THE COMPLETE Shipwright. Plainly and Demonstratively Teaching the Proportions used by Experienced Ship-Wrights, according to their Custom of Building; both Geometrically and Arithmetically performed. To which is added, Certain Propositions in Geometry, the use of a diagonal SCALE, to draw a Draught, with the Making, Graduating, or Marking of a Bend of Moulds, and ordering of the same. The Extraction of the Square Root, with a Table of Squares. Also, a way of Rowing of Ships, by heaving at the Capstane, useful in any Ship becalmed; with other things useful in that ART. By EDMUND bushnel Shipwright. LONDON, Printed by W. Leybourn for George Hurlock, and are to be sold at his Shop at Magnus' Church corner in Thames-Street, near London-Bridge, 1664. TO THE READER. Friendly Reader, THe Matter contained in this Treatise, is written only for the good and profit of my Countrymen, who are still in that capacity that once I myself was; that is, ignorant of what they should know in their Trades, and desire Instruction; not that I presume to teach those long experienced Ship-Wrights, whose actions hath declared their Abilities to the whole World, in their Building of so Gallant, and Serviceable a Fleet of Ships, as at present his Majesty, the King of England, is furnished withal, no King having the like, either for Offence, or Defence: yet their knowledge they desire to keep to themselves, or at least among so small a number as they can; for although some of them have many Servants, and by Indenture (I suppose) bound to Teach them all alike the same Art and Mystery that he himself useth; Yet it may be he may Teach some one, and the rest must be kept ignorant, so that those Ship wright's, although bred by such knowing Men, yet they are able to teach their Servants nothing, more than to Hue, or Dub, to Fay a Piece when it is Moulded to his place assigned, or the like: but if occasion require, that the greatest part of these Men, by being Carpenters of Ships, or the like, may be removed from England to Virginia, or New-England, or the like Countries, where Timber is plenty for their use, yet through their ignorance, they durst not undertake such a Work: For their sakes I have written this Book, wherein the Reader shall find instructions sufficient for Moulding of any Ship, or Vessel whatever, with the Masting of them, drawing of Draughts, and all in a very plain and exact Method, which I am confident will be understood by the meanest capacities, if they can but read English, and have the benefit of a little Arithmetic as Addition, Substraction, Multiplication, Division: be diligent, and I shall be thereby encouraged, if need be, to help thee farther in the Art. Farewell, Thy Friend, Edmund bushnel. THE CONTENTS. Ch. I. OF Geometrical Problems Page 1 Ch. TWO Of your SCALE Page 4 Ch. III Concerning the drawing your Draught upon Paper Page 6 Ch. IV. Showing, how to sweep out the Bend of Moulds upon a Flat Page 9 Ch. V. The Description of the Rising Lines aftward on, and forward on; with the Narrowing Lines, and Lines of Breadth: As also the Narrowing of Lines at the top of the Timbers Page 12 Ch. VI Showing the Making and graduating, or marking of the Bend of Moulds Page 15 Ch. VII. Arithmetically showing how to frame the body of a Ship by Segments of Circles: being a true way to examine the truth of a Bow Page 22 Ch. VIII. How to Extract the Square Rooot Page 31 Ch. IX. A Description of the Table of Squares Page 39 Ch. X. Showing, how to hang a rising line by several sweeps, to make it rounder aftward, then at the beginning of of the same Page 53 Ch. XI. Concerning Measuring of Ships Page 59 Ch. XII. Concerning the Masts of Ships Page 62 Ch. XIII. Concerning Rowing of Ships, when they are becalmed. Page 65 Advertisement. IF any Gentleman, studious in the Mathematics, have, or shall have occasion for Instruments thereunto belonging, they may be furnished with all sorts, useful both for Sea or Land, either in Brass, or Wood, by Walter Hayes, at the Cross-Daggers in Moorfields, next door to the Popes-Head Tavern; where they may have all sorts of Maps, Globes, and Mathematical Paper, Carpenter's Rules, Post and Pocket-dyalls' for any Latitude, Steel Letters, Figures, Signs, Planets, or Aspects, at reasonable Rates. ERRATA. PAge 7, Line 9, Read Halses 8 16 come not fowl 8 21 faied 11 2 at f or G 12 23 afore 13 2 afore 13 19 aftward 14 15 L 14 28 ●/● 15 25 deal off 16 7 Sirmarks 18 ● none, that 18 11 at the