The Sea Gunner Published By John Seller

  • a Cascable deck
  • b Base Ring
  • c Touchole
  • d The Chamber
  • e Reinfourd Ring
  • f Trunions
  • g Cornish Ring
  • h Trunion Ring

THE Sea-Gunner: Shewing the PRACTICAL PART OF GUNNERY, As it is used at SEA. AND, As an Introduction thereto, there is Exhi­bited two Compendiums, one of Vulgar, the other of Decimal ARITHMETICK, With necessary Tables relating to that ART. To which is added An APPENDIX, Shewing the Use of a Proportional Scale, for the ready working of any Question in Gunnery. And the Use of the Sea-Gunners Rule, of an excellent Contrivance; containing an Epitome of the Art of Gunnery in it self.

Composed by [...] S [...]R, Senior.

LONDON: Printed by H. Clerk for the Author, and are to sold by him at the Hermitage in Wapping, 1691.

THE PREFACE TO THE READER.

Courteous Reader,

HAving observed for several Years, that there hath been a great want of a Piece of Sea-Gunnery, that has been principally adap­ted for Sea-Service, in a Treatise by it self; (for those Books that were Extant, were chiefly inten­ded for Land Service;) and at this time most of them being out of Print, I judg'd this a fit op­portunity for the publishing this small Treatise, hoping it will be gratefully accepted by our Sea-Gunners.

And in regard that those who would be Students in that Art, ought (in some competent measure) to be acquainted with Arithmetick; for the sake of such, I have exhibited two Compendiums thereof, one in Vulgar, and the other in Decimal Arithmetick, as a necessary preparation for the working those Questions that are incident to that Art.

And for the ease of such as are not fully ac­quainted therewith, I have furnished them with a Proportional Scale, whereby they may perform all the Operations that are useful in Gunnery; as also, to extract the Square and Cube-Roots, and how to perform the same by Logarithms, and by Gunter's-Scale.

To which I have added several necessary Ta­bles useful in Gunnery, with proper Questions and their Answers, and useful Observations and Instructions,

And for the better accomplishing the Design of this Book, I have consulted with the best ap­proved Authors, that have written on this Subject.

Also at the end of this Treatise, I have pre­sented you with a small Tract as an Appendix, particularly of the use of the Proportional Scale; and of the use of a Rule of a new contrivance, fit for the Pocket, that hath upon it, an Epitome of the Practical part of Gunnery in it self, which I call the Sea-Gunner's Rule: All which I submit to the favourable construction of the Judicious,

And rest
your Friend to serve you, John Seller.

A TABLE Of the Principal Matters contained in this Book.

  • CHAP. I. Of Vulgar Arithmetick, Page 1
    • Of Notation of Numbers, ibid.
    • Numeration and Addition, 3
    • Subtraction, 5
    • Multiplication, 7
    • Division, 15
    • The Rule of Three Direct, 19
    • The Rule of Three Reverse, 22
    • Double Rule of Three 23
  • [Page]CHAP. II. Of Decimal Arithmetick, 28
    • Notations of Fractions, ibid.
    • Addition of Decimals, 33
    • Subtraction of Decimals, 34
    • Multiplication of Decimals, 35
    • Division of Decimals, 37
    • A Decimal Table of Pence and Farthings, 46
    • A Table of Decimals of one pound Sterling, 48
    • A Table of the Decimals of an English Foot to every Inch and eight parts of an Inch, 49
  • CHAP. III. The Extraction of the Square-Root by Arithmetick, 52
    • To extract the Sqaure-Root by Logarithms and Gunter's-Scale. 56
    • To extract the Cube-Root by Arithmetick, 57
    • To prepare a Cube Number for extraction, 59
    • To extract the Cube-Root by the Logarithms, 63
    • To extract the Cube-Root by Gunter's-Scale, 64
    • A Table of Square-Roots, 65
    • A Table of Cubick-Roots, 66
    • To make the Tables of Square and Cube-Roots, 67
    • A Table of Logarithms, 8
    • [Page]A Description and use of the Table of Loga­rithms, 82
    • Multiplication by Logarithms, 86
    • Division by Logarithms. 87
    • Of a Circle, 88
    • 1. The Diameter being given, to find the Cir­cumference by the Logarithms, ibid.
    • 2. The Circumference being given, to find the Diameter, 89
    • 3. The Diameter of a Circle being given, to find the Area or Superficial Content thereof, 90
    • 4 The Circumference being given, to find the Area, 91
  • CHAP. IV. Containing Geometrical Rudiments useful in the Art of Gunnery, 93
    • To raise a Perpendicular from the middle of a Line given, ibid.
    • To let a Perpendicular fall from a point assigned to the middle of a Line given, 94
    • To raise a Perpendicular upon the end of a Line given, 95
    • To let fall a Perpendicular from a point assigned unto the end of a Line given, 96
    • To draw a Line parrallel to a Line given, 97
    • A Geometrical way to find the Diameter of a Bullet that weigheth twice as much as a known Bullet, 98
    • [Page]The weight of a Shot given, to find the Dia­meter Geometrically, 100
  • Chap. V. Geometrical Theorems and Problems, 102
    • Arithmetical Problems appertaining to the Art of Gunnery, and wrought by Decimal Arithmetick, by Logarithms and Gunter's Scale.
    • Prob. 1. The Diameter of a Circle being, given to find the Circumference, ibid.
    • Prob. 2. The Circumference of a Circle being given, to find the Diameter, 105
    • Prob. 3. The Diameter of a Circle being given, to find the side of a Square equal to it, 107
    • Prob. 4. The Circumference of a Circle being given, to find the side of a Square equal in Content to that Circle, 108
    • Prob. 5. The Diameter of any Spherical body being known, to find the Circumference, 109
    • Prob. 6. The Circumference of any Spherical body be­ing known, to find the Diameter, 110
    • [Page]Prob. 7. The Diameter and Circumference of any Spherical body being known, to find the Superficial Content, 111
    • Prob. 8. The Axis or Diameter of a Globical body be­ing known, to find the solid Content, 112
    • Prob. 9. The Diameter of a Bullet being given with the weight, to find the weight of another Bullet of the same Metal, but of another Dia­meter, either greater or lesser, 113
    • Prob. 10. Having the weight of a Bullet of one kind of Metal, to find the weight of a Bullet of another kind of Metal, being equal in Magnitude, 116
    • Prob. 11. A Bullet of Iron that weighteth seventy two Pound, what will a Bullet of Lead weigh that is equal to it in bigness, 118
    • Prob. 12. The Diameter and Weight of any one Cylender of a Piece of great Ordnance taken at the base Ring being known, to find the weight of any other Piece of the same Metal and shape, either greater or lesser, its Dia­meter being only known, 119
    • [Page]Prob. 13. Having the Diameter and Weight of any Piece of great Ordnance, to find the Weight of another Piece of Ordnance of another [...], that is of the same sha [...]. 121
    • [...]ob. 14. To find the Sup [...]fi [...]l Content of the Convex face of any Piece of Ordnance, and also the solid Content of the Concavity thereof, 123
    • Prob. 15. To know how much of every kind of Metal is contained in any Brass Piece of Ord­nance, 126
    • Prob. 16. By knowing what quantity of Powder will load some one Piece of Ordnance, to find how much of the same Powder will load any other Piece of Ordnance greater or lesser, 130
    • A Table of the Weight of Iron shot in Pounds and Ounces, from one Inch Diameter to ten Inches to every eighth part of an Inch, 132
    • A Table shewing the Heighth and Weight of Iron, Lead and Stone Shot, according to their Diameters in Inches and Quar­ters, 134
  • [Page]CHAP. VI. Of the different Fortifications of most Pieces of Ordnance, 142
  • CHAP. VII. How much Powder is fit for Proof, and what for Action, for any Piece of Ord­nance.
    • To make Ladles to load your Guns with, 145
  • CHAP. VIII. To know what Bullet is fit to be used for any Gun, 147
    • To make Cartridges, Moulds and Formers for any sort of Ordnance, 148
  • CHAP. IX. Containing certain Theorems in Gunnery, 150
    • A Table of Right Ranges, or point blanck at several degrees of Mounture, 156
  • CHAP. X. Necessary Instructions for a Gun­ner, 157
  • CHAP. XI. Shewing an easie way to dispart a Piece of Ordnance. 164
  • CHAP. XII. To level a Piece of Ordnance to shoot point blank, 166
  • CHAP. XIII. How to search a Piece of Ord­nance, and to discover whether there be any [Page]Flaws, Cracks or Hony Combs in any Piece, 168
  • CHAP. XIV. How Moulds, Formers and Cart­ridges are to be made for any sort of Ordnance, 170
  • CHAP. XV. How much Rope will make Breech­ings, and Tackles for any Piece, 172
  • CHAP. XVI. How to know what Diameter eve­ry shot must be of, to fit any Piece of Ord­nance, 174
    • To tertiate a Piece of Ordnance, 175
    • How to make a Shot out of one Ship into ano­ther, in any weather whatsoever 181
    • In what order to place your great Guns in Ships, 183
  • CHAP. XVII. Several things necessary to be known by a Gunner, but especially of Powder, 188
    • To know good Powder, 189
    • To preserve Powder from decaying, 190
    • To find the experimental weight of Powder (Tower proof) that is found convenient for Service, to be used in Guns of several [Page]fortifications or thickness, and by conse­quence strength of metal, 191
    • To know whether the Trunions of any Gun are placed right, 192
    • The Practical way of making Gun-Powder, 193
    • To renew and make good again any sort of Gun-Powder, having lost its strength by moisture, long-lying, or by any other means, 195
  • CHAP. XVIII. How to make Hand-Granadoes to be hove by hand, 196
  • CHAP. XIX. How to make fine Pots of Clay, 197
    • How to make Powder-Chests and Stink-Balls, 198
  • CHAP. XX. The Properties, Office, and Duty of a Sea-Gunner, 199
In the Appendix.
  • CHAP. I. A Description of the Proportional Scale, and its use in the Art of Gunnery, Page 1
    • Numeration on the Lines, 2
    • Prop. 1. A whole number consisting of two, three or four places being given, to find the place on the Scale representing the same, ibid.
    • Prop. 2. Having two Numbers given, to find as many more as you please, which shall be in continual proportion one to another, as the two Numbers were, 4
  • CHAP. II. Multiplication upon the Propor­tional Scale, 6
  • CHAP. III. Division by the Proportional Scale 8
  • CHAP. IV. The Golden Rule Direct by the Pro­portional Scale, 10
  • CHAP. V. The Golden Rule Reverse by the Proportional Scale, 13
  • CHAP. VI. Of Duplicate proportion by the Pro­portional Scale, 15
  • [Page]CHAP. VII. To extract the Square Root by the Proportional Scale by Inspection, 17
  • CHAP. VIII. To extract the Cube Root upon the Proportional Scale, by Inspection, 19
  • CHAP. IX. Cubical Proportion by the Proporti­onal Scale, 20
  • CHAP. X. Of the Mensuration of divers Re­gular Superficial Figures, 21
  • CHAP. XI. Of Spherical Bodies, such as Globes or Bullets, 23
  • A Description of the Sea-Gunners Rule, being the Epitome of the Art of Gunnery, from p. 25 to the end.

THE Sea-Gunner.

A COMPENDIUM OF Vulgar Arithmetick. CHAP. I.

ARITHMETICK is the Science of Numbring, and Resolving all Questions of Numbers, Rational or Irrational.

Notation of Numbers.

1. Notation of Numbers, is the Descrip­tion and Explication of any Number by Fi­gures or Notes, whereof there are ten, and no more.

One two three four five six seven eight nine ten.
1 2 3 4 5 6 7 8 9 10

Notation of Numbers, consisteth of Names, Va­lues, Degrees, or Places and Periods.

As 1. Numbers are named, Ʋnites, Thou­sands, Millions, &c.

2. Their Values is reckoned from the Right-hand.

3. Their Degrees or Places, are ten-fold, &c.

4. Their Periods, are Ʋnites, Tens, Hun­dreds, which are Illustrated in the following Table.

A Table of Notations.
NamesMillionsThousandsƲnites.
ValueCXICXICXI
Degrees or Places987654321
Periods321
Integers.999999999
888888888
777777777
666666666
555555555
444444444
333333333
222222222
111111111

RƲLE.

Begin at the Right-hand and go backward, and say, 9 in the first place is 9. 9 in the the second place is 90. 9 in the third place is 900. 9 in the fourth place, is 9000, Nine Thousand; 9 in the fifth place, is 90000, Ninety Thousand; and so on; observing the Names above, their Values, Places and Periods.

NƲMERATION.

NƲmeration is the first part of Arthme­tick, and serveth to express the value of any Number given; The Integers of Num­bers, are the nine Figures and the Cypher, and begin to number them at the Right-hand, to the Left, increasing each Figure ten-times as before.

ADDITION.

ADition is the gathering of two, or more Numbers into one Sum, and hath two general Cases.

CASE I. In Addition of Tens, Hun­dreds, Thousands, &c.

RƲLE. Draw a line under the Numbers given, begin at the Right-hand, and first place; add up the Unites, carry the Tens to the next place, and let the remaining Works below; so do all along as you go backward, and in the last place, set down all that you have added, with that which you carry.

Example.

 Years
From the Creation of the World to Noah▪s Flood,1656
From Noah's Flood, to the giving of the Law,0875
From the Law, to the Birth of our Saviour,15 [...]8
From the Birth of Christ, to the Year1690
 5729

In Addition of Integers and Parts.

RƲLE. Draw a Line under the Numbers given, and begin as before, at the least De­nomination; add up right, and set the parti­cular Sums of the several rows, under every one, (in their proper place) according to their respective value, whether it be in Num­ber, Weight, or Measure.

Example.

There are several Men owe a Merchant several Sums of Monev; it is required to know the Sum of those Debts.

 l.s.d.
One Man owes,2301702
Another owes,1101604
Another,0741009
Another,9790811
The Total Debt is13951302

SƲBTRACTION.

SƲbtraction is the taking a lesser Number from a greater, or an Equal from an Equal.

What remains, is the Residue, or Excess, and bath two Cases.

CASE I. In Subtraction of Tens, Hun­dreds, Thousands, from Tens, Husdreds, Thousands, &c.

RƲLE. Set the greater Number above the lesser, and draw a line under them. Then begin at the Right-Hand, and take the lesser from the greater, or Equals from E­quals, and set the Differenee or Residue, un­der every one, in their due place.

Example.

 l.
A Man oweth to a Merchant9758
And he hath paid of that Debt,3514
There Remains due,6244

CASE II. When some of the inferiour Numbers are greater than the superiour Numbers.

RƲLE. Set your Numbers in order as before; draw a line under them, and begin at the Right-hand; and according to the Numbers respective value, borrow one of the next to the Left-hand above, out of which Subtract, what remains add to the superiour, and set their Sum under the line; then what you borrow, pay to the next Number on the Left-hand below, and so proceed through­out the work, according to this or the for­mer Rule.

 l.s.d.
As from529134
Take347167
Rests,171169

Proof. Add the two inferiours; their Sum is equal to the superiour.

MƲLTIPLICATION.

MƲltiplication serveth to perform that at once which Addition doth at many times.

And to multiply readily, it is necessary that the ensuing Table should be perfectly learned.

Pythagora's Table.
123456789
24681012141618
369121518212427
4812162024283236
51015202530354045
61218243036424854
71421283542495663
81624324048566472
91827364554637281

The Ʋse of this Table is to muliply any Num­ber in the outer Column to the left hand; by any Figure at the top, and in the common Angle of meeting, is the Answer to the Question; as 7 times 9, you will find to be 63.

In Multiplication, Note, that the uppermost Number is always the Multplicand, and the lower the Multiplier; and the Figures which remain when the Work is done, is called the Product.

Multiplication may be divided into six Cases.

CASE I. If the Multiplicand have dovers Figures, and the Multiplier but one;
Rule.

Draw the Multiplier into the first Figure of the Multiplicand, and subscribe the Units of the Product, but carry the Tens to the next place; then draw the Multiplier into the second Figure of the Multplicand, and add the Tens you carried to the Units of that Product, subscribe the Units of their Sam, and carry the Tens to the third place; accordingly proceed to the end of the work. As, if 5436 is to be mutiplied by 6, according to the following Example.

5436,Multiplicand.
6,Multiplier.
32616,Product.

CASE II. If the Multiplicand and the Multiplier have each of them more than one Figure;
Rule.

For the first Figure, do as before; and ha­ving drawn the second Figure of the Multi­plier into the first Figure of the Multiplicand, [...] the Units of the Product under that se­cond Figure of the Multiplier, and carry the Tens, setting all the rest of the Multiplica­tion as by the former Rule; and this direct­eth, making so many particular Rows of Products as you Figures in your Multiplier; at last add them together for a total Pro­duct.

Example.

4532,Multiplicand.
32,Multiplier.
9064,Particular
13596,Products.
145024,Total Product.

CASE III. If Cyphers are in the Multiplicand and Multiplier, or either of them;
Rule.

Set down to the right hand of the first Product as many Cyphers as are in the Mul­tiplicand and Multiplier, so that the first U­nit of the Product of the first Multiplier may stand under the first Figure of the Mul­tiplicand, and work the rest according to the other Rule.

Example.

As, [...]

CASE IV. If a Cypher or Cyphers be in the middle of the Multiplicand;
Rule.

Work according to the former Rules till you come to the Cyphers, then under the first o, subscribe the Tens you carried; but under the rest of the Cyphers set Cyphers, except under the last, where subscribe the Units remaining of the Product of the next Figure of the Multiplier drawn into the Multiplicand; the rest is according to the other Rules.

Example,

As, [...]

CASE V. If a Cypher or Cyphers be in the middle of the Multiplier;
Rule.

Multiply as before is taught, until you come to the Cyphers in the Multiplier, which subscribe in order before the particular pro­duct of the next Multipler, drawn into the Multiplicand; then set the Units of its Pro­duct under that Multiplier, and observe the other Rules for the rest.

As, [...]

CASE VI. If Cyphers be both in the middle of the Multipli­cand, and also in the Multiplier;
Rule.

When you come to the Cyphers in the [Page 13]Multiplicand, then under the first Cyphers place, set the Tens you carried (if any be) and after that, as many Cyphers as are in the Multiplier (no Figure intervening) then mul­tiply into the next Figure of the Multipli­cand, subscribe the Units of the Product, and carry the Tens in the same Row, and so do in every Row of the particular Pro­ducts, according as this or some of the other Rules require.

Example.

[...]

You may abbreviate Multiplication by the help of Subtraction; especially when to be multiplied by 5, or 9; As,

CASE I. To multiply any Number by 5.
Rule.

Subtract half the Number, and to it add a Cypher.

Example.

As,45276
Product226380

being to be multiplied by 5, halve the Num­ber, and add a Cy­pher at the latter end, and the Work is done.

CASE II. To multiply any Number by 9.
Rule.

Add a Cypher to the Number given to be multiplied by 9, and subtract the first Num­ber out of it, and the Remainer is the Pro­duct or Answer of the Question.

Example.

Let the Number be 6789, to which add o Cypher, and the Number is thus, 67890; [Page 15]out of which subtract the first Number, and the Remainder is 61101, the Product or Answer of the Question.

DIVISION.

DIvision serveth to divide any Number into as many parts as you please, and consisteth of three Numbers, the Divisor, the Dividend, and the Quotient; for see how often the Divisor is contained in the Divi­dend, so many Figures it produceth in the Quotient; or see how often 1 is contained in the Divisor, so many times the Quotient is contained in the Dividend, which is all one.

If you were to divide 888 pound amongst 4 men, the Question is, what each man must have? Or­der your Work as in this Example.

[...]

The first demand is, how many times 4 can you have in 8? The answer is 2, which 2 place in the Quotient; then multiply the 2 in the Quotient by 4, (the Divisor) and that makes 8; place 8, under the 8 on the lest Figure of the Dividend, and draw a line un­der it, and subtract 8 from 8, and there remains o. Then take down the next 8, and demand how many times the Divisor is con­tained in the Dividend (8) which is 2 times; set that 2 in the Quotient, and multiply the Divisor 4 by that 2, which makes 8; set that 8 under the second Figure of the Dividend, and draw a line as before, and subtract it from the 8 in the Dividend, and there re­mains 0. Proceed in the same manner as you have done with the rest, and you will find 222 in the Quotient, and 0 remains of the Work; so that you see, according to the former Proposition, that 4 the Divisor is contained in 888 (the Dividend) 222 times; and the Quotient is contained in the Divi­dend, as often as 1 is contained in the Divi­sor, which is 4 times: So that it appears by the Work, that 888 Pounds being divided between 4 Men, there is 222 Pounds comes to each Mans share.

If 28770 Pounds is to be Divided amongst 84 Men; the Question is, what each man must have?

Note that Men is the Divisor, Mo­ney the Dividend, and Quotient is the Answer.

[...]

For the first work, say how many times 84 can you have in 28? which cannot be; there­fore you must find the Divisor in 287, over which last figure always place a Prick, as in the Example: Then say how many times 8 (the first figure in the Divisor) is there in 28, the two first Figures in the Dividend, which is 3 times; which 3 place in the Quotient, and multiply the Quotient by the Divisor, and it makes 252; which place under the prick'd Number, and Subtract it from 287, and there will remain 35: then draw down the next Figure 7, which makes 357, and say, how many times 8 can you have in 35? [Page 18]which is 4 times; place 4 in the Quotient; then multiply 4 the Quotient by 48 the Di­visor, makes 336, which place under 357, (as in the work;) then draw a line and subtract, and there rests 21; then take down 0 to the Remainder 21, makes 210; then say, how many times 8 can you have in 21? the Answer is 2; which 2 place in the Quotient, then multiply the 2 by the Divisor, makes 168, which place under 210, as in the Example; then draw a line and subtract it, and there rests 42.

So that it appears, that if 28770 Pounds is to be divided amongst 84 men, that there is 342 Pounds comes to each man's share, and 41/4 [...] of a Pound more.

