines are great Cir∣cles of a Sphere, which being projected on a plain Superficies, become streight-lines; which lines do continually va∣ry, according as the Planes on which they are descri∣bed, or projected, do lie situated in respect of the Horizon of the Place.

Now a Dial may be made on any Plain Super∣ficies, for all Plain Superficies are Posited either Perpendicular, Parallel, or Oblique, to the Horizon of the Place, in which the Plane is seated.

In the delineation of all which Dials in this Chapter described, (which are the most Emi∣nent, and usefull Dials now used) I have used this Method: First, I have shewed how to de∣lineate them by Geometrical Projection, by Scale, and Compass only: and secondly how they may be described by Arithmetical Calculation, of both which I have been very plain and large.

SECT. I. Of the Delineation and Projection of sundry most usefull Dials.

PROP. I. How to draw the Hour-lines on an Equinoctial Plain.

AN Equinoctial Plane, is such which lieth Parallel unto the Equinoctial, and is an Horizontal Plane, under the Pole. This is the first and plainest kind of Dials, and is made after this manner: First describe the Circle AE, W, E, R, for your Planes, then Cross it with the two Diameters EW, and AER. Then* 1.298 divide the Semicircle E, W, R, into 12 equal parts in the points ☉, ☉, ☉, &c. Then from the Center Q, and through the said points

draw streight lines, which shall be the true Hour-lines belonging unto this Equinoctial Plane. Now because these Planes are capable of re∣ceiving all the Hour-lines from Sun-rising unto the Sun-setting, in Summer; therefore the Hour∣lines of 4, and 5, in the Morning; and 7, and 8, in the Evening; must be delineated as you see done in the Figure: These Hours may be sub-di∣vided into half Hours, and Quarters: The* 1.299 Stile of this Dial, must be a streight Pin, or Wyre set Perpendicular, to the Plain, on the Cen∣ter Q. and of any convenient length. This Dial may be made for any Latitude, and is of good use for Seamen, and others.

PROP. II. How to draw the Hour-lines on a Polar Plane.

A Polar Plane is one that lies Parallel unto the Pole, and under the Equinoctial is an Hori∣zontal Dial: the way to make this Dial is thus. First draw the line AB, for the Horizontal line of the Plane; and cross it at the Middle at right angles, with the line 12, Q, 12, which is the* 1.300 Meridian or Hour line of 12; Then upon the line 12, Q 12, either above or below the point Q, assume any point as S, then setting one foot of your Compasses in S, describe the Semicircle CED, which divide into 12 Equal parts, in the points ☉, ☉, ☉, &c. Then lay a Ruler un∣to S, and unto the several points ☉, ☉, ☉, &c. and it will cross the line AB, in the points x, x, x, &c. Then through those points draw (by prop. 4. §. 1. chap. 4.) right lines all Parallel

unto 12 Q 12, and so is your Dial finished. Then according unto the breadth of the Plane, you may proportion* 1.301 your Stile,* 1.302 Whose height must be equal to the di∣stance between the two Hour-lines 12, and 9, or 12, and 3, and then will the shadow of the upper edge thereof shew the Hour of the day: The height of the Stile, is also found thus.

As the Tangent of the Hour-line 4 or 5,

To the Distance hereof from the Meridian.

So is the Radius,

To the Height of the Stile.

Then for the other Hour-line, say,

As the Radius,

To the Height of the Stile.

So is the Tangent of any other Hour-line,

To the Distance thereof from the Meridian line.

PROP. III. How to draw the Hour-lines on a Meridian Plane, which is an East, or West Dial.

A Meridian Plane stands upright directly in the Meridian, and hath two Faces, one to∣wards the East, and the other towards the West.

Now admit it be required to make a direct East Dial, in the Latitude of 51° 32': let A, B,* 1.303 C, D, be a Dial-plane, on which you would de∣scribe a Direct East Dial, on the point D, de∣scribe

an obscure Arch HG, with the Radius of ••our line of Chords, then take 38° 28', the Complement of your Latitude, place it from G to L; then draw DL quite through the Plane; Then to proportion your Stile unto your Plane, so that all the Hours may be placed thereon, from Sun-rising to 11 a Clock. Assume two* 1.304 points in the line LD, as K, for 11; and I for the 6 a Clock Hour lines; then draw 6, 16, and 11, K 11, Perpendiculur to LD. This done, with the Radius of your line of Chords on L, strike the Arch OP, and from P, to O, place 15° 00'; and draw OK, to cut 6 I 6, in M, so shall IM be the height of the Stile proportioned unto this Plane; which may be a Plate of Brass, whose breadth must be equal to the distance between the Hour-lines of 6, and 9, which must be placed Perpendicular to the Plane, on the line 6, I 6, whose shadow of the upper edge, shall shew the Hour of the day. Now to draw the Hour-lines, with the Radius of your line of Chords, on M strike the Arch QN, which divide into 5 equal parts in the points •, •, •, &c. Then lay a Ruler from M un∣to each of those points, and it will cut the line JK in the points *, *, *, &c. through which points (by prop. 4. § 1. chap. 4.) draw Parallels to 6 I 6, as the lines 77, 88, &c. which shall be the true Hour-lines of an East Plane, from 6* 1.305 in the Morning, till 11 before Noon. Then for the Hour-lines of 4, and 5, you must prick off 5 as far from 6, as 6 is from 7; and 4, as far as 6 is from 8; and draw the Hour-lines 55, and 44, as before. Thus is your Dial compleat∣ed, and in the forming of which, you have 〈1 page duplicate〉〈1 page duplicate〉

〈1 page duplicate〉〈1 page duplicate〉

made both an East, and a West Dial; which is the same in all respects, only whereas the Arch H G, through which the Equinoctial passed in the East Dial, was described on the right hand of the Plane, in the West it must be drawn on the left hand, and the Hour-lines 4, 5, 6, 7, 8, 9, 10, and 11, in the Forenoon in the East Dial, must be 8, 7, 6, 5, 4, 3, 2, and 1, in the West in* 1.306 the Afternoon; as in the Figure plainly appear∣eth: Now you may find the distance of the Hour-lines from the Substile, by this Analogy or Proportion.

As the Radius,

To the Height of the Stile.

So is the Tangent of any Hours distance from 6,

To the distance thereof from the Substile.

PROP. IV. How to draw the Hour-lines on a direct South, and North Plane,

This Plane or Dial must stand upright, ha∣ving his face or Plane, if it be a South Dial, di∣rectly opposite unto the South; but if a North Plane, directly opposite unto the North; now admit it be required to make a Direct South Dial,* 1.307 for the Latitude of 51° 32': To make which first describe the Circle ABCD, to represent an E••ect direct South Plane, cross it with the Dia∣meters CB, and AD, then out of your Line of Chords take 38° 28', the Complement of the La∣titude, and set it from A, unto a, and from B, unto b, Then lay a Ruler from C unto a, and it will cut the Meridian ARD, in P, the Poles of

[illustration]

Plate V Page 261

* 1.308 * 1.309* 1.310* 1.311

other hath the North Pole of the World eleva∣ted above it, and beholdeth the North part of the Meridian. The Hour-lines of 9, 10, 11, or 1, 2, and 3, is not expressed on this Plane, because 12, representeth 12, at Midnight; neither are the other said Hours expressed, because the Sun is never above the Horizon, at those Hours; Therefore the North Dial is capable only to* 1.312 receive these Hours, namely 4, 5, 6, 7, and 8, in the Morning; and 4, 5, 6, 7, and 8, at Night; as doth plainly appear in the Figure: Now the distance of the Hour-lines from the Meridian, may be found by this Analogy, or Propor∣tion.

As Radius or S. 90°,

To Sc. of the Latitude.

So is T. of the Hour from Noon,

To T. of the Hour-line from the Meridian.

PROP. V. How to draw the Hour-lines on an Horizontal Plane.

This Horizontal Plane, or Dial, is one of the best and most usefull Dials in our Oblique He∣misphere: Admit it be required to make an Horizontal Dial, for the Latitude of 51° 32': To make which, first describe the Circle AB* 1.313 CD, which representeth your Horizontal Plane, Then cross it with the two Diameters ARC, and BRD, Then take 51° 32' out of your Line of Chords, and set it from B, to a, and from C, to b, Then lay a Ruler from A, unto a, and it will cut the Meridian BD, in P, the Pole of the

World, Then lay a Ruler from A, unto b, and it will cut ABD the Meridian, in the point AE, where the Equinoctial cutteth the Meridian, then through the three points A, AE, and C, draw* 1.314 the Equinoctial Circle, whose Center is at H; (and found as in the former proposition) Then divide the Semicircle ADC into 12 equal parts, in the points •, •, •, &c. Then lay a Ruler to R the Center of the Plane, and on those points, so shall the Equinoctial Circle AAeC, be by it divided into 12 unequal parts in the points *, *, *, *, &c. Then a Ruler laid unto P the Pole of the World, and those Points, shall cut the Semicircle CDA in those Points I, I, I, &c. Lastly, from the Center R, and through those Points, let there be drawn right lines, which shall be the true Hour-lines of such an Horizontal Plane, from 6 in the Morning, untill 6 at Night; but for the Hours of 4 and 5 in* 1.315 the Morning; and 7 and 8 in the Evening; they are delineated by producing 4 and 5 in the Evening, through the Center R, and 7 and 8 in the Morning; extending them out, unto the other side of the Plane, so shall you have those Hour-lines also on your Plane delineated as you see in the Figure. The Stile of this Plane may be a thin Plate of Brass, cut exactly unto the Quantity of an Angle of 51° 32', and set Per∣pendicular on the Meridian line, for the forming of this Stile take out of your Line of Chords 51° 32', and set it from D, unto e, and draw Re, which shall be the Axis of the Stile, you may also prefix the Halves, and Quarters of Hours, in the very same manner as the Hours themselves were drawn.

Now to find out the distance of the Hour-lines from the Meridian, say,

As the Radius or S. 90°,

To the S. of the Latitude.

So is the T. of the Hour from Noon,

To the T. of the Hour-line, from the Meridian Line.

These kinds of Dials being so frequently used* 1.316 with us, in this Oblique Sphere, for the help of the speedy delineating of them, I have an∣nexed hereunto the Table of Longomontanus, wherein the Hour-lines, for many Latitudes, are calculated.

A Table shewing the Distance of the Hour∣lines from the Meridian, in these Degrees of Latitude.

An Horizontal Dial, Latitude.	The Hours from the Meridian.	A South Erect Dial, Latitude.
xi.	i.	x.	ii.	ix.	iii	viii.	iv.	vii	v.	vi.
D	M	D	M	D	M	D	M	D	M	D	M
30	7	38	16	6	26	34	40	54	61	49	90	00	60
31	7	51	16	34	27	14	41	42	62	28	90	00	59
32	8	4	17	1	27	53	42	30	63	6	90	00	58
33	8	17	17	27	28	34	43	17	63	45	90	00	57
34	8	30	17	54	29	13	44	5	64	42	90	00	56
35	8	43	18	20	29	49	44	46	64	56	90	00	55
36	8	56	18	45	30	25	45	28	65	27	90	00	54
37	9	9	19	9	31	1	46	9	65	58	90	00	53
38	9	21	19	34	31	37	46	50	66	29	90	00	52
39	9	33	19	57	32	9	47	26	66	55	90	00	51
40	9	46	20	20	32	40	48	1	67	20	90	00	50
41	9	58	20	43	33	14	48	37	67	45	90	00	49
42	10	10	21	7	33	47	49	13	68	11	90	00	48
43	10	22	21	29	34	17	49	44	68	32	90	00	47
44	10	24	21	50	34	46	50	14	68	52	90	00	46
45	10	43	22	12	35	15	50	45	69	14	90	00	45
46	10	54	22	33	35	44	51	16	69	37	90	00	44
47	11	5	22	33	36	10	51	43	69	53	90	00	43
48	11	16	23	12	36	35	52	9	70	10	90	00	42
49	11	26	23	32	37	1	52	35	70	28	90	00	41
50	11	36	23	51	37	27	53	1	70	43	90	00	40
51	11	46	24	9	37	50	53	24	70	58	90	00	39
52	11	56	24	26	38	13	53	46	71	12	90	00	38
53	12	5	24	44	38	36	54	8	71	27	90	00	37
54	12	14	25	2	38	59	54	30	71	41	90	00	36
55	12	23	25	18	39	18	54	50	71	53	90	00	35
56	12	32	25	33	39	38	55	9	72	4	90	00	34
57	12	46	25	49	39	58	55	28	72	16	90	00	33
58	12	48	26	5	40	18	55	46	72	27	90	00	32
59	13	56	26	19	40	36	56	1	72	38	90	00	31
60	13	58	26	30	40	53	56	15	72	47	90	00	30

PROP. VI. How to draw the Hour-lines, on an Erect declining Plane.

These Planes are made to set on the sides of Houses, wherein the Meridian is always a Per∣pendicular,* 1.317 drawn on the Plane, in whose top is the Center, where the Substile, and the Hour∣lines all meet.

Now before we can delineate the Hour-lines on any such Planes, two things must be given: As the Latitude of the Place, and the Planes De∣clination; by having which we must find these three things: viz. The Poles height above the Plane. The distance of the substile from the Meri∣dian. And the Plane's difference of Longitude.

For the finding of which Requisites, by Geo∣metrical Projection, we describe on the Dial Plane, these Circles of the Sphere, viz. The Horizon, Meridian, and Equinoctial, which be∣ing described in their true Position, on the Plane, we proceed thus.* 1.318

Admit it be required to make a Direct South Dial, on an Erect, Direct South Plane, Declining Westward 24° 20', in the Latitude of 51° 32'.