top 19 2 stick 20 29 you h●w to 22 5 make a 23 3 & 5 3/5 54 16 12 foot 57 1 two last numbers. Books Printed and Sold by George Hurlock, at Magnus' Church-Corner in Thames-Street, near London Bridge. Seamans' Calendar, or Ephemerideses of the Sun, Moon, and certain of the most notable fixed Stars, etc. Norwoods' Doctrine of Triangles, with Logarithmes, lately printed, exactly corrected, and much enlarged by the Author himself. Norwoods' Epitome, applied to plain and Mercators sailing Norwoods' Sea man's Practice, containing a Fundamental Problem in Navigation, experimentally verified. Safeguard of Sailors, or Great Rutter, by Ro. Norman. Sea man's secrets. A Table of Gauging all manner of Vessels, by Jo. Goodwin. Pathway to perfect sailing, by Richard Poltar▪ Pitiscus his Doctrine of Triangles, with Canons. Navigator by Cap. Charles Saltonstal, newly printed, with additions, showing the Deceits of the plain Charts used in our time, and a way to prove the projection of any Plain Chart. Dary's description and use of a Universal Quadrant. Seaman's Dictionary, or the Exposition and Demonstration of all the parts and things belonging to a ship▪ together with an Explanation of all the Terms and Phrases used in the practic part of Navigation, by Sir Henry Manwarring. Exact Tables of Natural and Artificial Sincs, Tangents, Secants and Logarithms, and an Institution Mathematical, containing their constructions and use in the solution of all Triangles Plain and Spherical, and the application thereof in Astronomy, Dialling, and Navigation. Seaman's Glass, now newly published, with the addition of many propositions in Navigation, Astronomy, and Dialling, not before printed. The Complete Cannoneer, showing the principles and grounds of the Art of Gunnery, as also several Fire-Works for Sea and Land. The Advancement of the Art of Navigation, or Seaman's Canon of Triangles, showing by a new Canon of Sines, Tangents, and Secants, how to resolve all Cases of right lined Triangles, only by looking into the Tables, without any Calculation. Particularly applied to all the three kinds of Sailing, viz. by the Plain Chart, Mercators' Chart, by a Great Circle; and to the Art of Surveying. Trigonometria Brittannica, or the Doctrine of Triangles, in folio, exhibiting the Logarithms of all numbers, from one to a hundred thousand, and the Sins and Tangents to the hundred part of a Degree, with Mr. Gellibrands Doctrine of Triangles, faithfully translated from the Latin Copy. The Sector on a Quadrant, containing the Description and use of three general Quadrants, accommodated for the ready finding the Hour and Azimuth universally in the equal Limb. The Complete Modellist, showing how to raise the Model of any Ship or Vessel, either in proportion, or out of proportion▪ and to find the length and bigo●ss of every Rope, in all Vessels exactly, with the weight of their Anchors and Cables. There is a new Book▪ called the Pilots Sea-Mirror, which is a Compendium of the largest Wagoner, or the lightning Sea-Collumbe; Containing all Distances or thwart Courses of the Eastern, Northern, and Western Navigations, with a general Tide Table, for every day, and the Change and Full of the Moon exactly for eight years, also Courses and Distances throughout the straits. Printed for George Hurlock, at Magnus' Church Corner, by London Bridge. The Saints Anchor-hold in all storms and Tempests, Published for the support and comfort of God's people in all times of Trial, by John Davenport Pastor of the Church in New-Haven in New-Ingland. There will shortly be made public a Book, Entitled, The Mariner's Compass Rectifled, containing, First, a Table showing the hour of the day, the Sun being upon any point of the Compass. Secondly, Tables of the Suns rising and setting. Thirdly, Tables showing the points of the Compass, that the Sun and Stars rise and set with. Fourthly, Tables of Amplitudes; all which Tables are Calculated from the Equinoctial, to 60 degrees of Latitude, with Tables of Latitudes and Longitudes, after a new order, with the description and use of all those Instruments that are in use in the Art of Navigation, either for Operation or Observation. THE COMPLETE Shipwright. CHAP. 1. Of Geometrical Problems. BEfore we proceed to draw the Draught of any Ship or Vessel, it will be necessary to be acquainted with some terms in Geometry: as to know what a Point and a Line meaneth, which every Book treating of Geometry plainly teacheth, and