man lighting a cannon, parts labelled 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: Showing 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 ARITHMETIC, With necessary Tables relating to that ART. To which is added An APPENDIX, Showing 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 itself. 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 itself; (for those Books that were Extant, were chief intended for Land Service;) and at this time most of them being out of Print, I judged 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 Arithmetic; for the sake of such, I have exhibited two Compendiums thereof, one in Vulgar, and the other in Decimal Arithmetic, 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 itself, 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 Arithmetic, 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 CHAP. II. Of Decimal Arithmetic, 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 Arithmetic, 52 To extract the Sqaure-Root by Logarithms and Gunter's-Scale. 56 To extract the Cube-Root by Arithmetic, 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 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 parallel 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 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 Arithmetic, 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 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 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 showing the Height and Weight of Iron, Led and Stone Shot, according to their Diameters in Inches and Quarters, 134 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 Former's for any sort of Ordnance, 148 CHAP. IX. Containing certain Theorems in Gunnery, 150 A Table of Right Ranges, or point blank at several degrees of Mounture, 156 CHAP. X. Necessary Instructions for a Gunner, 157 CHAP. XI. Showing an easy way to dispart a Piece of Ordnance. 164 CHAP. XII. To levelly 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 Flaws, Cracks or Honey Combs in any Piece, 168 CHAP. XIV. How Moulds, Former's 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 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 Gunpowder, 193 To renew and make good again any sort of Gunpowder, having lost its strength by moisture, long-lying, or by any other means, 195 CHAP. XVIII. How to make Hand-Granadoes to be have 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 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 Arithmetic. CHAP. I. ARITHMETIC is the Science of Numbering, 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, Unites, Thousands, Millions, etc. 2. Their Values is reckoned from the Right-hand. 3. Their Degrees or Places, are tenfold, etc. 4. Their Periods, are Unites, Ten, Hundreds, which are Illustrated in the following Table. A Table of Notations. Names Millions Thousands Unites. Value CXI CXI CXI Degrees or Places 987 654 321 Periods 3 2 1 Integers. 999 999 999 888 888 888 777 777 777 666 666 666 555 555 555 444 444 444 333 333 333 222 222 222 111 111 111 RULE. 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. NUMERATION. NVmeration 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 cipher, and begin to number them at the Right-hand, to the Left, increasing each Figure ten-times as before. ADDITION. Addition is the gathering of two, or more Numbers into one Sum, and hath two general Cases. CASE I. In Addition of Ten, Hundreds, Thousands, etc. RULE. Draw a line under the Numbers given, begin at the Right-hand, and first place; add up the Unites, carry the Ten 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 Year 1690 5729 In Addition of Integers and Parts. RULE. 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 own a Merchant several Sums of Monev; it is required to know the Sum of those Debts. l. s. d. One Man owes, 230 17 02 Another owes, 110 16 04 Another, 074 10 09 Another, 979 08 11 The Total Debt is 1395 13 02 SUBTRACTION. SVbtraction is the taking a lesser Number from a greater, or an Equal from an Equal. What remains, is the Residue, or Excess, and bathe two Cases. CASE I. In Subtraction of Ten, Hundreds, Thousands, from Ten, Husdreds, Thousands, etc. RULE. 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 Merchant 9758 And he hath paid of that Debt, 3514 There Remains due, 6244 CASE II. When some of the inferior Numbers are greater than the superior Numbers. RULE. 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 Lefthand above, out of which Subtract, what remains add to the superior, and set their Sum under the line; than what you borrow, pay to the next Number on the Lefthand below, and so proceed throughout the work, according to this or the former Rule. l. s. d. As from 529 13 4 Take 347 16 7 Rests, 171 16 9 Proof. Add the two inferiors; their Sum is equal to the superior. MULTIPLICATION. MVltiplication 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. 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 6 9 12 15 18 21 24 27 4 8 12 16 20 24 28 32 36 5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81 The Use 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 Ten to the next place; then draw the Multiplier into the second Figure of the Multplicand, and add the Ten you carried to the Units of that Product, subscribe the Units of their Sam, and carry the Ten 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 Ten, 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 Ciphers are in the Multiplicand and Multiplier, or either of them; Rule. Set down to the right hand of the first Product as many Ciphers 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 cipher or Ciphers be in the middle of the Multiplicand; Rule. Work according to the former Rules till you come to the Ciphers, then under the first o, subscribe the Ten you carried; but under the rest of the Ciphers set Ciphers, 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 cipher or Ciphers be in the middle of the Multiplier; Rule. Multiply as before is taught, until you come to the Ciphers 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 Ciphers be both in the middle of the Multiplicand, and also in the Multiplier; Rule. When you come to the Ciphers in the Multiplicand, then under the first Ciphers place, set the Ten you carried (if any be) and after that, as many Ciphers 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 Ten 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 cipher. Example. As, 45276 Product 226380 being to be multiplied by 5, halve the Number, and add a cipher at the latter end, and the Work is done. CASE II. To multiply any Number by 9 Rule. Add a cipher to the Number given to be multiplied by 9, and subtract the first Number out of it, and the Remainder is the Product or Answer of the Question. Example. Let the Number be 6789, to which add o cipher, and the Number is thus, 67890; 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 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 Man's 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 pricked 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? 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, etc. so respectively must the other be; and the like 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 RULE 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 RULE 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: W Miles l. s. d. W. Miles 600 50 1 6 6 2500 100 To work this, you must first reduce the Money into the lowest Denomination expressed, 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, than 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. 