Now in order to find the requisites before mentioned, describe the Circle ZHNO, and cross it with the two Diameters ZQN, and H QO: now Z is the Zenith, N the Nadir, ZQN the Hour-line of 12, HQO the Horizon. Now seeing the Plane declines S. W. 34° 20': make Na, and Ob, each equal to 34 20: Then a Ruler layed from Z, to a, will cut the Horizon in S, the

South point of the Horizon, through which draw the Meridian ZSN, whose Center is at Y, found as in the fourth Proposition aforegoing: Then a Ruler laid from Z to b, will cut the Hori∣zon in W, the West point thereof. Now the Horizon and the Meridian being projected on the Plane, take out of your line of Chords 51° 32', which place from H, unto c, and from N, unto d; then lay a Ruler from W, unto c, and it cutteth the Meridian in P, the Pole of the World. Then through P and Q,* 1.319 draw the line PQD, which representeth the Axis of the World, and the Substilar line of the Dial, then lay a Ruler from W, to d, it cutteth the Meridian in AE, so is W AE two points through which the Equinoctial must pass, whose Center is found as afore to be at M, (being always in the Axis of the World) so have you on your Plane the Horizon HQO, the Meridian ZPSAe N, and the Equinoctial LAeKWG, described on the Plane as required.

Now first to find the Poles height above the Plane, which in this Scheme is represented by BP, Lay a Ruler from G, unto P, and it shall cut the Plane in V, then measure the di∣stance BV, on your line of Chords, and you* 1.320 will find it to contain 34° 33', which is the Poles height above the Plane.

Secondly, To find the distance of the Substile from the Meridian represented in the Scheme by the Arch ZB, or ND, which measured as afore will appear to be 18° 08', the distance of the Substile from the Meridian.

Thirdly, To find the Plane's Difference of Lon∣gitude, which in the Scheme is represented by

the Angle AEPK, lay a Ruler from P, unto AE, and it cutteth the Plane in X, then measure the Arch DX, as afore, and so will you find the Planes Difference of Longitude, to be 30° 00': Thus by Geometrical Projection have we found* 1.321 all the three Requisites: Now to find them by Arithmetical Calculation observe these Analogies or Proportions.

1. For the Poles height above the Plane, say,

As Radius or S. 90°,

To Sc. of the Latitude 38° 28'.

So is Sc. of the Declination 65° 40',

To S. of the Poles height above the Plane 34° 33'.

2. For the Distance of the Substile, from the Meridian, say,

As the Radius or S. 90° 00',

To the S. of the Plane's Declination 24° 20'.

So is Tc. of the Latitude 38° 28',

To the T. of the Substilar Distance from the Meridian 18° 10'.

3. For the Plane's Difference of Longitude, say,

As the Sc. of the Latitude 38° 28',* 1.322

To the Radius or S. 90° 00'.

So is S. of the Substilar Distance 18° 10',

To the S. of the Difference of Longitude 30 Deg.

Or, it may be found thus.

As the S. of the Latitude,

To the Radius.

So is the T. of the Declination,

To the T. of the Difference of Longitude requi∣red.

These things found, we come now to shew how the Hour-lines may be projected. To pro∣ject which observe, First, to lay a Ruler from P the Pole of the World, to AE the Intersection of* 1.323 the Equinoctial with the Meridian, and it will cut the Plane in x, where begin to divide the Semicircle L x G, into 12 Equal parts in the Points •, •, •, •, &c. Then lay a Ruler from Q, to every of those parts, and it shall cut the Equinoctial; and divide it into 12 unequal parts, in the points *, *, *, *, &c. Then a Ruler laid from P the Pole of the World unto each of these points, it will divide the Plane into 12 unequal parts in the Points I, I, I, I, &c. Then by a Ruler laid from the Center Q, to those points, draw right lines, which shall be the true Hour-lines proper unto such a Declining Plane, as you see plainly demonstrated by the Scheme.

Now the Substilar line falleth in this Dial, just on the Hour-line of 2, in the Afternoon, be∣cause the Plane declineth Westerly. The Angle of the Stile is DQR 34° 33'. which may be either a Plate or Wyre, brought into such an Angle, which must be placed Perpendicular to the Plane, and directly over the Substilar line QD 2.

Now the distance of the Hour-lines, from the Substilar line, may also be found by this Analogy or Proportion.

As the Radius,* 1.324

To the S. height of the Pole above the Plane.

So is the T. of the Hour-line from the Meridian of the Plane,

To the T. of the Hour-line from the Substile.

Thus have you compleated your Dial; as you see in the Scheme, and here you may take notice that having finished a West Decliner, you have also made an East Decliner; if you only convert the Hour-lines of the West Decliner, in such manner as you see in Fig. 72. on the East* 1.325 Decliner, and compleat all as you see in that Scheme.

Thus I have explained the making and de∣lineating of the best and most usefull Dials both by Geometrical Projection, and also by A∣rithmetical Calculations, in as brief and compen∣dious a manner as possible. There are sundry other kind of Dials, as Incliners, Decliners, and Recliners, which being not so usefull, for bre∣vity sake, they are here omitted: As for Instru∣mental Dials, as Quadrants, Rings, Cylinders, &c. Which depend on the Sun's height, I refer you to Mr. Edm. Gunter's Book, wherein they are largely described.

As for the Beautifying and Adorning of those Dials, &c. by describing on them the Equi∣noctial, Tropicks, Parallels of Declination, Paral∣lels of the Sun's Place, Length of Days, the Sun's Rising and Setting, Jewish, Italian, and Babylo∣nish Hours, Almicanthars, Azimuths, Circles of Position, the Signs Right Ascending, Descending, Culminating, &c. I do advise you to consult Mr. Gunter, Mr. Foster, Mr. Wells, and Mr. Holwel's Works, all which Authors have very learnedly shewed the describing of them, by several large Schemes, and Figures, for the plainer Illustration thereof.

Now seeing the Latitude of a Place must be first known, before a Dial can be made to it,

[illustration]

Plate VI P〈…〉〈…〉

* 1.326

I have therefore hereunto annexed a Table of the Latitudes of all the principal Cities, Towns, and Islands, in and about Great Britain and Ire∣land; so that if you are to make a Dial, for any of those parts, you may have recourse to this Table, and make your Dial to the Latitude of that place, which you find to be the nearest to the Place, for which you are to make your Dial.

A Table of the Names and Latitudes of all the Principal Cities, Towns, and Islands, in and about Great Britain and Ireland.

ENGLAND.

D.	M.
ARundel	51	00
Bedford	52	15
Barwick	55	54
Bristol	51	35
Buckingham	52	10
Cambridge	52	20
Canterbury	51	25
Carlisle	55	20
Chichester	50	48
Chester	53	18
Colchester	52	08
Dover	51	20
Derby	53	00
Dorchester	50	50
Durham	54	56
Exeter	50	48
D.	M.
Falmouth	55	22
Glocester	51	57
Guilford	51	12
Hartford	51	54
Hereford	52	17
Huntington	52	30
Ipswich	52	20
London	51	30
Lincoln	53	20
Leicester	52	45
Lancaster	54	15
Northampton	52	24
Norwich	52	45
Nottingham	53	00
Newcastle	55	12
Oxford	51	50
Portsmouth	51	08
Plimouth	50	36
Reding	51	40

D.	M.
Salsbury	51	12
Stafford	52	50
Stanford	54	44
Shrewsbury	52	50
Truero	50	30
Winchester	51	03
Worcester	52	25
Warwick	52	30
York	54	00

WALES.

D.	M.
ANglesey	53	28
Barmonth	52	50
Brecknock	52	01
Cardigan	52	12
Caermarthen	51	56
Carnarvan	53	16
Denbigh	53	13
Flint	53	17
Landaffe	51	35
Monmouth	51	51
Montgomery	51	56
Pembroke	51	46
Radnor	52	19
St. David	52	00

SCOTLAND.

D.	M.
ABerdeen	57	30
Dunblain	56	21
Dunkel	56	48
Edenbrough	56	00
Glascow	55	58
Kinsaile	57	44
Orkney	60	06
D.	M.
St. Andrews	56	40
Skyrassin	58	38
Sterling	56	12

IRELAND.

D.	M.
ANtrim	54	38
Arglas	54	10
Armagh	54	14
Carterlagh	52	41
Clare	52	34
Corke	51	55
Droghedagh	53	58
Dublin	53	55
Dundalke	53	52
Galloway	53	02
Kenney	52	30
Kildare	53	00
Kings Town	53	08
Knockfergns	54	40
Kynsale	51	41
Lymerick	52	30
Queens Town	52	52
Waterford	52	09
Wexford	52	18
Youhall	51	53

ISLANDS.

D.	M.
WIght	50	48
Portland	50	30
Man	54	24
Limdey	51	22
Jerzey	49	12
Garnzey	49	〈◊〉〈◊〉

CHAP. XIII. Of FORTIFICATION.

THE Utility of this Mathematical Art called Fortification, or Military Archi∣tecture, is so well known, that it needs not my commendation, and therefore to speak any thing thereto, were but to light a Candle before the Sun.

In the handling of this part of the Mathema∣ticks, I shall be as brief as possible, yet as plain as can be desired: In the prosecution of which, I shall use this Method. As First, I shall give you the most principal Definitions or Terms be∣longing to this Art. Secondly, I shall prescribe the most conducing Maxims or Rules herein observed. Thirdly, I shall shew how to deline∣ate the Ground-line of any Fortification, accor∣ding to the several Proportions, used by the best and most experienc'd Inginiers of Italy, France, Holland and England; Fourthly, I shall describe the Construction of the chief and principal Out∣works now in use; and Lastly, lay down some general Maxims or Rules, by most Modern Au∣thors observed in Irregular Fortifications.

SECT. I. Of the Definitions of the Lines, and Angles, belonging to the Principal Ground work of any Regular Fortification.

• 1. THE Exterior or outward Line, which* 1.327 boundeth the Rampart, at the Foot next the Ditch, is the principal and only Line to be regarded 〈…〉〈…〉 all Regular, or Irregular For∣tifications, being the Basis on which all the o∣ther Lines, and parts of the Fortification doth depend.
• 2. The Exterior Polygon, is the outward side of any Regular Figure, as in the Hexagon (which Figure I shall make use of through this Tract) the side AA, is the Exterior Polygon.* 1.328
• 3. The Interior Polygon, is the inward side of any Regular Figure, as in the Hexagon is noted by any of the sides between P and P.
• 4. The Bastion or Bulwork, is that great work of any Fort, that advanceth its self towards the Campaigne, and here are six all marked with B, the lines which terminate them, are two Gor∣ges, two Flanks, and two Faces.* 1.329
• 5. The Demi-Gorge or Gorge-line PC, is half the Entrance into the Bastion, and terminates the point C, whereby the Flank shall be raised.
• 6 The Flank is another Out-line of the Ba∣stion as CF, which terminateth the Curtain, and Face.

• 7. The Face is the utmost line of the Bastion, as FA, two lines thereof doth form the Angle of the Bastion A, or the Flanked Angle.
• 8. The line forming the Flank FF, is a prick∣ed line, made use of by the Dutch Inginiers, and others.
• 9. The Capital is AP, part of the line com∣ing* 1.330 from the Center ☉, terminated at the point of the Bastion A.
• 10. The Curtain is that part of the Interior Polygon CC, which lieth betwixt the two Basti∣ons B, and B.
• 11. The line of Defence is AC, passing from A, the point of the Ba∣stion, to C the Angle of* 1.331 the Flank, and Curtain, and ought never to ex∣ceed 800 English Feet* 1.332.
• 12. The line Stringent, is the line coming from the point of the Bastion A, and prolonged on the Face AF, to the Curtain D, which shew∣eth that DC, the part of the Curtain, (by some called the second Flank will scour the Face.
• 13. The Diameter of the Interior Polygon, is the line ☉P, coming from the Center thereof ☉.
• 14. The shortest line from the Center unto the Curtain, is ☉ m. These are the Definitions of the principal lines, appertaining to the Ground∣work of any Regular Fortification, the Angles followeth.
• 15. The Angle of the Center of the Polygon* 1.333 is P☉P
• 16. The Angle of the Polygon PPP, is always the Complement of the Angle at the Center, or remainer unto 180 Degrees.

• 17. The Angle of the Triangle PPO is always the one half of the Angle of the Polygon PPP.
• 18. The Angle of the Bastion, or the Flanked Angle FAF, is exposed unto the Batteries of the* 1.334 Besiegers, and formed by the two Faces, FA, and FA, which ought never to be less than 60, nor much above 100 Degrees.
• 19. The Angle of the Espaule, or Shoulder, is formed by the Face, and Flank, as AFC.
• 20. The Angle of the Flank CCF, is formed by the Curtain, and the Flank, and is most com∣monly a Right Angle, but by some later Ingi∣niers, is made Obtuse, or more than a Right Angle, or 90 Degrees.
• 21. The Angle made by the two lines Sitchant, At A is called the Angle of the Tenaile.
• 22. The Angle forming the Flank, is CPF, which Angle is made use of by most of the Dutch Inginiers.* 1.335

SECT. II. Of General Maxims or Rules observed in Fortifications.

• 1. THat all the parts of the Place, be of Cannon Proof flanked, i. e. defended from another place, which place is no farther

• distant than the reach of a Musket-shoot, from the place to be Flanked or defended.* 1.336
• 2. That in all the Place, there may be no part of the Wall, or outside of the Rampire, that is not seen from the top to the bottome of the Mote, or Ditch.
• 3. That the Bastions are large, and full of Earth, and not empty; the bigger they are, they are the more to be esteemed, there being the more room to intrench, in case of necessity: whose Gorge let be at least 35 fadoms, and their Flank at least 18 fadoms.
• 4. That the Angle of the Bastion, or Flanked Angle, be not much above 90, nor much less than 60 Degrees, for in the former it would lie too very Obtuse, and open, at the Point; and in the latter it would be too slender, and so easily to be battered down, by the Ene∣mies Cannon.
• 5. That the Angle of the Flank may be some∣what Obtuse; neither is there any more virtue in a Right-angle, than in any other, for the defence of the Fort.
• 6. That the length of an extended Curtain be not above 135 Fadoms, nor the single above 80 Fadoms, nor be less than 40 Fadoms, to be well defended from two Flanks.