therefore we shall pass that by, supposing that none will endeavour to study the Art of a Shipwright, that is ignorant of these things; and therefore leaving these Definitions, I will proceed to some Geometrical Problems necessary to this Art. PROB. 1. How to draw a Parallel Line. PArallel lines are such lines as are equidistant one from another in all parts, and are thus drawn. Draw a line of what length you please, (according to your occasion) as the line A B, then open the compasses to what distance you pleas, or as your occasions require, and set one foot of the compasses▪ towards one end of the given line, as at A, with the other foot make a piece of an arch of a circle, over or under the given line, as the arch C, keeping the compasses then at the same distance, make such another arch towards the other end of the line, setting one foot in B, and with the other describe the arch D, then laying a Ruler to the outside of these two arches, so that it may exactly touch them, draw the line C D, which will be parallel to the given line A B, or equidistant, for so signifieth the word Parallel, to be of equal distance. PROBL. 2. How to erect a Perpendicular, from a point in a right line given. LEt there be a point given in the line A B, as the point C, whereon to raise a perpendicular. Set one foot of the compasses in the given point C, and open them to what distance you please, as to the point E, make a little mark at E, and keeping the compasses at the same distance, turn them about, and make a mark at the point F, in the line A B: Then remove the compasses to one of those marks at E or F, and setting one foot fast therein, as at the point F, open the other foot wider, and therewith draw a small arch over the point C, as the arch D, then keeping the compasses at the same distance, remove them to E, and setting one foot in E, with the other foot draw another little arch, so as to cross the former arch in the point D, through the crossing of these two arches A D, draw a line to the given point C, as the line D C, which shall be perpendicular to the line A B. Divers other ways there are to raise a perpendicular, which I shall leave to the farther practice of such as desire diversity of ways, and proceed to the raising of a Perpendicular on the end of a line. PROBL. 3. To raise a Perpendicular on the end of a line. DRaw a line at pleasure, or according to your work, as the line A B: On the end thereof as at B, set one foot of the Compasses, and open them to what wideness you please, as to C, and keeping fast one foot at B, pitch one foot by adventure in C, then keeping one foot of the compasses in C, and at the same distance, remove the foot that was in B, to the point D in the line A B: then (keeping the compasses still at the same distance) lay a ruler to the points D and There are other ways to effect this, which I shall leave to farther practice of the learner, this being the properest for our purpose. PROB. 4. From a Point given, to let fall a Perpendicular upon a Line given. FRom the point C, let it be required to let fall a perpendicular upon the line A B, proceed thus: Fix one foot of the compasses in the point C, and open them to a greater distance then just to the line A B, and make with the same extent the two marks E and F, in the given line A B, then divide the distance between the two points E and F into two equal parts in the point D, then lay a Ruler to the given point C, and to the point D, and draw the line C D, which will be perpendicular to the given line A B. CHAP. II. Of your SCALE. BEing perfect in the raising and letting fall of perpendiculars, and in the drawing of Parallel lines, you may proceed to draught: but first I will unfold unto you the use of a diagonal Scale of Inches and Feet, whose use is to represent a foot measure, or a Rule so small, that a Ship of 100 foot by the Keel, may be demonstrated on a common sheet of paper, really and truly to be so many foot long, and so many foot broad, of such a depth, and of such a height between the Decks. And therein, the first thing to be considered is, the length of the platform, and of the Vessel you intent to demonstrate, to the end you may make your Scale as large as you can, because the larger the Scale is, the larger will the draught be, and so the measure of the demonstration will be the larger, and more easy to unfold. The Scale adjoining consisteth (as you see) of 12 feet in all, 11 thereof are marked