11 00 10 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 Lefthand 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 Pioners 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. Pioners. Months. 16 24 4 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 24 by 16, and divide the Product by 4, the Quotient will be 96; as doth appear by the work. Which shows that 96 Pioners must be employed to finish the Moat in 4 Months. CHAP. II. A COMPENDIUM OF Decimal Arithmetic. Note 1st. Notation of FRACTIONS. Numerators, 5 15 150 1070 Denominators, 10, 100, 1000, 10000 Note 2d. Of how many places soever the Numerator of a Decimal Fraction doth consist, of so many Ciphers with a Unite before them, do the Donominators consist. So the Denominator of 5 is 10, of 15 is 100, of 005 is 1000, etc. Note 3d. When the Numerator of a Decimal Fraction consists not of so many places as the Denominator hath Ciphers, prefix so many Ciphers on the left hand as is directed in Note 2 d. So 5/100 is written thus, 05; 15/1000 is writ thus, 015; 50/10000 thus, 0050; 6/1000 thus, 006. Note 4th. Ciphers 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 cipher or Ciphers cut off equal Ciphers in both; so will the Fraction be reduced into lesser terms, 2 0 10 0 2 00 10 00 2 000 10 000 Thus, 20/100 200/1000 or 2000/10000 Are reduced as in the Table. Note 5th. Ciphers 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 Ciphers 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 Ciphers 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 expressed, 1/140. Therefore (as before) add Ciphers 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; 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 Farthing. 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, etc. 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 Ciphers, or at least supposed to be filled up. Example. Suppose ,47,309 is to be subtracted from 54, you are to put so many Ciphers as will make up the Fraction, and then Subtract, and the work will stand Thus, Or thus, 54,000 38,000 ,47 309 0, 130 07691 37,860 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 supplied with a cipher or Ciphers, 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 Ciphers 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 Arithmetic, 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, etc. 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 cipher or Ciphers 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 Ciphers 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 Ciphers; so the general 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 a cipher 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 cipher 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 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 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 Ciphers 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. 1 0010416 2 0020833 3 0031250 I 0041666 1 0052083 2 0062500 3 0072916 TWO 0083333 1 0093750 2 0104166 3 0114583 III 0125000 1 0135416 2 0145833 3 0156250 IV 0166666 1 0177083 2 0187500 3 0197916 V 0208333 1 0218750 2 0229166 3 0239583 VI 0250000 1 0260416 2 0270833 3 0281250 VII 0291666 1 0302083 2 0312500 3 0322916 VIII 0333333 1 0343750 2 0354166 3 0364583 IX 0375000 1 0385416 2 0395833 3 0406250 X 0416666 1 0427083 2 0437500 3 0447916 XI 0458333 1 0468750 2 0479166 3 0489583 XII 0500000 1 0510416 2 0520833 3 0531250 XIII 0541666 1 0552083 2 0562500 3 0572916 XIV 0583333 1 0593750 2 0604166 3 0614583 XV 0625000 1 0635416 2 0645833 3 0656250 XVI 0666666 1 0677083 2 0687500 3 0697916 XVII 0708333 1 0718750 2 0729166 3 0739583 XVIII 0750000 1 0760416 2 0770833 3 0781250 XIX 0791666 1 0802083 2 0812500 3 0822916 XX 0833333 1 0843750 2 0854166 3 0864183 XXI 0875000 1 0885416 2 0895833 3 0906250 XXII 0916666 1 0927084 2 0937500 3 0947916 XXIII 0958333 1 0968750 2 0979166 3 0989583 XXIV 1000000 A Table of Decimals of one Pound Sterling in Shillings. Sh. Decim. 1 050000 2 100000 3 150000 4 200000 5 250000 6 300000 7 350000 8 400000 9 450000 10 500000 11 550000 12 600000 13 650000 14 700000 15 750000 16 800000 17 850000 18 900000 19 950000 20 100000 21 105000 22 110000 23 115000 24 120000 25 125000 26 130000 27 1350000 28 1400000 29 1450000 30 1500000 31 1550000 A Table of the Decimals of a Foot to every Inch and Eighth part of an Inch. Inches. 8 Part. Decimal. 1 001041 2 002083 3 003125 4 004166 5 005208 6 006250 7 007291 I 008333 1 009375 2 010416 3 011458 4 012500 5 013541 6 014583 7 015625 TWO 016666 1 017708 2 018750 3 019791 4 020833 5 021875 6 022926 7 023958 III 025000 1 026041 2 027208 3 028125 4 029166 5 030200 6 031299 7 032291 IV 033333 Inches. 8 Part. Decimal. 1 034385 2 035416 3 037395 4 037499 5 038541 6 039583 7 040625 V 041666 1 042610 2 043750 3 044718 4 045833 5 046875 6 047927 7 048854 VI 050000 1 051104 2 052083 3 053125 4 054166 5 055207 6 056250 7 057291 VII 058333 1 059375 2 051041 3 061457 4 062500 5 063531 6 064583 7 065625 VIII 066000 1 067610 2 068750 3 069896 4 070833 5 071875 6 072916 7 073958 IX 075000 1 076041 2 077083 3 078125 4 079166 5 080208 6 081250 7 082291 X 083333 1 084375 2 085416 3 086457 4 087500 5 088541 6 089687 7 090625 XI 091666 1 092708 2 093750 3 094791 4 095833 5 096875 6 097926 7 098958 XII 100000 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 shows the Inches and Eight parts of a Foot, and the Second Column shows the Decimal Number answering thereto. Example. Seek for 11 Inches, and 8/4 or a half in the First Column, 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; 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 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 writ 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 writ 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 writ 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 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 is 2,79588 The half thereof is 1,39794 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 CUBE 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 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, 1 8 27 64 125 216 343 512 729 1000 Squar. 1 4 9 16 25 36 49 64 81 100 Roots, 1 2 3 4 5 6 7 8 9 10 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 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 Ten) 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 the First Figure (towards the Left hand) of the Divisor is contained in the correspondent part of the Dividend, writ 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 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 erroneous, 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 is 3,23754 The Third part thereof is 1,07918 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. 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 16 256 17 289 18 324 19 361 20 400 21 441 22 484 23 529 24 576 25 625 26 676 27 729 28 784 29 841 30 900 31 961 32 1024 33 1089 34 1156 35 1225 36 1296 37 1369 38 1444 39 1521 40 1600 41 1681 42 1764 43 1849 44 1936 45 2025 46 2116 47 2209 48 2304 49 2401 50 2500 51 2601 52 2704 53 2809 54 2916 55 3025 56 3136 57 3249 58 3364 59 3481 60 3600 61 3721 62 3844 63 3969 64 4096 65 4225 66 4356 67 4489 68 4624 69 4761 70 4900 71 5041 72 5184 73 5329 74 5476 75 5625 76 5776 77 5929 78 6084 79 6241 80 6400 81 6561 82 6724 83 6889 84 7056 85 7225 86 7396 87 7569 88 7744 89 7921 90 8100 91 8281 92 8464 93 8649 94 8836 95 9025 96 9216 97 9409 98 9604 99 9801 100 10000 A Table of Cubick Roots from One to an Hundred. R. Cube. 