Now to know what part of a Pound this or any other Fraction is, after the Remainder of any Division; Observe this Rule.

Multiply the Remainder 42 by 20, to bring it into Shillings; then divide it by 48, the Divisor and the Quotient will answer the Question, which in this Example, is 10 Shil­lings more to each man's share, as appears by the work.

[...]

The Rule of Three Direct.

IT is called the Rule of Three, because in all Questions in this Rule, you have al­ways three Terms given to find a fourth.

It is called the Rule of Proportion for this reason; see what proportion is be­tween the first Term and the second, the same proportion is between the third Term and the fourth.

It is called the Golden Rule for the Ex­cellency in its Operations.

  • It is known by At and How
  • It is known by If and What
  • It is known by As and So

To work this Rule, you must multiply the second Term by the Third, for the Di­vidend; and divide the Product by the first, the Quotient will give you the fourth Term demanded.

Here Note; That the first and third Number must always be of the same Deno­mination; As if one be Pounds, Pence, Yards, Tuns, Hours, Men, &c. so re­spectively must the other be; and the like [Page 20]is to be understood of the second and the fourth, as in the following Example:

If 12 Yards of Karsey cost 3 Pound, what shall 435 Yards cost?

[...]

Reduce the Shillings into Pounds, by di­viding the same by 20; and the Answer is 108 Pound, 15 Shillings, the price of 435 Yards.

If 7 Inches Diameter gives 22 Inches in Circumference what Circumference shall 36 Inches Diameter require?

[...]

The RƲLE of THREE REVERSE.

TO work this Rule, you must multiply the first Term by the second, and di­vide the Product by the third, and the Quo­tient will give you the fourth Term de­manded.

If 30 men require 25 Weeks to build a Fort, in how many Weeks will 20 Men build the like?

[...]

The Double RƲLE of THREE.

IF 600 Pounds weight for 501 Miles Car­riage, cost 1 l. 6 s. 6d. what shall 2500 Weight cost 100 Miles Carriage? State the Question thus:

WMilesl.s.d.W.Miles
600501662500100

To work this, you must first reduce the Money into the lowest Denomination ex­press'd, which is 318 Pence; then multi­ply the 2500 by 100, and also by the Num­ber of Pence: All that Product must be divided by the two first Numbers multi­plied together (which is the Divisor) to divide the other Product by. When the Operation is done, then you must reduce the Pence into Shillings, and Shillings in­to Pounds; and in the Conclusion you will find the Answer of the Question to be

l.s.d.
110010

Note the Work.

[...]

[...]

The Work being finished, the Answer of the Question is 11 l. 10 d. the 11 l. is ap­parent, but the 10 d. is included in the re­maining Fraction 0416. To find the Value of this Fraction in Pence, multiply the Fraction by 20, cutting off 4 Figures, (be­cause there is so many in the Fraction.) The Remainder multiply by 12, cutting off still 4 Figures, and there will remain to the Left-hand 10, which is 10 Pence, the va­lue of the Fraction.

The Rule of Three Reverse.

THE Reverse, or backward Rule of Three, is to be used when the third Number requires less, or less requires more.

The Rule.

Multiply the First Number by the Second, and Divide the Product by the Third, the Quotient will be the Fourth Number sought; which always shall be of the same denomi­nation with the Second Number.

For Instance.

If 24 Pionecrs require 16 Months to dig a Moat about a Town, how many Pioneers must there be employed to dig the same Moat in 4 Months?

In stating this Question, you must note, That 24, though it be the First named, is not to be the First Number in the work; because the Middle term must always be of the same Denomination with that which is sought; and the Three Numbers put in or­der stand thus.

Months.Pioneers.Months.
16244

Here 'tis plain, less requires more; that is, less time more hands.

Therefore it must be wrought by the Re­verse Rule; and accordingly you may mul­tiply [Page 27]24 by 16, and divide the Product by 4, the Quotient will be 96; as doth appear by the work.

[...]

Which shews that 96 Pioneers must be employed to finish the Moat in 4 Months.

CHAP. II. A COMPENDIUM OF Decimal Arithmetick.

Note 1st. Notation of FRACTIONS.

Numerators,5151501070
Denominators,10,100,1000,10000.

Note 2d.

Of how many places soever the Numera­tor of a Decimal Fraction doth consist, of [Page 29]so many Cyphers with a Unite before them, do the Donominators consist.

So the Denominator of 5 is 10, of 15 is 100, of 005 is 1000, &c.

Note 3d.

When the Numerator of a Decimal Fra­ction consists not of so many places as the Denominator hath Cyphers, prefix so many Cyphers on the left hand as is directed in Note 2d. So 5/100 is written thus, 05; 15/1000 is writ thus, 015; 50/10000 thus, 0050; 6/1000 thus, 006.

Note 4th.

Cyphers at the end of a Decimal Fraction do neither augment nor diminish the value thereof; so that 2. 20. 200. 2000, are De­clmals of one and the same value: For when the Numerator and Denominator do each end with a Cypher or Cyphers cut off equal Cyphers in both; so will the Fraction be reduc'd into lesser terms,

20
100
200
1000
2000
10000

Thus, 20/100 200/1000 or 2000/10000 Are reduc'd as in the Table.

Note 5th.

Cyphers added to the left hand of any Number in Decimals, decrease it ten fold thus 015/1000.

Note 6th.

To Reduce a Vulgar Fraction to a Decimal.

The Rule.

To the Numerator of the given Fraction, add what number of Cyphers you please, and Divide it by the Denominator, the Quotient is the Decimal Fraction.

Example 1.

I desire to know what the Decimal Fra­ction of Sixteen Shillings is, which in a Vulgar Fraction is 16/20; now you may add to the Numerator 16, what Cyphers you please: Suppose Four, and the work stands as follows, [...] and the Quotient is 8000 for Decimal Fractions of 16 Shillings.

Example 2d.

What is the Decimal of one Peny, which as it is the Fraction of 20 Shillings, (in Vul­gar Fractions,) it is thus exprest, 1/140. Therefore (as before) add Cyphers to the Numerator 1, and divide by 240, as in this following Example.

[...]

Note 7th. To reduce a Decimal Fraction hnno a Vulgar.

Rule. Let the Fraction be multiplied by 20; (if it be the Fraction of a Pound Ster­ling,) and the remaining Decimals by 12; [Page 32]and if any more remain, then multiply by 4, to bring them into Farthings; noting this, that in all your Multiplications, you must observe to cut off so many Figures of your Products as there are Figures in the Decimal Fraction.

Example.

I would know the quantity of this Fra­ction, 396875 of a Pound Sterling; pro­ceed according to the foregoing Rule, and the work will appear as in the following Table to be 11 Shillings, 11 Pence, 1 Far­thing.

[...]

I would know the quantity of this Fra­ction, 396875 of a Pound Sterling; pro­ceed according to the foregoing Rule, and the work will appear as in the followin work, to be 7 Shillings, 11 Pence, 1 Fa [...] thing.

[...]

Where you see that I multiply the Fra­ction by 20, to bring it into Shillings; and that Product by 12, to bring it into Pence; and that Product by 4, to bring it into Farthings.

Addition of Decimals.

Note 8th.

ADdition of Decimals is the same as with whole Numbers, only you must observe an Order in placing them; (that is,) to place every number under its proper Denomination, whole Numbers under one another, Tenths or Primes under Tenths or Primes, and Seconds under Seconds, &c. distinguishing the whole Numbers from the Fractions by a Point or Comma, and ad­ding them together as whole Numbers, still setting down the Excess above Ten, and so carrying the Tenths to the next place to­wards the Left hand.

Examples.

[...]

Subtraction of Decimals.

IN Subtraction of Decimals, observe the same order in placing them, as is di­rected in Addition; and then subtract the Lesser from the Greater as in the whole Numbers.

Note 9th.

When the Decimals in both Numbers given, consist not of the same number of Places, that Decimal that is defective in places towards the right hand, must be fil­led up with Cyphers, or at least supposed to be filled up.

Example.

Suppose ,47,309 is to be subtracted from 54, you are to put so many Cyphers as will make up the Fraction, and then Subtract, and the work will stand

Thus,Or thus,
54,00038,000
,47 3090, 130
0769137,860

Multiplication of Decimals.

Note 10th.

IN any of the Cases which can happen in Multiplication of Decimals, multi­ply the Numbers given, as if they were whole Numbers, then cut off or separate as many Figures from the Product, by a Point or Comma, as there are Fractions Multiplicand, Multiplicator, or both; which Figures so cut off or separated, are the Fraction of the Product. And the Figures toward the left hand of the point or Comma shall be the Integers or whole number of the Product; and if they do not make so many, they are to be supplyed with a Cy­pher or Cyphers, which may happen when the Product is a Fraction.

Example.

[...]

Note 11th.

In Multiplication of whole Numbers, the Product is always increased so many times more than the Multiplicand as the Multi­plicator contains Unites, as 5 times 4 make 20: But in Multiplication of Fractions, the Product is always less than either of the two Numbers alone, as in Example the IV, where you see one Number is 75, the Deci­mal of 15 Shillings, and the other 0125, the Decimal of three Pence; yet the Product of the Multiplication is but the Decimal of 2 Pence Farthing, as you may see if you look forward in the Decimal Table of Pence and Farthings, pag. 46.

The Reason is, because 1 being multi­plied by one, can produce but one; there­fore that which is less than 1, as (are all proper Fractions,) being Multiplied by that which is less than 1, must needs be dimi­nished by the Multiplication. And this Di­minution bears the same Proportion to the Multiplicator, as the Multiplicand beareth to a Unite.

For as 15 Shillings the Multiplicand is [...] of a Pound, so Two Pence Farthing the Product is [...] of the Multiplicator 3 Pence.

Division of Decimals.

Note 12th.

IN Division of Decimals, the Dividend must sometimes be prepared, by adding a competent number of Cyphers to make room for the Divisor to find out a Fraction, and for the Reduction of Vulgar Fractions into Decimals.

Note 13th.

In the whole Doctrine of Decimal Arith­metick, there is no part so difficult as this of Division, in regard to the variety of o­peration, in respect of the Quotient, what part of it to cut off in the various Divi­sions of whole Numbers with Fractions, and Fractions with Fractions, &c. all which varie­ties shall be solved with this ensuing Rule.

A General Rule to know the true value of the Quotient.

THere must be so many Figures cut off in the Quotient, as will make those in the Divi­sor (if any be) equal to the Number of Decimal parts in the Dividend.

Note 14th.

If the Quotient doth not consist of as many places as are required by the General Rule to be cut off, you may supply that defect by prefixing a Cypher or Cyphers before the Quotient toward the left hand.

Example 1. To Divide a whole Number by a Fraction.

Suppose the whole Number to be 82, which is required to be divided by this Fraction, 056, because there is a defect of Figures in the Dividend 82; therefore I add 5 Cyphers thereto, and place them in their due order, and when the work is finished, you will find 6 Figures come in the Quo­tient. (Now the Question is,) how many of these Figures are proper to be cut off for a Fraction; therefore note, that there being three Decimal Fractions in the Divisor, and 5 in the Dividend, therefore I cut off the last Figures in the Quotient, which being added to the 3 Figures in the Divisor, makes them equal to the Fraction in the Dividend, which is 5 Cyphers; so the ge­neral [Page 39]Rule is made good, as you may see in the work.

[...]

Example 2. To divide a Fraction by a whole Number.

Here (according to the 9th Note,) I pre­fix [Page 40]a Cypher before the Quotient, there being (after the Division is finished) only Four Figures in the Quotient; so then there are 5 Figures in the Dividend and 5 in the Quotient, according to the general Rule; as you may see in the work.

[...]

Example 3.

To Divide a whole Number, and a Fraction by a Fraction.

[...]

Here you see 4 Figures are cut off in the Quotient, which with the 2 in the Divisor, makes 6, which is equal to the Decimal parts in the Dividend; according to the General Rule in pag. 37, aforegoing.

Example 4.

To divide a Fraction by a whole Number and a Fraction.

[...]

Here are 7 Decimals in the Dividend, and when the Division is finished, there are 4 Figures in the Quotient, which with the 2 in the Divisor, makes but 6; Therefore according to the 9th note, I prefix a Cypher before the Quotient on the left hand, and then they are equal.

Example 5.

To divide a Fraction by a Fraction.

[...]

According to the General Rule I cut off 4 Figures to the Right hand in the Quotient, which makes those in the Divisor equal to those in the Divi­dend.

Example 6.

To divide a whole Number and a Fraction by a whole Number.

[...]

Here are only 2 Figures to be separated in the Quotient; there being no Decimals [Page 44]in the Divisor, and only 2 in the Divi­dend.

Example 7. To Divide a whole Number by a whole Number and a Fraction.

[...]

There being 7 Decimals in the Divi­dend, I therefore cut off 5 Figures in the [Page 45]Quotient, which with the 2 in the Divisor, make 7 according to the General Rule. p. 37.

Example 8. To divide a whole Number and a Fraction, by a whole Number and a Fraction.

[...]

According to Note 9th (in pag. 34,) add Cyphers to the Dividend, and when the work is finished, I find 5 Figures in the Quotient, 3 of which must be cut off, that they may make those of the Divisor 6, equal to the Decimals in the Dividend, according to the Rule.

A Decimal Table of Pence and Farthings.

Pence. Farth.Decimal.
10010416
20020833
30031250
I0041666
10052083
20062500
30072916
II0083333
10093750
20104166
30114583
III0125000
10135416
20145833
30156250
IV0166666
10177083
20187500
30197916
V0208333
10218750
20229166
30239583
VI0250000
10260416
20270833
30281250
VII0291666
10302083
20312500
30322916
VIII0333333
10343750
20354166
30364583
IX0375000
10385416
20395833
30406250
X0416666
10427083
20437500
30447916
XI0458333
10468750
20479166
30489583
XII0500000
[Page 47]10510416
20520833
30531250
XIII0541666
10552083
20562500
30572916
XIV0583333
10593750
20604166
30614583
XV0625000
10635416
20645833
30656250
XVI0666666
10677083
20687500
30697916
XVII0708333
10718750
20729166
30739583
XVIII0750000
10760416
20770833
30781250
XIX0791666
10802083
20812500
30822916
XX0833333
10843750
20854166
30864183
XXI0875000
10885416
20895833
30906250
XXII0916666
10927084
20937500
30947916
XXIII0958333
10968750
20979166
30989583
XXIV1000000

A Table of Decimals of one Pound Sterling in Shillings.

Sh.Decim.
1050000
2100000
3150000
4200000
5250000
6300000
7350000
8400000
9450000
10500000
11550000
12600000
13650000
14700000
15750000
16800000
17850000
18900000
19950000
20100000
21105000
22110000
23115000
24120000
25125000
26130000
271350000
281400000
291450000
301500000
311550000

A Table of the Decimals of a Foot to every Inch and Eighth part of an Inch.

Inches. 8 Part.Decimal.
1001041
2002083
3003125
4004166
5005208
6006250
7007291
I008333
1009375
2010416
3011458
4012500
5013541
6014583
7015625
II016666
1017708
2018750
3019791
4020833
5021875
6022926
7023958
III025000
1026041
2027208
3028125
4029166
5030200
6031299
7032291
IV033333

[Page 50] Inches. 8 Part.Decimal.
1034385
2035416
3037395
4037499
5038541
6039583
7040625
V041666
1042610
2043750
3044718
4045833
5046875
6047927
7048854
VI050000
1051104
2052083
3053125
4054166
5055207
6056250
7057291
VII058333
1059375
2051041
3061457
4062500
5063531
6064583
7065625
VIII066000
1067610
2068750
3069896
4070833
5071875
6072916
7073958
IX075000
1076041
2077083
3078125
4079166
5080208
6081250
7082291
X083333
[Page 51]1084375
2085416
3086457
4087500
5088541
6089687
7090625
XI091666
1092708
2093750
3094791
4095833
5096875
6097926
7098958
XII100000

The Calculating of this Table, is by Di­viding every Inch and 8 Parts by 96, be­cause there are so many parts in the Foot, every Inch being divided into 8 Parts, serving to Reduce Inches and 8 Parts to the Decimals of a Foot, or the contrary.

An Explanation of this Table.

The First Column shews the Inches and Eight parts of a Foot, and the Second Co­lumn shews the Decimal Number answering thereto.

Example.

Seek for 11 Inches, and 8/4 or a half in the First Collumn, and in the next you will find the Decimal thereof 095833.

CHAP. III. THE EXTRACTION OF THE Square Root.

THe Extraction of the Square Root is that by which having a number gi­ven, another number may be found, which being Multiplied by itself, produceth the number required.

Any Square number being given to be Extracted, thus it may be prepared. Ac­cording to this Rule, put a Point over the first place thereof to the Right hand (being the place of Unites;) then proceeding to­wards the left hand, pass over the second place, and put a Point over the third place; [Page 53]also passing over the Fourth place, put ano­ther Point over the Fifth, and so forward in such manner, that between every Two Points which are next one to another; so that one place may be intermitted accor­ding to this Example, 630436. Suppose the Square Root of this Number be required; the First Point is to be placed over 6, and the Second over 4, and so of the rest as you see in the Example; and note, that as ma­ny Points as are placed in that manner, of so many Figures will the Root be.

To fit it for operation, draw a crooked Line on the Right hand of the Number propounded for Extraction, then find the Root of the First Square, and place it in the Quotient, which in this Example is found to be 7; [...] Then Square the Quotient which is 49, and place it under the first Square of the Num­ber [Page 54]given, (viz.) 63, and Subtract the 49 from the First Square; and place the Re­mainder orderly underneath the Line, which is 14, to which Remainder being down, the next Squares of the Number propound­ed, and place them on the Right hand of the said Remainder; (and may now be cal­led the Resolvend.) Then double the Root, being the Number placed in the Quotient, which is 14, and place them on the Left hand of the Resolvend (like a Divisor,) parted off with a Crooked Line.

Then demand how often that Divisor is contained in the Resolvend, which may be now called the Dividend (proceeding in all respects as you do in Division,) and write the answer in the Quotient on the Right hand of the Divisor; then if you ask how often the Divisor 149 is found in the Di­vidend 1404, the Answer is 9 times: There­fore write 9 in the Quotient, and also after the Divisor 14.

Then Multiply all the Numbers which stand on the Left hand of the Resolvend, viz. (before the Crooked Line,) and write the Product orderly underneath the Resol­vend; then having drawn a Line under the said Product, subtract it from the Resol­vend, and subscribe the Remainder under the Line which is 63: unto which Num­ber [Page 55]bring down the remaining Figures of the Resolvend, and then there will be 6336 at the Left hand, of which number draw another Crooked Line; then double the Quotient, which is 158, and set it on the Left hand of the said Crooked Line; then demand how often you may have 158 in 633: the Answer is 4, which 4 must be placed in the Quotient; then multiply that by each Figure of the Divisor, and subscribe the Pro­duct orderly under the Dividend, and sub­tract it therefrom, and there remains 16; so the work is finished, and the Square Root of that Number 630436 is 794, and 16 which remains, intimates that the Root is something greater than 794, but less than 795; yet how much greater than 794 is not yet discovered by any Rules of Art. But farther Progress may be made for a nearer discovery of the truth; but in this case it being but a small difference, I shall wave it.

To Extract the Square Root by the Logarithms.
The Rule.

HAlf the Logarithm of any Number, is the Logarithm of the Square Root thereof.

Example.

Let the Square Number given be 5625,

The Logarithm of 625 is2,79588
The half thereof is1,39794

which is the Logarithm of 25, the Root of the said Number.

By Gunter's Scale.

To Extract the Square Root, is to find a mean proportional Number between I and the Number given; therefore divide the Space between them into Two Equal parts, and that shall be the Root sought.

Example.

Let it be required to find the Square Root of 144; Divide the distance betwixt I and 144 equally, and the Compasses will fall on 12, the Root sought.

The EXTRACTION of the CƲBE ROOT.

THe Extraction of the Cube Root is that by which having a Number gi­ven, another may be found, which being first Multiplied by itself, and then by the Pro­duct produceth the Number given.

[...] the Extraction of the Cube Root, the [...]ber propounded is always conceived to be a Cubical Number; that is, a certain Number of little Cubes, comprehended within one entire great Cube, so that the Root of any perfect Cubical Number is a Right Line of a Solid Body, containing 6 Equal Sides, which constitutes as many Square Superficies, or a Number Multiplied twice in itself, which in the Solid, hath length, breadth and depth, as may more [Page 58]plainly appear in this Annexed Cubical Figure.

[cube]

A Cube Number is either Single or Com­pound.

A Single Cube Number is that which is produced by the Multiplication of one single Figure, first by itself, and then by the Product, and is always less than 100; so 64 is a single Cube Number produced by the Multiplication of 4, First by itself, and then by the Product as in the Margin.

[...]

A Compound Cube Number, is when there are Two Figures in the Root.

All the Single Cube Numbers and Square Numbers, together with their respective Roots, are expressed in this Table follow­ing.

Cubes,1827641252163435127291000
Squar.149162536496481100
Roots,12345678910

To prepare a Cube Number for Extraction.
The Rule.

PUt a Point over the First place thereof, towards the Right hand, (viz.) the place of Unites, then passing over the Se­cond and Third places, put another over the Fourth, and passing over the Fifth and Sixth, put another over the Seventh, al­ways observing the same order in inter­mitting Two Places (between every Two Adjacent Points) place as many Points as the Number will permit, as may plainly appear in this Example. Let 1728 be [Page 60]the Number given, place the Points accor­ding to this Rule.

[...]

Which done, draw a Crooked Line on the Right hand of the Number to signify a Quotient; then find the Cube Root of the First Cube which is 1, as you may see in the Table, which 1 set in the Quotient. Then subscribe the Cube of the Root placed in the Quotient, under the First Cube of the Number given, which in this Example is 1.