• 7. That the Rampire be so wide, that so a Parapet of Earth Cannon proof may be erected thereon, and a Teraplane left, full wide for the Ordnance to be recoiled.
• 8. That the Mote or Ditch be at least 20 Fadoms broad, and as deep as possible. Now dry Motes in great Cities are to be preferred before others, that are full of Water, to facili∣tate the Sallies, the relief, and retreat of the Besieged; and in small Fortifications the Motes full of Water are the most Esteemable, because in such Sallies are not necessary, and Surprises are very much to be feared.
• 9. that the Parts that are most remote from the Center, be commanded by those which are nearest to it.
• 10. That the Defence of a Face is much stronger, when the Angle made by the Face, and Exterior Polygon is a great Angle; this Maxim is so very essential, that it will try the goodness of any Fortification whatsoever: Thus I have described the 10 chiefest Maxims, ne∣cessary for good Fortifications.

SECT. III. Of the Construction and making of the prin∣cipal Ground-line of a Fort, according to the most Modern ways, used by the Ita∣lian, Dutch, French, or English Ingi∣niers.

I. Of the Italian Fortifications.

GEnnaro Maria, Mathematician to the Ca∣tholick King, wrote at Florence, his Ele∣ments of Military Architecture entituled, Breve Trattato delle Moderne Fortificazioni. This Ita∣lian Author was a very Learned and Skilfull Mathematician, and famous in his Nation. In his said Book Printed 1665, he makes the Interior Polygon 800, and not less than 600 Feet, his De∣mi-Gorge, he makes ⅛ of it, and so for the Flank of the Quadrangle. But for the Pentagon, and all Figures above, he makes the Flanks 1/10 part of the Gorge more, and he placeth his Flank at Right Angles with the Curtain.

Supposing his Interior Polygon 1000 parts, his Gorges will be 125, and in the Quadrangle the Flanks will be 125, but of the Pentagon, and all above, 138 parts. For the Faces, he makes them to fall on the third part of the Curtain, unless in the Square, which he allows no second Flank.

PROP. I. To fortifie a Hexagon according to this Author's Proportion.

First describe the Hexagon PPP, &c, then* 1.337 divide the Interior Polygon PP, into 1000 equal parts, take 125 for the Gorges, and set it from P to C. Then on C raise a Perpendicular, make it equal to 138 parts, for your Flanks CF, then draw the Face AF, falling on the third part of the Curtain CC, at D, and so do on every Ba∣stion, untill the work is compleated.

II. Of the French Fortifications.

Monsieur De la Mont, in his Fortifications Offensive, and Defensive, printed 1671: And Monsieur Manesson Mallet in his late work, in∣tituled Travaux de Mars, printed 1672, assign∣eth these proportions for the laying down the Ground-line of a Fort.

Both these Authors make the Interior Polygon 768 English Feet, which they divide into 5 parts, and taking one for the Gorge 153½ Feet. Both divides it into 3 parts, and takes one for the Capital, that is 256 Feet.

Now our first Author De la Mont, makes the* 1.338 Flank to stand at Right-angles and takes 115½ Feet for it, which is ¼ of the Curtain, and so draws the Bastions, in all save the Quadrangle, and Pentagon, which he makes to have no second Flank.

PROP. II. To fortifie a Hexagon according to the Proportion of De la Mont.

First describe your Hexagon P, P, P, &c. Now supposing your Interior Polygon PP, 1000 parts, the Capital 333, the Gorge 200, and the Flank 150 parts, take out of your Triangular Scale Fig.* 1.339 75, (which is made for the more speedy de∣lineation according to this proportion of De la Mont) PA for the Capital, and prick it off from PA, on all the Bastions. Then take PC, and prick off all the Gorges from P to C. Then take FC and prick it off at Right Angles, from C to F. Lastly draw all the Faces AF, AF, &c. so is your Hexagon compleat, as required.

PROP. III. To fortifie a Hexagon according to Manesson Mallet's Proportion.

Now our Authour Monsieur Manesson Mallet, in his Works intituled Travaux de Mars, de∣viates from our former Authour, only in this: that as De la Mont did place his Flanks at Right Angles, he places them at 98 Degrees with the Curtains, and leaves no second Flank in all his Fortifications.

Therefore having described the Polygon PP,* 1.340 &c. divide PP into 1000 parts, prick off the Capitals PA 333, and the Gorges PC 200, then lay off the Flanks CF, 150 parts, at an Angle

of 98 deg. with the Curtain CC (by prop 5. §. 1. chap. 4.) and draw all the Faces, AF, AF, &c. Falling on C the point of the Flank and Curtain, so shall your Hexagon be fortified as was required.

III. Of the Dutch Fortifications.

The Emperour Ferdinand III. hath learnedly* 1.341 altered the Method of Fritach, Dogen, Gold∣man, and Faulhaberus, all which were Dutch Inginiers, and wrote large Volumes on this Subject; in his Works intituled Amussis Ferdi∣nandea, published 1654; by turning their way of working by Angles, into working by Sides.

Thus he setteth down a Catholick way of delineating the Sides, or Lines of any Fort by his 60 prop. thus, the Interior Polygon to be 66, the Capital 24, the Gorge 15, and the Flank 12. Or in making the Interior Polygon 22, the Capital 8, the Gorge 5, and the Flank 4. Or yet making the Interior Polygon 1000, the Ca∣pital 363, the Gorge 227, and the Flank 181, this is an Epitome of all the Dutch Fortifications, and is general excepting for the Square, which must have no second Flank.* 1.342

PROP. IV. To fortifie a Hexagon according to the Emperour's Proportion.

First describe the Polygon PPP, &c. divide P P, &c. into 22 parts, take 8 for the Capitals

PA, which prick off all round from P to A, take 5 for the Gorges; which prick off all round from C to P, then take 4 for the Flank CF,* 1.343 which prick off all round at Right-angles from C to F, lastly draw the Faces AF, AF, AF, &c. So is the Hexagon compleated as was required.

IV. Of the English Fortifications.

His late Majesty of Great Britain Carolus II. of ever blessed Memory, hath much facilitated the Method of Count Pagan, who in his Fortifications printed at Paris 1645, did place the Flanks at Right-angles with the Line of De∣fence, and he works by the Exterior Polygon. Now His Majesty places the Flank, at Right-angles with the line of defence of the Interior Polygon, and works after another manner: Count Pagan makes the proportion of the Grand Royal Fort. Supposing the Exterior Polygon to be 1000 parts, will make the Perpendicular MT to be 150, and* 1.344 the Complement of the line of Defence TC to be 185, which may serve for a general proportion be the length what it will, only in a Square the proportions must thus be altered in the Grand Royal Fort, the Perpendicular MT must be 162, in the Mean R 144, and in the Petty Royal 126, the Complement of the Line of Defence for the Grand Royal Fort is 228, and for the Mean Royal Fort 198, also for the Petty Royal Fort 198.

PROP. V. How to fortifie a Hexagon according to Count Pagan's Proportion.

To delineate this Work draw a line, about the middle whereof as at M, set off MA, the half of the Exterior Polygon 500 parts, which makes the Exterior Polygon 1000, then on M (by prop. 1. §. 1. chap. 4.) raise the Perpendi∣cular* 1.345 Mm, which make Mt, MT Equal to 150, then draw ATC, and ATC, then take 185, and place it from T to C, and to C, and draw CC for the Curtain, then on the points C raise Perpendiculars CF, to the line of defence CA, for the Flanks, so have you also the Faces FA. Then on the Points A set off half the Angle of the Figure, to wit 60° (as you see in the Ta∣ble in page 38) and draw the lines OA and O A, so shall O be the Center of the Figure, and PC the Gorge, and AP the Capitals: then finish* 1.346 each Bastion at your own discretion, and the Work is finished as required.

PROP. VI. To fortifie a Hexagon according to the way prescri∣bed by His Majesty Carolus II.

His late Majesty C. II. hath much facilitated this Work, as will appear in this following Ex∣ample, by making the line of Defence, stand at Right-angles with the Flank of the Interior Polygon, by this Table, which supposes the Inte∣rior* 1.347 Polygon to be 1000. Then

Polygons	4	5	6	7	8	9	10	Strait∣lines.
Capital	398	437	367	333	312	300	291	233
Gorge-line	155	196	203	242	252	260	263	300

Now describe the Hexagon PP, &c. Then divide the Interior Poligon PP, into 1000 parts, take 367 and prick off all the Capitals PA; Then take 203 and prick off all the Gorges from P to C. Now draw the lines of defence AC and* 1.348 AC, &c. Then at C, set the Flanks at Right Angles with the line of Defence AC, so shall FC be the Flank, and FA the Faces, then finish every Bastion, and your Hexagon is for∣tified as was required.

☞ Thus have I set down the several Ways and Rules, for laying the fundamental Ground∣line, from the most con∣siderable Inginiers of* 1.349 this last Age, out of all which it's most agreeable to those Authors, and to practice, to take ⅓ of the Interior Polygon for the Capital, ⅕ for the Gorge, and Flank, which leaves 6/10 for the Cur∣tain, and let this be taken for a general Rule, where the Flank, and Curtain, stand at Right Angles.

PROP. VII. By the Semicircle to lay down on the Ground, any of the former Fortifications.

Having drawn the Plot of your Fort on Im∣perial* 1.350 paper, or Vellom, and if it be a Regular Fort you need not describe it but two half Ba∣stions from the Center, for that will be sufficient. Having such a Plate whose length is set down on each respective line, and all proper Angles expressed, will not only be usefull for laying down the Work, but for finding the Solidity of the Ramparts, Parapets, and the other Earth Works See Fig 76.

If it be in such a Place, that from the Center* 1.351 of the Fort, all the Angles may be seen, place your Semicircle at Z, and lay off all the An∣gles of the Center, which here is 60°; then mark out the Diametrical lines, and making them their due length, as by your Plate they appear to be, set Piquets, on all the P, P'^s upright with the Plane, Then take up your Instrument and place a Piquet at Z. Then lock-spit out all the Polygons PP. Then mark out the Gorges CP, then set out the Flanks CF, either at Right* 1.352 Angles, or as otherwise required. Then lock∣spit out the Flanks CF, and the Faces AF, ha∣ving first set off the Capital PA, so is the Fort lined out for the Ground-line.

But if there be Houses and Obstacles in the* 1.353 way, that from the Center all may not be seen, then must you mark out any one side and mea∣sure it, and at each End set off the Angles of

the Polygon, (which here is 120°) and draw side after side, untill all be finished: Then fi∣nish the Bastions as before, and here great care must be had, or else you will run into infinite Errours.

☞ But you have liberty Experimentally to alter any of the former proportions, as you have occasion, and as will best serve the Place; as you see by the fortifying a streight lined Figure: Fig. 77. wherein Count* 1.354 Pagan's or in Manesson's way it may not be al∣lowed without some alterations.

SECT. IV. Of the Dimensions, and Measures of the Rampires, Parapets, Mote, Coridor, or Covert-way, and its Esplanade, or Breast∣work.

THE Rampire's thickness and height, must receive its Determination from the Judg∣ment of the Inginiers, and Purse of the Prince.

The Height TS, must not exceed 18 Feet,* 1.355 not be less than Ten; the thickness may be from 50, to 80 RA, in all Royal Works, and ac∣cording as Earth is to be had. The slope of the inward side of the Rampire TR, is commonly a foot for a foot, therefore RS, the Talu, will be equal to the Height TS, so if TS be 18, RS will be 18, if 15 then 15 feet.

The outward Slope QA, is generally pro∣portioned ½ a Foot for a Foot, so if the Height OQ be 18, the Talu OA, will be 9 Feet, &c.

The Height of the Parapet ZD, must always* 1.356 be 6 Feet, the Exterior Height PM must be 4 Feet, the thickness of the Parapet DQ, in light Earth must be 20 Feet, in stiff Earth 16, and in Solid Rough Clay 14 Feet; suppose it be 18 Feet, PM will be 4 Feet, MQ 2 Feet, LD 1 Foot, so will the lower thickness LQ be 21 Feet.

The Height of the Banquet VX is 1½ Foot, and thickness VL, 3 Feet.

The Lizier must be made so wide, as to support the Rampire from slipping into the Ditch, and is taken from 3, to 10 Feet; the Mote or Ditch may be from 70, to 130 Feet broad, that is, from E to G, and the depth IF may be 8, 10, or 12 Foot deep, the little Ditch at the bottom of the Mote represented by c q g, must be as large and deep as the Earth and Work will give leave.

The Coridor and the Esplanade or Breast-work* 1.357 on it, is left about 18 Feet wide, from G to C; on which is placed a Parapet, and Banquet, like that on the Rampire, which Parapet or Espla∣nade, must slope so into the Campaigne, that a streight line drawn from Z, the Top of the Ram∣pire, may terminate OFd, the Slope thereof.

PROP. VIII. How to lay down the Profile of the Work, according to this Table.

Feet.
The Base of the Rampart RA	70
Height TS and QO	16
Interior Talu RS	16
Exterior Talu OA	8
Base of the Parapet LQ	21
Interior Height ZD	6
Exterior Height MP	4
Exterior Talu MQ	2
Interior Talu QD	1
Breadth of the Banquet V, L	3
Height of it V, X	1½
The Terra Plana TV	25
The Lizier AE	3
The Mote's breadth EG	112* 1.358
The Depth of it IF	12
Breadth at its bottom FH	88
The Talus EI, or KG	12
The Breadth of the little Ditch c g.	18
The Depth of it	5
The Coridor GC	18
The Seat of the Esplanade	60
The Height CF	6

Now to lay down this Profile draw a line of a convenient length as RSOACGD for the level or Ground-line, then by your Scale of 20, o•• 30, at most in an Inch, representing Feet. Take out of it 70 for RA, 16 for R S, 8

for OA, 3 for AE, 112 for EG, 18 for GC, and 60 for CD, and mark them off on your Paper (as in Fig. 78.) at S, O, I, K, C, raise or* 1.359 let fall Perpendiculars (by prop. 1, 2, or 3. §. 1. chap. 4.) then take 16 for ST, and OQ, 12 for IF and KH, and 6 for Cf, and draw RT, TQ, QA, EF, HG, cf, fd: Then from Q set off QL 12, LV 3, QM 2, and LD 1, and raise the Perpendiculars MP 4, DZ6, and VX 1½ Then draw VX, XY, YZ, ZP, and PQ, and make the little Ditch by its measure, so is th•• Profile perfected: as for the Faus-Bray, they ar•• now out of use, therefore I omit them.