with figures downwards, beginning at 1, 2, 3, 4 and so to 11: the first at the top is subdivided into inches by diagonal lines, as the distance between the first line of the Scale and the first diagonal line is one inch, the second is 2, and the third 3 inches, and so to Six. The way to demonstrate the Scale, you see, is very easy: Draw Seven lines parallel to each other, and of what length you please, to retain what number of Feet you please, then beginning at the top, set off with the compasses the length of your Feet both allow and aloft, then draw lines thwart the parallel lines, to every foot of the Scale, and set numbers to them, beginning at the second foot 1, and to the third 2, to the fourth 3, and so forward, leaving the first Foot to be divided into Inches by the diagonal lines, as you see in the foregoing Scale. CHAP. III. Concerning the drawing your Draught upon Paper. HAving fitted your Scale ready, draw a line to represent the Keel of the Ship, as you see in the draught following of 60 foot long by the Keel, and 20 foot broad: the straight line that representeth the Keel is marked with A B. Then draw a line underneath of equal length to signify the bottom of the Keel. Then next you may proceed to the Stern-post, as the line A C will signify the foreside or the inside thereof, racking the one quarter of his length aft, and for the length of the Stern-post it must be directed to the built of the Ship, as whether she be to be a deep Ship or a shallow Ship, so that the draught of the water ought to be respected first, and then the lying of the Ports for the convenience of Ordnance, for that the upper transome of the Buttock, commonly is just under the Gun-Room ports, to the upper edge of the said transome we understand the length of the Stern-post, although if the Stern-post were continued to the height of the tiler, and another Transome fared there for the tiler to run on, it would steady the quarters of the Vessel very much, and do good service. The Stern-post being drawn, we may proceed to draw the Stem, which in the following Draught is not so much racked as was the old proportion of England, which was the whole breadth of the Ship, for than it should be 20 foot, but it is no more than 15 foot, just ¾ of the breadth, for too much rack with the Stem doth a great deal of damage to any Ship, if we consider that in this small Vessel, had we given 5 foot more Rack, all the weight of the Ships Head, and Boltspreet, Foremast, Manger, Halsps, Brest-hooks aloft, had been so much farther forward, where there would have been want of Body to lift it, so that it must of necessity be detriment to the Vessel when she saileth against a head sea, and a great strain to her. Now it will be very good to spend as much of this rack as we can under the water, for it will help the Ship to keep a good Wind, by giving her something more Body in the water. Next draw the Water-line, in the following draught signified by the pricked line; it is drawn to 9 foot height afore, and to 10 foot height abaft from the upper edge of the Keel, and higher abaft than afore, for the most Ships sail by the Stern, and also for that the Guns should lie something higher abaft than afore from the water. Then proceed to hanging of the Waals, and here you see the lower Waalle drawn from the head of the Sterne-post, to signify that it should lie against the end of the Transome, that the Transome Knees might be bolted to the Waals without board to one foot and an half under the Water-line, a little before the middle of the Water line, and at 9 foot high on the Stem, and the next Waale parallel to the lower Waale, one foot and an half asunder, so that the upper Waale will lie just at the water's edge, in the mid-ships, the upper edge of the Gun-deck will lie one foot aboye the water line abaft, and half a foot above water on the Stem; so then letting the lower sell of the Ports be two foot from the Gun-decks▪ the lower edge of the Ports will be three foot from the water abaft, and two foot and an half afore, in the middle of the Gun-deck 2 foot 9 inches, sufficient for so small a Vessel, a greater Vessel would require to have the Guns something farther from the water, then if another Waale be required, first set off the Ports in their places, that the Waale may lie above the Ports, or else he would be cut with the ports in pieces, the upper Deck with height respecting the bigness of the Ship, having respect to not over building