Then draw a Line under the Cube sub­scribed aforesaid, and subtract this Cube from the First Cube, and place the Re­mainder orderly underneath the Line, which in this Example is nothing; to which Re­mainder, bring down the ne x Cube, which is 728, placing it on the Right hand of the Remainder, which number so placed, may be called the Resolvend; having drawn a Line underneath the Resolvend, Square the Root in the Quotient, that is, multiply it in itself, and subscribe 3 the Triple of the said Square or Product under the Resolvend, and place it under 7, the place of Hun­dreds.

Then Triple the Root or Number in the Quotient, which is 3, and subscribe this Triple Number in such a manner, that the First place thereof, (the place of Unites,) may stand under the Second place, (the place of Tens) in the Resolvend, which Triple is Three which I place under 2: Then the Triple Square of the Root, and the Triple of the Root being so placed, draw a Line under them, and add them together, the Sum is 33 for a Divisor.

Then let the whole Resolvend, except the First place thereof towards the Right hand, (viz.) the place of Unites, be esteemed as a Dividend; then demanding how often [Page 62]the First Figure (towards the Left hand) of the Divisor is contained in the correspon­dent part of the Dividend, write the An­swer in the Quotient; for if I ask how many times Three in 7, the Answer is twice, therefore I place 2 in the Quotient.

Then draw another Line under the work, and multiply the Triple Square before sub­scribed (under 7) by the last Figure placed in the Quotient, which is 2, and say, 2 times 3 is 6; which Product I subscribe un­der the said Triple Square (viz.) under the 3, which stands under the 7, as you may see in the work.

Then Multiply the Figure last placed in the Quotient, namely 2, by the Triple Number before subscribed under 2 in the Resolvend; for 2 being multiplied by it­self, produceth 4, which being multiplied by the Triple Number 3, the Product is 12, which I subscribe with the 1 under 6, and the 2 under 3; as in the work may appear.

Then Cube the last Figure in the Quo­tient which is 8, which place in such man­ner, that it may stand under the place of Unites in the Resolvend, as you may see in the work.

Lastly, Draw a Line under all, and add up the Three last Numbers together in the [Page 63]same order as they are placed, and the Sum is 728, which being Subtracted from the Resolvend, and there remaineth o; so the Cubic Root is found to be 12.

Note when the Sum of the Three last Num­bers before mentioned, is greater than the Re­solvend, the work is erronious, and then you may reform it, by placing a Figure less in the Quotient.

To Extract the Cube Root by the Logarithms.
The Rule.

DIvide the Logarithm of the given Num­ber by 3, so shall you have the Loga­rithm of the Root required.

Example.

Let the Cube Number given be 1728 as before,

The Logarithm of 1728 is3,23754
The Third part thereof is1,07918

which is the Logarithm of 12, the Cube Root required.

Likewise Multiply the Logarithm of any Number by Three, and it produceth the Logarithm of the Cube thereof.

To Extract the Cube Root by Gunter's Scale.

TO Extract the Cube Root, is to find the First of Two Mean Proportio­nals between 1, and the Number whose Cube Root you require; therefore you must Divide the space between those Two Numbers into Three equal parts.

Example.

Let it be required to find the Cube Root of 1728, as before: Divide the distance between 1 and 1728, into Three Equal parts, one Third part of that distance shall reach from 1 to 12, which is the Cube Root required.

A Table of Square Roots from One to an Hundred.

R.Sq.
11
24
39
416
525
636
749
864
981
10100
11121
12144
13169
14196
15225
16256
17289
18324
19361
20400
21441
22484
23529
24576
25625
26676
27729
28784
29841
30900
31961
321024
331089
341156
351225
361296
371369
381444
391521
401600
411681
421764
431849
441936
452025
462116
472209
482304
492401
502500
512601
522704
532809
542916
553025
563136
573249
583364
593481
603600
613721
623844
633969
644096
654225
664356
674489
684624
694761
704900
715041
725184
735329
745476
755625
765776
775929
786084
796241
806400
816561
826724
836889
847056
857225
867396
877569
887744
897921
908100
918281
928464
938649
948836
959025
969216
979409
989604
999801
10010000

A Table of Cubick Roots from One to an Hundred.

R.Cube.
11
28
327
464
5125
6216
7343
8512
9729
101000
111331
121728
132197
142744
153375
164096
174913
185832
196859
208000
219261
2210648
232167
2413824
255625
2617576
2719683
2821972
2924389
3027000
3129791
3232768
3335937
3439304
3542825
3648656
3750653
3854872
3955419
4064000
4168921
4274088
4379507
4485184
4591125
4697336
47103823
48110592
49117649
50125000
51135651
52140608
53148877
54157464
55167375
56175616
57185193
58195112
59205379
60216000
61226981
62238328
63293047
64262244
65274625
66287496
67300753
68314432
69329199
70333000
71357011
72373348
73389017
74405224
75411875
76438976
77456533
78474522
79493039
80512000
81531441
82550408
83571787
84592604
85614125
86636056
87648303
88681472
89705669
90729000
91753571
92778688
93804357
94830584
95857375
96884736
97915673
98941192
99970299
1001000000

To make the Table of Square Roots.

The Table of Square Roots is made by Multiplying each Figure into itself; the Product is the Square of the Number required. As for Example in the Root 29, which being Multiplied in itself, produ­ceth 841, the Square of that Number is 29.

To make the Tables of Cubick Roots.

The Table of Cubick Roots, are made by Multiplying the Root in itself; and that Product again by the Root, and the last Number is the Cube Number required. As for Example in the Root 12, which being Multiplied in itself, produceth 144, that being Multiplied by 12, produceth 1728, the Cube Number of 12.

A TABLE OF LOGARITHMS OF Absolute Numbers, from One to a Thousand.

Num.Logar.
10,00000
20,30103
30,47712
40,60206
50,69897
60,77815
70,84510
80,90309
90,95424
101,00000
111,04139
121,07918
131,11394
141,14613
151,17609
161,20412
171,23045
181,25527
191,27875
201,30103
211,32222
221,34242
231,36173
241,38021
251,39794
261,41497
271,43136
281,44716
291,46239
301,47712
311,49136
321,50515
331,51851
341,53148
351,54407
361,55630
371,56820
381,57978
391,59106
401,60206
411,61278
421,62325
431,63347
441,64345
451,65321
461,66276
471,67210
481,68124
491,69020
501,69897
511,70757
521,71600
531,72428
541,73239
551,74036
561,74819
571,75587
581,76343
591,77085
601,77815
611,78533
621,79239
631,79934
641,80618
651,81291
661,81954
671,82607
681,83251
691,83885
701,84510
711,85126
721,85733
731,86332
741,86923
751,87506
761,88081
771,88649
781,89209
791,89763
801,90309
811,90848
821,91381
831,91908
841,92428
851,92942
861,93450
871,93952
881,94448
891,94939
901,95424
911,95904
921,96379
931,96848
941,97313
951,97772
961,98227
971,98677
981,99123
991,99563
1002,00000
1012,00432
1022,00860
1032,01284
1042,01703
1052,02119
1062,02531
1072,02938
1082,03342
1092,03743
1102,04139
1112,04532
1122,04922
1132,05308
1142,05690
1152,06070
1162,06446
1172,06819
1182,07188
1192,07555
1202,07918
1212,08278
1222,08636
1232,08990
1242,09342
1252,09691
1262,10037
1272,10380
1282,10721
1292,11059
1302,11394
1312,11727
1322,12057
1332,12385
1342,12710
1352,13033
1362,13354
1372,13672
1382,13988
1392,14301
1402,14613
1412,14922
1422,15229
1432,15534
1442,15836
1452,16137
1462,16435
1472,16732
1482,17026
1492,17319
1502,17609
1512,17898
1522,18184
1532,18469
1542,18752
1552,19033
1562,19312
1572,19590
1582,19866
1592,20140
1602,20412
1612,20683
1622,20951
1632,21219
1642,21484
1652,21748
1662,22011
1672,22272
1682,22531
1692,22789
1702,23045
1712,23300
1722,23553
1732,23805
1742,24055
1752,24304
1762,24551
1772,24797
1782,25042
1792,25285
1802,25227
1812,25768
1822,26007
1832,26245
1842,26482
1852,26717
1862,26951
1872,27184
1882,27416
1892,27646
1902,27875
1912,28108
1922,28330
1932,28550
1942,28780
1952,29003
1962,29226
1972,29447
1982,29666
1992,29884
2002,30103
2012,30320
2022,30535
2032,30750
2042,30963
2052,31175
2062,31387
2072,31597
2082,31806
2092,32015
2102,32222
2112,32428
2122,32634
2132,32828
2142,33041
2152,33244
2162,33445
2172,33646
2182,33846
2192,34044
2202,34223
2212,34439
2222,34635
2232,34830
2242,35025
2252,35218
2262,35411
2272,35603
2282,35793
2292,35983
2302,36173
2312,36361
2322,36549
2332,36736
2342,36922
2352,37107
2362,37291
2372,37475
2382,37658
2392,37840
2402,38021
2412,38202
2422,38381
2432,38561
2442,38739
2452,38917
2462,39093
2472,39270
2482,39445
2492,39620
2502,39794
2512,39967
2522,40140
2532,40312
2542,40483
2552,40654
2562,40824
2572,40993
2582,41162
2592,41330
2602,41497
2612,41664
2622,41830
2632,41996
2642,42160
2652,42325
2662,42488
2672,42651
2682,42813
2692,42975
2702,43136
2712,43297
2722,434 [...]
2732,436 [...]
2742,43775
2752,43933
2762,44091
2772,44248
2782,44404
2792,44560
2802,44716
2812,44871
2822,45025
2832,45179
2842,45332
2852,45484
2862,45636
2872,45788
2882,45939
2892,46090
2902,46240
2912,46389
2922,46538
2932,46687
2942,46835
2952,46982
2962,47129
2972,47276
2982,47422
2992,47567
3002,47712
3012,47857
3022,48001
3032,48144
3042,48287
3052,48430
3062,48572
3072,48714
3082,48855
3092,48996
3102,49136
3112,49276
3122,49415
3132,49554
3142,49693
3152,49831
3162,49969
3172,50106
3182,50243
3192,50379
3202,50515
3212,50650
3222,50786
3232,50920
3242,51054
3252,51188
3262,51322
3272,51455
3282,51587
3292,51720
3302,51851
3312,51983
3322,52114
3332,52244
3342,52375
3352,52504
3362,52634
3372,52763
3382,52892
3392,53020
3402,53148
3412,53275
3422,53403
3432,53529
3442,53656
3452,53782
3462,53908
3472,54033
3482,54158
3492,54282
3502,54407
3512,54531
3522,54654
3532,54777
3542,54900
3552,55023
3562,55145
3572,55267
3582,55388
3592,55509
3602,55630
3612,55751
3622,55871
3632,55991
3642,56110
3652,56229
3662,56348
3672,56467
3682,56585
3692,56703
3702,56820
3712,56937
3722,57054
3732,57171
3742,57287
3752,57403
3762,57519
3772,57634
3782,57749
3792,57864
3802,57978
3812,58092
3822,58206
3832,58320
3842,58433
3852,58346
3862,58659
3872,58771
3882,58883
3892,58995
3902,59106
3912,59218
3922,59329
3932,59439
3942,59549
3952,59660
3962,59769
3972,59879
3982,59988
3992,60097
4002,60206
4012,60314
4022,60423
4032,60530
4042,60638
4052,60745
4062,60853
4072,60959
4082,61066
4092,61172
4102,61278
4112,61384
4122,61490
4132,61595
4142,61700
4152,61805
4162,61909
4172,62014
4182,62118
4192,62221
4202,62325
4212,62428
4222,62531
4232,62634
4242,62737
4252,62839
4262,62941
4272,63043
4282,63144
4292,63246
4302,63347
4312,63448
4322,63548
4332,63649
4342,63749
4352,63849
4362,63949
4372,64048
4382,64147
4392,64246
4402,64345
4412,64444
4422,64542
4432,64640
4442,64738
4452,64836
4462,64933
4472,65031
4482,65128
4492,65225
4502,65321
4512,65418
4522,65514
4532,65610
4542,65706
4552,65801
4562,65896
4572,65991
4582,66086
4592,66181
4602,66276
4612,66370
4622,66464
4632,66558
4642,66652
4652,66745
4662,66838
4672,66932
4682,67024
4692,67117
4702,67210
4712,67302
4722,67394
4732,67486
4742,67578
4752,67669
4762,67761
4772,67852
4782,67943
4792,68033
4802,68124
4812,08214
4822,68305
4832,68395
4842,68484
4852,68574
4862,68664
4872,68753
4882,68842
4892,68931
4902,69020
4912,69108
4922,69196
4932,69285
4942,69373
4952,69460
4962,69548
4972,69636
4982,69723
4992,69810
5002,69897
5012,69984
5022,70070
5032,70157
5042,70243
5052,70329
5062,70415
5072,70501
5082,70586
5092,70672
5102,70757
5112,70842
5122,70927
5132,71012
5142,71096
5152,71181
5162,71265
5172,71349
5182,71433
5192,71517
5202,71600
5212,71684
5222,71767
5232,71850
5242,71933
5252,72016
5262,72099
5272,72181
5282,72263
5292,72346
5302,72428
5312,72509
5322,72591
5332,72673
5342,72754
5352,72835
5362,72916
5372,72997
5382,73078
5392,73159
5402,73239
5412,73320
5422,73400
5432,73480
5442,73560
5452,7364 [...]
5462,73719
5472,73799
5482,73878
5492,73957
5502,74036
5512,74115
5522,74191
5532,74272
5542,74351
5552,74429
5562,74507
5572,74585
5582,74663
5592,74741
5602,74819
5612,74896
5622,749 [...]3
5632,75051
5642,75128
5652,75205
5662,75282
5672,75358
5682,75435
5692,75511
5702,75587
5712,75664
5722,75740
5732,75815
5742,75891
5752,75967
5762,76042
5772,76118
5782,76193
5792,76268
5802,76343
5812,76418
5822,76492
5832,76567
5842,76641
5852,76716
5862,76790
5872,76864
5882,76938
5892,77011
5902,77085
5912,77159
5922,77232
5932,77305
5942,77379
5952,77452
5962,77525
5972,77597
5982,77670
5992,77743
6002,77815
6012,77887
6022,77960
6032,78032
6042,78104
6052,78175
6062,78247
6072,78319
6082,78390
6092,78462
6102,78533
6112,78604
6122,78675
6132,78746
6142,78816
6152,78887
6162,78958
6172,79028
6182,79099
6192,79169
6202,79239
6212,79309
6222,79379
6232,79449
6242,79518
6252,79588
6262,79657
6272,79727
6282,79796
6292,79865
6302,79934
6312,80003
6322,80072
6332,80140
6342,80208
6352,80277
6362,80346
6372,80414
6382,80482
6392,80550
6402,80618
6412,80656
6422,80753
6432,80821
6442,80889
6452,80956
6462,81023
6472,81090
6482,81157
6492,81224
6502,81291
6512,81358
6522,81425
6532,81491
6542,81558
6552,81624
6562,81690
6572,81756
6582,81822
6592,81888
6602,81954
6612,82020
6622,82086
6632,82151
6642,82217
6652,82282
6662,82347
6672,82413
6682,82478
6692,82543
6702,82607
6712,82672
6722,82737
6732,82801
6742,82866
6752,82930
6762,82995
6772,83059
6782,83123
6792,83187
6802,83251
6812,83315
6822,83378
6832,83442
6842,83506
6852,83569
6862,83632
6872,83696
6882,83759
6892,83822
6902,83885
6912,83948
6922,84011
6932,84073
6942,84136
6952,84198
6962,84261
6972,84323
6982,84385
6992,84448
7002,84510
7012,84572
7022,84634
7032,84695
7042,84757
7052,84819
7062,84880
7072,84942
7082,85001
7092,85065
7102,85126
7112,85187
7122,85248
7132,85301
7142,85370
7152,85431
7162,85491
7172,8 [...]552
7182,85612
7192,85673
7202,85733
7212,85793
7222,85854
7232,85914
7242,85974
7252,86034
7262,86094
7272,86153
7282,86213
7292,86273
7302,86332
7312,86392
7322,86451
7332,86510
7342,86570
7352,86629
7362,86688
7372,86747
7382,86806
7392,86864
7402,86923
7412,86982
7422,87040
7432,87099
7442,87157
7452,87216
7462,87274
7472,87332
7482,87390
7492,87448
7502,87506
7512,87564
7522,87622
7532,87679
7542,87737
7552,87795
7562,87852
7572,87910
7582,87967
7592,88024
7602,88081
7612,88138
7622,88195
7632,88252
7642,88309
7652,88361
7662,88423
7672,88479
7682,88536
7692,88592
7702,88649
7712,88705
7722,88762
7732,88818
7742,88874
7752,88930
7762,88986
7772,89042
7782,89093
7792,89154
7802,89209
7812,89265
7822,89321
7832,89376
7842,89431
7852,89487
7862,89542
7872,89597
7882,89653
7892,89708
7902,89763
7912,89818
7922,89872
7932,89927
7942,89982
7952,90037
7962,90091
7972,90146
7982,90200
7992,90255
8002,90309
8012,90363
8022,90417
8032,90472
8042,90526
8052,90580
8062,90633
8072,90687
8082,90741
8092,90795
8102,90848
8112,90902
8122,90956
8132,91005
8142,91062
8152,91116
8162,91169
8172,91222
8182,91277
8192,91328
8202,91381
8212,91434
8222,91487
8232,91540
8242,91593
8252,91645
8262,91698
8272,91751
8282,91803
8292,91855
8302,91908
8312,91960
8322,92012
8332,92064
8342,92117
8352,92169
8362,92221
8372,92272
8382,92324
8392,92376
8402,92428
8412,92480
8422,92531
8432,92582
8442,92634
8452,92686
8462,92737
8472,92788
8482,92840
8492,92891
8502,92942
8512,92993
8522,93044
8532,93095
8542,93146
8552,93197
8562,93247
8572,93298
8582,93349
8592,93399
8602,93450
8612,93500
8622,93551
8632,93601
8642,93651
8652,93701
8662,93752
8672,93802
8682,93852
8692,93902
8702,93952
8712,94001
8722,94052
8732,94102
8742,94151
8752,94201
8762,94250
8772,94300
8782,94349
8792,94399
8802,94448
8812,94498
8822,94547
8832,94596
8842,94645
8852,94694
8862,94743
8872,94792
8882,94841
8892,94890
8902,94939
8912,94988
8922,9503 [...]
8932,95085
8942,95134
8952,95182
8962,95231
8972,95279
8982,95328
8992,95376
9002,95424
9012,95472
9022,95521
9032,95569
9042,95617
9052,95664
9062,95713
9072,95761
9082,95809
9092,95856
9102,95904
9112,95952
9122,95999
9132,96047
9142,96095
9152,96142
9162,96189
9172,96237
9182,96284
9192,96331
9202,96379
9212,96426
9222,96473
9232,96520
9242,96567
9252,96614
9262,96661
9272,96708
9282,96755
9292,96802
9302,96848
9312,96895
9322,96941
9332,96988
9342,97035
9352,97081
9362,97128
9372,97174
9382,97220
9392,97267
9402,97313
9412,97359
9422,97405
9432,97451
9442,97497
9452,97543
9462,97589
9472,97635
9482,97681
9492,97727
9502,97772
9512,97818
9522,97864
9532,97909
9542,97955
9552,98000
9562,98046
9572,98091
9582,98137
9592,98182
9602,98227
9612,98272
9622,98317
9632,98363
9642,98408
9652,98453
9662,98498
9672,98543
9682,98587
9692,98632
9702,9867 [...]
9712,98722
9722,98767
9732,98811
9742,98856
9752,98900
9762,98945
9772,98989
9782,99034
9792,99078
9802,99113
9812,99167
9822,99211
9832,99255
9842,99299
9852,99344
9862,99388
9872,99432
9882,99476
9892,99520
9902,99563
9912,99607
9922,99651
9932,99695
9942,99739
9952,99782
9962,99826
9972,99869
9982,99913
9992,99956
10003,00000

A Description and use of the Table of Logarithms.

THe Table contains all absolute Num­bers from One, to One Thousand, (sufficient for any operation in the Art of Gunnery.) In each Page of the Table is contained Six Columns; in the First, the Third and Fifth (towards the Left hand,) are contained all absolute Numbers begin­ning at 1, and so on by 2, 3, 4, 5, 6, &c. to 1000; (having the Letter N. at the Head of each Column.)

Then in the Second, Fourth and Sixth Co­lumn of every Page are contained the Lo­garithmical Numbers, answering each ab­solute Number, against which it standeth, and these Columns have at the head of them the word Logar. The Numbers being thus disposed in the several Pages of the Table, it is easie to find the Logarithmical Number that answers there to any absolute Number that shall be required.

Or on the contrary, if any Logarithmi­cal Number be given, it will be easie to find the Absolute Number to which it belong­eth.

For if you find your Absolute Number in any Column of the Table under the Letter N. that Number that standeth in the next Column to it on the Right hand under the Title Logar. is the Logarithmical Num­ber thereunto belonging.

And on the contrary, in what part of the Table soever you find any Logarithmical Number, that Number which standeth in the next Column on the left hand thereof, is the Absolute Number so found.

And note further, that all the Logarith­mical Numbers between 1 and 10, have a Cypher before them; all Numbers between 10 and 100 have the Figure 1 before them; all Numbers between 100 and 1000, have [Page 84]the Figure 2 before them; which 1 and 2 Figures are called the Characteristiques of those Numbers.

And to the end what I have here deli­vered may be made plain, I shall give exam­ples thereof in the Two following Propo­sitions.

Prop. 1.

Let it be required to find the Logarith­mical Number belonging to 16; turn to the Table in the First Column of the First Page, where you will find 16, under the Letter N. and right against it towards the Right hand, you shall find this Number, 1,20412, which is the Logarithm thereof.

Likewise in the same Page and Column a­gainst 25, you will find 1,39794, which is the Logarithm thereof.