The Solid Content of those Earth-Works may easily be attained by the former Rules which Content being got in Feet, divide that product by 324, the Quotient shall be the Soli•• Flores contained therein, a Flore being 18 foo•• square and 1 Foot deep.

SECT. V. Of the Dimensions and Construction of Pla〈…〉〈…〉 forms, Caveleers and Cazemats in t〈…〉〈…〉 Flanks.

1. PLatforms are Plantations where G〈…〉〈…〉 are to be placed, and are common〈…〉〈…〉 made of Plank, and Sleepers, there neede•• for one Gun, to be but one Platform, whi〈…〉〈…〉

must be 8 Feet broad next the Parapet, and 14 Feet wide at the other End, and their length should be 18 Feet.

2. An Embrasure is the Port-Hole made in the Parapet, which towards the Gun must be 4 Feet wide, and towards the Campaigne 8 Feet wide, whose height must be proportioned unto the Wheel of the Carriage; and are 16, 18 or 20 Feet assunder.

3. Cavaleers or Mounts are Massy pieces of Earth raised on or near the Rampart, above the Parapet, on which Ordnance and small shot may be planted. As to their Construction I shall follow the Method of Manesson, who pla∣ces them in the Gorge of the Bastion, and gives this Rule for it, [saith he].

Lengthen out* 1.360 the line of defence to E, untill it cut the Capi∣tal, the Center of your Cavaleer shall be the middle point betwixt P and E, to wit at F, then with the distance of 84 Feet on the Center F strike a Circle, which shall be the Base of your Cavaleer:

Now its height ought to be at least 20 Feet; and if the Work be to be faced with Stone, or Brick it needs not not have a Talu above 3½ Feet, so that the Dia∣meter at the top will be about 153 Feet, whereon, set a Parapet of 20 Seat, and high, and other Demensions as aforesaid in the Rampire, and there will be a Terra-plana at the top of above 100 Feet, whereon six pieces of Ordnance may be planted, making Embrasures and Platforms as was last directed.

4. Cazemats are made in the Corners of the Flanks, and are several Platforms for Guns to be planted on, thereby to be hid from the 〈1 page duplicate〉〈1 page duplicate〉

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Battery of the Enemy: As to the Construction I shall follow Manesson's Directions, first as to the form, and also to the measure: [saith he]

The Caremate shall take up one half of the Flank, and no more; The Grand Caremate D B is about 7, 8 or 9 Feet from the Level of the Plane of the Fort, and hath a passage into it from within the Fort A, C is its Parapet of 20 or 22 Foot Seat, and in it let there be 3 or 4 Embrasures; D is the part thereof most hid from the Enemies Cannon; F is the Magazine* 1.361 for this Battery; H is the second Caremate, G the Ladder, and L the Magazine, and M the Parapet; this is to hold but one Gun; M is the third Caremate on the level of the Bast. which let be all firm, in which let there be no void place.

The Dimensions and Construction accor∣ding to the Method of this our Authour are thus [saith he]

Lengthen the Line of Defence from C to G some 40 Feet, then draw CD, parallel to Cf, (by prop. 4. §. 1. chap. 4.) let CF be half of cf, so that cF may be equal to Ff, then from the middle of the Face opposed, draw KF, and let it cut GD, in I, then make I* 1.362 L, and FM equal to 6 Feet, then make MN 66 Feet, and draw NO parallel to the Flank, which let be 24 Feet: Lastly [saith he] for the Orillon or Blind, prolong the Face FT 36 Feet, and also FV 36 Feet, then joyn TV, and make that part all solid: So is your Caremate finished: Let the height of the lower Caze∣mat, be 6 Feet as before, and let all the rest be compleated as you see in the Figure.

SECT. VI. Of the Dimensions, and Constructions of those Out-Works, called Ravelins, Horn, Crown∣works, &c.

THE Ravelin is a certain Work lying be∣yond the Mote, or Ditch, for the co∣vering the Curtain, Bridge, and Gate; the Angle of the Ravelin must not be less than 60, nor much above 100 deg. the manner of deline∣ating it is thus. Lengthen out the middle line of the Curtain OM unto a convenient length,* 1.363 then take with your Compasses the length of the Curtain CC, and setting on Foot in F, the point of the Face and Flank, cross the middle line in q; then laying your Ruler at q, and to the points F, draw the lines of the Ravelin q R and qS, which shall be the Ground-lines of the Ravelin: The M••te surrounding it must be half the breadth of the Great Mote; the Ram∣pert may be 30 Feet thick, and some 6, 7 or 8 Feet high, on which may stand a Parapet equal to that of the Rampire.

Now if from the points F you raise streight lines into the Campaign, at Right Angles to the Curtain, and from the points F set off FE, and* 1.364 FE 720 or 750 Feet, then may you joyn EE either with, A;

Single Tenaile: which is done by joyning EE, and dividing it into four equal parts, take one and place from D to N, and so draw EN

and EN, so have you a Single Tenail IENEL, which must have a Mote Rampire, and Parapet like the Ravelin. Secondly it may if occasion require be fortified with, A;

Horn-Work: which is done by joyning the points E E, and fortifying the Exterior Polygon EE as is afore taught: Or divide EE into three* 1.365 parts; make ME, and EN equal to MO; then draw N M, which divide so likewise at O and P; then draw E O and E P; then at P and O raise Perpendiculars O Q and P R, so shall M, E, Q, O, P, R, E, N, be the Horn-Work which was desired: which must likewise have small Rampires and Parapets, as afore.

For the Crown-Work: From the Center of the Fort O draw O M B of a convenient length, then from the middle of the Ravelin set off 1000 or 800 Feet to B, then on q, strike the Arch D B E, set off the Curtain, and Demi∣Gorge P C C, from B to F, and G both ways, then draw C F and C E, to terminate the points I and H on the Counterscerp; then toke* 1.366 ⅓ part of B F or B E, and set it from B to M; and srom F to L, and from E to M; then draw L M, and M N; then for your Demi∣Bastions make N P and L O equal to N E, &c. Then for the Demi-Gorges of the whole Bastion in the middle, let them be equal to ⅕ of the In∣teriour Poligon L M or M N, viz. M Y or M X; then finish the Bastions by drawing the lines of Defence, and raising Perpendiculars, or making Angles of 98° at O, X, Y, and P, then the Crown work is finished as desired. You may make Ravelins and other Works (beforementi∣oned) before these Curtains if occasion require.

There are some other Works which are used; as Half Moons, Bonnets, Double Tenails, Counter∣gards, Horseshoes, Priests-Caps, &c. which would be superfluous to speak of in this place.

5. Cittadels, are Castles or Forts of the least sort, and are the Out∣works lastly used, which are* 1.367 commonly of 4 or 5 Bastions, and are pla∣ced in such Order, that there may be two Faces, and a Curtain towards the Town: the Construction whereof is after this manner. Lengthen out the line OM, and therein find the Center of the Cittadel, the In∣terior Polygon of the Pentagon may be ¾ of the Curtain adjoyning, or a little more; the Cen∣ter of the Square may be on P the point of the Interior Polygon, the Center of the Hexagon may be near the outward point of the Bastion of the Town, taken away to make the Cittadel in, which may be delineated as afore: The Motes and other Works in proportion accordingly, and the Rampires as high as those of the City or Town.

SECT. VII. Of some Maxims or Rules necessary to be known in Irregular Fortification.

IRregular Fortifications is when any Town or Place is to be fortified, which lieth in an Irregular form; i. e. whose Sides and Angles are unequal in the forti∣fying of Irregular Fi∣gures* 1.368. I shall here say very little, only I shall lay down some Precepts that are of immediate concern in fortifying of Irregular Figures, and shall refer you to peruse Marlois, Dogen, Fritach, Taurnier, Dilichius, &c. which will greatly satisfie and help you: To this end know,

1. That the same Laws and Maxims for Re∣gular Fortifications stand and be in force for Irregular; i. e. that the line of Defence must not exceed the Port of a Musquet, nor the Angles of the Bastion be less than 60°, nor much above 90°, &c.

2. That no inward Angle of the Place be less than 90°, if it be so it must be altered, and that point may be made the outward point of a Bastion.

3. That between Regular and Irregular For∣tifications, there is no other difference, but by rectifying the sides that are too short, or too long, and altering the Angles that are too little; as for the sides, if they be above 500, and un∣der 1000 Feet, they may be fortified by Ba∣stions placed according to the usual manner, at the extreme points thereof; But if the sides be between 1000 and 1700 Feet, then in the midst you may place a Plat Bastion, and at the Extreme Points, place two Bastions, as before: But if the line be less than 500 Feet, you may lengthen it, by producing it into the Plane: As for the Angles, they are made greater or lesser according as occasion requireth. For the Raising the Rampires, Parapets, and other Out∣works, they are to be as in the Regular, and the Out-work may be placed before the Cur∣tains as was before mentioned.

4. That the Capital, in any Regular or Irre∣gular Bastion, is found by dividing the Angle of the Polygon into two equal parts (by prop. 7. §. 1. chap. 4.) and by producing the line of Angular Division or Separation, on which the due length of the Capital must be placed, which observe for a general Rule.

SECT. VIII. Of the Dimensions and Construction of small Forts, or Scones, which are built for the Defence of some Pass, River, or other place.

WHEN they are made Regular, of 4, 5, or 6 Bastions, then they may be fortified by the precedent Rules, but there are others of smaller Dimensions fit for the same purpose: viz. Triangle with Demi-Bastions, Square with Demi-Bastions, Parallelograms with Demi-Bastions and Tong, Star Redoubts of four, five or six points, and Plain Redoubts.

PROP. IX. To fortifie a Triangle, with Demi-Bastions.

This Triangle may consist and be compre∣hended of three equal or unequal sides in this Example: let it be an Equilaterial Triangle PPP* 1.369 Now divide PP into three parts, then take 1, and prick off the Capitals PA, &c. and the Gorges make equal thereunto, as PC, PC, &c. then make the Flanks FC to stand at Right Angles, and to be ½ of PC or PA, then draw the Faces AF, AF, &c. and the Work is finished as required.

PROP. X. To fortifie a Square with Demi-Bastions.

The sides of the Square may be from 100 to 200 Feet, let PP be 180 Feet, which divide in∣to 3 parts, take one for the Gorges PC, and for* 1.370 the Capitals PA, and prick them off all round as you see, then take ⅙ of PP, and at Right Angles prick off the Flanks CF, then draw the Faces AF, AF, &c. and the Figure is compleated.

PROP. XI. To fortifie a Parallelogram with Demi-Bastions, and Tong.

First describe the Parallelogram, or Long∣Square, PPPP, then divide PP into 6 parts (the side on which the Tong, or Tenaile, is placed) and make MC equal unto ⅙ thereof, and also MG, and MH. Draw CG, GC, and CH, HC,* 1.371 then finish the Demi-Bastions as before, so shall the Work be compleated as was required. A Long Square may also be fortified as Fig. the 77.

PROP. XII. To fortifie a Star Redoubt of 4, 5, or 6 Points

• 1. A Star Redoubt of four points may have his side from 40 to 60 Feet: First describe the Square PPPP, then divide PP into two parts at M, take ¼ of PM, (and by prop. 1. §. 1. ch. 4.)* 1.372

• raise Perpendiculars round at M, make MA equal to ¼ of PM, and draw all as in the Fi∣gure.
• 2. A Star Redoubt of five points is thus fortified. Describe the Pentagon PP, &c. then divide PP into halves at M, raise the Perpendiculars MA, make MA equal to ⅓ of PM, and draw the Fort* 1.373 in all respects as the Figure representeth.
• 3. A Star Redoubt of six points is thus fortified. Describe the Hexagon PPP, &c. divide PP into two equal parts at M, then raise Perpendiculars at the M'^s, then make MA equal to ½ of PM,* 1.374 or ¼ of PP, and draw every respective line as you see in the Figure.

PROP. XIII. To Delineate a Plain Redoubt

Plain Redoubts are called Grand Redoubts, which are used as Batteries in Approaches, whose side may be from 60 to 80 Feet, or Petit Redoubt,* 1.375 which are used for a Court of Guards in the Trenches, and may be from 20 to 50 Feet, and are framed and delineated in all respects as you see in Fig 90.

The Profile's to be set on these several Works, and the Motes, are alterable and uncertain, for they being sometimes used in Approaches; then they do require the Breast-work at the Bottom to be 7 or 8 Foot wide, and the Interior Height 6, and the Exterior 5 Feet, and the Mote to be either 8, 10 or 12 Feet, sometimes 14 or 20 Feet wide at the bottom, and the height of 7, 8 or 9 Feet, to have two, or three

ascents to rise to the Parapet. There are many other things belonging to this Art, which the limitation I am bound to, will not permit here to be treated of.

CHAP. XIV. Of Military Orders, or the Embat∣telling and Encamping of Souldiers.

SECT. I. Of the Embattelling and Ordering of Soul∣diers.

BATTAILS are considered either in re∣spect of the number of Men, or in respect of the form of Ground. In the respect of the number of Men, it is either a Square Battail, a Double Battail, a Battail of the Grand Front, or a Battail of any proportion, of the number in Rank to the number in File. In respect of the form of the Ground, the Battail is either a Geome∣trical Square of Ground, or a long Square of Ground. For the Distance, or Order of Souldiers,

martialled in Array, is distinguished either into Open Order, which is when the Centers of their places are 7 Feet distant assuunder, both in Rank and File, or Order; which is when the Centers of the places are 3 ½ Feet distant both in Rank and File; or else 3 ½ Feet in Rank, and 7 Feet in File.