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000 11 1331 12 1728 13 2197 14 2744 15 3375 16 4096 17 4913 18 5832 19 6859 20 8000 21 9261 22 10648 23 2167 24 13824 25 5625 26 17576 27 19683 28 21972 29 24389 30 27000 31 29791 32 32768 33 35937 34 39304 35 42825 36 48656 37 50653 38 54872 39 55419 40 64000 41 68921 42 74088 43 79507 44 85184 45 91125 46 97336 47 103823 48 110592 49 117649 50 125000 51 135651 52 140608 53 148877 54 157464 55 167375 56 175616 57 185193 58 195112 59 205379 60 216000 61 226981 62 238328 63 293047 64 262244 65 274625 66 287496 67 300753 68 314432 69 329199 70 333000 71 357011 72 373348 73 389017 74 405224 75 411875 76 438976 77 456533 78 474522 79 493039 80 512000 81 531441 82 550408 83 571787 84 592604 85 614125 86 636056 87 648303 88 681472 89 705669 90 729000 91 753571 92 778688 93 804357 94 830584 95 857375 96 884736 97 915673 98 941192 99 970299 100 1000000 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. 1 0,00000 2 0,30103 3 0,47712 4 0,60206 5 0,69897 6 0,77815 7 0,84510 8 0,90309 9 0,95424 10 1,00000 11 1,04139 12 1,07918 13 1,11394 14 1,14613 15 1,17609 16 1,20412 17 1,23045 18 1,25527 19 1,27875 20 1,30103 21 1,32222 22 1,34242 23 1,36173 24 1,38021 25 1,39794 26 1,41497 27 1,43136 28 1,44716 29 1,46239 30 1,47712 31 1,49136 32 1,50515 33 1,51851 34 1,53148 35 1,54407 36 1,55630 37 1,56820 38 1,57978 39 1,59106 40 1,60206 41 1,61278 42 1,62325 43 1,63347 44 1,64345 45 1,65321 46 1,66276 47 1,67210 48 1,68124 49 1,69020 50 1,69897 51 1,70757 52 1,71600 53 1,72428 54 1,73239 55 1,74036 56 1,74819 57 1,75587 58 1,76343 59 1,77085 60 1,77815 61 1,78533 62 1,79239 63 1,79934 64 1,80618 65 1,81291 66 1,81954 67 1,82607 68 1,83251 69 1,83885 70 1,84510 71 1,85126 72 1,85733 73 1,86332 74 1,86923 75 1,87506 76 1,88081 77 1,88649 78 1,89209 79 1,89763 80 1,90309 81 1,90848 82 1,91381 83 1,91908 84 1,92428 85 1,92942 86 1,93450 87 1,93952 88 1,94448 89 1,94939 90 1,95424 91 1,95904 92 1,96379 93 1,96848 94 1,97313 95 1,97772 96 1,98227 97 1,98677 98 1,99123 99 1,99563 100 2,00000 101 2,00432 102 2,00860 103 2,01284 104 2,01703 105 2,02119 106 2,02531 107 2,02938 108 2,03342 109 2,03743 110 2,04139 111 2,04532 112 2,04922 113 2,05308 114 2,05690 115 2,06070 116 2,06446 117 2,06819 118 2,07188 119 2,07555 120 2,07918 121 2,08278 122 2,08636 123 2,08990 124 2,09342 125 2,09691 126 2,10037 127 2,10380 128 2,10721 129 2,11059 130 2,11394 131 2,11727 132 2,12057 133 2,12385 134 2,12710 135 2,13033 136 2,13354 137 2,13672 138 2,13988 139 2,14301 140 2,14613 141 2,14922 142 2,15229 143 2,15534 144 2,15836 145 2,16137 146 2,16435 147 2,16732 148 2,17026 149 2,17319 150 2,17609 151 2,17898 152 2,18184 153 2,18469 154 2,18752 155 2,19033 156 2,19312 157 2,19590 158 2,19866 159 2,20140 160 2,20412 161 2,20683 162 2,20951 163 2,21219 164 2,21484 165 2,21748 166 2,22011 167 2,22272 168 2,22531 169 2,22789 170 2,23045 171 2,23300 172 2,23553 173 2,23805 174 2,24055 175 2,24304 176 2,24551 177 2,24797 178 2,25042 179 2,25285 180 2,25227 181 2,25768 182 2,26007 183 2,26245 184 2,26482 185 2,26717 186 2,26951 187 2,27184 188 2,27416 189 2,27646 190 2,27875 191 2,28108 192 2,28330 193 2,28550 194 2,28780 195 2,29003 196 2,29226 197 2,29447 198 2,29666 199 2,29884 200 2,30103 201 2,30320 202 2,30535 203 2,30750 204 2,30963 205 2,31175 206 2,31387 207 2,31597 208 2,31806 209 2,32015 210 2,32222 211 2,32428 212 2,32634 213 2,32828 214 2,33041 215 2,33244 216 2,33445 217 2,33646 218 2,33846 219 2,34044 220 2,34223 221 2,34439 222 2,34635 223 2,34830 224 2,35025 225 2,35218 226 2,35411 227 2,35603 228 2,35793 229 2,35983 230 2,36173 231 2,36361 232 2,36549 233 2,36736 234 2,36922 235 2,37107 236 2,37291 237 2,37475 238 2,37658 239 2,37840 240 2,38021 241 2,38202 242 2,38381 243 2,38561 244 2,38739 245 2,38917 246 2,39093 247 2,39270 248 2,39445 249 2,39620 250 2,39794 251 2,39967 252 2,40140 253 2,40312 254 2,40483 255 2,40654 256 2,40824 257 2,40993 258 2,41162 259 2,41330 260 2,41497 261 2,41664 262 2,41830 263 2,41996 264 2,42160 265 2,42325 266 2,42488 267 2,42651 268 2,42813 269 2,42975 270 2,43136 271 2,43297 272 2,434●● 273 2,436●● 274 2,43775 275 2,43933 276 2,44091 277 2,44248 278 2,44404 279 2,44560 280 2,44716 281 2,44871 282 2,45025 283 2,45179 284 2,45332 285 2,45484 286 2,45636 287 2,45788 288 2,45939 289 2,46090 290 2,46240 291 2,46389 292 2,46538 293 2,46687 294 2,46835 295 2,46982 296 2,47129 297 2,47276 298 2,47422 299 2,47567 300 2,47712 301 2,47857 302 2,48001 303 2,48144 304 2,48287 305 2,48430 306 2,48572 307 2,48714 308 2,48855 309 2,48996 310 2,49136 311 2,49276 312 2,49415 313 2,49554 314 2,49693 315 2,49831 316 2,49969 317 2,50106 318 2,50243 319 2,50379 320 2,50515 321 2,50650 322 2,50786 323 2,50920 324 2,51054 325 2,51188 326 2,51322 327 2,51455 328 2,51587 329 2,51720 330 2,51851 331 2,51983 332 2,52114 333 2,52244 334 2,52375 335 2,52504 336 2,52634 337 2,52763 338 2,52892 339 2,53020 340 2,53148 341 2,53275 342 2,53403 343 2,53529 344 2,53656 345 2,53782 346 2,53908 347 2,54033 348 2,54158 349 2,54282 350 2,54407 351 2,54531 352 2,54654 353 2,54777 354 2,54900 355 2,55023 356 2,55145 357 2,55267 358 2,55388 359 2,55509 360 2,55630 361 2,55751 362 2,55871 363 2,55991 364 2,56110 365 2,56229 366 2,56348 367 2,56467 368 2,56585 369 2,56703 370 2,56820 371 2,56937 372 2,57054 373 2,57171 374 2,57287 375 2,57403 376 2,57519 377 2,57634 378 2,57749 379 2,57864 380 2,57978 381 2,58092 382 2,58206 383 2,58320 384 2,58433 385 2,58346 386 2,58659 387 2,58771 388 2,58883 389 2,58995 390 2,59106 391 2,59218 392 2,59329 393 2,59439 394 2,59549 395 2,59660 396 2,59769 397 2,59879 398 2,59988 399 2,60097 400 2,60206 401 2,60314 402 2,60423 403 2,60530 404 2,60638 405 2,60745 406 2,60853 407 2,60959 408 2,61066 409 2,61172 410 2,61278 411 2,61384 412 2,61490 413 2,61595 414 2,61700 415 2,61805 416 2,61909 417 2,62014 418 2,62118 419 2,62221 420 2,62325 421 2,62428 422 2,62531 423 2,62634 424 2,62737 425 2,62839 426 2,62941 427 2,63043 428 2,63144 429 2,63246 430 2,63347 431 2,63448 432 2,63548 433 2,63649 434 2,63749 435 2,63849 436 2,63949 437 2,64048 438 2,64147 439 2,64246 440 2,64345 441 2,64444 442 2,64542 443 2,64640 444 2,64738 445 2,64836 446 2,64933 447 2,65031 448 2,65128 449 2,65225 450 2,65321 451 2,65418 452 2,65514 453 2,65610 454 2,65706 455 2,65801 456 2,65896 457 2,65991 458 2,66086 459 2,66181 460 2,66276 461 2,66370 462 2,66464 463 2,66558 464 2,66652 465 2,66745 466 2,66838 467 2,66932 468 2,67024 469 2,67117 470 2,67210 471 2,67302 472 2,67394 473 2,67486 474 2,67578 475 2,67669 476 2,67761 477 2,67852 478 2,67943 479 2,68033 480 2,68124 481 2,08214 482 2,68305 483 2,68395 484 2,68484 485 2,68574 486 2,68664 487 2,68753 488 2,68842 489 2,68931 490 2,69020 491 2,69108 492 2,69196 493 2,69285 494 2,69373 495 2,69460 496 2,69548 497 2,69636 498 2,69723 499 2,69810 500 2,69897 501 2,69984 502 2,70070 503 2,70157 504 2,70243 505 2,70329 506 2,70415 507 2,70501 508 2,70586 509 2,70672 510 2,70757 511 2,70842 512 2,70927 513 2,71012 514 2,71096 515 2,71181 516 2,71265 517 2,71349 518 2,71433 519 2,71517 520 2,71600 521 2,71684 522 2,71767 523 2,71850 524 2,71933 525 2,72016 526 2,72099 527 2,72181 528 2,72263 529 2,72346 530 2,72428 531 2,72509 532 2,72591 533 2,72673 534 2,72754 535 2,72835 536 2,72916 537 2,72997 538 2,73078 539 2,73159 540 2,73239 541 2,73320 542 2,73400 543 2,73480 544 2,73560 545 2,7364● 546 2,73719 547 2,73799 548 2,73878 549 2,73957 550 2,74036 551 2,74115 552 2,74191 553 2,74272 554 2,74351 555 2,74429 556 2,74507 557 2,74585 558 2,74663 559 2,74741 560 2,74819 561 2,74896 562 2,749●3 563 2,75051 564 2,75128 565 2,75205 566 2,75282 567 2,75358 568 2,75435 569 2,75511 570 2,75587 571 2,75664 572 2,75740 573 2,75815 574 2,75891 575 2,75967 576 2,76042 577 2,76118 578 2,76193 579 2,76268 580 2,76343 581 