Also you shall (by the same Rule) find that

  • The Logarithm of 4 will be 0,60206
  • The Logarithm of 51 will be 1,70757
  • The Logarithm of 321 will be 2,50650

and by the Converse of what is here deliver­ed, you may find the Absolute Number an­swering to any given Logarithms as in the following Proposition.

Prop. 2. A Logarithmical Number being given, to find the Absolute Number thereunto belonging.

Let it be required to find the Abso­lute Number belonging to this Logarithm, 1,20412; look in the Table in the First Page thereof, and casting your Eye down among the Numbers, under the word Logar. you will find this Number 16, to stand just against it, on the Left Hand which is the Absolute Number of that Loga­rithm.

The same is to be understood of all o­ther Numbers comprised in the foregoing Table.

Observing this Caution; when you have a Logarithmical Number given, (which when you look for) you cannot find in the Table, you must then take the nearest Num­ber thereto, and the Absolute Number which stands against it, is the nearest (less) whole Number, which you must take.

As for Example.

If you have this Logarithmical Num­ber, 0,63258, which if you look for in the [Page 86]Table, you cannot find it; therefore you must take the nearest less Number which you will find to be 0,60206; and right against it (on the Left hand), you will find to be 4, the nearest Absolute Number to that Lo­garithm.

Let this suffice for the Description; next follows the Use.

The Ʋse of the Table of Logarithms in Arith­metick, which shall be exemplified in Questions of Multiplication, Division, and the Extract­ing the Square and Cube Roots, being such parts of Arithmetick which tend wholly to the matter intended in this Treatise; and there­fore I shall begin with Multiplication.

Multiplication by the Loga­rithms.

YOu must add the Logarithms of the Two Numbers, (to be Multiplied to­gether,) and the Sum of them will be the Logarithm of the Number produced by that Multiplication.

Example.

Let it be required to Multiply 48 by 5; First set down the Two Numbers to be Multiplied One under another, and to them set their respective Lo­garithms, as in the Margin; which being added toge­ther, the Sum of them (which is the Loga­rithm of the Product) being sought in the Table, the Absolute Number answering thereto is 240, the Product of those Two Numbers Multiplied together.

481,68124
50,69897
2402,38021

Division by the Logarithms.

AS Multiplication (by the Logarithms) was performed by Addition, so Di­vision is performed by Subtraction: Where­fore to perform Division, you must Subtract the Logarithm of the Number, by which you are to Divide from the Logarithm of the Number, which is to be Divided, and the Number which remains shall be the Logarithm of the Quotient.

Example.

Let it be required to Divide 228 by 12;

228.2,35793
12.1,07918
19.1,27875

First set down the Logarithm of 228, and under it set the Logarithm of 12, and Subtract the Lesser from the Greater, the Remainder is the Lo­garithm of the Quotient; which being sought in the Table, you will find 19 to be the Answer of the Question, being the Quotient sought: And so many times is 12 contained in 228.

Of a CIRCLE.

1. The Diameter being given, to find the Circum­ference by the Logarithms.

THe Proportion is as 7 to 22, so is the Diameter to the Circumference.

Wherefore to find the Circumference of any Circle, whose Diameter is given,

Add the Logarithm of the Diameter given to the Logarithm of 22, and from the Sum of them Subtract the Logarithm of 7, the [Page 89]Remainder shall be the Logatithm of the Cir­cumference sought.

Example.

If the Diameter of a Circle be 113, what is the Circumference?

First set down the Logarithm of 22, which is—

1,34242
2,05308
3,39550
0,84510
2,55040

Add the Logarithm of 113 which is from which Subtract the Loga­rithm of 7, which is— which being sought in the Tables is the nearest Logarithm of 355; and so much is the Circum­ference of a Circle, whose Dia­meter is 113.

2. The Circumference of a Circle being given, to find the Diameter.

The Proportion is as 22 is to 7; so is the Circumference to the Diameter.

Wherefore to the Logarithm of 7, add the Logarithm of the Circumference given, and from the Sum, Subtract the Logarithm of 22, the Remainder shall be the Loga­rithm of the Diameter.

Example.

If the Circumference of a Circle be 355, what is the Diameter thereof?

First set down the Logarithm of 7. which is—

0,84510
2,55023
3,39533
1,34242
2,05291

and to it add the Logarithm of 355 from which Subtract the Loga­rithm of 22— and the Remainder is the nearest Logarithm of 113, — which is the Diameter required.

3. The Diameter of a Circle being given, to find the Area or Superficial Content thereof.

The Proportion is as 28 is to 22, so is the Square of the Diameter to the Area.

Wherefore to the Logarithm of 22, add the Logarithm of the Diameter doubled, and from the Sum subtract the Logarithm of 28, the Remainder shall be the Logarithm of the Area required.

Example.

If the Diameter of a Circle be 12, what is the Area or Superficial Content thereof?

First set down the Logarithm of 22, which is —

1,34242
1,07918
1,07918
3,50078
1,44716
2,05362

and to that the Logarithm of 12, the given Diameter, set down— Twice — Add all Three together,— from which Subtract the Loga­rithm of 28,— The Remainder is the nearest Lo­garithm to the Number 113, and some small matter more is the Area of that Circle.

4. The Circumference of a Circle being given, to find the Area.

The proportion is as 88 is to 7; so is the Square of the Circumference to the Area.

Wherefore to the Logarithm of 7, add the Logarithm of the Circumference Twice, and from the Sum Subtract the Logarithm of, 88; the Remainder shall be the Logarithm of the Area required.

Example.

If the Circumference of a Circle be 38, what is the Area thereof?

First set down the Logarithm of 7, which is0,84510
To which add the Logarithm of1,57978
the Circumference Twice.1,57978
The Sum4,00466
Subtract the Logarithm of 88,1,94448
the Remainder is the nearest Lo­garithm of 115 the Area sought.2,06018

CHAP. IV. CONTAINING Geometrical Rudiments Useful in the Art of GUNNERY.

How to raise a Perpendicular from the middle of a Line given.

LEt the Line given be A. B. and let C be a Point therein given, from which it is required to raise a Perpendicu­lar. First therefore open the Compasses to any convenient distance; and setting one [Page 94]Foot in the Point C, with the other set off on either side thereof the equal distances C A, and C B; then opening the Com­passes to any convenient wider distance, setting one Foot in the Point A, with the other strike the Occult Arch at F,

[geometrical diagram]

the [...] with the same distance, set one Foot in the Point B, and with the other draw the Arch F, crossing E in the Point D; from whence draw the Line DC, which Line is a Perpendicular unto the given Line A, B, as was required.

To let a Perpendicular fall from a Point assigned, to the middle of a Line given.

Let the Line given whereupon you would have a Perpendicular let fall, be the Line B C D, and the Point A to be the Point as­signed [Page 95]from whence you would have the Perpendicular let fall from the given line B C D; First set one Foot of your Com­passes in the Point A, and opening them to any convenient distance, so that it be more than the line A C; Describe one Arch of a Circle with the other Foot, so that it may cut the line B C D, twice, that is, at E and at F;

[geometrical diagram]

then find the middle between these, which will be the Point C; from which Point draw the line at C, which is the Perpendicular which was to be let fall.

To raise a Perpendicular upon the end of a Line given.

Suppose the line whereupon you would have the Perpendicular raised, be the line [Page 96]A B; first open your Compasses to a conve­nient distance, and set one Foot in the Point B, and let the other Foot fall any where a­bove the line, as at the Point D; and in that Point, let one Foot of your Compas­ses remain, turning the other about un­til it touch the line A B, in the Point E,

[geometrical diagram]

then turn the same Foot of the Compasses towards C, and draw an Occult Arch, and lay the Edge of a Ruler to those Two Points E and D, and where the same edge of the Ruler doth cut the Arch C, from that Point draw the line C B, which shall be a Perpen­dicular at the end of the line A B.

To let fall a Perpendicular from a Point assigned, unto the end of a Line given.

Let the line A B be given, unto which it is required to let a Perpendicular fall from [Page 97]the assigned point D unto the end A. First, from the assigned point D, draw a line unto any point of the given line A B, which may be the line D C E; then find the middle of the line D E, which is at C, place one foot of your Compasses in that point, and extend the other foot unto D or E, with which distance draw the Semicircle D A E, which shall cut the given line A B, in the point A,

[geometrical diagram]

from which point draw the Line D A, which is the Perpendicular let fall from the assign­ed point D, on the end of the given line A B, as was required.

To draw a Line Parallel to a Line given.

Let A B, be a Line given, whereunto it is required to draw a Parallel. First, set [Page 98]one Foot of the Compasses in the point, C, and opening the other Foot at pleasure, draw the Arch E, then with the same distance set one Foot in the point D, and draw the other Arch F.

[geometrical diagram]

Lastly, lay a Rule to the convexities of both those Arches, and draw the line G H, which shall be a Parallel to A B, as was required.

A Geometrical Problem useful in the Art of Gunnery.

A Geometrical way to find the Diameter of any Bullet that weighteth twice as much as a known Bullet.

TAke the Diameter of the lesser Bullet, whose weight you know, and square that Diameter. (viz.) Make a Geometrical [Page 99]Square, each side to be equal to the Diame­ter of the Bullet given, then draw a Diago­tal line from either of the Two opposite Angles, and that Diagonal shall be the Dia­meter of a Bullet twice the weight of the other; then divide the said Diagonal into Two equal parts, setting one Foot of the Compasses in the midst of that Diagonal, and with the other Foot describe a Circle, and that Circumference will represent a Bullet twice as much weight as the other.

The sight of the Annexed Figure, is a sufficient Explanation.

[geometrical diagram]

A B is the Diameter of the lesser Bullet A C, the Diameter of the greater.

Performed by Arithmetick.

Suppose the Diameter of the lesser Bullet be Five Inches, the Square thereof is Twen­ty Five, the Double of it is Fifty, the Root thereof is 7 1/7 and so much is the Diameter of the greater Bullet.

The weight of any Shot given, to find the Diameter Geometrically.

Suppose a Shot be One, Two or Three Pound weight of Metal, or Stone assign­ed, if one Pound divide the Diameter in­to Four parts, and Five such parts will make the Diameter of a Shot of the said Metal or Stone, that shall weigh just Two Pound.

Divide the Diameter of a Shot weighing just Two Pound in Seven equal parts, and Eight such parts will make a Diameter of a Shot of Three Pound. And divide the Diameter of a Shot of Three Pound into Ten equal parts, and Eleven such maketh a Shot of Four Pound.

Divide the Diameter of a Shot of Four Pound into Thirteen parts, Fourteen such parts will make a Diameter for a Shot of Five Pound.

And so dividing each next Diamter in­to Three equal parts more, the next Lesser was divided into; and it will with one part added from a Diameter of a Shot, that will weigh just one Pound more. So you may proceed infinitely increasing or decreasing, by taking one part less than it is appointed to be divided into.

CHAP. V. Geometrical Theorems AND PROBLEMS.

Theorem 1.

ALL Circles are equal to that Right Angled Triangle, whose contain­ing sides, the one is equal to the Semidiameter, and the other to the Circum­ference thereof.

Theorem 2.

The proportion of the Diameter of a Cir­cle to the circumference, is as 1,000000 to 3,141593 fere, or as (Archim.) 7 to 22.

Theorem 3.

The proportion of the Diameter to the side of the Square equal to the Circle, is as 1,000000 to 886227 fere.

Theorem 4.

The proportion of the Diameter to the side of the inscribed Square, is as 1,000000 to 707107 fere.

Theorem 5.

The proportion of the Circumference to the Diameter, is as 1 to .318310 fere; or as 22 to 7.

Theorem 6.

The proportion of the Circumference to the side of the Square equal to the Circle, is as 1 to .282095.

Theorem 7.

The proportion of the Circumference to the side of the inscribed Square, is as 1 to .225078.

Arithmetical Problems apper­taining to the Art of Gun­nery, and wrought by Deci­mal Arithmetick, by the Logarithms, and Gunter's Scale.

PROB. 1. The Diameter of a Circle being given, to find the Circumference.

The Analogy.

AS 1 is to the Diameter, so is 3.142 to the Circumference; or as 7 to 22, so is the Diameter to the Circumference.

If the Diameter of a Circle be 15 Inches, what is the Circumference by Gunter's Scale?

By the Logarithms.
As the Log. of 15 (the Diameter)1,17609
is to the Logarithm of 3,1420,49720
so is the Logarithm of0,00000
to the Logar. of the Answer.47,13|67329

Extend the Compasses (upon the Line of Numbers) from 1 to the Diameter, the same extent will reach from 3.142 to 47.13 the Circumference.

PROB. 2. The Circumference of a Circle being given, to find the Diameter.

The Analogy.

AS 3,142 is to 1, so is the Circumfe­rence 47:13 to the Diameter 15 Inches.

If the Circumference of a Circle be 47 Inches, and 13 parts of a 100 (supposing every Inch to be divided into 100 parts,) what is the Diameter? or as 22 to 7, so is the Circum­ference to the Diameter.

By the Logarithms.
As the Logarithm of3,1420,49720
is to the Logarithm of10,00000
so is the Logarithm of47.131,67329
to the Logar of the Answer. 1,67329
 151,17609
By Gunter's Scale.

Extend the Compasses upon the line of Numbers from 47.13 the Circumference, the same extent, the same way shall reach from 3,142. to the Diameter 15.

PROB. 3. The Diameter of a Circle being given, to find the side of a Square equal to it.

If the Diameter of a Circle be 15 Inches, what shall be the side of a Square equal to it?

The Analogy.

AS 1 is to 15, so this Number 8862 to 13.29 the side of a Square equal in con­tent to that Circle.

By the Logarithms.
As the Logarithm10,00000
is to the Logarithm151,17609
so is the Logarithm of88620,94753
to the Answer13,292,12362
By Gunter's Scale.

Extend the Compasses from 1 to 8862, the same extent shall reach from 15 to 13.29.

PROB. 4. The Circumference of a Circle being given, to find the side of a Square, equal in content to that Circle.

If the Circumference of a Circle be 47, 13, the side of a Square equal to it is required.

The Analogy.

AS 1 is to 47.13 so is this Number 2821, to 13.29 the side of the Square re­quired.

By the Logarithms.
As the Logarithm of10,0000
is to the Logarithm of47130,67329
[...]o is the Logarithm of28210,45040
to the Answer13,291,12369
By Gunter's Scale.

Extend the Compasses upon the Line of Numbers from 1 to 2812, the same extent shall reach the same way from 47.13 to 13.29 the side of the Square required.

PROB. 5. The Diameter of any Spherical body being known, to find the Circumference.

Let the Diameter of a Bullet be 9 Inches, and the Circumference demanded.

The Analogy.

AS 1 is to 3,142, so is 9 to 28,28 fere, the Circumference sought.

By the Logarithms.
The Log. of 3,124—0,49720Being Added, gives the Log, of 28,28.
and the Log. of 9 Inch. 0,95424Being Added, gives the Log, of 28, 28.
1,4 5144Log. Required.
By Gunter's Scale.

Extend the Compasses from 1 to 9, the same extent shall reach from 3,142 to 28,28 Inches the Circumference required.

PROB. 6. The Circumference of any Spherical body being known, to find the Diameter.

Let the Circumference of a Bullet be 28,28 Inches, and 28 Hundred parts, the Diame­ter is required.

The Analogy.

AS 3.142 is to 1, so is 28,28 to 9 Inches, the Diameter required.

By the Logarithms.
Log. 28.28145144
Log. 3,142049720 Subtracted.
 ,95424 Log. 9. Required.
By Gunter's Scale.

Extend the Compasses upon the Line of Numbers from 3,142 to 1, the same extent the same way shall reach from 28.28 to 9 the Diameter required.

PROB. 7. The Diameter and Circumference of any Spheri­cal Body being known, to find the Superficial Content?

Let the Diameter of a Shot be 9 Inches, and the Circumference 28 Inches and 2800 parts of an Inch, how many Square Inches is there contained on the Superficies of that Shot.

The Analogy.

AS 1 is to 9 Inches the Diameter, so is 28,28 the Circumference to the Super­ficies 254,5.

So that there is contained in the Super­ficies of the same Bullet 254 Inches and an half.

By the Logarithms.
Log. 9.95424
Log. 28,28145144
S.254,512-40568 Log. Required.
By Gunter's Scale.

Extend the Compasses from 1, to 28 28 on the Line of Numbers, the same extent the same way shall reach from 9 to 254,5, the Superficial Content required.

Or else by knowing the Diameter, work thus; Extend the Compasses from 1 to 81, the Square of the Diameter, and the same extent will reach from this Number 3, 142, to 254, 5 the Superficial content as before.

PROB. 8. The Axis or Diameter of a Globical body being known, to find the Solid Content.

If the Diameter of a Shot be 9 Inches, what is the Solid Content in Square Cubical Inches?

☞ The Rule for this and the like Questions is this; as the Diameter is to the Cube itself, so is 11 to the Solid Content.

The Analogy.

AS the Diameter 9 is to the Cube thereof 729, so is 11 to the Solid Content in Cubical Inches.

By the Logarithms.
As Logar.90,95424
is to Logar.7292,86272
so is Logar.111,04139
to the Cubical Content.3,90411
 8912,94987 Log. found.
By Gunter's Scale.

Extend the Compasses from 9 to 11, the same extent shall reach from 729 to 891, the Cubical Inches contained in that Bullet, or the extent from 1 to the Diameter, being thrice repeated from. 5238, will reach the Solid Content required.

PROB. 9. The Diameter of a Bullet being given with the weight, to find the weight of another Bullet of the same Metal, but of another Diameter, either greater or lesser.

Let there be propounded an Iron Bullet of 4 Inches Diameter, weighing 9 Pound, and let the Question be put to know what ano­ther [Page 114]Bullet (of the same Metal) will weigh that is of 8 Inches Diameter.

The Analogy.

AS the Cube of 4 the First Diameter which is 64, is to 9 l. so is the Cube of 8 the last Diameter, which is 512, to 72 l. the weight required.

By the Logarithms.
The Rule.

Triple the difference of the Logarithms which belong to the Two Terms, which have the same denomination; then if the First Term be less than the Second, add that Sum to the Logarithm of the other Term: so you shall have the Logarithm of the 4th Term desired.

Diameter 4 Inches, Logar.0,60206
Diameter for 8 Inches, Logar.0,90309
Difference,30103
Difference tripled0,90309
Weight given 9 l. Logar.0,95424
Weight required 72 l. Logar.1,85735
By Gunter's Scale.

Extend the Compasses from 4 to 8, the same extent from 9 thrice repeated, will reach to 72, the Answer required.

So if a Bullet of 4 Inches Diameter weigh 4 l, a Bullet of 6 Inches Diameter, shall weigh 30 l, and a Bullet of 7 Inches Diame­ter shall weigh 47 [...]. l, and a Bullet of 3 In­ches Diameter, shall weigh 4 l.

But here it is necessary to shew what Proportions there are between several Metals used for this purpose; as of Brass, Iron, Lead and Stone, according to the best Approved Authors.

1. The proportion between Lead and Iron, is as 2 to 3; so that a Leaden Bullet of 3 Pound weight, is equal in Diameter with an Iron Bullet of 2 Pound weight.

2. The proportion between Iron and Stone, is as 3 to 8; therefore a Stone of 6 Pound weight is equal in bigness to a piece of Iron of 16 Pound weight.

3. The proportion between Lead and Stone, is as 4 to 1; so that a Bullet of Lead of Eight Pound, and a Stone Bullet of Two Pounds, are equal in Diameter.

4. The proportion between Iron and Brass, is as 16 to 18; and the proportion between Lead and Brass, is as 24 to 19.

And here note, that some Stone is heavier than other, and so likewise of Metals, the finer they are, the heavier they be, being of the same magnitude.

PROB. 10. Having the weight of a Bullet of one kind of Metal, to find the weight of a Bullet of another kind of Metal, being equal in magnitude.

Example. If a Leaden Bullet weigh 106 Pounds, what will a Bullet of Marble weigh?

By the Third Rule aforegoing, it is found that a Bullet of Lead to the Bullet of Stone, bears such proportion as 4 to 1.

The Analogy.

AS 4 ∶ 1 ∷ 106 ∷ 26,5.

Performed by the Logarithms.
The Logarithm of 106 is,02530
The Logarithm of 4 is,60206
The Logarithm of 26,5 found42324
By Gunter's Scale.

Extend the Compasses upon the Line of Numbers from 4 to 1, the same extent from 106 shall reach the same way to 26,5 the weight of a Stone Bullet that is equal in bigness to that Leaden one of 106 Pound.

On the contrary, having the weight of a Stone Bullet, to find the weight of a Lea­den Bullet of the same magnitude; extend the Compasses from 1 to 4, the same extent shall reach from 26,5 to 106.

PROB. 11. A Bullet of Iron that weigheth 72 Pound, what will a Bullet of Lead weigh that is equal to it in bigness?

The Analogy.

AS ∶ 2 ∷ 3 ∷ 72 ∷ 108.

By the Logarithms.
Logarithm 2,30103
Logarithm 3,,47713
Logarithm 72,,85733
 1,33476
Logarithm 108,03343
By Gunter's Scale.

Extend the Compasses from 2 to 3, on the Line of Numbers, the same extent shall reach from 72 to 108 the weight sought.

But if the weight of the Leaden Bullet be given, (viz.) 108, then to get the weight of the Iron Bullet.

Extend the Compasses from 3 to 2, the same extent shall reach from 108 to 72, the weight of the Irom Bullet.

PROB. 12.

The Diameter and Weight of any one Cylinder or Piece of great Ordnance taken at the Base Ring being known, to find the weight of any other piece of the same Metal and Shape, either greater or lesser, its Diameter being only known.

As for Example.

If a Brass Saker whose Diameter is 11,5 In­ches, what will another Piece weigh, whose Diameter is 8,75 Inches?

By Arithmetick.
The Analogy.