small Ships, to damage their bearing of Sail. Then for the Head, the length of the Knee would be two thirds of the breadth, so then the Knee of the Head in this Draught will be 12 foot 8 inches long, and for his place, as low as conveniently he can, provided that the Rails of the Head, fall not fowl of the ha●shols, because that in placing of the Knee low, giveth room to round the Head, and steeve it to content: The place of the Knee will be at, or very near, the upper Waal, the upper edge of the Knee against the upper edge of the uper harping, which will be very well for the lower Cheeks of the head to be faced against, for by that means they will be clear of any Seam to avoid Leakings, and will very well bolt the end of the harping, if a Brest-hook be fastened also within board against them, will very well fasten all together. Then for the steeving of him, and rounding the Knee, a regard must be had to the lying of the Boltspreet, leaving room enough for the Lion and Scroll under the Boltsprit. Then▪ for the rounding of the Rails, round them most at the after ends. For the heights between Decks and Steeridge, Cabin, Forecastle, those heights are commonly mentioned in contract by the Master or Owners building. Place this Draught at Page 8 cross-section of a ship's hull CHAP. IU. Showing, how to sweep out the Bend of Moulds upon a Flat. FIrst, draw a line, as the line AB, then in the middle thereof, as at the point C, raise a perpendicular, as is the line CM, perpendicular to the line AB; then set off the half breadth, on either side, at the Points AB, and draw the two lines IA, and KB, parallels to CD, signifying the breadth of the Vessel 20 foot; then draw the two lines OF, and HG, signifying the breadth of the Floor thwart Ships, 8 Foot, more than one third part of the breadth, which was formerly an old Proportion; so that according to that it should have been but 6 Foot 8 Inches. Herein any may do as they please, give more or less; my judgement is, rather more than less: for, that it maketh a Vessel to carry more in Burden, and I conceive it may, if it be well ended forward, it will not damage the Sailing: I also think, it doth stiffen a Vessel on this account. Our English Vessels have been used to have their breadth lying at the height of the Half Breadth, then observing 1/3 breadth for the length of the Floor Thwart Ships, it maketh the Vessels Body to be very near a Circle, as is a Cask, which causeth such Vessels to be easy to slew in the Water; yet I would not exceed neither, or run into extremes herein, but if I were to make a Vessel stiff, I would that the Half Breadth be more than the draught of Water, which causeth that the Body be stronger in the Water, and will not Slew so easily. Now to sweep out the Sides under Water, I draw the diagonal lines DA, and DB; then I divide the diagonal lines into 9 parts, and set off 2 of them from the Corners A and B, to the points e, than I set off the Dead Rising, which is 4 Inches, one Inch to a Foot, for half the breadth represented in the Figure above, by the little line parallel to FG: from which Dead Rising, take with the Compasses the Distance that will draw a piece of an Arch from ● to ●, and so as one foot of the Compasses stand in the line OF, and exactly touch the points at the Dead-Rising, at f or g, and touch also the points e, over which point falls at ⊙, in OF, or ⊙ in HG, wherewith I describe the Arch e F, or e G, which is by the Scale in the Draught 4 Foot 8 Inches: then for the other part of the Side upwards, seek for a Point in the breadth line IK, at which, if one foot of the Compasses be set, and the other foot opened to the Extreme Breadth, will also draw, or signify an Arch to meet with the other Lower Arch, on the diagonal line at e, which is at the points ⊙ and ⊙; thus the point ⊙, between D and K, near H, Sweepeth the contrary Side I e, and so the point ⊙, between DIEGO, near E, Sweepeth the contrary side at K, extend the same Sweep also above the Breadth line above Water 3 or 4 Foot, the length of this Sweep is 12 Foot 9 Inches: then set off the Tumbling Home, at the Height of the two first Haanses, at the Main Mast and Foarcastle, 2 foot of a side; then draw a line from the said 2 Foot of Narrowing, at the points o v, till it break off on the back of the Sweep, on either side. This kind of Demonstration I conceive most suitable to our following discourse of Arithmetical Work, I could have cited other ways, but