2,76418 582 2,76492 583 2,76567 584 2,76641 585 2,76716 586 2,76790 587 2,76864 588 2,76938 589 2,77011 590 2,77085 591 2,77159 592 2,77232 593 2,77305 594 2,77379 595 2,77452 596 2,77525 597 2,77597 598 2,77670 599 2,77743 600 2,77815 601 2,77887 602 2,77960 603 2,78032 604 2,78104 605 2,78175 606 2,78247 607 2,78319 608 2,78390 609 2,78462 610 2,78533 611 2,78604 612 2,78675 613 2,78746 614 2,78816 615 2,78887 616 2,78958 617 2,79028 618 2,79099 619 2,79169 620 2,79239 621 2,79309 622 2,79379 623 2,79449 624 2,79518 625 2,79588 626 2,79657 627 2,79727 628 2,79796 629 2,79865 630 2,79934 631 2,80003 632 2,80072 633 2,80140 634 2,80208 635 2,80277 636 2,80346 637 2,80414 638 2,80482 639 2,80550 640 2,80618 641 2,80656 642 2,80753 643 2,80821 644 2,80889 645 2,80956 646 2,81023 647 2,81090 648 2,81157 649 2,81224 650 2,81291 651 2,81358 652 2,81425 653 2,81491 654 2,81558 655 2,81624 656 2,81690 657 2,81756 658 2,81822 659 2,81888 660 2,81954 661 2,82020 662 2,82086 663 2,82151 664 2,82217 665 2,82282 666 2,82347 667 2,82413 668 2,82478 669 2,82543 670 2,82607 671 2,82672 672 2,82737 673 2,82801 674 2,82866 675 2,82930 676 2,82995 677 2,83059 678 2,83123 679 2,83187 680 2,83251 681 2,83315 682 2,83378 683 2,83442 684 2,83506 685 2,83569 686 2,83632 687 2,83696 688 2,83759 689 2,83822 690 2,83885 691 2,83948 692 2,84011 693 2,84073 694 2,84136 695 2,84198 696 2,84261 697 2,84323 698 2,84385 699 2,84448 700 2,84510 701 2,84572 702 2,84634 703 2,84695 704 2,84757 705 2,84819 706 2,84880 707 2,84942 708 2,85001 709 2,85065 710 2,85126 711 2,85187 712 2,85248 713 2,85301 714 2,85370 715 2,85431 716 2,85491 717 2,8●552 718 2,85612 719 2,85673 720 2,85733 721 2,85793 722 2,85854 723 2,85914 724 2,85974 725 2,86034 726 2,86094 727 2,86153 728 2,86213 729 2,86273 730 2,86332 731 2,86392 732 2,86451 733 2,86510 734 2,86570 735 2,86629 736 2,86688 737 2,86747 738 2,86806 739 2,86864 740 2,86923 741 2,86982 742 2,87040 743 2,87099 744 2,87157 745 2,87216 746 2,87274 747 2,87332 748 2,87390 749 2,87448 750 2,87506 751 2,87564 752 2,87622 753 2,87679 754 2,87737 755 2,87795 756 2,87852 757 2,87910 758 2,87967 759 2,88024 760 2,88081 761 2,88138 762 2,88195 763 2,88252 764 2,88309 765 2,88361 766 2,88423 767 2,88479 768 2,88536 769 2,88592 770 2,88649 771 2,88705 772 2,88762 773 2,88818 774 2,88874 775 2,88930 776 2,88986 777 2,89042 778 2,89093 779 2,89154 780 2,89209 781 2,89265 782 2,89321 783 2,89376 784 2,89431 785 2,89487 786 2,89542 787 2,89597 788 2,89653 789 2,89708 790 2,89763 791 2,89818 792 2,89872 793 2,89927 794 2,89982 795 2,90037 796 2,90091 797 2,90146 798 2,90200 799 2,90255 800 2,90309 801 2,90363 802 2,90417 803 2,90472 804 2,90526 805 2,90580 806 2,90633 807 2,90687 808 2,90741 809 2,90795 810 2,90848 811 2,90902 812 2,90956 813 2,91005 814 2,91062 815 2,91116 816 2,91169 817 2,91222 818 2,91277 819 2,91328 820 2,91381 821 2,91434 822 2,91487 823 2,91540 824 2,91593 825 2,91645 826 2,91698 827 2,91751 828 2,91803 829 2,91855 830 2,91908 831 2,91960 832 2,92012 833 2,92064 834 2,92117 835 2,92169 836 2,92221 837 2,92272 838 2,92324 839 2,92376 840 2,92428 841 2,92480 842 2,92531 843 2,92582 844 2,92634 845 2,92686 846 2,92737 847 2,92788 848 2,92840 849 2,92891 850 2,92942 851 2,92993 852 2,93044 853 2,93095 854 2,93146 855 2,93197 856 2,93247 857 2,93298 858 2,93349 859 2,93399 860 2,93450 861 2,93500 862 2,93551 863 2,93601 864 2,93651 865 2,93701 866 2,93752 867 2,93802 868 2,93852 869 2,93902 870 2,93952 871 2,94001 872 2,94052 873 2,94102 874 2,94151 875 2,94201 876 2,94250 877 2,94300 878 2,94349 879 2,94399 880 2,94448 881 2,94498 882 2,94547 883 2,94596 884 2,94645 885 2,94694 886 2,94743 887 2,94792 888 2,94841 889 2,94890 890 2,94939 891 2,94988 892 2,9503● 893 2,95085 894 2,95134 895 2,95182 896 2,95231 897 2,95279 898 2,95328 899 2,95376 900 2,95424 901 2,95472 902 2,95521 903 2,95569 904 2,95617 905 2,95664 906 2,95713 907 2,95761 908 2,95809 909 2,95856 910 2,95904 911 2,95952 912 2,95999 913 2,96047 914 2,96095 915 2,96142 916 2,96189 917 2,96237 918 2,96284 919 2,96331 920 2,96379 921 2,96426 922 2,96473 923 2,96520 924 2,96567 925 2,96614 926 2,96661 927 2,96708 928 2,96755 929 2,96802 930 2,96848 931 2,96895 932 2,96941 933 2,96988 934 2,97035 935 2,97081 936 2,97128 937 2,97174 938 2,97220 939 2,97267 940 2,97313 941 2,97359 942 2,97405 943 2,97451 944 2,97497 945 2,97543 946 2,97589 947 2,97635 948 2,97681 949 2,97727 950 2,97772 951 2,97818 952 2,97864 953 2,97909 954 2,97955 955 2,98000 956 2,98046 957 2,98091 958 2,98137 959 2,98182 960 2,98227 961 2,98272 962 2,98317 963 2,98363 964 2,98408 965 2,98453 966 2,98498 967 2,98543 968 2,98587 969 2,98632 970 2,9867● 971 2,98722 972 2,98767 973 2,98811 974 2,98856 975 2,98900 976 2,98945 977 2,98989 978 2,99034 979 2,99078 980 2,99113 981 2,99167 982 2,99211 983 2,99255 984 2,99299 985 2,99344 986 2,99388 987 2,99432 988 2,99476 989 2,99520 990 2,99563 991 2,99607 992 2,99651 993 2,99695 994 2,99739 995 2,99782 996 2,99826 997 2,99869 998 2,99913 999 2,99956 1000 3,00000 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, etc. 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 easy 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 easy 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 cipher before them; all Numbers between 10 and 100 have the Figure 1 before them; all Numbers between 100 and 1000, have 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 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 Use of the Table of Logarithms in Arithmetic, which shall be exemplified in Questions of Multiplication, Division, and the Extracting the Square and Cube Roots, being such parts of Arithmetic 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. 48 1,68124 5 0,69897 240 2,38021 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 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 is 0,84510 To which add the Logarithm of 1,57978 the Circumference Twice. 1,57978 The Sum 4,00466 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 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 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 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 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 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 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 Arithmetic. 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 Arithmetic, 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 3,142 0,49720 so is the Logarithm of 0,00000 to the Logar. of the Answer. 47,1367329 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 of 3,142 0,49720 is to the Logarithm of 1 0,00000 so is the Logarithm of 47.13 1,67329 to the Logar of the Answer. 1,67329 15 1,17609 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 Logarithm 1 0,00000 is to the Logarithm 15 1,17609 so is the Logarithm of 8862 0,94753 to the Answer 13,29 2,12362 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 of 1 0,0000 is to the Logarithm of 4713 0,67329 ●o is the Logarithm of 2821 0,45040 to the Answer 13,29 1,12369 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— 0,49720 Being Added, gives the Log, of 28,28. and the Log. of 9 Inch. 0,95424 Being Added, gives the Log, of 28, 28. 1,4 5144 Log. 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. 28.28 145144 Log. 3,142 049720 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. 28,28 145144 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. 9 0,95424 is to Logar. 729 2,86272 so is Logar. 11 1,04139 to the Cubical Content. 