AS 11,5 is to 1900 ∷ so is 8,75 to almost 8,37.

By the Logarithms.
As the Log. greatest Diam.11,5306069
The Log. of the least,8,75294200
Difference Increasing 11869
Multiplyed by 3
Produceth this difference 35607
Which being Subtracted from the Logarithm of the weight given, 1900 327853
There remains the Log. 837 2,92245
By Gunter's Scale.

Extend the Compasses from 11,5 to 8,75, the same distance will reach from the weight given, 1900 Pound being thrice repeated to 837 Pound,

If a Piece of Ordnance of 4 Inches Dia­meter weigh 1600 Pound, what will ano­ther Piece weigh, being of the same shape and metal of 2 Inches Diameter? Answer, 200 Pound.

PROB. 13.

Having the Diameter and weight of any Piece of great Ordnance of one Metal, to find the weight of another Piece of Ord­nance of another Metal that is of the same shape.

In this Problem there will be required a double o­peration to find out its weight.

Example.

Let there be a Brass Piece of Ordnance of 11,5 Inches Diameter at the Base Ring, weighing 1900 Pound (as before,) and let the Question be to find the weight of an Iron Piece of Ordnance of the same shape; viz. 8,75 Inches Diameter.

In this and the like cases, you must in the First place find the weight of the Piece 8,75 Inches Diameter, as in the last Theorem, as if it were a Brass Piece; and having found the weight to be 837 Pound, you must next seek the proportional Numbers, as in Page 116, at the latter end of the Ninth Problem, whose proportion is there found to be as 16 to 18, which is the proportion between [Page 122]Brass and Iron, Brass being the heavier Metal.

Therefore having found the weight,

The Analogy is

AS 18 is to 16, so is 837 to 744.

By the Logarithms.
Log. of 18 1,25527
Log. of 16 1,20412
  ,92272
 Sum,12684
Log. found,744,87157
By Gunter's Scale.

Extend the Compasses from 18 to 16, the same extent, the same way shall reach from 837 to 744.

PROB. 14. To find the Superficial Content of the Convex face of any Piece of Ordnance, and also of the Solid Content of the Concavity thereof.

Suppose the Circumference of the Conca­vity be 22 Inches, and the length of it 12 Foot, or 144 Inches, the Question is, what is the Superficial Content of the Convex face, or what the Solid Content of the Con­cave Bore.

For the Superficies the Analogy is,

AS 1 ∶ 22 ∷ 144 ∶ 3168, Square In­ches.

By the Logarithms.
Logarithm22,34242
Logarithm144,15836
Logar. found,3168,50078
By Gunter's Scale.

Extend the Compasses from 1 to 22, on the Line of Numbers, the same extent, the same way shall reach from 144, to 3168, the Square Inches required.

To find the Solid Content.

First get the Semidiameter, which in this Example is 3, 5 Inches, and also the Semi­cumference, which here is 11, these be­ing had,

The Analogy is thus;

AS 1 is to 3.5 ∷ 11 ∶ 38,5.

So many Square Inches are contained in the Base or Plain of the Concavity of the Mouth.

By the Logarithms.
Logarithm3554407
Logarithm1104139
Logarithm38,558546
By Gunter's Scale.

Extend the Compasses from 1 to 3, 5 the Diameter of the Concave assumed, the same extent will reach the same way from 11 to 38,5, the Base of the Cylinder required.

The Base of the Cylinder being thus found, to find the Solidity of the Cylinder.
The Analogy.

AS 1 is to 38,5 (the Area of the Base of the Cylinder,) so is the length of the Cylinder 144 Inches to 5544 Cubical Inches.

By the Logarithms.
Logarithm385,58546
Logarithm144,15836
Logarithm5544,74382
By Gunter's Scale.

Extend the Compasses on the Line of Numbers, from 1 to 38,5, the same extent, the same way shall reach from 144 to 5544.

PROB. 15. To know how much of every kind of Metal is contained in any Brass Piece of Ordnance.

If the proportions of Metals used by Gun­founders is supposed to be thus, that for e­very 100 Pound of Copper, to put in 10 Pound of Brass, and 8 Pound of Pure Tin; now supposing this Mixture to be true, let it be required how much of every sort of these Metals is in a Gun of 5600 Pound weight.

For Answer to this and the like Questions, first joyn all the several mixtures together, that 100, 10, and 8, and this must be the First Number in the Rule of Proporti­on; the weight of the Piece, the Second Number, which here is 5600, and the Third Number is each several sort of Me­tal in the mixture, which is here 100, 10, and 8.

The Operation.

The Sum of the common Mixtures are 118.

And then the Analogies are thus,

As 118 is to 5600,

    100 Copper,
    10 Latten,
    8 Tin.
So is1004745,7Copper, 
So is10474,6Brass, 
So is8379,7Tin.118
Analogy for the Copper is,

As 118 to 5600, so is 100 to 4745,7 Copper.

Analogy for Brass.

As 118 to 5600, so is 10 to 474,6 fere, Brass.

Analogy for Tin.

As 118 to 5600, so is 8 to 379, 7 fere Tin.

Which Three Sums thus found,

4745,7
474,6
379,7
56000

being added toge­ther, they make, the just weight of the piece pro­pounded.

By the Logarithms.

The Proportions are thus wrought.

For the Copper.
Logarithm118071882
Logarithm5600748188
Logarithm100000000
  748188
Log. found,474,57676306

Here you are referred to a larger Table of Lo­garithms, than is in this Book, for this opera­tion and the next following.

For Brass.
Logarithm118071882
Logarithm5600748188
Logarithm10000000
  748188
Logarithm found,474,57676306
For Tin.
Logarithm118071882
Logarithm5600748188
Logarithm8903090
  651278
Logarithm379,7579396
By Gunter's Scale. For the First Operation for Copper.

Extend the Compasses from 118 (upon the Line of Numbers) to 5600, the same [Page 130]extent, the same way, shall reach from 100 to 4745,7.

For Brass.

Extend the Compasses from 118 to 5600, the same extent shall reach from 10, to 4746, being one place less than the former.

For Tin.

Extend the Compasses from 118 to 5600, the same Extent, the same way shall reach from 8 to 379,7.

PROB. 16.

By knowing what quantity of Powder will load some one Piece of Ordnance, to find how much of the same Powder will load any other Piece of Ordnance, Greater or Lesser.

Example.

If a Saker of 3,75 Inches Diameter in the Bore requires Four Pound of Powder for its Load, what will a Demy Cannon of 6, 5 In­ches Diameter in the Bore require?

The Analogy.

AS 4,75 is to 4, so is 6, 5 to 20 [...] fore.

☞ But note, that it is here under­stood, that the Demy-Cannon ought to be as well Fortified as the Saker is; (viz.) it should bear the same proportion to the Saker, both in weight and thickness of Metal that the Bore thereof beareth to the Saker; for the Demy-Cannon in this Example, ought to be 8351 Pounds, which would be of a Proportion to the Saker, to carry a proportional weight of Powder.

But if the Demy-Cannon be found to want of its proportional weight with the Saker, as if it weigh but 6000 Pounds, then to find its due load in Powder answerable to its strength and weight of Metal,

Multiply the weight thereof 6000 by 20,8 the Charge already calculated, and divide the Product by 8351, the weight it ought to have had, and the Quotient is 14,9; there­fore 14,9 Founds is a sufficient Charge for such a Gun.

A Table of the weight of Iron Shot in Pounds and Ounces, from One Inch Diameter, to Ten Inches, to every Eighth part of an Inch.

Shot.b.oz
1 [...]0 [...]2
1 [...]003
20004
30005
40007
50009
60012
70014
II0102
10105
20109
30114
40203
[...] [...]208
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
[...] [...] [...]
10913
21012
31112
41213
51314
61501
71604
V1709
11814
22005
32113
42326
52500
62611
72808
VI3008
13205
23405
33606
43809
54014
64304
74511
VII4803
15013
25309
35606
45905
56205
66507
76810
VIII7200
17506
27815
38209
48605
59003
69403
79804
IX10208
110613
211104
311513
412009
512505
613005
713506
X14004

The foregoing Table was Calculated from the Directions in this Chap Prob. 9. page 113.

One Example will shew the use of this Table.

Example.

Inquire the weight of a Shot whose Diame­ter is 6.

Look for 6 Inches in the Column under Title Shor, and right against it in the Co­lumns under Title lb. and oz. you will find 45 lb. and Eleven Ounces, the weight re­puired.

A Table shewing the height and weight of Iron, Lead and Stone shot, according to their [...] in Inches and Qu [...]ters, and their respective weights in Pounds and Ounces.

  Iron.Lead.Stone.
Inches.Quarters.Pounds.Ounces.Pounds.Ounces.Pounds.Ounces.
10 [...]00301
11 [...]00603
12 [...]00904
131001305 [...]
201111107
21192009
222230012
232144310
303125014
314126918
326181129
337591427
[Page 135]40815115213
4110101515310
421210171543
43141421559
50175241263
5120130078
522323510814
532663991010
60300450114
613405101212
62380570143
634206301512
704807201710
715307901914
725808702412
73640960240
807210106 2614
81780117 2808
82873130 3408
83950142 3510

[Page 136]

  Iron.Lead.Stone.
Inches.Quarters.Pounds.Ounces.Pounds.Ounces.Pounds.Ounces.
901010150 3710
911096161 404
9212110181 442
9313211198 498
1001380207 5110
1021642246 600
1101840275 698
1122160324 810
1202400360 900
1303050451 1140
1403892583 1468
One Example will shew the use of this Table.

A Shot is 7 Inches [...] Diameter, which Number seek in the First Column; in the next, you have the weight of the Iron Shot, 64 Pound; and in the Third Column, you find the Leaden Shot to weigh 96 Pound; and in the 4th Column, the weight of the Stone Shot to be 24 Pound.

A General Table of Gunnery shewing the Length and Weight of most of our En­glish Ordnance, the Diameter of their Bere, the weight of their Shot, the Ladles length, and their weight of Pow­der to Charge them.

Names of the Pie­ces of [...].Diameter of the Bore.Length of the Piece.Weight of the Piece in Pounds.Breadth of the Ladle.Length of the Ladle.
 Inches.Parts.Feet.Inches.PoundsInches.Parts.Inches.Parts.
Basc.12462002040
Rabanet.14563002441
Falconets.22604004074
Falcon.26707504482
Minion Ordinary30708005084
Minion Large.328010005090
Saker Lowest.348014006496
Saker Ordinary.3690150066104
Saker Eldest.40100180072110
Demy-Culv. Low.42100200080120

[Page 138]

Names of the Pie­ces of Ordnance.Wetght of Pow­der.Diameter of the Shot.Weight of the Shot.Piece Shoots point blank.
 PoundsOuncesInches.Parts.FoundsOuncesPaces.
Base.08110360
Rabanet.012130570
Falconets.14221990
Falcon.242528120
Minion Ordinary.282735120
Minion Large.3430312125
Saker Lowest.3632413150
Saker Ordinary.403460160
Saker Eldest.503676163
Demy-Culv. Low.644090174

[Page 139]

Names of the Pie­ces of Ordnance.Diameter of the Bore.Length of the Piece.Weight of the Piece in Pounds.Breadth of the Ladle.Length of the [...]addle.
 Inches.Parts.Feet.Inches.PoundsInches.Parts.Inches.Parts.
Demy-Calv. Ord.44110270080126
Demy-Culv. Eld.46110300084134
Culverins Best.50110400090142
Culv. Ordinary.52110450094160
Culv. Largest.541104800100160
Demy-Can Low.621105400104200
Demy Can. Ord.641205600120220
Demy Can. Lar.661206000120220
Cannon-Royal.801208000140240

Four Dutch Pieces.
A 3 Pounder.29411075055180
A 6 Pounder.370100150065140
A 12 Pounder.46190300090106
A 24 Pounder.57970500011090

[Page 140]

Names of the Pie­ces of OrdnanceWeight of Pow­der.Diameter of the Shot.Weight of the Shot.Piece Shoots point blank.
 PoundsOuncesInches.Parts.PoundsOuncesPaces.
Demy-Culv. Ord74421012175
Demy-Culv. Eld.88441213178
Culverins Best.10046151180
Culv. Ordinary.11650179181
Culv. Largest.11852205183
Demy-Can. Low.14060308156
Demy-Can. Ord.17861325162
Demy-Can. Lar.180654014180
Cannon-Royal.32874595185

Four Dutch Pieces.
A 3 Pounder.108556240120
A 6 Pounder.60440120160
A 12 Pounder.3834960178
A 24 Pounder.11027730189

One Example of the use of the foregoing Table is sufficient, which shall be of the Saker Ordinary, where you will find the Diameter of the Bore to be 3 Inches and [...] of an Inch, the length of the Piece to be 9 Foot, the weight of the Piece 1500 Pound the breadth of the Ladle to be 6 Inches [...] of an Inch, and the length of the Ladle to be 10 Inches and [...], of which is half an Inch, and the weight of Powder to Charge that Piece is 4 Pounds, the Diame­ter of the Shot to be 3 Inches [...], which is 3 Inches and [...], the weight of the Shot to be 6 Pound, and that the Piece shoots point blank 160 Geometrical Paces.

[...]
[...]

CHAP. VI. Of the Different Fortifications of most Pieces of Ordnance.

THere are Three Degrees used in Fortifying each sort of Ordnance, both Cannons and Culverings.

First, Such as are ordinarily Fortified are called Legitimate Pieces.

Secondly, Such whose Fortification is les­sened, are therefore called Bastara Pieces.

Thirdly, Those that are Extraordinary Pieces, are called Double Fortified.

The Fortification is reckoned by the thickness of the Metal at the Touch-hole, at the Trunnions, and at the Muzzle, in proportion to the Diameter of the Bore.

The Cannons double Fortified, have full one Diameter of the Bore, in thickness of [Page 143]Metal at the Touch-hole, and [...] at the Trunnions, and in their Muzzle [...].

The Lessened Cannons have at their Touch-hole ¾ or 12/16 of the Diameter of their Bore, in thickness of Metal, and [...] at the Trunnions, and [...] at the Muzzle.

The Ordinary Fortified Cannons have 7/8 at the Touch-hole, 5/ [...] at the Trunnions, and 3/ [...] at the Muzzle.

All the Double Fortified Culverings, and all Lesser Pieces of that kind, have 1 Diame­ter, and [...] at the Touch-hole, [...] at the Trun­nions, and 9/16 at the Muzzle.

The Ordinnary Fortified Culverings are Fortified every way as your Double For­tified Cannons; and the Lessened Culve­rings as the Ordinary Cannons in all points.

CHAP. VII. How much Powder is fit for Proof, and what for Action for any Piece of Ordnance.

FOR Cannons 4/ [...] of the weight of the Iron Shot for Proof, but for Ser­vice, half the weight of the Shot is enough, especially for Iron Ordnance, which will not endure so much Powder as Brass Guns by one quarter.

For Culverings their whole weight of their Shot for proof, and for Service 2/ [...], for the Saker and Falcon 4/ [...] of the weight of their Shot.

And for Lesser Pieces, the whole weight of the Shot may be used in Service, till they grow hot, for then you must abate by discretion.

For proof these Lesser Pieces, you may take one, and ⅓ of the weight of the Shot, therein also must be respect had to the strength and goodness of the Powder, which is to be ordinary Corn Powder.

To make Ladles to Load your Guns with.

THe Ladles ought to be so proportioned for every Gun, that Two Ladles full of Powder may Charge the Piece; which in General Terms is thus.

The breadth of all Ladles are to be Two Diameters of the Shot, that so a Third may be left open for the Powder to fall freely out of the Ladle, when you turn it bottom upwards; the length of the Ladles must be somewhat different, according as the Piece is Fortified.

For Double Fortified Cannons, the length of the Ladle may be Two Diameters and One half of their Shot, besides so much as is necessary to fasten it to the Head of the Ladle-Staff, which will require One Diameter more of Plate; (but this is not reckoned to the length of the Ladle, be­cause it holds no Powder. For Ordinary Can­nons [Page 146]the Ladle must not exceed Two Dia­meters of their Shot in length.

For Culverings and Demy-Culverings, the Ladle may be Three Diameters of their Shot, and Three and a half for Lesser Guns to load them at Twice.

If you would load them at once, you must double the length of the Ladle.

☞ Observe this for a General Rule, that a Ladle Nine Balls in length, and Two Balls in breadth, will hold the just weight of the Shot in Powder.

But note, that Iron Ordnance must have but Three Quarters of the Charge of Brass Ordnance.

CHAP. VIII. To know what Bullet is fit to be used for any Gun.

IT is convenient that the Bullet be some­what less than the Bore of the Gun; that it may have vent in the Dis­charge, and not stick and break the Piece.

Now some think one Quarter of an Inch less than the Bore, will serve for all Guns, but this vent is too little for a Cannon, and too much for a Falcon.

It is more Rational and Artificial to di­vide the Bore of the Gun into Twenty equal parts, and let the Diameter of the Bullet be Nineteen of those parts, accor­ding to which proportion the Table afore­going, in page 137 is Calculated.

To make Cartridges, Moulds and Formers for any sort of Ord­nance.

THe matter of which Cartridges a made, are either Canvas or Paper Roy­al, either of which being prepared, take the height of the Bore of the Piece, and let the piece of Cloth or Paper be Three times the Diameter of the Bore or Chamber of the Piece for the Breadth, and for the length according as your Piece is; (that is to say,) for the Cannon the length of the Cartridge must be Three Diameters, in the length for Culverins, Saker, Falcons, &c. Four Diame­ters, leaving at the top or bottom one Dia­meter more for the bottom of the Cartridge, cutting each side somewhat larger for the sewing and glewing them together, having a due respect for the augmenting or dimi­nishing of your Powder, according to the goodness or badness thereof, and to the ex­traordinary over-heating of your Piece; and according to what you are to have your Car­tridges made, you must have a Former of Wood turned to the height of the Shot, and a convenient length longer than the Car­tridge; [Page 149]before you paste or glew your Paper on the former, first tallow it, so will the Canvass or Paper slip off with­out starting or tearing; if you make Car­tridges for Taper-bored Guns, your for­mer must be accordingly tapered; if you make your Cartridges of Canvass, allow one Inch for the Seams, but of Paper [...] of an Inch, more than your 3 Diameters for past­ing; when your Cartridges are upon the former, having a bottom ready fitted, you must past the bottom close and hard round about, then let them be well dryed, and mark every one with black or red Lead, or Ink, how high they ought to be filled: And if you have no Scales nor Weights, these Diameters of Bullets make a reason­able Charge; for the Cannon two and a quarter, for the Culvering 3, and for the Saker 3 and a half, for the lesser Pieces 3 and a quarter of the Diameter of the Bullet, and let some want of their weight against the time they are over-hot, or else you endanger your self and others.

CHAP. IX. Containing certain THEOREMS IN GUNNERY.

THEOREM I.

THere are Three material causes of the greater violence of any Shot made out of a great Gun, viz. the Powder, the Piece, and the weight of the Bullet.

THEOREM II.

Powder is compounded of Three Princi­ples or Elements, Salt-Petre, Sulphur and Coal, whereof it is that which causeth the greater violence.

THEOREM III.

Although Salt-Petre be indeed the only and most material cause of the violence, and that Powder is made more forcible, where­in is the greater quantity of Petre; and of those forementioned Ingredients, there is a certain proportion to be used, as to ren­der it the most fit for Service upon several considerations; of which more hereafter.

THEOREM IV.

Although Powder is the principal and efficient cause of the Force and violence of any Shot, yet such due consideration ought to be had to the proportions therein used in the Art of Gunnery, as giving more or less than the due proportion, it may diminish the force of the Shot.

THEOREM V.

There is such a convenient weight to be found of the Bullet, in respect of the Pow­der and Piece, as the Bullets Metals being heavier or lighter than that weight, shall rather hinder than farther the violence of the range of the Shot.

THEOREM VI.

There is such a convenient Proportion to be found for the Length of every Piece to its Bore, or the Diameter of the Bullet, in respect of the Powder and weight of the Ball; as either increasing or diminishing that Proportion, it shall abate or hinder the violence of the Shot.

THEOREM VII.

Besides these three most material Causes of violence, the several Randoms or dif­ferent Mountures of Pieces will cause a great Alteration, not only in the far shooting of all Pieces, but also of their violent Bat­tery.

THEOREM VIII.

Besides these aforementioned, there are many other accidental Alterations which may happen, (especially at Sea,) sometimes by reason of the Wind, the Rarity or Con­densation of the Air, the heating or cool­ing of the Piece; The different charging by ramming the Powder fast or loose, by close or loose lying of the Bullet; By the unequal recoil of the Piece, or by reason of the Ship being upon a Tack, and the Gun standing on the wind-ward or Lee-ward side of the Ship, or by the uneven lying of the Piece in the Carriage, with divers such like Accidents, whereof no certain Rules can be prescribed to reduce those uncertain Differences to any certain Proportions: but all these by Practice, Experience and a good judgment are to be performed.

THEOREM IX.

Any Piece being mounted 90 degrees a­bove the Horizon directly to the Zenith, the violent Motion, (being in that situa­tion directly opposite to the Natural) car­ries the Bullet in a perfect right-line dire­ctly upward, till the form of the violence [Page 154]is spent, and the natural Motion gotten the victory; then doth the Bullet return down again by the same perpendicular Line.

THEOREM X.

But if any Piece is discharged upon any Angle of Mounture; although the vio­lent Motion contend to carry the Bullet directly by the Diagonal Line, yet as the natural Motion prevails, it constrains it to a Curvity; and in these two Motions is made that mixt Compound or Helical Curvity. And here note, that although the last de­clining Line of the Bullets Circuit seemeth to approach somewhat to the Nature of a right Line; yet it is indeed Helical, and mixt so long as there remaineth any part of the violent motion; but after that is spent, then his motion is absolutely perpendicular to the Horizon.