PROP. I. To Order any number of Souldiers into a Square Battail of Men.

Admit it were required to Martial into a Square Battail 16129 Men: To doe which ex∣tract the Square Root of 16129 (by prop. 8. §. 1. chap. 1.) which is 127, therefore you are to place 127 Men in Rank, and also in File.

PROP. II. To Order any number of Souldiers into a Double Battail.

Admit 16928 Men were to be Martialled into a Double Battail, extract the Root of half the number of Men; i. e. of 8464, whose Root is 92, therefore I say that 92 Men must be pla∣ced in File, and 184 in Rank, to order that number of Men propounded into a Double Bat∣tail.

[illustration]

Plate VII Page 302

PROP. III. To Order any number of Souldiers into a Battail of the Grand Front.

Admit 16900 Souldiers were to be Martialled into a Battail of the Grand Front, that is Qua∣druple. Extract the Square Root of 4225 (that is ¼ of the Men) the Root is 65; therefore I say 65 must be placed in File, and 260 in Rank, to form a Battail of the Grand Front.

PROP. IV. Any number of Men, together with their distance in Rank and File, being propounded, to Order them into a Square Battail of Ground.

Admit 2500 Souldiers were to be Martialled into a Square Battail of Ground, in such sort that their distance in File should be 7 feet, and in Rank 3 feet, and 'tis required to know how many Men must be placed in Rank and in File to draw up 2500 Men into Square Battail of the Ground. According to prop. 1. §. 1. ch. 1. say, As — 7 to 3, So is 2500 to 1071, &c. whose Square Root is 32, &c. Therefore I say 32 Men are to be placed in File. Now to find how many Men are to be placed in Rank, di∣vide 2500 by 32, the Quotient is 78, which are the number of Men to be placed in Rank, and 4 Men to be disposed elsewhere.

PROP. V. Any number of Souldiers propounded, to Order them in Rank and File, according to the reason of any two Numbers given.

Admit 6400 Souldiers are to be Martialled into Array, in such Order that the number of Men placed in File, shall bear such proportion to the number in Rank as 7 to 13; (according to prop. 1. §. 1. chap. 1.) say as 7 to 13, so is 6400 to 11885, &c. whose Square Root is 109, &c. the number of Men to be placed in Rank, by which divide 6400, it produces, 58, &c. the number of Men to be placed in File, and 78 Men to be employed elsewhere.

SECT. II. Of Castermetation, or Quartering and En∣camping of Souldiers.

IN Quartering and Encamping of Souldiers, it is requisite, the Quarter-Master General, and all other under Quarter-Masters, be skilled at Foot measure, that so they may lay out their Quarters as directed.

The common allowance for the depth of Ground, that a Regiment of Horse or Foot will take up, the wideness must be answerable to

the Number of Men 200 Feet for the Huts in length, and 100 for the Commanders, and Sut∣lers, before them; every two Souldiers to a Hut, 8 Feet broad, and 8 Feet deep, 2 Feet Hut, from Hut, so that there may stand 20 Huts in the 200 Feet, the Ally betwixt Hut, and Hut, may be 8 Feet, that is 16 Feet in width, and 200 in length for 40 Men, which is 3200 Feet, and for the 100 Feet more, 1600 Feet, in all 4800 Feet, and there must be 25 Rows of Huts, for 1000 Men; so that for a Regiment of Foot containing 1000 Men, with Officers, and Sutlers, will take up 120000 Feet, which is 2 Acres and 3 Roods, which because of Ways may be allowed 3 Acres of Ground, for every Regiment, which may be 350 Feet deep, and 370 Feet wide, or near 360 Feet square: Now if 1000 Men, Officers, Sutlers, High-ways and all take up a Square of 360 Feet, how many Feet shall the Side of a Square be wherein 10000 Footmen, &c. may be encamped? say (by prop. 1. chap. 1.) as 1000, to 10000, so is the Square of 360, viz. 129600, to 1296000, the Square of 1138 Feet, which is very near 30 Acres of Ground.

For the Quartering of Horse, you must keep the same depth of 300 Feet for all, and take 200 Feet for the Huts, the Horse Huts must be 10 Feet deep, and 4 wide; so that 12 Horses may stand in one Hut together, which is 48 Feet long, and 10 wide, and 6 Feet a Street; The Huts for the Troops, will be 6, for 12 Troops; now conceive a Regiment to consist of 8 Troops, 50 to a Troop, it will take up leaving 20 Feet Streets, and Cross-ways, very near as much Ground as a Regiment of Foot, Ways and all must be allowed

3 Acres, near 360 Feet square, so that 10 Regi∣ments of Horse will take up 30 Acres: More∣over, it will be needfull and you may very well allow, as much ground as both Horse and Foot will take, for the Train of Artillery, Victuallers, Pa∣rade Places, &c. From these considerations the young beginner, nay even the better practised Souldier may receive help, and thereby be ena∣bled to Encamp an Army if required.

CHAP. XV. Of GUNNERY.

SECT. I. Of the Names of the Principal Members of a Piece of Ordnance.

1. * 1.376ACANNON is a long round Body, either of Brass, or Iron, formed and made hollow by Art, and proportion, to offend afar off, with a Ball of Iron, Stone, or any Artificial Substance, charged with Gun-Powder, in its charged Cilin∣der, which being fired, in an instant performs

its desired Effect. This Machine was invented by* 1.377 an Englishman, and first put in practice by the Venetians against the Ge∣noveses at Chiezza, Anno 1376.

2. The Superficies of the Mettal, is the out∣side round about the Piece.

3. The Body is the Substance of the whole Mass of Mettal.

4. The Chase is the Concavity of the Piece, in which they put the Charge.

5. the Muzzel is the Extremity of the Chase by which you load, and unload the Piece.

6. The Calibre is AB the Diameter of the* 1.378 Muzzel or Mouth.

7. The Touch-hole, is that little vent, which passeth from the Convex Superficies, to the very Chamber of the Piece, made to give fire to the Powder within as C, that which encloseth the Extremity of the Chase about the Touch-hole is called the Breech or Coyl.

8. The Cascabel is the Pammel at the Breech or Coyl as D.

The Trunnions, are pieces of Metal fixed unto the Exterior Superficies of the Gun on which he moves in the Carriage as E, E.

The Body of the Piece, is that which is com∣prehended betwixt the Center of the Trunnions and the Cascable EG.* 1.379

The Vacant Cylinder, is comprehended betwixt the Cent: of the Trunnions & the Muzzel as EB.

The Frees, or Muzzel Ring is that thick Cor∣nish which, incompasseth the Convex Superficies of the Piece at I, The Base Ring is KLG, The Reinforced Ring is M, The Trunnion Ring is N, and the Cornish Ring is O.

The Line of the Cylinder, is a direct line ima∣gined to be described along the Chase Parallel unto the middle of the Chase as XZ.

The Line of Metal, is a line touching both* 1.380 Cornishes, as MNI.

The Dispart line of the Piece, is the difference betwixt the Semidiameter of the Muzzel, and Base Ring as the line IH.

The Vent of the Piece is the difference betwixt the Diameter of the Shot, and the Mouth of the Piece, as e d.

The Chamber, or Charged Cylinder, is that part of the Chase towards the Touch-hole equally large, nor narrower in one place than in ano∣ther, and doth contain the Powder and Ball.

SECT. II. Of the Dimension of our Usal English Can∣non, and other Ordnance, &c.

IN the following Table I have set down the length and weight of our most usual English Ordnance, the Diameters and Weight of their Bullets, the length and breadth of their Ladles, the Weight of Powder to Charge them, &c.

The Names of the several Pie∣ces of Ordnance.	Guns length	Guns weight	Guns bore	Bullets diamet.	Bullets weight	Ladles length	Ladles breadth	Powder weight	Shoots Level	Utmost Random
Feet	Inches	Pounds	Inches	8 parts	Inches	8 parts	Pounds	Ounces	Inches	8 parts	Inches	8 parts	Pounds	Ounces	Paces	Paces
A Base.	4	6	200	1	2	1	1	0	5	4	0	2	0	0	8	60	600
A Rabinet.	5	6	300	1	4	1	3	0	8	4	1	2	4	0	2	70	700
A Falconet.	6	0	400	2	2	2	1	1	5	7	4	4	0	1	4	90	900
A Falconi	7	0	750	2	6	2	5	2	8	8	2	4	4	2	4	130	1300
Minion ordinary.	7	0	800	3	0	3	7	3	4	8	4	5	0	2	8	120	1200
Minion largest.	8	0	1000	3	2	3	0	3	12	9	0	5	0	3	4	125	1250
Saker leaft.	8	0	1400	3	4	3	2	4	12	9	6	6	4	3	6	150	1500
Saker ordinary.	9	0	1500	3	6	3	4	6	0	10	4	6	6	4	0	160	1600
Saker old sort.	10	0	1800	4	0	3	6	7	5	11	0	7	2	5	0	163	1630
Demiculver least.	10	0	2000	4	2	4	0	9	0	12	0	8	0	6	4	174	1740
Demiculver ordinary.	11	0	2700	4	4	4	2	10	11	12	6	8	0	7	4	175	1750
Demiculver old sort.	11	0	3000	4	6	4	4	12	11	13	4	8	4	8	8	178	1780
Culverin least.	11	0	4000	5	0	4	6	15	0	14	2	9	0	10	0	180	1800
Culverin ordinary.	12	0	4500	5	2	5	0	17	5	16	0	9	4	11	6	181	1810
Culverin largest.	12	0	4800	5	4	5	2	20	0	16	0	10	0	11	8	183	1830
Demicannon least	11	0	5400	6	2	6	0	30	0	20	0	11	4	14	0	156	1560
Demicannon ordin.	12	0	5600	6	4	6	1	32	0	22	0	12	0	17	8	162	1620
Demicannon large	12	0	6000	6	6	6	3	36	0	22	6	12	6	18	0	180	1800
Cannon Royal	12	0	8000	8	0	7	4	58	0	24	0	14	0	32	8	185	1850

PROP. I. How to know the different Fortification of a Piece of Ordnance.

In fortifying any Piece of Ordnance there are three degrees observed, as first Legitimate Pieces, which are those that are ordinarily fortified; secondly Bastard Pieces, which are such whose Fortification is lessened; thirdly Double fortified Pieces, or extraordinary Pieces.

The Fortification of any Piece of Ordnance, is accounted by the thickness of the Metal at the Touch-hole, Trunnions, and at the Muzzel, in proportion to the Diameter of the Bore.

The Legitimate Pieces, or the ordinary fortifi∣ed Cannons, have ⅞ at the Touch hole, ⅝ at the Trunnions, and ⅜ at the Muzzel of the thickness of the bore, in thickness of Metal. Bastard Cannons, or lesned Cannons, have ¾ at their Touch-hole, or 12/16, and 9/16 at their Trunnions, and 7/16 at their Muzzel: the Double fortified Cannons have full one Diameter of the Bore in thickness of Me∣tal at the Touch-hole, and 11/16 at the Trunnions; and 7/16 at their Muzzel. Now all double fortified Culverins, &c. are 1 ⅛ at the Touch-hole, 15/16 at the Trunnions, and 9/16 at the Muzzel, and the Ordi∣nary fortified Culverins, are fortified every way as double fortified Cannons, and lesned Culverins as Ordinary Cannons in all respects.

PROP. II. How to know how much Powder is fit for proof, and what for service, for any Piece of Ordnance.

For Cannons take ⅘ of the weight of their Iron Bullet of good Corn Powder for Proof, and for service ½ the weight of the Iron Bullet is sufficient, especially for Iron Ordnance, which will not en∣dure so much Powder, as Brass ones will receive by ¼ in Weight, for Culverins allow the whole Weight of the Shoot for Proof, and ⅔ for Ser∣vice. For Sakers, and Falcons, take ⅘ of the Weight of the Shoot, and for lesser Pieces the whole weight may be used in service, untill they grow hot, but then there must be some abate∣ment made at discretion, and take 1 ⅓ of the weight of their Iron Bullet for Proof.

PROP. III. To know what Bullet is fit to be used in any Piece of Ordnance.

The Bullet must be somewhat less than the Bore of the Gun, that so it may have vent in the discharge, some Authors affirm ¼ of an Inch less than the Bore will serve, all Ordnance, but this vent is too much for a Falcon, &c. and too little for a Cannon: therefore I approve them not, but commend Mr. Phillipes's proportion* 1.381 to your Use, which is to divide the Bore of the Gun into 20 equal parts; and let the Diameter of the Bullet be 19/20 thereof, according to which* 1.382 proportion the precedent Table is calculated.

PROP. IV. By knowing the proportion of Metals one to another, and by knowing the Weight of one Ball, to know what any other shall weigh.

The common received proportions for Me∣tals are these.

Lead is to Iron as 2, to 3.

Lead is to Brass as 24, to 19.

Lead is to Stone as 4, to 1.

Iron is to Lead as 3, to 2.

Iron is to Brass as 16, to 18.

Iron is to Stone as 3, to 8.

The more exact proportion betwixt Metals are thus known. Admita Cube, or Ball of Gold, weigh 100 l. A Cube of any of those Metals ensuing of the same bigness, shall bear such pro∣portion, as followeth, to the said Cube of Gold.

li.	pts.	li.	pts.
Gold.	100	00	Iron.	42	10
Quicksilver.	71	43	Tinn.	38	95
Lead.	60	53	Stone.	15	80
Silver.	54	39	Water.	05	68
Brass.	47	37

It is the opinion of Dr. Wybard in his Tactome∣tria, that a Bullet of Cast Iron, whose Diameter is 4 Inches, doth weigh 9 l. Averdupoize weight.

Now to find what any other Bullet, or Cube shall weigh; say (as in prop. 4. chap. 1.) As the Cube of the Bullet propounded, is to his weight, so is the Cube of another Bullet given, to his weight, and so observe still this propor∣tion.