3,90411 891 2,94987 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 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 tripled 0,90309 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 show what Proportions there are between several Metals used for this purpose; as of Brass, Iron, Led 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 found 42324 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 Arithmetic. The Analogy. AS 11,5 is to 1900 ∷ so is 8,75 to almost 8,37. By the Logarithms. As the Log. greatest Diana. 11,5 306069 The Log. of the least, 8,75 294200 Difference Increasing 11869 Multiplied 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 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 Boar. For the Superficies the Analogy is, AS 1 ∶ 22 ∷ 144 ∶ 3168, Square Inches. By the Logarithms. Logarithm 22 ,34242 Logarithm 144 ,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. Logarithm 35 54407 Logarithm 11 04139 Logarithm 38,5 58546 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. Logarithm 385 ,58546 Logarithm 144 ,15836 Logarithm 5544 ,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 Gun-founders 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 join 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 Latin, 8 Tin. So is 100 4745,7 Copper, So is 10 474,6 Brass, So is 8 379,7 Tin. 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. Logarithm 118 071882 Logarithm 5600 748188 Logarithm 100 000000 748188 Log. found, 474,57 676306 Here you are referred to a larger Table of Logarithms, than is in this Book, for this operation and the next following. For Brass. Logarithm 118 071882 Logarithm 5600 748188 Logarithm 10 000000 748188 Logarithm found, 474,57 676306 For Tin. Logarithm 118 071882 Logarithm 5600 748188 Logarithm 8 903090 651278 Logarithm 379,7 579396 By Gunter 's Scale. For the First Operation for Copper. Extend the Compasses from 118 (upon the Line of Numbers) to 5600, the same 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 Boar requires Four Pound of Powder for its Load, what will a Demy Cannon of 6, 5 Inches Diameter in the Boar 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 Demi-canon 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 Demi-canon in this Example, aught to be 8351 Pounds, which would be of a Proportion to the Saker, to carry a proportional weight of Powder. But if the Demi-canon 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 Found'st 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 ●0 03 2 00 04 3 00 05 4 00 07 5 00 09 6 00 12 7 00 14 TWO 01 02 1 01 05 2 01 09 3 01 14 4 02 03 ● ●2 08 ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● 1 09 13 2 10 12 3 11 12 4 12 13 5 13 14 6 15 01 7 16 04 V 17 09 1 18 14 2 20 05 3 21 13 4 23 26 5 25 00 6 26 11 7 28 08 VI 30 08 1 32 05 2 34 05 3 36 06 4 38 09 5 40 14 6 43 04 7 45 11 VII 48 03 1 50 13 2 53 09 3 56 06 4 59 05 5 62 05 6 65 07 7 68 10 VIII 72 00 1 75 06 2 78 15 3 82 09 4 86 05 5 90 03 6 94 03 7 98 04 IX 102 08 1 106 13 2 111 04 3 115 13 4 120 09 5 125 05 6 130 05 7 135 06 X 140 04 The foregoing Table was Calculated from the Directions in this Chap Prob. 9 page 113. One Example will show 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 showing the height and weight of Iron, Led and Stone shot, according to their 〈◊〉 in Inches and Quarter's, and their respective weights in Pounds and Ounces. Iron. Lead. Stone. Inches. Quarters. Pounds. Ounces. Pounds. Ounces. Pounds. Ounces. 1 0 ● 0 0 3 0 1 1 1 ● 0 0 6 0 3 1 2 ● 0 0 9 0 4 1 3 1 0 0 13 0 5● 2 0 1 1 1 11 0 7 2 1 1 9 2 0 0 9 2 2 2 2 3 0 0 12 2 3 2 14 4 3 1 0 3 0 3 12 5 0 1 4 3 1 4 12 6 9 1 8 3 2 6 1 8 11 2 9 3 3 7 5 9 14 2 7 4 0 8 15 11 5 2 13 4 1 10 10 15 15 3 10 4 2 12 10 17 15 4 3 4 3 14 14 21 5 5 9 5 0 17 5 24 12 6 3 5 1 20 1 30 0 7 8 5 2 23 2 35 10 8 14 5 3 26 6 39 9 10 10 6 0 30 0 45 0 11 4 6 1 34 0 51 0 12 12 6 2 38 0 57 0 14 3 6 3 42 0 63 0 15 12 7 0 48 0 72 0 17 10 7 1 53 0 79 0 19 14 7 2 58 0 87 0 24 12 7 3 64 0 96 0 24 0 8 0 72 10 106 26 14 8 1 78 0 117 28 08 8 2 87 3 130 34 08 8 3 95 0 142 35 10 Iron. Lead. Stone. Inches. Quarters. Pounds. Ounces. Pounds. Ounces. Pounds. Ounces. 9 0 101 0 150 37 10 9 1 109 6 161 40 4 9 2 121 10 181 44 2 9 3 132 11 198 49 8 10 0 138 0 207 51 10 10 2 164 2 246 60 0 11 0 184 0 275 69 8 11 2 216 0 324 81 0 12 0 240 0 360 90 0 13 0 305 0 451 114 0 14 0 389 2 583 146 8 One Example will show 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 showing the Length and Weight of most of our English Ordnance, the Diameter of their Bear, the weight of their Shot, the Ladles length, and their weight of Powder to Charge them. Names of the Pieces of 〈◊〉. Diameter of the Boar. Length of the Piece. Weight of the Piece in Pounds. Breadth of the Ladle. Length of the Ladle. Inches. Parts. Feet. Inches. pounds' Inches. Parts. Inches. Parts. Basc. 1 2 4 6 200 2 0 4 0 Rabanet. 1 4 5 6 300 2 4 4 1 Falconets. 2 2 6 0 400 4 0 7 4 Falcon. 2 6 7 0 750 4 4 8 2 Minion Ordinary 3 0 7 0 800 5 0 8 4 Minion Large. 3 2 8 0 1000 5 0 9 0 Saker Lowest. 3 4 8 0 1400 6 4 9 6 Saker Ordinary. 3 6 9 0 1500 6 6 10 4 Saker . 4 0 10 0 1800 7 2 11 0 Demy-Culv. Low. 4 2 10 0 2000 8 0 12 0 Names of the Pieces of Ordnance. Wetght of Powder. Diameter of the Shot. Weight of the Shot. Piece Shoots point blank. Pounds Ounces Inches. Parts. Found'st Ounces Paces. Base. 0 8 1 1 0 3 60 Rabanet. 0 12 1 3 0 5 70 Falconets. 1 4 2 2 1 9 90 Falcon. 2 4 2 5 2 8 120 Minion Ordinary. 2 8 2 7 3 5 120 Minion Large. 3 4 3 0 3 12 125 Saker Lowest. 3 6 3 2 4 13 150 Saker Ordinary. 4 0 3 4 6 0 160 Saker . 5 0 3 6 7 6 163 Demy-Culv. Low. 6 4 4 0 9 0 174 Names of the Pieces of Ordnance. Diameter of the Boar. Length of the Piece. Weight of the Piece in Pounds. Breadth of the Ladle. Length of the ●addle. Inches. Parts. Feet. Inches. pounds' Inches. Parts. Inches. Parts. Demy-Calv. Ord. 4 4 11 0 2700 8 0 12 6 Demy-Culv. Eld. 4 6 11 0 3000 8 4 13 4 Culverins Best. 5 0 11 0 4000 9 0 14 2 Culu. Ordinary. 5 2 11 0 4500 9 4 16 0 Culu. Largest. 5 4 11 0 4800 10 0 16 0 Demy-Can Low. 6 2 11 0 5400 10 4 20 0 Demy Can. Ord. 6 4 12 0 5600 12 0 22 0 Demy Can. Lar. 6 6 12 0 6000 12 0 22 0 Cannon-Royal. 8 0 12 0 8000 14 0 24 0 Four Dutch Pieces. A 3 Pounder. 2 94 11 0 750 5 5 18 0 A 6 Pounder. 3 70 10 0 1500 6 5 14 0 A 12 Pounder. 4 61 9 0 3000 9 0 10 6 A 24 Pounder. 5 79 7 0 5000 11 0 9 0 Names of the Pieces of Ordnance Weight of Powder. Diameter of the Shot. Weight of the Shot. Piece Shoots point blank. Pounds Ounces Inches. Parts. pounds' Ounces Paces. Demy-Culv. Ord 7 4 4 2 10 12 175 Demy-Culv. Eld. 8 8 4 4 12 13 178 Culverins Best. 10 0 4 6 15 1 180 Culu. Ordinary. 11 6 5 0 17 9 181 Culu. Largest. 11 8 5 2 20 5 183 Demy-Can. Low. 14 0 6 0 30 8 156 Demy-Can. Ord. 17 8 6 1 32 5 162 Demy-Can. Lar. 18 0 6 5 40 14 180 Cannon-Royal. 32 8 7 4 59 5 185 Four Dutch Pieces. A 3 Pounder. 10 8 5 56 24 0 120 A 6 Pounder. 6 0 4 40 12 0 160 A 12 Pounder. 3 8 3 49 6 0 178 A 24 Pounder. 1 10 2 77 3 0 189 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 Boar 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 Touchhole, at the Trunnions, and at the Muzzle, in proportion to the Diameter of the Boar. The Cannons double Fortified, have full one Diameter of the Boar, in thickness of Metal at the Touchhole, and ●● at the Trunnions, and in their Muzzle ●●. The Lessened Cannons have at their Touchhole ¾ or 12/16 of the Diameter of their Boar, in thickness of Metal, and ●● at the Trunnions, and ●● at the Muzzle. The Ordinary Fortified Cannons have 7/8 at the Touchhole, 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 Touchhole, ●● 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 than 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 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 Boar 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 Boar, 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 Boar 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 Former's 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 Boar of the Piece, and let the piece of Cloth or Paper be Three times the Diameter of the Boar 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, etc. 