From whence may be collected this Corro­lary, That any Piece being mounted to any degree of Random, shall make the Horizon­tal range proportional to the Degree of Elevation, of which you have a Resemblance in the Annexed Scheme, Plate I.

Any Piece therefore discharged at any Mounture or Random, first throweth forth [Page 155]her Bullet directly to a certain distance, called the Point-blank Range, and then af­terward maketh a Curve, or declining Arch, and lastly finisheth in a direct Line, or nigh inclining towards it; therefore the farther any Piece shooteth in her direct Line (commonly called Point-blank) the more force she hath in the Execution; and the more ponderous the Bullet is, the more it shaketh in battery, although it pierceth not so deep.

THEOREM XI.

The utmost Random of any Piece of Ord­nance, is generally judged to be at 45 De­grees of Elevation; and if you mount your Piece to a greater Angle, the Random of the Bullet will be shorter; and to know the right Range of most Pieces, you may see in this annexed Table, as the Title may inform you, where you may see the Horizontal Range or Point blank, and the utmost Ran­dom of each respective Piece, the latter be­ing commonly ten times the distance of the right Ranges.

And for the Right Ranges and Random to several Degrees of Mounture, you may note these ensuing Tables, which is measu­red by Paces, 5 Foot to a Pace.

A Table of Right Ranges or Point­blanks at seve­ral Degrees of Mounture.A Table of Ran­doms at seve­ral Degrees of Mounture.
The Degrees of Mounture.Right Ranges.The Degrees of Mounture.Right Randons.
0190192
12091289
22272404
32443510
42614610
52785712
62856828
73027934
832081044
933791129
10354101214
20454201917
30693302185
40855402289
501000502283
601140601792
701220701214
801300801000
901350900000
A Diagram for Randoms upon each first Six points of the Gunners Quadrant

A Scale of Paces

to face Page 156

CHAP. X. Necessary Instructions for a Sea-Gunner.

1. THE First thing is, that when a Gunner cometh into a new Ship, that he diligently and carefully measure his Guns, to know they are full fortified, be reinforced or lessened in Me­tal.

2. Then he must with a Ladle and Spunge, draw and make clean all his Guns within, that there may be no old Powder, Stones, Iron, or any thing that may do harm.

3. That he search all the Guns within, to see if they are taper Chamber'd, or true bored, or whether they be Crack'd, Flaw'd, or Honey comb'd within; and finding what Ball she shoots, to mark the Weight of the Ball over the Port; that thereby he may [Page 158]see the Mark or Number upon the Carriage and Case; so that in time of service they may not go wrong.

4. The Guns being dimensioned and clean as aforesaid, take half a Ladle of Powder for every Gnn, and blow them off, spunge them well; and finding them clean, you may load them with their respective Car­tridges and Powder, they being ramm'd home with a strait Wadd after it.

Then let the Ball role home to the Wadd, and set a Wadd close home to the Ball, that the Ball may not roul out with the motion and tumbling of the Ship.

Then must you Tomkin that Piece at the Muzzle, with a wooden Tomkin, which you must tallow round about, to preserve the Powder from wetting.

Likewise make a little Tapon of Ockam for the Touch-hole, which must be tallow­ed also, to prevent any wet coming to the Powder that way; then let your leaden A­pron be put over it; then make your Piece fast, as occasion presents.

5. The Piece being loaded and fast, then provide to every Piece 24 Cartridges at least, ready made; that is to say, 12 fill'd, and 12 empty.

Likewise you must be careful, so long as the Gunner's Crew are busie with the Pow­der, [Page 159]that there be no burning Match or Fire in the Ship; Also to lay his Cartridges in Barrels or Chests, that when there is occa­sion to use them, they may be without a­buse.

6. The Gunner must see that he sorts his Ball very well, and lay every sort by them­selves in several Cases; and upon every Case set the Weight of one of the Shot, which is in them.

Also you ought to make the Bags for Hail for the Guns above, and fill them with Stones, small Shot, or Pieces of old Iron, which may be a great annoyance to the Enemies Men.

7. If it falls out that any new Ports must be cut out in the Ship, you must be care­ful that it be made over a Beam, or as near one as possible you can; Also that they be not higher or lower than the Ports be­fore; likewise that there be room for the Guns to play, because if one Gun be dis­mounted, there might be another brought to her place: And observe that the Car­riage stand on her Trucks. The up­permost part of the Carriage must stand in the middle of the Port, up and down, that a Man may lay his Piece as you please.

8. You must be careful that the Powder in the Powder-Room be well covered [Page 160]with Hides: And also that the Ropes, Ram­mers, and Spunges be ready at hand. And you must not let the Powder be unturned above a Month, because the Salt-Petre will be apt to sink to the lower part of the Barrel, which would be dangerous to make use of that Powder; And you must every Month draw your Guns; if you think they have got any wetness or moisture in the Powder; Also for fear of the Salt Petre dis­solving, which may prejudice the Piece. You must also be careful of the Candle and Fire about the Gun Room, and espe­cially the Powder Room, that there may come no disafter.

Likewise a Gunner must keep a good Account of all Materials that belong to the Guns, as Ball, Match, and Powder. What part thereof he spends, also what remains.

9. A Gunner must use all diligence be­fore they engage with an Enemy, to set a Barrel of Water betwixt every two Guns, that when they have conveniency they may dip the Spunges for the cooling of the Guns, and for fear of Fire remaining in the Piece, which may do hurt.

10. Also you must be sure that there be no melted Fire-works done in the Ship, but ashore; for it is dangerous, and a great ha­zard to the Ship, and Goods; and Men's Lives may thereby be destroyed.

Also that in time of service, no Fire­works be brought up in the Round-house, or great Cabbin, to stand, for fear of Shot coming from the Enemy may fire it, and so destroy the Ship. But rather to have them kept below in the Powder-Room, or Stew­ard-Room, to prevent Danger.

11. Necessaries that a Gunner ought to have for his Ordnance, and the quantity thereof according to the Length of the Voyage, the Quantity and Quality of his Guns.

Also if you go in a Man of War, or a Merchant-man, then there is difference of Provisions; only I will here name them all that belong to a Sea Gunner, that he may take such a Proportion of each, as the oc­cusion may require, and at the End of the Voyage to give an Account what Stores are spent, and what there is yet remaining.

Gunners Stores.

Powder and Match.

Round-shot of every sort.

Double-headed Shot.

Cut Iron of a Foot, or a Fcot and a half long.

Wooden Tomkins for each sort of Gun.

Cartridge-Paper and Glew.

Threed, Needles, Twine and Starch.

Mallets, Handspikes, Rammer heads.

Worms, Ladles, Spunge-heads, & Spunge­staves, Beds and Quoins of several sorts.

Old Shrouds for Breeching, and twice lay'd Stuff for Tackles.

Lashers, double and single Blocks, new Rope for double Tackles.

Some old Shrouds for Spunges, some Lines, Marline, Tarr'd Twine, Port-Ropes.

Moulds for Cartridges for each fort of Gun, Axle-Trees and Trucks.

Pouch-Barrels and Linstocks, Crows, Splice-Irons, Primes, Staples and Rings, Tackle-Hooks, Nails, Thimbles, Port-Bands, Sheet-Lead and Leaden-shot, old Canvass, Scales and Weights.

Lanthorns, Muscovia-Lights with a large Bottom to put Water in, to prevent dan­ger from the Sparks of the Candle flying upon the Powder-dust, that may get into the Lanthorn, Dark-Lanthorns, Powder-Measures, Sope, Powder-Horns, Priming-Irons, Nippers, Plyers, Moulds to cast leaden Bullets.

And for Instruments such as follow, which every Gunner of a Ship ought to be fur­nished withal.

Callaper Compasses large and small, for taking the Diameters of the Base Ring, Bo­dy or Muzzle of a Gun, and the Diameters of Shot.

A New Rule called the Sea-Gun­ners Rule, whose use is shewed at the End of this Book.

Brass Heights for Shot.

A Gunners Scale and Quadrant.

Brass Compasses with Steel-points,

Which Instruments, and any other be­longing to the Art of Navigation you may be furnished with, by John Seller, at the Hermitage in Wapping; with all sorts of Books, and Maritime Charts, and Atlasses, for any of the known Parts of the World.

CHAP. XI. Shewing an Easie way to dispart a Piece of Ordnance.

FIrst take the Diameter of the Piece upon the thickest Part, at the Breech of the Gun, with a Pair of Callaber Compasses, and see upon the Quadrant of your Callabers, how many Inches that is; the half of which Diameter take between a Pair of Compasses, and put that distance off upon a Sheet of Cartridge-Paper, which will make two Points upon the Paper, as A and B; then take the Diameter of the thickest part with your Callabers, and see how many Inches that Diameter is, And take the half thereof between your Com­passes, and set one Foot in A, and the other Point in C upon the said Line AB, at C.

[...]

Then take the Distance from C, to B, on the Line, and that is the true Dispart of the Piece; and if you take a Stick or Straw of that length, and set on the Muz­zle fastned with Wax, it will be a true Dispart for that Piece.

CHAP. XII. To Level a Piece of Ordnance to shoot Point-Blank.

TO shoot Point-Blank is to be un­derstood, that when the Cylender of the Piece lyeth level with the Horizon, so that the Ruler of the Gun­ners Quadrant being put into the Mouth of the Piece, the Line and Plummet hangeth Perpendicular, then that Piece lyeth in its true Position, to shoot Point Blank.

And to make a good shot at a Mark, within Point-blank reach of the Piece, The Piece lying in that Position, as is before shewn; then set up your Dispart upon the Muzzle; then if you put your Eye down to the highest part of the base Ring (as you took the Diameter of) and bring the top of the Dispart in a righe-line, with the [Page 167]Object at a Distance, that ought to be of the same Heighth from the Horizon at your Breech of the Gun and the Dispart, then is your Sight or visual Line also parallel to the Horizon, and if there be nothing de­fective in the Piece or Carriage, you will make a good Shot.

But if you intend to elevate your Piece, discharge it of some of the Quoins at the Breech, and by your Quadrant applyed to the Muzzle, you may elevate the Piece to what Angle you please; as may be perfor­med by the New Sea Gunners Rule, whose Use is shewn at the latter End of this Book.

CHAP. XIII. How to search a Piece of Ordnance, to discover whether there be any Flaws, Cracks or Hony-combs in the Piece.

IN a clear Sun-shiny-day, take a Piece of Looking-glass, and reflect the Beams of the Sun into the Cavity of the Piece, by the means of which a clear Light will appear within the Piece, by which you may discover any Flaw or Honey-Comb therein.

Another Way.

Take a long Stick with a slit at the End of it, and put an End of Candle lighted, and put it into the Cylender, turning the [Page 169]Stick every way; and you may very well dis­cover Flaws or Honey-Combs, if there be any in the Piece.

Another Way to discover Cracks.

Immediatly after you have discharg'd your Piece, let one be ready with a Tom­kin to clap into the Mouth of the Piece, with a Piece of Sheep-skin wrapped about the Muzzle of the Piece, and the same time let one stop the Touch-hole; and if there be any Crack through the Metal a visible Smoak will appear.

Another Way.

If you strike a Piece of Ordnance with a smart stroke, with a Hammer on the Out­side, and if you hear a hoarse sound, it is an evident Sign the Piece is not sound, but there is some Crack in it.

But if after every stroak with the Ham­mer you hear a clear sound, you may cer­tainly conclude the Piece to be sound.

CHAP. XIV. How Moulds, Formers and Car­tridges are to be made for any sort of Ordnance.

CArtridges are usually made of Can­vass, or Royal Paper; to make them first take the heighth of the Bore of the Piece, and allow 1/21 part of the Diameter for the Vent, and make the breadth of the Cartridges three Diameters of the Chamber of the Piece, besides the sewing or pasting, and from the Cannon to the whole Culvering is allowed about two Diameters for the length, from the Culve­ring to the Minion, the Cartridge is two Diameters and a half, and from the Minion to the Base three Diameters.

To every sort of Ordnance you must have a Former turn'd to the heighth of the Cartridge, which is 1/ [...] parts of the Di­ameter of the Bore, and half an Inch lon­ger than the Cartridge.

Before you paste the Paper on the For­mer, tallow it, that the Canvass or Paper may slip off, without starting or tearing.

If you make your Cartridges for Taper­bored Guns, your Former must be Ta­per'd accordingly; if you make your Car­tridges of Canvass, allow an Inch for the Seams, but if you make them of Paper, al­low ¾ of an Inch (more than three Dia­meters) for the pasting.

When your Cartridges are upon the For­mer, having a Bottom ready fitted, you must paste the Bottom close and hard round about; then let them be well dried, and mark every one with black or red Lead, or Blacking, how high they ought to be filled; and if you have no Scales nor Weights, these Diameters of the Bullets make a rea­sonable Charge for a Cannon, 2 and ¼ for a Cannon, three Diameters for a Culvering, and 3½ for the Saker; And for the lesser Pieces 3 and ¾ of the Diameter of the Ball, and let some want of their weight against the time the Piece may be over-hot, or else you may endanger your self and o­thers: [Page 172]Note that at Sea the Guns are never charged with a Ladle, but with Car­tridges.

CHAP. XV. How much Rope will make Britch­ings and Tackles for any Piece.

IN Ships that carry Guns, the most ex­perienced Gunners take this Rule; look how many Foot your Piece is in length, four times so much is the length of your Tackle, and your Britchings twice the length; and if the Ropes are suspected of strength, then you may nail down Quoins to the four Trucks of heavy Guns, that they may have no play; and if Breechings and Tackles should give way in foul Wea­ther, it is best immediacely to dismount your Gun; that is the surest way.

What Powder is allowed for Proof, and what for Action.

FOR the biggest sort of Pieces, as Can­non, take for Proof ⅘ of the weight of the Iron-shot, or for service ½ the weight, for the Culvering almost the weight of the Shot for Proof and for Action; for the Saker and Falcon, take for Proof the weight of the Shot, and for Action 4/ [...], and for les­ser Pieces the whole Weight of the Shot for service; and for Proof give them one, and [...] of the Weight of the Ball in Pow­der.

CHAP. XVI. How to know what Diameter every Shot must be of, to fit any Piece of Ordnance.

DIvide the Bore of the Piece into twenty equal Parts, and one of these Parts is sufficient vent for a­ny Piece, the rest of the nineteen Parts must be the heighth of the Shot: But most Gunners now-a-days allow the Shot to be just one quarter of an Inch lower than the Bore of the Piece, which rule makes the Shot too big for a Cannon, and too little for a Faulcon; but if the Mouth of the Piece be grown rounder than the rest of the Cylender within by often shooting; to choose a Shot for such a Piece, you must [Page 175]try with several Rammer-heads, until you find the Diameter of the Bore in that Place where the Shot useth to lye in the Piece, and a Shot of one twentieth part lower than that Place, is sufficient.

Every Gunner ought to try his Piece, whether it be not wider in the Mouth than the rest of the Chase, and then proceed to chuse his Shot.

To tertiate a Piece of Ordnance.

This word Tertiate is a Term principal­ly used by foreign Gunners, meaning there­by only the measuring and examining the Fortification of Metals in a Piece, tertiating; because it is chiefly to be measured and ex­amined in three principal Parts of a Piece, Viz. at the Breech, the Trunions and the Mouth: And there are three Differences in Fortification of each sort of Ordnance, ei­ther Cannon or Calverings, for they are either double fortified, ordinary fortified or lessened, as Legitimate, Bastard, or ex­traordinary Pieces: For the Cannon double fortified or re-inforced, hath fully one Dia­meter of the Bore in Thickness of Metal at her Touch-hole, and 11/16 at the Trunions, and 7/16 at her Muzzle; and the ordinary Can­nons [Page 176]have ⅞, at the Chamber ⅝, at the Trunions 3/ [...] The lessened Cannons have ¼ at the Chamber, and 9/16 at the Trunions, at the Muzzle 5/16, &c.

Now that every Gunner may be assu­red of the Fortitude of any Piece of Ord­nance, and so may the more safely and boldly allow her a due Loading and Pro­portion of Powder, both for Proof and Service, that she may without danger per­form her utmost Execution, you may ob­serve this following Direction:

As for Example.

Suppose there is a Culvering that shoot­eth an Iron-shot of 17 l, with 13 l. of Corn-Powder, which is ⅘ of the Weight of the Shot; the Question is, whether she may be able to bear so much Powder, and if need were, more which question cannot be well answered without examining or tertiating her Metal, which may be thus performed.

First with a Ruler draw a Line upon a Paper or Slate, as you may see in the an­nexed Figure, as the Line AB.

Then with a Pair of Compasses with re­versed Points, take the Circumference of the Bore of the Piece, and Measure the same upon an Inch-Rule.

Then take the same Measure from any other Scale of equal parts of a competent size, and divide that distance into two e­qual parts with your Compasses, and ha­ving that distance in your Compasses, set one foot in the Point C, and describe the circle DEFG, which circle is equal to the bore of the Piece.

[figure]

Then with a pair of Calaber Compasses, take the Thickness or Diameter of the [Page 178]Metal at the Touch hole, and Measure the same upon a rule as before, and take that distance between your Compasses, and with half that distance setting one Foot in the point E describe the circle HIKL, which shall represent the circumference of the Me­tal at the Touch-hole, so that you may take the Compasses and Measure the Diameter of the bore GE, which is equal to the distance of LG or EI which shews, that there is one Diameter of Metal round the Concave Cylinder of the Piece; you may therefore be sure that it is an ordinary for­tified Culvering; but to know if it be a Bastard, or extraordinary Gulvering, it cannot be known by the fortification but by the length thereof, being longer than ordi­nary, it is therefore called an extraordina­ry Culvering, and being shorter than the ordinary, it is therefore called a Bastard Culvering.

Now this being found to be an ordinary Culvering, she will bear 4/ [...] of the weight of her shot in Cannon Powder, which is 13 l. 9 ounces.

But to be more assured of her fortitude, the measure of her Metal may be taken at her Trunions and Neck as followeth.

At the cornishing before her Trunions, with a pair of Calaber Compasses, you [Page 179]may take the Diameter of the body of her Metal there, as you did before at the Touch­hole, and measure the same Diameter upon a rule, then take your Compasses and from the same scale as you did use before, take that distance and divide it in two equal parts, and setting one Foot of the Compas­ses in C describe the circle M N, and if found ⅞ of the bore, it is the proportional fortification for an ordinary Culvering, and the like may be done with the Neck which the circle OP doth represent, and the distance from G to O being [...] of the height of her bore, and is the due thickness of her Metal, for an ordinary Culvering at her Neck.

But if in taking the measures aforesaid there had been found at the Touch-hole from G to L (the thickness of one Diame­ter at the bore, and ⅛ more, it would have signified that it had been a double fortified or a reinforced Piece, having also at the Trunions GM [...], and at the Neck GO [...] of the height of her bore, then she shooting an Iron shot of 17 l. would have en­dur'd 17 l. of Cannon Corn Powder to be loaded with, and to be fired without dan­ger, and would conveyed the shot further than the ordinary could have done upon the like degrees of Mounture.

Contrariwise, if the Circles there had been found that from G to L had been but ⅞ of the height of her bore at the Touch­hole, and at her Trunions but ¾ which is G M, and at the Neck from G to O but 7/8 [...], of the height of the bore, then she would appear to be one of the lessened or slender fortified Culverings, and must be allowed but 12 pound 9 ounces of Cannon Corn Pow­der, to convey her shot of 17 l. which upon like elevation will not carry a shot as far as the ordinary.

In this manner all other Guns are to be measured and tertiated only with this allow­ance withal that the Demy Culvering hath 1/24 and the Saker 1/23 and the Falcon [...] more Metal comparatively than the whole Culvering hath.

And if a Piece is found that it is not truly bor'd, you must always reckon that the Piece is no otherwise fortified than she is found to be, where her Metal is found to be thinest.

How to make a Shot out of one Ship unto another in any Wea­ther whatsoever.

IN time of service when you are on a suddain to make a Shot at a Ship, and know not what dispart will serve the Piece, then you must take your aim at what part of the Ship you judge to do most exe­cution, and look along by the side of the Piece, as near as you may at the middle of the Breech unto the middle of the Mouth of the Piece, and so place her to the best advantage, and quoin up the tayl of the Piece fast (for that giveth the true height of the mark) Then minding the steeridge take your best opportunity and give fire, and if the Sea be any thing grown, choose your Piece that is nearest the Main-Mast and in the lower Teer, if the Ship can keep her Ports open, for there she doth least la­bour; and when you are to make a Shot at a Ship, you must be sure to have a good Helms-Man that can steer steady.

And he that giveth level must lay his Piece directly with that part of the Ship that he doth mean to shoot at. And if the [Page 182]Enemy be to Leeward of you, then give fire when the Ship doth begin to ascend or rife upon a Sea, which is the best oppor­tunity that doth present.

But if the Enemy is on the weather-gage of you, then wait an opportunity when the Ships do right themselves; for if you should give fire at the heelding of your Ship, then you would shoot over the other Ship; and if the Sea be high, there is no better time to give fire than when your Enemies Ship begins to rise on the top of a Sea, for then you have a better mark than when she is in the trough of the Sea: All which several obser­vations must be managed, with a good judgment and discretion of the Gunner.

And he that is at the Helm must be Yare-Handed with the Helm, to observe the motion of the Enemy, to luff when the Enemy luffs, and to bear up when the Ene­my bears up; and it is always good to level the Piece rather under the place you shoot at than over.

And if in a fight, if you intend to lay your Enemy, aboard then call up your Company either to enter or defend.

And if you are resolved to enter, then be sure to level your Bases or other small Guns ready to discharge to the best advantage you can at the first boarding, at such a [Page 183]place where his Men have most recourse, and if you can possibly, at boarding endea­vour to take off his Rudder by a great shot, or at his Main Mast &c.

In what Order to place your great great Guns in Ships.