SECT. III. Of the Qualification of an able Gunner, and necessary Operations before shooting, and in shooting.

A Gunner ought to be a Man of Courage, Experience, and Vigilant; he ought to have good skill in Arithmetick, to know the Ex∣traction of the Roots, &c. He ought to have skill in Geometry, to take heights, distances, &c. to know the Divisions and Use of his Circle, Quadrant, and Quadret; to know how to level, and to lay Platforms, and to raise Batteries. He must know the Names of all sorts of Ordnance, their Weight, the Height of their Bore, the Height and Weight of their Shot, the length and breadth of their La∣dles, how much Powder to use for proof, and action; The Shoots Level, and the Shoots Ran∣dom; He must know the Names of all the Mem∣bers of a Piece of Ordnance, he must also know the length, thickness and breadth of all manner of Carriages, and must know all the parts thereof: Viz. the Cheeks or Sides, the Axtree, Spokes, Nave, Hoops, Transomes, Bolts, Plates, Drawing-Hooks, the Clout, the Hole for the Linspin, the Shafts, the Thill and Thill-bolt, the Fore-lock, and Fore-lock∣keys, Capsquares, the Fore-lock-pins and Chain, the Pintle and Bolt-hole, Fellows, Nayles, Fellow-bars, Stirropes, the Ruts of the Wheel, Dowledges, Beds, Coines, Leveres, Hand screws, &c. He must also know how to make his Ladles, Spunges, Cartrid∣ges,

whether of Paper or Canvas, and to have by him Formers of all sorts, Sheep-skins undrest to make Spunges, Powder, Shot, Needles, Thread, Paste and Starch, Marlin, Twine, Nails, Hand∣spikes, Crows of Iron, Granado-shells, and Mate∣rials for Composition, Fasces, Budg Barrels, Can∣non-Baskets, &c. These being general things he is to know, and at all times to have ready by him, and he is more particularly to know these following parts of his Art: As,

PROP. I. How to Tertiate, Quadrate, and to Dispart a Piece of Ordinance.

1. To Tertiate a Piece, is to find whether it hath its due thickness at the Trunnions, Touch∣hole, and Neck; if the Trunnions, and the Neck are in its due order, and the Chase streight.

2. To Quadrate a Piece mounted, is to see whether it be directly placed, and equally poi∣zed in the Carriage; which is known by find∣ing in the Convex Superficies of the Base, and Muzzel Ring; the point which is Perpendicular, over the Soul of the Piece which may be found by the Gunners Instrument, called a Level; an Instrument whose use is so vulgarly known, that it needeth not my Explanation.

3. To Dispart a Piece, is to fix, or elevate on the Convex point of the Muzzel Ring, a Mark, as far distant from the Cylinder, or Soul of the Piece, as is the point of the Base-Ring; to the end, that the Visail-ray which passeth by these marks, may be Parallel to the Chase, Soul, or

Cylinder of the Piece. Now the Dispart, i. e. the difference of the Semidiameters of the Cor∣nishes, may be by a pair of Calliper Compasses at∣tained. Which found, place on the Top of the Cornish-Ring, near the Muzzel, over the mid∣dle of the Inferior Cylinder.

PROP. II. To know how far any Piece of Ordnance will shoot, &c.

As to the several shootings in Artillery, Au∣thors differ much in their Judgments, and Opi∣nions, but they all unanimously agree that the Ball being shot forth flies through the Air, with a Violent, Mixt and Natural motion; de∣scribing a Parabolical line, in whose beginning and ending are lines sensibly streight, and in the middle carved: In the beginning the Imprest* 1.383 force driving forward by the Fire, the Natural gravity of the Ball doth describe a Right-line, called the Direct line, or Rangs of the Ball's Cir∣cute.

In the middle that force diminisheth, and the Natural Gravity prevaileth, so that it de∣scribeth a curved line, called the Ball's middle Helical or Conical Arch* 1.384; In the End the Natu∣ral Gravity overcoming the Imprest violence, (which becomes altogether weak and faint) de∣scribes a new right line, called the Ball's declining line, in which the Ball tends towards the Center of the Earth, as towards a Place natural unto all heavy bodies: See Figure the 92. These motions are somewhat longer, according as the Piece is

mounted from the level unto the Angle of 45 deg. which is called the Utmost Random: The Elevation of which, is regulated by the Gunners Quadrant, the Use of which Instrument is so generally known, and by so many Authors ful∣ly explained, that I here crave leave to omit it: But take these for General Rules.

• 1. That a Shoot at Right Angle, strikes more* 1.385 violent and furiously than at Oblique Angles, therefore Gunners use when they are to bàtter down a Tower, Wall, or Earth-work, to shoot point blank at the Object, Tire by Tire; by discharging all the Pieces in Battery against the self same. Object, in the same; Instant, holding it for a Maxim, that ten Cannons discharged together, do fan more Execution than discharged one after another. Now at Oblique Angles they shoot either Cross ways or by rebounding.
• 2. That the speediest way to make a B••••a••h in a Wall, &c. Is by shooting at the Object from two Batteries, which ruins for more speedily than by striking the Object, with one Battery, at Right Angles, although that one Battery, hath as many Cannon as the other two hath.
• 3. That if you were to batter a Flank, covered with an Orillion, (which because you cannot possi∣bly batter it right forward) you must therefore of necessity batter it Obliquely, by way of Rebounding, thus: Chuse a fit place in the Curtain to be your Object, on which you may play with your Battery obliquely, so that by a rebound the shoot may leap 〈…〉〈…〉 the Flanks, holding for a Maxim, in this operation,* 1.386 That the Angles of Incidence and Reflection are Equal.

  Now we come to shew the length of the

• ...

  Right Range, of all our Common English Ord∣nance, which is set down in the precedent Ta∣ble, in which the Cannon exceed not 185 Paces, &c. Esteeming the Pace 5 English Feet, nor his utmost Random above 1850 Paces, which Table so sheweth for all other Natures.

As for the Ranges, and Randoms, to the seve∣ral Degrees and points of Mounture of the Quadrant, I have hereunto annexed the Tables, calculated by the Experiments of sundry most Eminent Artists, whose Works will perpetuate their Worth and Name to succeding Genera∣tions.

A Table of Ranges, and Randoms, to the several Degrees of Mounture of the Quadrent.

A TABLE OF
Right Ranges or Points Blanks.	Randoms or the First Graze.
The Degrees of the Pieces Mounture.	0	The Right Range in Paces of 5 Feet.	192	The Degrees of Mounture.	0	The Paces of the Random 5 Feet a Pace.	192
1	209	1	298
2	227	2	404
3	249	3	510
4	261	4	610
5	278	5	722
6	285	6	828
7	302	7	934
8	320	8	1044
9	337	9	1129
10	354	10	1214
20	454	20	1917
30	693	30	2185
40	855	40	2289
50	1000	50	2283
60	1140	60	1792
70	1220	70	1214
80	1300	80	1000
90	1350	90

The Use of the Table of Randoms.

This Table is most agreeing to Cannons, and Culverins; and the greatest sort of Ordnance, the Use thereof is thus.

Admit a Saker to be mounted to 3 deg. shoots the Bullet 323 Paces, how far will it shoot be∣ing mounted unto 7 deg. Say (by prop. 1. chap. 1.)

As 510 the Tabular distance for 3 deg. of Mounture, to 323, the distance found,

So is 934 the Tabular distance for 7 deg. of Mounture, to 591 272/510, the distance required, which the Saker according to this Experiment shall shoot at 7 deg. of Mounture.

Mr. NYE in his Book of Gunnery printed Anno, 1647, saith he made an Experiment by a Saker of 8 Feet long, which he loaded with three pounds of Powder, of an exact weight, both Powder and Wad at every charge, every time ramming it down with three equal stroaks, as near as possible; but on the Bullet he put no Wad, because the Saker was mounted; And thus he made four Shoots, each of them half an Hour after the other, that so the Piece might be of equal temper, and mounted his Piece to these 4 degrees of Mounture, viz. 1 deg. 5 deg. 7 deg. 10 deg. and found these Randoms.

At 1 Deg. the Random was 225 Paces.

At 5 Deg. the Random was 416 Paces.

At 7 Deg. the Random was 505 Paces.

At 10 Deg. the Random was 630 Paces.

According to which Experiment, he framed this Table of Randoms.

Deg.	Paces	Deg.	Paces
0	206	6	461
1	225	7	505
2	274	8	548
3	323	9	589
4	370	10	630
5	416

Captain HEXAM in his Book of Gun∣nery, shews how by finding out the Random of a Cannon, for the first Degree of Mounture, thereby to find the Random for every Degree to 45 deg. or utmost Random, and this is his Rule to perform it. First find how many Paces the Cannon will shoot being laid level by the Metal, (which by him is accounted 1 deg.) Then di∣vide the distance found, by 50, then multi∣ply the Quotient by 11, so shall the product be the greatest Digression, or Difference betwixt Rangs, and Rang; which being divided by 44, the Quotient giveth the Number of Paces, which the Bullet will lose in the other Rangs, from Degree, unto Degree; according to this Rule, this Table is calculated.

A Table of Randoms to 45 Degrees, account∣ing a Pace 2 ½ Foot.

D. Moun	Paces.	Diff.	D. Moun.	Paces.	Diff.
0	0775	225	23	4685	110
1	1000	220	24	4795	105
2	1220	215	25	4900	100
3	1435	210	26	5000	95
4	1645	205	27	5095	90
5	1850	200	28	5185	85
6	2050	195	29	5270	80
7	2245	190	30	5350	75
8	2435	185	31	5425	70
9	2620	180	32	5595	65
10	2800	175	33	5560	60
11	2975	170	34	5620	55
12	3145	165	35	5675	50
13	3310	160	36	5725	45
14	3470	155	37	5770	40
15	3625	150	38	5810	35
16	3775	145	39	5845	30
17	3920	140	40	5875	25
18	4060	135	41	5900	20
19	4595	130	42	5920	15
20	4325	125	43	5935	10
21	4450	120	44	5945	5
22	4570	115	45	5950

I have hereunto also annexed the Table calculated by Alexander Bianco, for all sort of Ordnance, (which Table I account one of the best that was ever yet found Extant) In his Work printed 1648.

A Table of Randoms for the first six Points of the Gunner's Quadrant.

Points.	1	2	3	4	5	6
Falconet.	375	637	795	885	892	900
Falcon.	550	935	1166	1254	1309	1320
Minion.	450	765	954	1026	1071	1080
Saker.	625	1062	1325	1125	1487	1500
Demi-culver.	725	1232	1537	1653	1725	1740
Culverin.	750	1275	1590	1710	1785	1800
Demi-cannon.	625	1062	1325	1425	1487	1500
Cannon of 7.	675	1147	1431	1489	1606	1620
Double Cannon.	750	1275	1590	1710	1785	1800

SECT. IV. Of Shooting in Mortar-Pieces.

A Mortar-Piece is a short Piece, with which they shoot Bombs, Granado-Shells, Stone∣Balls, &c. not by a Right line but from a Cur∣ved, from on high; so that it may fall where it

should be desired: Now this Mortar is placed in the Carriage, in all respects as you see in Fig. 93. in which A signifies the Carriage, B the Mor∣tar,* 1.387 C the Course the shoot flies, and D the Place on which it falls.

Bombs are great hollow Balls of Iron, or Brass, in which are put fine Sifted Gun-Powder, which by a Fuse, they proportion to them a due Fire, that so they may break assoon as they fall amongst the Enemies. These Fuses are small Trunks of Wood, Tinn, or Iron, filled with a prepared Composition for that purpose. Granadoes are of the same form with Bombs, only smaller, and many times are cast by hand, and are made of Iron, Brass, Glass, or Earth.

Now in Order to the well shooting in those kind of Machines called Mortars, 'tis requisite to observe these following Rules: as,

• 1. That before you make a shoot at any Place, you find the distance thereof from your Mortar, which may be obtained by Prop. 3. §. 4. Chap. 9.
• 2. That the Bombs, or other Bodies that are to be shot, be of equal weight, otherwise the shoots will vary
• 3 That the Carriage in breadth be always on a Level, and without any descent, that so it may not leap in discharging.
• 4. That the Powder with which the Mortar is loaded, be always of the same force and weight.
• 5. That the Charge of the Mortar, as well in Powder as in Wadding, be always rammed in with blows equally heavy, and of equal number.
• 6 That the Wadds be always either of Wood, or Tampeons, or else of Okam, for the strongest drives it farthest.

• 7. That the Fuses be newly made, in those days that they are to be used, and that they be made of a Composition proportionable to the Range that the shoot shall make in the Air, so that the Bomb may break in the very moment of its fall; which Com∣position must be such, that though it fall in the Water, yet not to extinguish, but the Bomb there to break. Now before we proceed any farther, I think it necessary, to shew how to compose your Ingredients for your Fuse.

PROP. I. To make Fuses for Bombs, &c.

The Composition for Bombs must be of a slow motion, that so time enough may be given to throw either Bombs, Granadoes, Fire-balls, Thundring-Barrels, &c. They are compounded of these Ingredients, thus: Take a pound of Gun-Powder, 4/16 of Sulphur 4/16 of Salt-Peter, well beaten, dry, and sifted separately, then mix it, and make up your Fuse thereof: Or take Powder of Benjamin, and Small-Coles, all well beaten and mixed together with some Oyl of Piter, and so fill your Fuse therewith.

Now the use of Mortar-Pieces, being for the most part to shoot up at Random, therefore the Randoms of these Pieces is very necessary to be known: Therefore hereunto I have annexed the Tables of Randoms, calculated by the Experi∣ence of the best of Authors, which have wrote on this Subject; most of which do agree in their Randoms, although they are in a several dress.

Diego-Uffano-Zutphen in his Works printed 1621, hath calculated these two following Ta∣bles, the one for the 12 points of the Quadrant, the other for every Degree, taking the one Half of each Number, and so 'tis reduced into our English Paces of 5 Feet, which Tables were esteemed and made use of, both by Captain Hexam, and Mr. Norton, and are as followeth.