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; before you paste or glue your Paper on the former, first tallow it, so will the Canvas 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 Canvas, 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 dried, 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 Culverin 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 overhot, or else you endanger yourself 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 IU. 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 Boar, 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 lose, by close or lose 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 windward 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 judgement 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 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 mixed 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 mixed so long as there remaineth any part of the violent motion; but after that is spent, than 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 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 Point-blanks 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'. 0 19 0 192 1 209 1 289 2 227 2 404 3 244 3 510 4 261 4 610 5 278 5 712 6 285 6 828 7 302 7 934 8 320 8 1044 9 337 9 1129 10 354 10 1214 20 454 20 1917 30 693 30 2185 40 855 40 2289 50 1000 50 2283 60 1140 60 1792 70 1220 70 1214 80 1300 80 1000 90 1350 90 0000 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 Sponge, 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 Chambered, or true bored, or whether they be Cracked, Flawed, or Honey combed within; and finding what Ball she shoots, to mark the Weight of the Ball over the Port; that thereby he may 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, sponge them well; and finding them clean, you may load them with their respective Cartridges and Powder, they being rammed home with a straight 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 roll 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 Touchhole, 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 filled, and 12 empty. Likewise you must be careful, so long as the Gunner's Crew are busy with the Powder, 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 with Hides: And also that the Ropes, Rammers, and Sponges 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 Sponges 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 Fireworks 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 , 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 Merchantman, than 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. Gunner's 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 Glue. Thread, 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 laid Stuff for Tackles. Lashers, double and single Blocks, new Rope for double Tackles. Some old Shrouds for Sponges, some Lines, Marline, Tarred Twine, Port-Ropes. Moulds for Cartridges for each fort of Gun, Axletrees 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 Canvas, Scales and Weights. Lanterns, 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 Lantern, Dark-Lanthorns, Powder-Measures, Soap, 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 showed 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 Atlases, for any of the known Parts of the World. CHAP. XI. Showing an Easy 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 fastened with Wax, it will be a true Dispart for that Piece. CHAP. XII. To Levelly a Piece of Ordnance to shoot Point-Blank. TO shoot Point-Blank is to be understood, that when the Cylender of the Piece lieth 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, than that Piece lieth 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 shown; 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 Object at a Distance, that aught to be of the same Height 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 intent to elevate your Piece, discharge it of some of the Quoins at the Breech, and by your Quadrant applied 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 shown 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 Honeycomb 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 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. Immediately after you have discharged your Piece, let one be ready with a Tomkin to clap into the Mouth of the Piece, with a Piece of Sheepskin wrapped about the Muzzle of the Piece, and the same time let one stop the Touchhole; and if there be any Crack through the Metal a visible Smoke 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 stroke with the Hammer you hear a clear sound, you may certainly conclude the Piece to be sound. CHAP. XIV. How Moulds, Former's and Cartridges are to be made for any sort of Ordnance. CArtridges are usually made of Canvas, or Royal Paper; to make them first take the height of the Boar 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 Culverin is allowed about two Diameters for the length, from the Culverin 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 turned to the height of the Cartridge, which is 1/●● parts of the Diameter of the Boar, and half an Inch longer than the Cartridge. Before you passed the Paper on the Former, tallow it, that the Canvas or Paper may slip off, without starting or tearing. If you make your Cartridges for Taper-bored Guns, your Former must be Tapered accordingly; if you make your Cartridges of Canvas, 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 passed 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 Culverin, 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 overhot, or else you may endanger yourself and others: 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, than 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 Wether, 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 Culverin 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 Boar 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 height of the Shot: But most Gunner's allow the Shot to be just one quarter of an Inch lower than the Boar of the Piece, which rule makes the Shot too big for a Cannon, and too little for a Falcon; 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 try with several Rammer-heads, until you find the Diameter of the Boar in that Place where the Shot useth to lie 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 choose 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 chief 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 reinforced, hath fully one Diameter of the Boar in Thickness of Metal at her Touchhole, and 11/16 at the Trunions, and 7/16 at her Muzzle; and the ordinary Cannons 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, etc. 