IT is first to be considered that the car­riage be made in such sort that the Piece may lie right in the middle of the Port, and that the Trucks or Wheels are not too high, for if they are too high, then it will keep the carriage, that it will not go close to the Ships side, so that by that means the Gun will not go far enough out of the Port, except the Piece be of a great length; and also if the Ships heelds that way, the Trucks will always run close to the Ships side, so that if you have occasion to make a shot, you shall not bring the Trucks off the Ships side, but that will run too again; and the Wheel or Trucks being too high, it is not a small thing will stay it, but will run over it.

And another inconveniency is, if the Trucks are too high, it will cause the Piece to have a greater reverse or recoyl, therefore [Page 184]for these reasons it is good to have low Wheels or Trucks to a Gun aboard of a Ship.

The best position that the Gun can be in is, to place it in the very midst of the Port, that is to say, that the Piece lying level at point blank, and the Ship to be upright without any heelding, that it be as many Inches from the lower side of that Port beneath, as it is upon the upper part above; and the deeper or higher the Ports are up and down, it is the better for making of a shot, for the heelding of a Ship, whether it be on the Lee or Weather side; for if you have occasion to shoot forward or back­ward, the steeridge of the Ship will serve the turn.

It is also very bad to have the Orlope or Deck too low under the Port, for then the Carriage must be made very high, which is very inconvenient in several res­pects, for in firing the Piece it is apt to overthrow, as also in the working and la­bouring of the Ship in foul weather.

And also you have consideration in pla­cing your Ordinance in a Ship, for the shortest Ordinance is best to be placed out of the Ships side, for several reasons.

1. For the ease of the Ship, for the shorter they are the lighter, and if the [Page 185]Ship should heel with the bearing of a Sail, then you must shut the Ports, especi­ally those Guns on the lower deck; then the shorter the Piece is, the easier it is to be taken in both for the shortness and weight also.

2. In like manner, the shorter the Piece lyeth out of the Ships side, the less it shall annoy them in the tackling of the Ships Sails, for if the Piece lyeth far out the Sheets, Tacks or Bowlines, it will be apt to be foul of the Guns.

For your long Guns they are best to be placed in the Gun-Room or any place, after on for a Stern-Chase, for two Reasons.

1. The Piece had need to be long, or else it will not go far enough out that it may be no annoyance to the works of the Stern that may over-hang, and so may blow away the Counter of the Ships Stern.

2. The Pieces that are placed abaft, are required to be long, because of the raking of the Ships Stern from below, so that the Carriages cannot come so near the Ports as they do by the Ships side, which is more up and down.

Also for such like Reasons as these, it is as well required to have long Pices to be placed forward or in the Fore-Castle, &c.

And here note that there must be regard had to the making of the Carriages, both for Forward-on or After-on for the places of the foremost trucks, in taking notice if the Ships side do tumble in or out, and al­so the cumbering of the Deck or Orlope; in all these cases it must be left to a good judgment and experience, in the convenient placing of Guns in a Ship.

How much Rope will make Breechings and Tackles for Guns.

For the Tackles.

YOU may observe this Rule, that as many Feet as your Piece is in length, so many Fathom must your Rope be.

For the Breechings.

They must always be four times the length of the Piece with some overplus for fast­ning at both ends. If in foul weather your Breechings and Tackies should give way, you have no better way for the present [Page 187]to prevent danger, than immediately to dismount the Piece.

It is also approved by able Gunners, that the Rammers and Spunges made with small Hawser should be armed close and hard with strong and twisted Yarn, from the Rammers end quite to the Spunge, which would much stiffen and make it more use­ful and lasting to ram both Wad and Bullet close to the Powder.

Let the head of the Rammers be of good Wood, and the heighth one Diameter, and ¾ thereof in length, or very little less then the heighth of the shot next the Staff; it must be turned small that a ferril of Brass may be put thereon, to save the head from cleaving; when you ram home the shot, the heads must be bored ½, for the Staff to be put in and fastned with a Pin through, and the Staff­length a foot more then the concave of the Gun.

CHAP. XVII. Of Powder. Several things necessary to be known by a Gunner; but espe­cially of Powder.

THE efficient cause for expelling the Shot is the Fire that is made of Powder, that is compounded of Salt-Petre, Brimstone and Charcoal.

The Salt-Petre gives the Blow or Report.

The Sulphur takes Fire, and the Coal rarifies the other two, to make them Fire the better.

Two sorts of Gun-Powder are common­ly in use.

One is made of five Parts of Salt-Petre, one of Brimstone, and one part of Char­coal.

The other (being stronger) is made of six one and one.

That of five one and one is generally used for great Guns, the other for Muskets and small Arms.

And it hath been generally observed, that forty two pound of Powder of five one and one, is stronger than forty five pound of four one and one; and forty pound of six one and one works greater effect, than for­ty two pound of five one and one, although all contain thirty pound of Salt-Petre.

Anciently they made Powder of four one and one; but this Powder by experience being sound too weak, is not now in use.

That Powder which at this day is re­ceived into their Majesties Magazine at the Tower of London, is made of six one and one.

To know good Powder.

1. The harder the Corns are in feeling, by so much the better it is.

2. When the Powder is of a fair Azure or French Russet colour, is it judged to be a very good sort and to have all its Ingredi­ents well wrought, and the Petre to be well refined.

3. Lay five or six Corns upon a white piece of Paper three fingers distance one from another, then fire one, and if the Powder is good they will all fire at once and leave nothing but a white chalky co­lour on the Paper; neither will the Paper be toucht: But if there remains a grossness of Brimstone and Petre, it discovers the Powder to be bad.

And take this for a general Rule, for a sign of good Powder; that which gives fire soonest, smoaks least, and leaves least sign behind it, is the best sort of Gun-Pow­der.

To preserve Powder from decaying.

To preserve good Powder, Gunners ought to have that reason to keep their Store in as dry a place that can be had in the Ship, and every Fortnight or three Weeks to turn all the Barrels and Cartridges upside down, so that the Petre may be dispersed to every part alike; for if it stands long, the Petre will always descend downwards, and if it be not well shak'd and moved, it will want of its strength at the top, and 1 l. at bottom with long standing will be stronger then 3 at the top.

To find the Experimental Weight of Powder (Tower-Proof) that is found convenient for Service, to be used in Guns of several Fortifications (or thickness) and by consequence strength of Metal.

TO find the strength of Guns the brief Rule is thus, First find the Diameter of the bore (or Chamber of the Gun) where the shot lies, then the true fortified Iron Guns ought to be 11 of those Dia­meters in the circumference of the Gun at the Touch-hole, 9 at the Trunions, and 7 at the Neck, a little behind the Mouth or Muzzle-ring where the dispart is set.

But Brass Guns having the same weight of Powder are as strong at nine Diame­ters of the Chamber bore about the Gun at the Touch-hole, and seven Diameters at the Trunions, and five at the Neck.

This is the Rule of true bored and true fortified Guns; and for those more or less fortified, observe the Proportions in this following Table.

 Brass Iron
More Fortified11Diameters13
More Fortified12Diameters14
True Fortified9Diameters11
Less Fortified8Diameters10
Less Fortified7Diameters9

Weight of Powder for Service is pro­portioned by the Numbers of Diameters of the Bore about the Gun at the Touch­hole, for such Guns so qualified as in the foregoing Table, viz. and to load them accordingly.

To know whether the Trunions of any Gun are placed right.

Measure the length of the Cylender from the Muzzle to the Britch, and divide the Length by 7, and divide the Quotient by 3, and the Product will shew how many the Trunions must stand from the bottom of the bore of the Piece, and that they ought to be placed so that ⅓ of the Piece may be seen above the Center of the Trunions.

The Practical way of making Gun-Powder.

The Essential Ingredients for making Gun-Powder are three, viz. Salt-Petre, Brimstone and Charcoal, and of these there are to be three several quantities and pro­portions, according to the use intended for; and for the best Powder that is now made, there is commonly used these proportions.

Salt-Petre,4, 5, 6 Parts.
Brimstone,1 Part.
Charcoal,1 Part.

The Cannon Powder hath commonly of Salt-Petre four times so much as of Brim­stone and Charcoal, and for Musket Pow­der it is usually made five times as much Salt-Petre as of Brimstone and Coal.

Now having the Proportional quantity of each of these Ingredients, put all the Salt-Petre together into a Caldron, and boyl it with so much Water as will serve to dis­solve it with; being so dissolved, it ought to be washed and lay'd upon a clean place; this done, beat the quantity of Coal into dust, then put this Charcoal dust being finely bea­ten [Page 194]into the disolved Petre, and incorpo­rate them very well together, and as you mingle them, put in by little and little the Sulpher very well beaten; when this mix­ture of Salt-Petre Brimstone and Coal are well incorporated, lay it forth to dry a little; when the same mixture is somewhat dryed and is very well mixed, sift it well through a Sieve; then casting Water or Vinegar up­on it, corn it, and when you have so done, dry it against the Fire and the Gun-Pow­der is made: There are divers ways to grind Gun-Powder; the best way is to stamp it in Mortars with a Horse-mill or Water-mill, for the Powder is thereby most finely bea­ten and with least labour; and to know if it be well done, you may with a Knife cut in pieces some of this Composition, and if it appear all black it is well done, but if any of the Brimstone or Petre is seen, it is not incorporated enough.

The manner to fift Powder is thus,

Prepare a Sieve with a bottom of thick Vellom or Parchment, made full of round holes, then moysten the Powder which shall be corned with Water, put a little Bowl into the Sieve, then fift the Powder so as the Bowl rowling up and down in the Sieve may break the clods of Powder, and make it by runding through the little holes to corn.

To Renew and make good again any sort of Gun-Powder, ha­ving lost its Strength by moist­ure, long lying, or by any other means.

Having moistned the said Gun-Powder with Vinegar or fair Water, beat it well in a Mortar, then sift it through a Sieve or fine Searce; for every l. of Gun-Powder mingle one Ounce of Salt-Petre that hath been pulverised, and when you have so done beat and moisten this mixture again, until by so breaking or cutting with a Knife, there is no sign of Salt-Petre or Brim­stone in it: Also corn this Powder when it is incorporated with the Petre, as it ought to be, and you have done.

CHAP. XVIII. How to make Hand-Granadoes to be Hove by Hand.

THere is good use made of Hand-Gran­adoes in Assaults and Boarding of Ships; these are made upon a Mould made with Twine, and covered over with Cart­ridge Paper and Musket Bullets cut in two, put with Past and bits of Paper thick on the out-side. After you have doubled the Shells, past on some at a time, and let it dry, and put some more until it be quite full; then dip it in scalding Rossen or Pitch and hang it up and it is for your use: But you must have the innermost end of the Twine left out, and before you pitch it you must draw out the Twine and stop the hole, and then pitch it.

To load them, fill these Shells with Gun-Powder, then make a Fuze of one pound of Gun-Powder and six Ounces of Salt-Petre and one of Charcoal, and fill the Fuze; then knock it up to the head within one quarter of an inch, which is only to find it by night.

Stop the rest of the holes well with soft Wax; your first Shells must be coated with Pitch and Hurds lest it should break with the fall; and be sure when you have fired the Fuze, suddenly cast it out of your hand, and it will do good execution.

CHAP. XIX. How to make Fire-Pots of Clay.

FIre-Pots and Balls to throw out of Mens hands may be made of Potters-Clay with Ears to hang lighted Matches to them; if they light on a hard thing they break and the Matches fire the Powder, and the half Musket Bullets contrived on them, as in the last Chapter, disperse and do much mischief.

Their mixture is of Powder, Petre, Sul­pher, [Page 198]Sal Armoniack of each one pound, and four Ounces of Camphire pounded and searced and mixt well together, with hot Pitch, Linseed Oyl or Oyl of Petre; prove it first by burning a small quantity, and if it be too slow add more Powder, or if it be too quick then put more Oyl or Rosin, and then it is for your use.

SECT. I. How to make Powder-Chests.

You must nail two Boards together like the ridge of a House, and prepare one Board longer and broader for the bottom: Between these three Boards put a Cartridge of Pow­der, then make it up like a Sea-Chest and fill it with pibble Stones, Nails, Stubbs of old Iron; then nail on the Cover and the ends to the Deck, in such a place as you may fire the Powder underneath through a hole made to put a Pistol in: These are ve­ry useful to anoy an Enemy if they board you.

To make Stink-Balls.

Take Gun-Powder 10 l. of black Pitch 6 l. of Tarr 20 l. Salt-Petre 8l. Sulpher [Page 199]Calafornia 4 l. melt these over a soft Fire to­gether, and being well melted put 2 l. of Cole dust of the Filings of Horses Hoofs 6 l. Assa Faetida 3 l. Sagapenum 1 l Spatula Faetida half a l. Incorporate them well toge­ther and put into this matter so prepared old Linnen or Woollen Cloath, or Hemp or Tow as much as will drink up all this mat­ter, and of these make them up in Balls of what bigness you please, and being thrown between Decks will be a great annoyance to the Enemy.

CHAP. XX. The Properties Office, and Duty of a Sea-Gunner.

1. A Gunner ought to be a sober, wake­ful, lusty, patient, prudent and quick Spirited Man; he ought also to have a good eye-sight and a good judgment in the time of service, so to plant his Piece to do most hurt or execution, either to the Hull or rigging of a Ship, as may be most expe­dient according to the appoinment of the Commander.

2. A Gunner ought to be skilful in Arith­metick and Geometry, in the making of all kind of Artificial Fire-Works, especially for service.

3. A Gunner ought to procure with all his power the Friendship and Love of every Person, and to take great care of his charge for his own safety as well as the Ship and all the Mens lives, by having special regard un­to his Powder Room and to be well satisfied in the carefulness of those that he doth in­trust to manage the business there, and to see that the Yeoman is careful always to keep a good and large Lanthorn, and to be kept whole, that it may prevent the flying in of the dust of the Powder, for the neg­lect of which it hath sometimes been con­jectured that some Ships have been blown up and lost for want of care in the Powder Room.

4. A Gunner ought at the receipt of his charge, to make an Inventory of all such things as shall be committed to him, as well to render an account as to consider the want of such Materials as are necessary to the well performance of his duty.

5. A Gunner ought to have his Gun-Room always ready furnished with all ne­cessaries belonging to his Art, which ought al­ways to be in readiness, viz. Ladles, Ram­mers, [Page 201]Spunges, Gun-Powder, Balls, Tam­kins, Wadds, Chain-shot, Cross-bar-shot, Quoins, Crows, Tackles, Breechings, Pow­der-Horns, Canvass, and Paper for Cart­ridges, Forms for Ladles, Cartridges, Need­les and Threed to sow and bind the Cart­ridges, Candles, Lanthorns, Handspikes, Poleaxes, little Hand-Baskets, Glew and Past, with a sufficient Crew of able and ex­pert Seamen, being yare-handed to travers a Piece, to Charge, Discharge, Mount, Wadd, Ram, make Clean, Spunge, and Prime and Scoure, and readily to do and perform any thing belonging to the Practical Part of Gunnery.

6. A Gunner ought always to have a Ru­ler about him, and a pair of Compasses, and Callabers to measure the heighth and length of every part of his concavity, and the length depth and wideness of every Ladle whereby he may know whether his Piece is laden with too much Powder, or is charged with a less quantity than it ought to have.

7. A Gunner ought to know the length and weight of all manner of Pieces, and be able to give an account readily how much Powder is a due charge for every Piece, and how ma­ny times a Piece may be shot off without harm, and how each kind of Piece should be charged with the Powder, Tamkin, Ball and Wadd.

8. A Gunner also must be skilful to make Salt-Petre, to refine and sublime Salt-Petre, to make divers sorts of Gun-Powder to purifie Brimstone, to amend any sort of Powder when it hath lost its vertue and force, and to know how much Salt-Petre ought to be put to the said unserviceable Powder, and to make it strong as it was before, and how many times the Salt-Petre that is put into the Powder ought to be refined.

9. A Gunner that serves at Sea must be careful to see that all their great Ordance be fast breeched, and that all the furniture be handsome and in a readiness as was said be­fore, and that they are circumspect about their Powder in the time of service, and to have an especial care of the Linstocks and Candles for fear of their Powder and their Fire-works, and the Oacum, which is very dangerous, and to keep your Pieces (as neer as you can within): And also that you keep their Touch-holes clean without any kind of dross falling in them; and it is good for the Gunner to view his Pieces and to know their perfect dispart, and to mark it upon the Piece or else in a Book or Table, and name every Piece what it is and where she doth lie in the Ship, and note how many inches halfs and quarters of inches the dispart cometh unto.

A Representation of the Sea Gunners Rule; as it appears on both sides, and the Edge.

These Instruments are Sold by John Seller Sen. att the Hermitage in Wapping.

A Scale for the resolution of Lineal proportions.

A Scale for the resolution of Quadratique proportions

A Scale for the resolution of Cubique proportions

AN APPENDIX, Shewing the Use of a Proportional Scale In several Questions in ARITHMETICK, In Lineal, Quadratick and Cubi cal Proportions, in the Mensu­ration of Superficies and Solids, and the Extraction of the Square and Cube-Root. The Figure of which Scale is annexed to the Front of this Page. As also the use of a New Rule, called the Sea-Ganners-Rule, con­taining an Epitome of the Art of Gunnery in it self.

By John Seller.

London, Printed in the Year, 1691.

CHAP. I. A Description of the Proportional Scale and its Ʋse in the Art of Gunnery.

THere are three Lines upon the Scale,

  • One for Lineal, Proportions.
  • One for Quadratick, Proportions.
  • One for Cubical, Proportions.

The uppermost Line is for the Resolutions of all Lineal Proportions, between Num­bers Lines and Superficies, this Line being a single Line of Numbers which is broken in the midst and laid side by side, for the great­er facility in their Operations.

The second or middlemost is for the Re­solution of all Quadratick Proportions, be­tween Lines and Superficies and the ex­traction [Page 2]of the Square Root, several of which Questions may be answered by inspection only.

The third and lowest Line is for the Re­solutions of all Cubical Proportions between Numbers, Lines and Solids, and the ex­traction of the Cubick Root.

Numeration on the Lines. PROP. I.

A whole number consisting of two, three or four places, being given, to find the point on the Scale representing the same.

The upper Line (that is for Lineal Pro­portion) in two parts (i. e.) a line of Num­bers broken and put side by side, the upper Line begins at 1, and if that be called one then the next figure must be called 2, and the next 3 and so to 10; but if you call the first 1, 10 then the next figure 2 you must call 20, the next 30, and so to 100.

And if you call the first 1, 100 then the next is 200, and the next 300, and so to the furthest 10, and that will be a thousand, and all the other intermediate Divisions are the tenth parts of Integers. The same way of [Page 3]Numeration as is explained in this is to be understood of all the rest of the Lines.

Example 1.

I would find the Number 25 on the Rule, I call the first 1, 10 and the second figure 20, and tell 5 tenths more which are also Inti­gers, where you will find a long stroke and that is 25.

Note that every fifth of the grand inter­mediate Divisions, are drawn forth with a longer Line than the rest, for ease of count­ing.

Example 2.

Let it be required to find the place of 144 upon the upper Line, call the first 1 upon your Line 100 for your first figure 1, then for 40 tell 4 of the grand Divisions for your second figure, and for the third figure which is 4 count four of the small intermediate Divisions, that very point is the place upon the Line representing 144.

Example 3.

Let it be required to find the place of 1690, for your first figure 1, count the 1 at the beginning to be 1000 for your second [Page 4]figure 6 count 6 of the grand Divisions which is 600, and for 90 count 9 tenths more which is the very point representing 1690.

Note by these examples you may perceive that the figures 1, 2, 3, 4, 5, 6, 7, 8, 9. do sometimes signifie themselves alone, some­times 10, 20, 30. &c. sometimes 100, 200, 300, &c. as the works thereby doth re­quire.

And by this varation and change of the Powers of these Numbers from 1 to 10 or 100 or 1000, any proportion either Arith­metically or Geometrically may be wrought; one, whereof I will insert for your better exercising on the Scale, by the often pract­ice whereof you will find the work facile and delightful, which shall be this follow­ing.

PROP. II.

Having two Numbers given, to find as many more as you please which shall be in continual proportion one to another as the two numbers were

For the working this Proposition, this is the Rule.

Place one foot of the Compasses in the first given Number, on the upper line, and extend the other foot to the other given Number; then may you turn from that se­cond Number, to a third, to a fourth, (as far as you can go on the upper line) then to a fifth, sixth, if the rule will admit.

Example.

Let the too given Numbers be 2 and 4, place one foot of your Compasses on 2, and extend the other foot to 4, that foot which now standeth in 2, being turned over will reach to 8, and so far it will come upon the up­per line; then bring it to 8 in the lower line, and turn the Compasses from 8 to 16, and from 16 in the upper line to 32, and from 32 to 64, and from 64 to 128, to 256 to 512 in the upper line, and so you may pro­ceed until you come to 4096.

Again let the 2 Numbers be 10, and 12, which you will find in the lower line, and that Extent will reach from 12 to 1, 4, 4, and from thence to 17, 28.

But if the Numbers were 1 and 12, which you must find on the lower line, then the third proportional will be 144, and the fourth 1728, and all with the same extent of the Compasses.

CHAP. II. Multiplication by the Proportional Scale.

IN Multiplication the Proportion is this: As 1 upon the line is to one of the Num­bers to be Multiplied, so is the other Num­ber to be Multiplied to the Product of them, which is the Number sought.

Example. 1.

Let it be required to Multiply 5 by 7, the Proportion is as 1 to 5, so is 7 to 35.

Therefore set one Foot of the Compasses in 1, in the lower line and extend the other Foot to 5, with that extent of the Compasses, place one Foot in 7, and the other Foot will fall on 35, which is the Product, (which is performed on the lower line) by extending [Page 7]from 1 in the middle of the Line to 5 back­wards, the same extent the same way will reach from 7 to 35, the Product required.

Example. 2.

Let it be required to Multiply 34 by 9, the Proportion is, as 1 is to 9, so is 34 to 306, set one Foot of the Compassess (in the low­er Line) from 1 or 10, in the middle, to 9 the same extent, will reach from 34 to 306.