I Judge this way sufficient. CHAP. V. The Description of the Rising Lines aftward on, and forward on; with the Narrowing Lines, and Lines of Breadth: As also the Narrowing Lines at the top of the Timbers. DRaw a Hanging line on the Draught, from the Keel, from the middle of the Keel to the height of the Water line, on the Post which will be the Rising line, as the line DE; this line DE is supposed to be swept, or drawn by a Semidiameter of a Circle, extended on a Perpendicular raised at the point E, for if it be shorter than such a Semidiameter of the true Circle, it will make a fuller line than it should be, and so must not be so long, or else it will make a breach at the beginning of the line; this, if the Centre be supposed to be Abaft such a Perpendicular, that should draw a Rising line Abaft, I say, that it will shorten the Rising line, and make it fuller than it should be; or than if it be farther forward, it will be straighter than a Circle, and also be a breach at the beginning of the Rising line; therefore it should be a Circle, I say, whose Semidiameter will be on the Perpendicular line, at the beginning of any such Rising line, on the Heel, either Afoare, or Abaft, and the like aught to be for all other crooked lines, as the narrowing lines Abaft, or Afoare, or at the Narrowing of the Floor, or other Circular lines, as Hanging of Waals, and the like; the way whereof I shall describe, to find the lengths of all such Sweeps by Arithmetic; as also the true Rising, Narrowing of any Timber, according to exact pieces of Circles, very useful for the setting of Bows, to try whether they hang to a true Sweep or no: I shall demonstrate it, I say, in the following discourse, and in this place end what I intent to say. For Demonstration then, At ¾ of the Keel forward I draw a Rising line forward to the height of the Water line, forward on the Stem, as you see the line op; and the little line, between these two lines, parallel to the inside of the Keel, marked Eo, is the dead rising 4 inches high, as in the bend of Moulds it is parallel to FG, the height of the breadth from the Mid-Ship forward is the lower Edge of the upper Waale; but afterward on it is the pricked line, between the Water line and the lower Waale, on the Post, which runneth forward to the edge of the Waale, and hath Figures set to it, to signify the places of the Timbers marked 1, 2, 3, 4, 5, to 15; as you see answers to the Figures on the Keel: and the Letters set to forward on, signify the places of the Timbers forward, marked ABCD to L, in the middle of the Vessel: the places marked with a Cipher signify the Flats, which have only Dead rising, although they ought to have, some of them, something more Dead rising then each other; and those that have least, to be placed in the middle of the rest, that so there be no Clings in the Bulge, but that it have also a little Hanging in it, it will seem the fairer: Then I draw a strait line, parallel to the bottom of the Keel, as is the line FG, parallel to the line AB, the Keel, and distant 10 foot by the Scale, which is the half breadth of the Vessel; for this line signifieth a line stretched from the middle of the Sterne-Post to the middle of the Stem, called by Ship-wrightes, a Ram-line: Parallel to this Middle line I draw another line strait, marked nm, and is 4 foot asunder from the Middle line, to signify the half length of the Floor thwartships, as in the Bend of Moulds OF is distant from DC 4 Foot: then I draw a Crooked line Abaft, within this line nm, to signify the narrowing of the Floor, to bring, or form the Vessels way Abast, as you see the line ik; Abaft and Afoare it is represented by the line lo: then here in this Draught I draw a Sweep, or a piece of a Circle, from the point G, the mark of the Timber G, on the Keel, to the half breadth of the Stem, to the point G on the Stem, signifying the Sweep of the Harping, and is Swept by the breadth of the Vessel 20 Foot; the piece of the Pricked Circle Abaft at the Starne, which is drawn by a Centre on the line FG, is the length of the Transom thwart the Starne, as is the Arch FS, the length whereof is 8 Foot, which doubled is 16 Foot, for the whole length: which is ⅘ of the breadth 20 Foot, the length of the Sweep that sweepeth it is the length of the Starnpost to the bottom of the Keel 14 Foot ⅓, than the Crooked line, from the end of the Transom, or from the point S, and toucheth the Keel at the point p: this Arch