A Table of Randoms for Mortar-Pieces, to the 12 Points of the Gunner's Quadrant, cal∣culated by Diego-Uffano-Zutphen.

583	570	534	468	377	248	100
6	5	4	3	2	1	0
.	.	.	.	.	.	.	☉
6	7	8	9	10	11	12
583	570	534	468	377	248	000

Now suppose the Mortar to be placed at ☉, the Pricks in the middle line representeth the several Randoms, numbred with the Degrees of the Quadrant, forward and backward, unto which the several Randoms are set; so you see that the Mortar being levelled point blank, throweth the Bomb 100 Paces, if the Mortar be mounted one Point, it throws the Bomb 248 Paces, &c. untill 'tis mounted to the 6th. point,

583 Paces, which is the utmost Random: Now if the Mortar be mounted higher to 7, 8, 9, &c. Points, the Randoms decrease again as before they did increase: as you see in the Table.

But in those latter Randoms there lieth a great mistake, as shall be made palpably appear. For if as they are distant from the sixth Point you make them equal to one another, then the Random of the 12 points, must be equal to the Random of 0 point, or the Level Random, which is 100 Paces from the Mortar. Now it is con∣trary to all Art and Reason, to think that if the Mortar be elevated to the 12th. point, i. e. bolt upright, it should shoot the Bomb 100 Paces from the Mortar; no, it cannot be; but according to all Reason the Bomb must fall down either on, or near the Mortar, and not 100 Paces dis∣tant, as is most erroneously conceived; the like errour is in the following Table of our said Author; but because Mr. Phillipps in his Ma∣thematical Manual hath amply demonstrated their Errours, I therefore shall say no more to the Errours that have been a long time gene∣rally conceived and embraced as a truth, but now are removed.

A Table of Randoms for Mortar-Pieces, to every Degree of the Quadrant.

The Degrees of Mounture.	0	The Paces of the Random.	100	The Degrees of Mounture.	89	The Degrees of Mounture.	23	The Paces of the Random.	480	The Degrees of Mounture.	66
1	122	88	24	490	65
2	143	87	25	500	64
3	164	86	26	510	63
4	185	85	27	518	62
5	204	84	28	525	61
6	224	83	29	531	60
7	243	82	30	536	59
8	263	81	31	540	58
9	280	80	32	543	57
10	297	79	33	549	56
11	315	78	34	552	55
12	331	77	35	558	54
13	347	76	36	562	53
14	362	75	37	568	52
15	377	74	38	573	51
16	393	73	39	577	50
17	406	72	40	580	49
18	419	71	41	581	48
19	432	70	42	582	47
20	445	69	43	583	46
21	457	68	44	584
22	460	67	45	585

The most exact Tables of Randoms for the Mortar, that I have seen or can find in any An∣cient, or Modern Author, is this following Ta∣ble, calculated by the experience and trial of that Famous Inginier Tomaso Moretii of Brescia, Inginier to the most serene Republique of Venice, in his Works Intituled, Trattatu delle Artiglieria, printed 1665. Where he supposeth the utmost

Random, equal to 10000, according to which proportion he framed this following Table.

A Table of the several Randoms of each Degree of the Quadrant, the greatest Equal to 10000.

Elev.	Elev.	Elev.	Elev.
1°	349	89°	23°	7193	67°
2	698	88	24	7431	66
3	1045	87	25	7660	65
4	1392	86	26	7880	64
5	1736	85	27	8090	63
6	2079	84	28	8290	62
7	2419	83	29	8480	61
8	2756	82	30	8660	60
9	3090	81	31	8829	59
10	3420	80	32	8988	58
11	3746	79	33	9135	57
12	4067	78	34	9272	56
13	4384	77	35	9397	55
14	4695	76	36	9511	54
15	5000	75	37	9613	53
16	5299	74	38	9703	52
17	5592	73	39	9781	51
18	5870	72	40	9848	50
19	6157	71	41	9903	49
20	6428	70	42	9945	48
21	6691	69	43	9976	47
22	6947	68	44	9994	46
45	10000	45

The Use of the Precedent Table is explained by these following Propositions.

PROP. II.

Finding that a Mortar of 300, with a Tampeon of Wood, being elevated 45°, or 6 Points of the Quadrant, sends a Bomb 800 Paces, how many Paces shall the same shoot, at the Elevation of 54°?

Look at the said 54° of the Table, and you Demon∣stration.

will find thē proportional Number 9511, to correspond thereunto. Now you find the pro∣portional Number belonging to 45° is 10000, then by Prop. 1. Chap. 1. Say as 10000, to 800, so is 9511, to 760 88/100, which are the Paces, the Mortar will send the Bomb at the Elevation of 54 Degrees.

PROP. III. Finding that a Mortar of 300, being elevated 54°, sends his Bomb 760 88/100 Paces, what Degree of Elevation must that Mortar have, to shoot the Bomb 555 Paces?

This is but the Converse of the former, there∣fore (according to Prop. 1. Chap. 1.) say, as 760 88/160 Paces, gives the proportional part or number 9511; so doth 555 Paces, give the proportional part 6945. Which number sought among the proportional Numbers, in the Table, you will find 68 Degrees to correspond to that proportio∣nal Number 6945, so that the Mortar must be elevated to 68 Degrees to shoot the Bomb 555 Paces, which was required to be known. These Rules and Precepts here delivered, I esteem ne∣cessary to be known by every Gunner, who in∣tends to be serviceable for his Prince and Coun∣trey.

Vive, vale: Siquid novisti rectius istis, Candidus imperti: Si non his utere mecum. Hora. lib. 1. Epist

FINIS.

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.
• * 1.13

  Defin.
• * 1.14

  Signifies the Num∣ber, or figure sought.
• * 1.15

  Signifies the Loga∣rithm answering to the number Opposite.
• * 1.16

  ☞ Note this Rule well, for this explains the use of the Table of proportional parts, printed at the end of this Book.
• * 1.17

  Definition. The Rule to find the Characteristick or Index appertaining to any Lo∣garithm.
• * 1.18

  〈 math 〉〈 math 〉
• * 1.19

  〈 math 〉〈 math 〉
• * 1.20

  〈 math 〉〈 math 〉
• * 1.21

  And here'tis ne∣cessary to understand, that every Circle is supposed to be divided into 360 E∣qual parts which are cal∣led Degrees, and every of those Degrees into 60 Minutes, and every Mi∣nute into 60 Seconds, and every Second into 60 Thirds, &c. so that a Semi-Circle contains 180 Degrees, and a Qua∣drant 90 Degrees; Now an Arch or Angle of a Triangle, is the Inter∣section of its two sides, and the measure thereof, is an Arch of a Circle, which cutteth each of the two sides equidistant from the Angular point, (which is the Center.) Now the Logarithm Sine, or Tang∣of any such Arch of a Triangle, containing a∣ny Number of Degrees or Minutes of the Quadrant, may be found in the Tables, printed at the End of this Book, where they are plainly expressed, and are found as directed in the precedent Rules.
• * 1.22

  Defin.
• * 1.23

  Defin.
• * 1.24

  Quod quaeritur cog∣noscendi illius gratia, quod semper est, non & ejus quod oritur, quando∣que & interit. Geometria, ejus quod est semper, Cogni∣tio est. Ac tollet igitur (ô Generose vir) ad veritatem, animum: atque ita, ad Philosophandum prepar••vit cogi∣tationem, ut ad supera convertamus: quae nunc, contra quàm decet, ad inferiora dejicimus, &c. Plato lib. 7. de Rap.
• * 1.25

  Fig. 1,
• * 1.26

  Fig. 2,
• * 1.27

  Fig. 2.
• * 1.28

  Fig. 3.
• * 1.29

  Fig. 4.
• * 1.30

  Fig. 5.
• * 1.31

  Fig. ••.
• * 1.32

  Fig. 6.
• * 1.33

  Fig. 7.
• * 1.34

  Fig. 8.
• * 1.35

  Those two propositions well understood, doth de∣monstrate many other pro∣positions, and thereon is grounded the vse of the Sector.
• * 1.36

  〈…〉〈…〉 9.
• * 1.37

  But this in a Hexa∣gon need not be done, be∣cause the 3 sides of the Triangle are equal, but in all other Poligons it must be done.
• † 1.38

  But if the third line do exceed or be short of the side of the Poligon pro∣pounded, then by parallels on each side, cut the sides of the Triangle, till you have found by those Inter∣sections where to set the line proposed, in any Po∣ligon, &c.
• * 1.39

  Fig. 10.
• * 1.40

  Fig. 11
• * 1.41

  ☞ Observe these Rules well, for you will find them of infinite use in Fortification, &c.
• * 1.42

  Fig. 11.
• * 1.43

  Fig. 12.
• * 1.44

  Fig. 13.
• * 1.45

  Fig. 14.
• * 1.46

  Fig. 15.
• * 1.47

  Fig. 16.
• * 1.48

  And this of all other the Inventions of Plato, Apollonius, Sporus, Archi∣tas, Diocles, Nicomedus; & many other famous Geo∣metricians and Philoso∣phers, I like best for the ready performance of this Conclusion, whose several Methods I could here de∣scribe, but for brevity sake do omit them.
• * 1.49

  Fig. 17.
• * 1.50

  Fig. 18.
• * 1.51

  Fig. 18.
• * 1.52

  So after the same manner, may divers Cir∣cles be added into one by the help of the former proposition well understood.
• * 1.53

  Fig. 19.
• * 1.54

  Fig. 20.
• * 1.55

  The Rule.
• * 1.56

  Fig. 21.
• * 1.57

  Fig. 22.
• * 1.58

  Fig. 23.
• * 1.59

  Fig. 24.
• * 1.60

  Fig. 25.
• * 1.61

  Fig. 26.
• * 1.62

  Fig. 27.
• * 1.63

  Fig. 28.
• * 1.64

  Fig. 29.
• * 1.65

  Fig. 30.
• * 1.66

  Fig. 31.
• * 1.67

  Fig. 32.
• * 1.68

  Fig. 33.
• * 1.69

  ☞ Note that every Sphere is equal unto two Cones, whose Height and Diameter of the Base is the same with the Axis of the Sphere. And a Sphere is two thirds of a Cylinder, whose Height and Diameter of the Base is the same with the Axis of the Sphere; according unto the 9th. Manifestation of the first Book of Archimedes of the Sphere and Cylinder
• * 1.70

  Fig. 34.
• * 1.71

  Fig. 36.
• * 1.72

  Fig. 35.
• * 1.73

  Fig. 34.
• * 1.74

  〈 math 〉〈 math 〉
• * 1.75

  Observe this for a ge∣neral Rule in Trigonome∣try.
• * 1.76

  Fig. 34
• * 1.77

  Fig. 34.
• * 1.78

  Fig. 34.
• * 1.79

  Fig. 35.
• * 1.80

  〈…〉〈…〉
• * 1.81

  Fig. 35.
• * 1.82

  Fig. 35.
• * 1.83

  Fig. 36.
• * 1.84

  ☞ The Rule to find the Complement Arithmeti∣cal, of any Logarithm Number.
• * 1.85

  9. 962398.
• * 1.86

  0. 037602.
• * 1.87

  [〈 math 〉〈 math 〉]
• * 1.88

  Fig. 36.
• * 1.89

  Fig. 36.
• * 1.90

  Fig. 36.
• * 1.91

  Fig. 37.
• * 1.92

  Fig. 37.
• * 1.93

  That is equal unto the two Angles B 40° and B. 53° as afore in the former Proposition.
• * 1.94

  [〈 math 〉〈 math 〉]
• * 1.95

  Fig. 37.
• * 1.96

  〈 math 〉〈 math 〉
• * 1.97

  Fig. 37.
• * 1.98

  Fig. 37.
• * 1.99

  Fig. 37.
• * 1.100

  Fig. 38.
• * 1.101

  〈 math 〉〈 math 〉
• * 1.102

  Fig. 38.
• * 1.103

  Fig. 38.
• * 1.104

  Fig. 38.
• * 1.105

  Fig. 38.
• * 1.106

  Fig. 38.
• * 1.107

  Fig. 38.
• * 1.108

  Fig. 38.
• * 1.109

  Fig. 38.
• * 1.110

  Fig. 38.
• * 1.111

  Fig. 38:
• * 1.112

  Fig. 38.
• * 1.113

  Fig. 38.
• * 1.114

  Fig. 38.
• * 1.115

  Fig. 38.
• * 1.116

  Fig. 38.
• * 1.117

  Fig. 39.
• * 1.118

  Note that in thes•• Operations, for the more facility of the learner, I omit Seconds, which doth belong unto the Angles, &c.
• * 1.119

  Fig. 39.
• * 1.120

  Fig. 39.
• * 1.121

  〈 math 〉〈 math 〉
• * 1.122

  ☞ And here ob∣serve that if the Sum of the two contained Sides exceed a Semicircle, then substract each side severally from 180°, and proceed with those Complements, as with the sides given, the Operation produceth the Complements of the Angles sought, unto a Semicircle or 180 Degrees.
• * 1.123

  Fig. 39.
• * 1.124

  ☞ And here observe that if the Sum of the two given Angles excede a Semicircle or 180°, substract them from a Se∣micircle, and proceed with the Residues, the Operation will produce each side's Complement to a Semicircle, or 180 Degrees.
• * 1.125

  Fig. 39.
• * 1.126

  〈 math 〉〈 math 〉
• * 1.127

  Fig. 39.
• * 1.128

  〈 math 〉〈 math 〉
• * 1.129

  〈 math 〉〈 math 〉
• * 1.130

  Fig. 39.
• * 1.131

  〈 math 〉〈 math 〉
• * 1.132

  〈 math 〉〈 math 〉
• * 1.133

  Fig. 39.
• * 1.134

  〈 math 〉〈 math 〉
• * 1.135

  〈 math 〉〈 math 〉
• * 1.136

  Fig. 39.
• * 1.137

  Fig. 39.
• * 1.138

  〈 math 〉〈 math 〉
• * 1.139

  〈 math 〉〈 math 〉
• * 1.140

  Fig. 40.
• ☞ 1.141

  Note that if the Angles at the Base be both of one kind, that then the Perpendicular falls within the Triangle: if of diverse kinds, without the Triangle.
• * 1.142