Otherwise set one Foot in 1, and extend the other to 34, the same extent from 9 shall reach to 306.

Example. 3.

Let it be required to Multiply 8, 75/100 by 6, 45/100 the Proportion is as 1 to 8, 75, so is 6, 45, to 56, 48, set one Foot (in the lower Line) from 1 to 8, 75, the same extent the same way will reach from 6,45, to 56,44, fere.

Or if you set one Foot in 1, and extend the other to 6, 45, the same extent shall reach from 8,75, to 56,44 almost, or 56¾.

CHAP. III. Division by the Proportional Scale.

IN Division there are 3 things to be taken notice of, viz. The

  • Dividend, or Number to be Divided.
  • Divisor, the Number by which the Divi­dend is to be divided.
  • Quotient, which is the Number sought.

And as often as the Divisor is contained in the Dividend so often doth the Quotient con­tain Unity.

For the working of Division this is the Analo­gy or Proportion.

As the Divisor, is to Unity or 1, so is the Dividend to the Quotient.

Example. 1.

Let it be required to divide 35 by 7, The Proportion is as 7 to 1, so 35 to 5; set [Page 9]one Foot of the Compasses in 7, (in the lower Line) and extend the other Foot to 1, that same extent will reach from 35 to 5, which is the Quotient: otherwise extend the Compasses from 7 to 35, that same extent will reach from 1 to 5.

Example. 2.

Let it be required to Divide 34 by 306, the Proportion is, as 34 is to 1, so is 306 to 9; extend the Compasses from 34 to 1, (in the lower Line) the same extent the same way, will reach from 306 to 9 which is the Quotient; or if you extend the Com­passes from 34 to 306, the same extent shall reach from 1 to 9.

Example. 3.

Let it be required to divide 5644, by 8,75, the Proportion is as 8,75 is to 1, so is 56,44 to 6,45; extend the Compasses (in the lower Line) from 8,75 to 1, the smae extent the same way will reach from 56,44 to 6,45: or extend them from 8,75 to 56, 44, the same extent will reach from 1, to 6, 45, as before; now to know how many Fi­gures are to be separated for a Decimal Fraction in the Quotient; I refer you to the [Page 10]Eighth Note of the First Chapter of this Treatise.

CHAP. IV. The Golden Rule Direct.

THis Rule may well be termed the Golden Rule; it being the most useful of any others: for having 3 Numbers given, you may by it find a fourth, in proportion to them, as by divers Examples follow­ing, shall be made plain: And this Rule is performed upon the Scale, with ease and Exactness; And for the working of it upon the Scale of proportion, this is the general Analogy.

As the first Number given, is to the second Number given, so is the third Number gi­ven, to the fourth Number required: Or as the first Number given, is to the third Num­ber given, so is the second Number given, to the fourth Number Required.

Therefore, always extend the Compasses from the first Number to the second, and that distance or extent applyed the same way upon the Line, shall reach from the [Page 11]third to the fourth Number required. Or otherwise extend the Compasses from the first Number to the third, and that extent applied the same way shall also reach from the second to the fourth. Either of these ways will effect the same things; and it is necessary thus to vary the Proportion, some­times to avoid the opening of the Compasses too wide, for when the Compasses are open­ed to a very large extent, you can neither take off any Distance exactly, nor give so good an Estimate of any parts required, as you may do when they are opened to a lesser distance: But this you will find out best by Practice and therefore I will now proceed to examples.

Example. 1.

If 45 Yards of Cloth cost 30 l. what will 84 cost at the same rate?

Analogies.

As 45, to 30, so 84, to 56. Extend the Compasses from 45 to 30, (on the lower Line) the same extent the same way will reach from 84 to 56 l. the Price of 84 Yards. Or extend the Compasses from 45 to 84, the same extent will reach from 30 to 56, as be­fore.

[...]
[...]

Example. 2.

If 100 l. yeilds 6 l. Interest for one Year, or 12 Months, what shall 75 l. yeild?

The Analogy.

As 100 is to 6, so is 75, to 4, 50. Extend the Compasses (upon the lower Line of the Scale of Lineal proportion) from 100 to 6, the same extent will reach from 75, to 4, 50, which is 4 l. 10 s. and so much will 75 l. yeild Interest in one Year.

Example. 3.

If 75 l. yeilds 4 l. 10 s. or 4 [...]0/100 150 Interest for one Year or 12 Months, what will 105 l. yeild? As 75 to 4,50, so is 150 l. to 9 l. Ex­tend the Compases from 75 to 4, 50, the same extent will reach from 150, to 9 which is 9 Pounds, the Interest of 150 l. for one Year.

CHAP. V. The Golden Rule Reverse.

IN this Reverse or backward Rule of Three, this Note is especially to be Ob­served, That if the third Number be greater than the first, then the fourth Number will be less than the second. And on the Contra­ry, if the third Number be less than the first, then the fourth Number will be greater than the second; as by example.

Example 1.

If 12 Work-men do a piece of Work in in 8 days, how many Work-men shall do the same piece of work in 2 days?

Here it is to be noted that in the Question, 12 is not the first Number, (though it be first named) but 2, for the Middlemost Term of the three, must be of the same kind with the fourth Number, that is to be sought; [Page 14]as in this example it is Men, therefore 12 which are Men, must stand in the middle or second place, because the fourth Number which is to be sought is also Men, and there­fore the Numbers will stand thus.

Days,Men,Days,Men.
212848,

For if 8 days require 12 Men, then 2 days (which is but a fourth part of 8 days) shall require four times 12 Men, that is 48 Men; for here less requires more, that is, less time more Men; and hence the work is contrary to the Direct Rule: wherefore to effect it, extend the Compasses from 2 to 8, the same extent will reach from 12, (the contrary way on the lower Line) to 48, which is the Number of Men that will effect the same Piece of work in two days.

Example 2.

If 1 Close will graze 21 Horses for 6 weeks how many Horses will the same close graze for 7 weeks? Extend the Compasses from 6 to 7, for you must always extend your Com­passes to Numbers of one kind, or Denomi­nation, (as here 6 and 7 are both Horses) the same extent from 21 backwards to 18, [Page 15]and so many Horses will the same Close graze in 7 weeks,

CHAP. VI. Of Duplicate Proportion performed by the Scale.

DUplicate Proportion is such Proportion as is between Lines and Superficies, and between superficies and Lines.

1. Of the Proportion of Lines to Superficies,

In this Case extend the Compasses, from the first to the second Number of the same denomination, (taken upon the upper Line of the Scale of quadratique proportion) which shall give the distance (upon the low­er Line of the same Scale) from the third Number unto the fourth.

Example. 1.

If the Diameter of a Circle be 14 Inches, and the Area, or superficial Content thereof be 154 Square Inches, what will be the con­tent [Page 16]of another Circle, whose Diameter is 28 Inches; extend the Compasses from 14 to 28, (on the upper Line, of the Scale of Quadra­tique Proportion) that extent shall reach from 154 in the lower Line of the same Scale, to 616, and that is the Area or content of a Circle whose Diameter is 28.

II. Of the Proportion of Superficies to Lines.

In this case extend▪ the Compasses un­to half the distance, between the two Numbers of the same denomination; that same extent shall reach from the third Num­ber to the fourth required.

Example. 1.

Let there be two Circles given, the Area or content of one being 154, and its Diame­ter 14, the Area of the other Circle is 616, what is the length of its Diameter upon the lower Line of the Scale of Quadratique Pro­portion? divide the distance between 154, and 616 into two equal parts, then with that distance set one foot in 14, and the other shall fall upon 28.

Example. 2.

There is a piece of Land containing 20 Pole square worth 30 l. there is another piece worth 91 l. 16 s. how many Pole square ought that piece to contain? divide the space between 30 l, and 91 l. 16 s. into two equal parts, then set that foot in 20 Pole, and the other Foot will reach to 35 Pole, and so many Pole square must the Land be that is worth 91 l. 16 s. Note that 16 shillings upon the Line is 2/ [...]..

CHAP. VII. Of Cubical Proportion.

CUbical Proportion is such a proportion as is between Lines and Solids, or betwen Solids and Lines.

1. Of the Proportion between Lines and Solids.

In this case extend the Compasses from the first Number to the second, of the same de­nomination, [Page 18]that extent being placed (in the lower Line of the Scale of Cubical pro­portion) from the third Number to the fourth answereth the Question.

Example

There is a Bullet whose Diameter is 4 In­ches, weighing 9 l. (what shall another Bull­et of the same Metal weigh) whose Diame­ter is 8 Inches? Extend the Compasses from 4 to 8 (in the upper Line, the same extent will reach (in the lower Line, from 9 to 72, the weight of the Shot whose Diameter is 8 Inches.

2. Of the Proportion of Solids to Lines.

In this case extend the Compasses into the third part of the distance between the two Numbers of like denomination, the sme ex­tent shall reach from the third to the fourth Number.

Example.

The weight of a Cube being 72 Pound, the side thereof is 8 Inches, and the weight of another Cube of the same matter, weigh­ing 9 l. what must the side be? Upon the lower Line, divide the distance between 9, [Page 19]and 72, into three equal Parts; then set one Foot of that distance in 8, and the other Foot shall rest in 4, the length of the side of the Cube required.

CHAP. VII. To Extract the Square Root, by the Proportional Scale, by Inspection.

TO Extract the Square Root, is to find a mean Proportional Number, between 1 and the Number given, which is common­ly done, by dividing the Square between them into two equal Parts; but upon the Scale of Quadratique proportion, it is found by Inspection; therefore if you seek 36 (in the lower Line of Quadratique Proportions) you will find in the upper Line, right against it 6, which is the Square Root thereof; in the same manner you may find the Square-Root of 81 to be 9, and of 144 to be 12, and of 256 to be 16.

CHAP. IX. To Extract the Cube Root upon the Proportional Scale, by Inspection.
Example.

LEt it be required to find the Cube Root of 216; seek (in the lower Line of the Scale, of Cubical proportion) for 216, and in the upper Line, right against it you will find 6, the CubeRoot required. In like man­ner you will find the Cube Root of 729, to be 9, of 1728, to be 12.

CHAP. X. Of the Mensuration of Divers Re­guler Superficial Figures by the Proportional Scale.
I. Of the Circle.

Example. 1.

THE length of the Diameter of any Circle given, to find the Circumfer­ence thereof. The Proportion between the Diameter and the Circumference of any Circle, is, as 7 to 22, wherefore if the Di­ameter of a Circle given, be 12 Inches, the Circumference thereof may be found by the following Analogy. As 7 is to 22, so is 12 to 37, 69. (In the Line of lineal Proporti­on) extend the Compasses from 7 in the up­er Line, to 22 in the lower Line; the same [Page 22]extent shall reach from 12 to 37, 69, the Circumference required.

Example 2.

The Circumference of a Circle being giv­en, to find the length of the Diameter.

This is the Converse of the former Ex­ample, and the Analogy is the Converse also.

Let the Circumference of a Circle be 37 Inches, 69 Parts, what is the length of the Diameter? As 22 is to 7, so is 37, 69 parts to 12 Inches, the Diameter sought.

Extend the Compasses from 22 to 7, in the Scale of Lineal Proportion; the same Ex­tent will reach from 37, 69, to the Diameter required.

Example 3.

The Diameter of a Circle being given, to find the Area or Superficial Content thereof.

Let the Diameter of a Circle be 15 In­ches, extend the Compasses (upon the Scale of Quadratick Proportion) from 1 in the lower Line, to 15 in the upper Line; the same extent shall reach always from 78, 54, to 176, 25, on the same Line, which is the Area of that Circle.

CHAP. XI. Of Spherical Bodies, such as Globes or Bullets.

Example 1.

THE Circumference of a Globe or Bul­let being 28 Inches, 28, parts, to find Diameter.

The ANALOGY.

As 22 is to 7, so is 28, 28, the Circum­ference, to 9 Inches the Diameter.

Extend the Compasses from 22 to 7, (on the Scale of Cubical prportions in the lower Line) the same Extent will reach from 28, 28, the Circumference, to 9 Inches, the Di­ameter of the Bullet sought.

Example 2.

The Diameter of a Shot being given 9 Inches, and its Circumference is 28 Inches, 28 parts: how many square Inches is there in the Superficies of that Bullet?

ANALOGY.

As 1 is to 9 Inches, so is 28, 28, the Cir­cumference to 254 Inches 5 parts, the su­perficial Inches in that Bullet.

Extend the Compasses from 1 to 9, (in the lower Line of the Scale of Cubical Pro­portion) the same extent shall reach from 28, 28, the Circumference to 254 Inches, 5 parts, the superficial Inches of that Bullet.

Example 3.

The Diameter of a Bullet being 9 Inches; how many Cubical Inches are therein con­tained?

The ANALOGY.

I. As 1 is to 9 the Diameter, so is 9 to a 4th Number, and that 4th Number to 729 the Cube of the Diameter.

II. As 9 the Diameter is to 729, its Cube. so is 11 to 891 Cubical Inches in that Bul­let.

Extend the Compasses from 1 to 9 in the Cubical-Scale, that extent will reach to 81, and from 81 to 729 the Cube of the Diame­ter; then extend the Compasses from 9 the Diameter to 729 its Cube, that extent will reach from 11 to 891 Inches, the solid Con­tent of the Bullet.

The Description of the Sea-Gun­ners RULE, being the Epitome of the Art of Gun­nery.

WHich takes the Convex Diameter of any Shot, the Concave Diameter of the Bore of any Gun, from the Base to the Cannon-Royal, on which is inserted a Line of Numbers for the ready working of any Question in Gunnery.

As also several Lines, shewing the Weight and Diameter of any Shot, with the weight of Powder and Shot for any Piece of Ord­nance, [Page 26]the Weight of the Piece, the Length and Breadth of the Ladle, and several other useful matters, as a Line of Chords, a Line of Rhumbs, Leagues and Longitude, (sup­posing that every Sea-Gunner is also a Na­vigator;) there is also a Circle that taketh any Angle, and will also shew the Degrees of Mounture of any Piece of Ordnance.

The Description of the Sea-Gun­ners Rule.

The Rule is a Foot in Length when open; on the Edg is a Line of Inches for the Measuring any thing necessary to be Measured, and may be of any other Length at pleasure.

Through the Head goeth a Brass Semi­circle fixed to the standing part of the Rule, on which are cut the Divisions that give the Diameter of a Shot, whose In­ches are shewed by the inside of the moveable leg of the Rule.

And the Concave Diameter of a Gun is cut upon the Brass Semicircle by the out­side of the moveable Leg.

There is also a Circle of Degrees upon the outward Limb or Edg of the Semi­circle, which is to give the Quantity of any Angle cut by the inside of the Move­able Leg of the Rule; by which you may observe the quantity of any Angle to give the Degree of Mounture, with the help of a Plummet that is to hang upon the movable Leg of the Rule.

The Ʋses follow.

A Description of one side of the Rule.

On one side of the Rule is placed a Line of Inches, abutting against another Line, which shews the weight of Iron-shot.

And on the same side is placed a Line of Numbers for the answering any Question in Gunnery, with the help of a pair of Com­passes.

A Description of the other side of the Rule.

There are several Lines on this side which shew,

The Names of the Guns.

The Weight of Powder.

The Weight of Shot fit for each Gun.

The Length of the Ladle for each Piece.

The Breadth of the Ladle.

The Weight of each sort of Guns.

Upon the same side is also plac'd the Lines of the plain Scale as,

The Line of Leagues.

The Line of Rhumbs.

The Line of Longitude.

And the Line of Chords.

All which is for the Accommodation of the Sea-Gunner, who is also a Mariner as well as Gunner.

In the inside of the Rule there lies two Brass Legs, which being taken out and opened as far as they will open, the two Points will do the Office of a pair of Ca­labers.

To take the Diameter of a Shot, cut by the inside of the moveable Leg upon the edge of the Brass Semicircle. The Uses fol­low.

Ʋse 1. To find the Concave Diameter of the Mouth of any Piece of Ord­nance.

This is performed by the two outer Cor­ners of the Rule, being put to the inside of the Concavity of any Gun, and open the Rule as wide as it will permit, then on the Brass Semicircle will be cut (by the outer edge of the Rule) the Diameter of the Concavity of the said Piece.

Ʋse 2. To find the Diameter of a Shot.

To perform this you must open the Brass Legs in the inside of the Rule, to their utmost Extent, then open the Points of them to the Diameter of the Shot, and the quantity of Inches and Parts will be cut upon the Semicircle, by the inside of the Leg.

Ʋse 3. To find the quantity of an Angle, or to find the degree of Moun­ture of any Piece of Ordnance.

To find the quantity of any Angle upon the Legs of the Rule to any Angle required, and the inside of the moveable Leg will cut the quantity of the Angle upon the Brass Semicircle.

To find and to set the degree of Mounture of a Piece of Ordnance.

You must hang a Threed and Plummet upon the moveable Leg, and put the fixed [Page 31]Leg into the Muzzle of the Piece, and open the moveable Leg until the Plummet falls perpendicular upon the Line, upon which the pin is fixed (upon which the Plummet is hang'd) and on the edge of the Brass Semi­circle, will be cut the Angle of the degree of Mounture by the inside of the moveable Leg.

Ʋse 4. The Ʋses of that side of the Rule, upon which the Line of Num­bers is placed.

On this side is placed a Line of Numbers for the ready operation of any Question in Gunnery, with the help of a pair of Com­passes. The manner of working on the Line of Numbers is shewn in the use of the Pro­portional Scale, in this Treatise, to which I refer you.

On this side of the Rule is plac'd a Line of Inches, on which may be found the Dia­meter of any Shot: And upon the Line ad­joyning to it is shewn the Weight of any Shot whose Diameter is given.

Example 1.

A Shot of four Inches Diameter the Weight is required.

Seek 4 in the Line of Inches (upon this flat side) and just against it you will find 9, which shews that a Shot of four Inches Diameter weighs nine Pounds.

Example 2.

A Shot of eight Inches Diameter the Weight is required.

Seek for 8 on the Line of Inches (on the flat side) and right against it (on the adjoyning Line) you will find 72 which is the Weight of the Shot that is eight Inches Diameter.

Ʋse 5. The Ʋse of the other side of the Rule.

On this side of the Rule are six Lines,

The First shews the Names of the Pieces, exprest by the several Names as, F for [Page 33]Faulcon, M for Minion, S for Saker, &c.

The Second, the Weight of Powder.

The Third, the Weight of Shot.

The Fourth, the Length of the Ladle.

The Fifth, the Breadth of the Ladle.

The Sixth, the Weight of the Piece.

One Example will shew their several uses, which shall be the Minion, which you will find upon the Line with the Letter M, where under that Line you will find 3 which is, three Pound of Powder for Ser­vice, and 4 under that, which is four Pound, the Weight of the Shot.

And 15 in the next Line under that, which shews that the Length of the Ladle is fifteen Inches, and in the next Line un­der that you find 6, which is six Inches, the Breadth of the Ladle; and in the last Line under 6 you will find 1100, the Weight of the Piece.

On this side are plac'd all the Lines of the Plain Scale, the Uses of which I shall not handle in this place, but refer you to my Book of Practical Navigation, where the use of every one of them is shewn at large.

FINIS.

A Catalogue of Books and Instruments belonging to the Art of Navigation and Gunnery; Sold by John Seller, Senior, at the Hermitage in Wap­ping.

Books in Folio.
  • ENglish Pilot for the Northern, Southern, Eastern, and Western Navigation.
  • Sea-Atlas, describing the Sea-Coasts in all the known parts of the World.
  • The Coasting Pilot, describing the Coasts of England, Holland, Zeeland, and France, as far as Silly, and Ʋshant.
  • A description of the Sands, Buoyes, Bea­cous and Sea-marks upon the Coast of Eng­land, from Dover to Orfordness.
  • Atlas Coelestis; shewing all the Constella­tions of the Stars, and other Phenomena's of the Coelestial Bodies, with Maps of the Sun and Moon, &c.
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Books in Quarto.
  • PRactical Navigation; being an Intro­duction to the whole Art; containing many Geometrical Desinitions and Problems, [Page]the Doctrin of plain and spherical Triang­les, plain Mercator and great Circle sayl­ing; sundry useful Problems in Astronomy, the use of Instruments in Navigation, the Azimuth Compass, Ring-dyal, the Forestaff, Quadrant, Plain-Scale, Gunters-Scale, Sini­cal Quadrant, plain Chart, Mercators Chart, both Globcs: Useful Tables of the Moons Age, of the Tides, and of the Sun's Place and Declination, Tables of Lines; Tangents and Sea-carts, and Logarithms, &c.
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Books in Octavo.
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  • Atlas Maritimus, describing the Sea-Coasts in all the known parts of the World.
  • Sea-Gunnery, containing the whole Art of Gunnery as it is used at Sea; containing two Compendiums of Arithmetick in Vulgar and Decimal, and necessary Tables belonging to that Art.
  • Tabular Dyalling, shewing the making Horizontal and Vertical Dyals for all Lati­tudes in the whole World.
  • An Almanack for an Age, shewing the Primes and Dominical Letters to the end of the World; as also, the day of the Month, the rising and setting of the Sun, and several other useful remarks.
  • Atlas Minimus, showing all the Empires and Kingdoms in the World, with Geogra­phical Tables, explaining the Divisions in each Map, in Twelves.

Insteuments of Navigation.

Meridian Compasses, Azimuth and Ampli­tude Compasses, Sea-quadrants, Cross-staves, Nocturnals, Gunter's Scales, Plain-Scales; Pocket-Compasses, Brass-Compasses of all sorts, Sinical-quadrants, Sectors, Brass-ring­dials, Load-stones, Hour-glasses, [...] Hours, ½ Minute, Watch-glasses, and ½ Watch-glasses, Black-lead Pencils, Slates and Slate-Pens, Pens, Telescopes, Perspective-glasses, Pock­et-Globes.

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