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 Culverin 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 Boar 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. Then with a pair of Calaber Compasses, take the Thickness or Diameter of the 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 Touchhole, 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 shows, 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 Culverin; 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 Culverin, and being shorter than the ordinary, it is therefore called a Bastard Culverin. Now this being found to be an ordinary Culverin, 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 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 Culverin, and the like may be done with the Neck which the circle OPEN 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 Culverin at her Neck. But if in taking the measures aforesaid there had been found at the Touchhole 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, than she shooting an Iron shot of 17 l. would have endured 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, than 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 Culverin hath 1/24 and the Saker 1/23 and the Falcon ● more Metal comparatively than the whole Culverin hath. And if a Piece is found that it is not truly bored, 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 Wether whatsoever. IN time of service when you are on a sudden to make a Shot at a Ship, and know not what dispart will serve the Piece, than 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 tail 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 Mainmast 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 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, than you would shoot over the other Ship; and if the Sea be high, there is no better time to give fire than when your Enemy's Ship gins to rise on the top of a Sea, for than 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 judgement 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 levelly the Piece rather under the place you shoot at than over. And if in a fight, if you intent 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 levelly your Bases or other small Guns ready to discharge to the best advantage you can at the first boarding, at such a 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 etc. 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, than 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 recoil, therefore 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 Ship should he'll with the bearing of a Sail, than 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 lieth out of the Ships side, the less it shall annoy them in the tackling of the Ships Sails, for if the Piece lieth far out the Sheets, Tacks or Bolins, 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 Forecastle, etc. 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 judgement 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 Breeching. They must always be four times the length of the Piece with some overplus for fastening at both ends. If in foul weather your Breeching and Tackies should give way, you have no better way for the present to prevent danger, than immediately to dismount the Piece. It is also approved by able Gunners, that the Rammers and Sponges made with small Hawser should be armed close and hard with strong and twisted Yarn, from the Rammers end quite to the Sponge, 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 height one Diameter, and ¾ thereof in length, or very little less than the height 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 fastened with a Pin through, and the Stafflength a foot more than 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 rarefies the other two, to make them Fire the better. Two sorts of Gunpowder 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 Majesty's 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, than 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 touched: 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 soon, smokes least, and leaves least sign behind it, is the best sort of Gunpowder. 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 shaked and moved, it will want of its strength at the top, and 1 l. at bottom with long standing will be stronger than 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, than the true fortified Iron Guns ought to be 11 of those Diameters in the circumference of the Gun at the Touchhole, 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 Touchhole, 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 Fortified 11 Diameters 13 More Fortified 12 Diameters 14 True Fortified 9 Diameters 11 Lesle Fortified 8 Diameters 10 Lesle Fortified 7 Diameters 9 Weight of Powder for Service is proportioned by the Numbers of Diameters of the Boar 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 show 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 Centre of the Trunions. The Practical way of making Gunpowder. The Essential Ingredients for making Gunpowder 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 Cauldron, and boil it with so much Water as will serve to dissolve it with; being so dissolved, it ought to be washed and laid upon a clean place; this done, beat the quantity of Coal into dust, than put this Charcoal dust being finely beaten into the disolved Petre, and incorporate them very well together, and as you mingle them, put in by little and little the Sulphur 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 dried 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 Gunpowder is made: There are divers ways to grind Gunpowder; 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 Vellum or Parchment, made full of round holes, then moisten the Powder which shall be corned with Water, put a little Bowl into the Sieve, than fift the Powder so as the Bowl rolling 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 Gunpowder, having lost its Strength by moisture, long lying, or by any other means. Having moistened the said Gunpowder with Vinegar or fair Water, beat it well in a Mortar, then sift it through a Sieve or fine Searce; for every l. of Gunpowder 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 outside. After you have doubled the Shells, passed 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 Gunpowder, then make a Fuze of one pound of Gunpowder 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 men's 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, Sulphur, Shall Armoniac of each one pound, and four Ounces of Camphire pounded and seared and mixed well together, with hot Pitch, Linseed Oil or Oil 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 Oil or Rosin, and then it is for your use. SECT. I. How to make Powder-Chests. You must nail two Board's together like the ridge of a House, and prepare one Board longer and broader for the bottom: Between these three Board's put a Cartridge of Powder, then make it up like a Sea-Chest and fill it with pebble 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 annoy an Enemy if they board you. To make Stink-Balls. Take Gunpowder 10 l. of black Pitch 6 l. of Tarr 20 l. Salt-Petre 8 l. Sulphur Calafornia 4 l. melt these over a soft Fire together, and being well melted put 2 l. of Cole dust of the Filings of Horse's 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 Linen or Woollen Cloth, 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, patiented, prudent and quick Spirited Man; he ought also to have a good eyesight and a good judgement 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 Arithmetic 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 men's 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 Lantern, 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, Sponges, Gunpowder, Balls, Tamkins, Wadds, Chain-shot, Cross-bar-shot, Quoins, Crows, Tackles, Breeching, Powder-Horns, Canvas, and Paper for Cartridges, Forms for Ladles, Cartridges, Needles and Thread to sow and bind the Cartridges, Candles, Lanterns, Handspikes, Pole-axes, little Hand-Baskets, Glue 