  Fig. 41.
• * 1.143

  Fig. 40.
• * 1.144

  Fig. 40.
• * 1.145

  〈 math 〉〈 math 〉
• * 1.146

  Fig. 40.
• * 1.147

  Fig. 40.
• * 1.148

  〈 math 〉〈 math 〉
• * 1.149

  Fig. 40.
• * 1.150

  Defin.
• * 1.151

  Defin.
• * 1.152

  Fig. 42.
• * 1.153

  Fig. 42.
• * 1.154

  Fig. 42.
• * 1.155

  Fig. 42.
• * 1.156

  Fig. 42.
• * 1.157

  Which is by the Greeks called 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, i e. bring Life; because the life of all Creatures depend on the cause of that Circle, for the Sun ascending in it and moving towards us, brings the Generation of things, and in descending the Corruption of all things sensible, and insensible, which are below the concavity of the Moon, &c.
• * 1.158

  Fig. 42.
• * 1.159

  Fig. 42.
• * 1.160

  Fig. 42.
• * 1.161

  Fig. 42
• * 1.162

  Fig. 42.
• * 1.163

  Fig. 42.
• * 1.164

  Fig. 42.
• * 1.165

  Fig. 42.
• * 1.166

  Fig. 43.
• * 1.167

  Fig. 43.
• * 1.168

  If the Sun's Declina∣tion be North, and in∣creasing, this Proportion finds the Sun's distance from ♈; but if decreasing from ♎ in the northern Sines. But if the Sun's Declinati∣on be South, and increasing from ♎; if decreasing from ♈, among the Southern Sines.
• † 1.169

  From the next Equi∣noctial point either ♈, or ♎.
• * 1.170

  As in Case 11 of Oblique Spherical Tri∣angles.
• * 1.171

  Watched the Time after Sun-setting when the Twilight in the West was shut in, so that no more Twilight than in any other part of the Skie near the Horizon appea∣red there: then by one of the known fixed Stars, having found the true Hour of the Night, he found the length of the Twilight, to be as in the rule is mentioned.
• * 1.172

  Or ½ Diurnal Arch.
• * 1.173

  To find the length of the least Crepusculum or Twilight.
• * 1.174

  Defin.
• * 1.175

  Defin.
• * 1.176

  Europe.
• * 1.177

  Which Pliny hath adorned in these words (saith he) Italia terra∣rum omnium alumna, ea∣dem & parens, numine Deûm electa, qua Coelum ipsum clariùs faceret, spar∣sa congregaret imperia, ri∣tus molliret, tot populo∣rum discordes linguas sermonis commercio ad Colloquia distraheret, & humanitati hominem da∣ret, i. e. Italy (saith Pli∣ny) is the Nurse and the Parent of all Religion, was elected by the Providence of the Gods, to make (if possible) the Heavens more famous; to gather the scattered Empires of the World into one Body, to temper the Barbarous rites of the Nations, to unite so many disagreeing Languages of Men by the benefit of one common Tongue: and in a word to restore Man to his Hu∣manity.
• * 1.178

  Europe.
• * 1.179

  Europe.
• * 1.180

  Europe.
• * 1.181

  Europe.
• * 1.182

  Europe.
• * 1.183

  Europe.
• * 1.184

  Europe.
• * 1.185

  Europe.
• * 1.186

  Europe.
• * 1.187

  Europe.
• * 1.188

  Europe.
• * 1.189

  Europe.
• * 1.190

  Europe.
• * 1.191

  Europe.
• * 1.192

  Europe.
• * 1.193

  Europe.
• * 1.194

  Europe.
• * 1.195

  Europe.
• * 1.196

  Europe.
• * 1.197

  Europe.
• * 1.198

  Europe.
• * 1.199

  Europe.
• * 1.200

  Europe.
• * 1.201

  Asia.
• * 1.202

  Asia.
• * 1.203

  Asia.
• * 1.204

  Asia.
• * 1.205

  Asia.
• * 1.206

  Asia.
• * 1.207

  Asia.
• * 1.208

  Asia.
• * 1.209

  Asia.
• * 1.210

  Asia.
• * 1.211

  Asia.
• * 1.212

  Asia.
• * 1.213

  Africa.
• * 1.214

  Africa.
• * 1.215

  Africa.
• * 1.216

  Africa.
• * 1.217

  Africa.
• * 1.218

  Africa.
• * 1.219

  Africa.
• * 1.220

  Africa.
• * 1.221

  America.
• * 1.222

  America.
• * 1.223

  America.
• * 1.224

  America.
• * 1.225

  America.
• * 1.226

  America.
• * 1.227

  America.
• * 1.228

  America.
• * 1.229

  America.
• * 1.230

  America.
• * 1.231

  America.
• * 1.232

  America.
• * 1.233

  The Do∣ctrine of Rightli∣ned Trian∣gles, both Right, and Oblique∣Angled, applied to Propositi∣ons of Plain Sai∣ling.
• * 1.234

  Fig. 44.
• * 1.235

  Fig. 44.
• * 1.236

  Fig. 44.
• * 1.237

  Fig. 44.
• * 1.238

  Fig. 45.
• * 1.239

  Fig. 45.
• * 1.240

  Fig. 46.
• * 1.241

  Fig. 46.
• * 1.242

  Fig. 47.
• * 1.243

  Fig. 47.
• * 1.244

  Fig. 48.
• * 1.245

  Fig. 49'.
• * 1.246

  But is indeed the In∣vention of our Learned Countryman Mr. Edw. Wright, although this Stranger hath almost got the Name and Praise thereof.
• * 1.247

  Fig. 50.
• * 1.248

  Fig. 50.
• * 1.249

  〈 math 〉〈 math 〉
• * 1.250

  Fig. 51.
• * 1.251

  Fig. 51.
• * 1.252

  Fig. 51.
• * 1.253

  Fig. 51.
• * 1.254

  Fig. 51.
• * 1.255

  Fig. 51.
• * 1.256

  Which Instru∣ment, and the Plain Table, I esteem as the two aptest In∣struments for Survey∣ing of Land; i. e. the Plain Table for small En∣closures, and the Semicircle for Cham∣pain Plains, Woods, and Moun∣tains.
• * 1.257

  Fig. 52.
• * 1.258

  Fig. 52.
• * 1.259

  Fig. 53.
• * 1.260

  Fig. 53.
• * 1.261

  Fig. 54.
• * 1.262

  Fig. 54.
• * 1.263

  Fig. 55.
• * 1.264

  Fig. 55.
• * 1.265

  Fig. 55.
• * 1.266

  Fig. 55.
• * 1.267

  Fig. 55.
• * 1.268

  Fig. 55.
• * 1.269

  Fig. 56.
• * 1.270

  Fig. 56.
• * 1.271

  Fig. 56.
• * 1.272

  Fig. 56.
• * 1.273

  Fig. 57.
• * 1.274

  Fig. 57.
• * 1.275

  Fig. 58.
• * 1.276

  〈 math 〉〈 math 〉
• * 1.277

  Fig. 58.
• * 1.278

  Fig. 58.
• * 1.279

  Fig. 58.
• * 1.280

  Is a Quadrangle, whose sides are not Paral∣lel, nor equal.
• * 1.281

  Euclides postulat hant fabricam'Trapezium, tanquam mensulam vo∣cari: & sanè nominis ejus ratio Geometrica nulla est: P. Rami lib. 14 pag. 94.
• * 1.282

  Fig. 59.
• * 1.283

  Fig. 60.
• * 1.284

  Fig. 61.
• * 1.285

  Which to do is no more than thus; with a Thread and Plummet fastened at the Center of the Semicircle, so that it hath liberty to play, move the Semicircle until the Thread playeth against 90 deg. then screw it fast, and it is Horizontal.
• * 1.286

  Fig. 61.
• * 1.287

  Fig. 61.
• * 1.288

  Fig. 62.
• * 1.289

  Fig. 63
• * 1.290

  4.
• * 1.291

  63.
• * 1.292

  〈◊〉〈◊〉.
• * 1.293

  Fig. 63.
• * 1.294

  Whose Surface is bounded by a Line called by Proclus a Helicoides, but it may also be called a Helix, a Twist or Wreath, &c.
• * 1.295

  See Procl. lib. 2. cap. 3. & Viturvius lib. 9. cap 3.
• * 1.296

  See Mr. Oughthred in his Book of the Cir∣cles of Proportion, page the 57. and Mr. Edm. Gunter in his Book of the Cross-staff, part 21. chap. the 4.
• * 1.297

  Which doth appear to have been in use a∣bove this 2400 Years, for King Achaz had a Dial: This Art requireth good skill in Geometry, and Astronomy: Now Cresibi∣us that famous Philoso∣pher measured the Hours and Times by the orderly running of Water. Then by Sand was the Hours measured. After that by Trochilike with Weights, and of late with Trochi∣like with Springs.
• * 1.298

  Fig. 64.
• * 1.299

  Fig. 64.
• * 1.300

  Fig. 65.
• * 1.301

  Fig. 65.
• * 1.302

  Which may be ei∣ther a Pin of the length of Q S, placed on Q, and Perpendicular unto the Plane, or it may be a piece of brass or elsewhat of the breadth of 12, to 3, or 9.
• * 1.303

  Fig. 66.
• * 1.304

  Fig. 66.
• * 1.305

  Fig. 66.
• * 1.306

  Fig. 67.
• * 1.307

  Fig. 68.
• * 1.308

  Fig.
• * 1.309

  Fig. 68.
• * 1.310

  Fig.
• * 1.311

  69.
• * 1.312

  Fig. 69.
• * 1.313

  Fig. 70.
• * 1.314

  Fig. 70.
• * 1.315

  Fig. 70.
• * 1.316

  Fig. 70.
• * 1.317

  Fig. 71.
• * 1.318

  Fig. 71.
• * 1.319

  Fig. 71.
• * 1.320

  Fig. 71.
• * 1.321

  Fig. 71.
• * 1.322

  Fig. 71.
• * 1.323

  Fig. 71.
• * 1.324

  Fig. 71.
• * 1.325

  Fig. 72.
• * 1.326

  Fig. 72.
• * 1.327

  Defin.
• * 1.328

  Fig. 73.
• * 1.329

  Fig. 73.
• * 1.330

  Fig. 73.
• * 1.331

  Fig. 73.
• * 1.332

  Because the length of the part of a Musket doth not much exceed that Mèasure.
• * 1.333

  Fig. 73.
• * 1.334

  Fig. 73.
• * 1.335

  Fig. 73.
• * 1.336

  Because the Defence ought to be easie, quick, certain, and of little charge, all which qualities the Musket hath and the Can∣non hath not, therefore the Defence of Fortification ought to be measured by the Port of a Musket, and not by that of a Cannon.
• * 1.337

  Fig. 74.
• * 1.338

  Fig. 74.
• * 1.339

  Fig. 74.
• * 1.340

  Fig. 74.
• * 1.341

  Fig. 74.
• * 1.342

  Fig. 74.
• * 1.343

  Fig. 74.
• * 1.344

  Fig. 74.
• * 1.345

  Fig. 74.
• * 1.346

  Fig. 74.
• * 1.347

  Fig. 74.
• * 1.348

  Fig. 74.
• * 1.349

  Observe this for a ge∣neral Rule in Regular Fortification.
• * 1.350

  Fig. 76.
• * 1.351

  Case 1.
• * 1.352

  Fig. 76.
• * 1.353

  Case 2.
• * 1.354

  Fig. 77.
• * 1.355

  Fig. 78.
• * 1.356

  Fig. 78.
• * 1.357

  Fig. 78.
• * 1.358

  Fig. 78.
• * 1.359

  Fig. 78.
• * 1.360

  Fig. 79.
• * 1.361

  Fig. 79.
• * 1.362

  Fig. 79.
• * 1.363

  Fig. 80.
• * 1.364

  Fig. 81.
• * 1.365

  Fig. 83.
• * 1.366

  Fig. 84.
• * 1.367

  Built to bridle the Town or the Place, left the Burghers should be rebellious, and to be the last refuge or place of retreat.
• * 1.368

  The Inginier must first form a Map of the Town or Place, with all the Ways, Passages, Old Walls, Rivers, Pools, Enclosures, and all o∣ther matters fit to be known in the draught, and then he is to design what Works he findeth most agreeing to the place to be Fortified.
• * 1.369

  Fig. 84.
• * 1.370

  Fig. 85.
• * 1.371

  Fig. 86.
• * 1.372

  Fig. 87.
• * 1.373

  Fig. 88.
• * 1.374

  Fig. 89.
• * 1.375

  Fig. 90.
• * 1.376

  Defin.
• * 1.377

  This Military Engine Bombarda, Gun, Can∣non, &c. So called from Bombo, a resounding Noise, Cannone, or Can∣non, from the likeness it holds with his Canna, Bore, or Concavity; Ar∣tigleria, from Artiglio, the Talons, or Claws of Ravenous Fowls, because its shot flying afar off tears and defaces all that it doth meet; from whence some Natures of this Machine are called Smeriglii, long winged Hawks, Falconi, Falconets; Passa volanti, swift flying Arrows, &c.
• * 1.378

  Fig. 91.
• * 1.379

  Fig. 91.
• * 1.380

  Fig. 91.
• * 1.381

  In his Mathe∣matical Manual.
• * 1.382

  page 165.
• * 1.383

  Fig. 91.
• * 1.384

  See Mr. Diggs in his Pantome∣tria, page 179.
• * 1.385

  General Rules to be observed in the bat∣tering down of a Place, or making of Breaches.
• * 1.386

  Accord∣ing to learned D'Chales, on the 4th Prop. of the first Book of Euclid.
• * 1.387

  Fig. 93.