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, Sponge, and Prime and Scour, 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 height 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 Gunpowder to purify Brimstone, to amend any sort of Powder when it hath lost its virtue 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 Ordnance 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 Fireworks, and the Oacum, which is very dangerous, and to keep your Pieces (as near 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, Showing the Use of a Proportional Scale In several Questions in ARITHMETIC, In Lineal, Quadratick and Cubi call 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 itself. By John Seller. London, Printed in the Year, 1691. CHAP. I. A Description of the Proportional Scale and its Use 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 Number's 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 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 gins 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 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 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 signify themselves alone, sometimes 10, 20, 30. etc. sometimes 100, 200, 300, etc. as the works thereby doth require. And by this variation 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, than 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 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 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 Eighth Note of the First Chapter of this Treatise. CHAP. IU. 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 applied the same way upon the Line, shall reach from the 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. yields 6 l. Interest for one Year, or 12 Months, what shall 75 l. yield? 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. yield Interest in one Year. Example. 3. If 75 l. yields 4 l. 10 s. or 4 ●0/100 150 Interest for one Year or 12 Months, what will 105 l. yield? 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, than the fourth Number will be less than the second. And on the Contrary, if the third Number be less than the first, than the fourth Number will be greater than the second; as by example. Example 1. If 12 Workmen do a piece of Work in in 8 days, how many Workmen 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; 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. 2 12 8 48, For if 8 days require 12 Men, than 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, 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 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 aught that piece to contain? divide the space between 30 l, and 91 l. 16 s. into two equal parts, than 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 between 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, 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, 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 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 Boar 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, showing the Weight and Diameter of any Shot, with the weight of Powder and Shot for any Piece of Ordnance, 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 show 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 Edge 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 showed by the inside of the movable leg of the Rule. And the Concave Diameter of a Gun is cut upon the Brass Semicircle by the outside of the movable Leg. There is also a Circle of Degrees upon the outward Limb or Edge of the Semicircle, which is to give the Quantity of any Angle cut by the inside of the Movable 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 Uses 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 shows 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 show, 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 placed 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 movable Leg upon the edge of the Brass Semicircle. The Uses follow. Use 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. Use 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. Use 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 movable 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 Thread and Plummet upon the movable Leg, and put the fixed Leg into the Muzzle of the Piece, and open the movable Leg until the Plummet falls perpendicular upon the Line, upon which the pin is fixed (upon which the Plummet is hanged) and on the edge of the Brass Semicircle, will be cut the Angle of the degree of Mounture by the inside of the movable Leg. Use 4. The Uses 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 shown in the use of the Proportional Scale, in this Treatise, to which I refer you. On this side of the Rule is placed a Line of Inches, on which may be found the Diameter of any Shot: And upon the Line adjoining to it is shown 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 shows 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 adjoining Line) you will find 72 which is the Weight of the Shot that is eight Inches Diameter. Use 5. The Use of the other side of the Rule. On this side of the Rule are six Lines, The First shows the Names of the Pieces, expressed by the several Names as, F for Falcon, M for Minion, S for Saker, etc. 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 show 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 shows 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 placed 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 shown 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 Seacoasts 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 Vshant. A description of the Sands, Buoys, Beacous and Sea-marks upon the Coast of England, from Dover to Orfordness. Atlas' Coelestis; showing all the Constellations of the Stars, and other Phenomena's of the Celestial Bodies, with Maps of the Sun and Moon, etc. Atlas-Terrestr is; containing a Collection of choice Maps of all the Empires and Kingdoms in the whole World. Books in Quarto. PRactical Navigation; being an Introduction to the whole Art; containing many Geometrical Desinitions and Problems, the Doctrine of plain and spherical Triangles, plain Mercator and great Circle sailing; 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 Moon's Age, of the Tides, and of the Sun's Place and Declination, Tables of Lines; Tangents and Sea-carts, and Logarithms, etc. Mr. Street Eoclesiastick and Civel Calendar. Books in Octavo. A Pocket-Book, containing choice Collections in Astronomy, Geography, dialing, Navigation. Atlas' Coelestis, containing the Systems and Theories of the Planets, the Constellations of the Stars, and other Phenomena's of the Heavens, with necessary Tables relating thereto. A new System of Geography, designed in a most plain and easy Method for the better under standing of that Science; accommodated with new Maps of the Countries, Regions, Empires, Monarchies, Kingdoms and Goverament in the whole World; with Geographical Tables explaining the Divisions in each Map. Atlas Maritimus, describing the Seacoasts 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 Arithmetic in Vulgar and Decimal, and necessary Tables belonging to that Art. Tabular dialing, showing the making Horizontal and Vertical Dial's for all Latitudes in the whole World. An Almanac for an Age, showing 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, Loadstones, Hour-glasses, ● Hours, ½ Minute, Watch-glasses, and ½ Watch-glasses, Black-lead Pencils, Slates and Slate-Pens, Pens, Telescopes, Perspective-glasses, Pocket-Globes.