Sparke, is the narrowing line Abaft at the breadth, and the Crooked pricked line within the Keel, marked with TR, is a Rising line, to order a hollow Mould by the Timbers, are placed at 2 Foot Timber and Room, as you may see by the Scale, the line drawn from the Poop to the Foarcastle, marked by the letters VW, is a line signifying the breadth of the Vessel, at the top of the side, from the top of the Poop to the Forecastle, the top of the Poop is in breadth 10 Foot, half the breadth at the beam; the use of this line is in ordering of the Moulds, to steady the Head of the Top-Timber Mould, to find his breadth aloft. CHAP. VI Showing the Making and graduating, or marking of the Bend of Moulds. Repair to some House that hath some Room or other broad enough to demonstrate the breadth of the Vessel, and height enough for the top of the Poop in the length of the Room; or else if you cannot find such a Room convenient, lay boards together, or planks, that may be large enough for your business, as in the following Scheame you see; First, a long square made for the breadth of the Vessel, as in the following Figure JABK: then make the Moulds by their Sweeps, and make Sirmarks to them for the laying of them together in their true places, off first the Mould, for the Floor being made, you may make a Sirmarke by the line OF, on the head of the Floor Mould, and another on the foot of the Navel Timber Mould, at the same place, to signify, that those two marks put together, they are in their true places, and will compare so when any Timbers are Moulded by them: those Sirmarks must also be marked off on the Timbers, and so in putting the Timbers up in the frame, a regard being had to compare Sirmarks with Hirmarks, each Timber will find his own place, and come to his own breadth, and give the Vessel that form assigned her by your Draught, if it be wrought by it, and so for all the other Moulds. In making your Moulds, that they may be smaller and smaller upwards, and not all of a bigness, you may measure the depth of the Side in the Mid Ships Circular, as it goeth from the Keel to the top of the Side, as here the Side, as it Roundeth, is 26 foot, and in depth at the Rounheads, or at the end of the Floor, is one Foot, as m m; and at the other end, at the head of the Timber is but half a Foot, as at n n, so then drawing two lines, as the lines n m, represents the diminishing of the Moulds in thickness upwards, as those two lines representeth; as if you would find the thickness of the Timbers at the breadth, take your 2 Foot Rule, and measure the length from the end of the Floor at the point F to I, at the breadth in the crooked body, and it is 11 Foot 9 Inches, signified at the Sirmarks there, those two lines show the thickness to be 9 Inches; and so thick aught the Moulds to be at the breadth of the Vessel. Now I have briefly touched the Demonstration of a Ship, by Projection, I shall now come to an Arithmetical way, far surpassing any Demonstration for exactness. CHAP. VII. Arithmetically showing how to frame the body of a Ship by Segments of Circles: being a true way to examine the truth of a Bow. LEt A B represent the length of a Rising line 12 foot long, or 144 inches, the height whereof let be B C, 5 foot, or 60 inches, to find the side D E, or D A, the radius of the circle A C, whereto A D is the Semidiameter; multiply the side A B 144 inches in itself, and so cometh 20736, which sum divide 144 144 576 576 144 20736 by the side B C, the height of the rising 60 inches, and so cometh 345, and 3●6/60, which is abreviated ³; unto this 345 ●/● must be added again the height of the Rising, the side B e, 60, which make 405 of an inch, which is the whole Diameter of the Circle, the half whereof is 202 1/● inches, and something more, near ●/4, therefore we will avoid the fraction, and account it 203▪ inches, or 16 foot 11 inches, which is the length of the Sweep, or the side D E, and so in all other Sweeps given whatsoever; the Rule is general, and holds true in all things: as to find the Sweep at once, that will round any Beam, or other piece of Timber that is to be Swept; remembering, that if it be a Beam, you are to find the Sweep you take but the half of his length. 23 (3 20736 (345 6000 66 Example, As if the Beam be 30 foot in length, and to round one foot, you must Work by 15, the half length of the Beam; and turn 15 foot into inches, by multiplying 15 by 12, so cometh 180 inches: remember the length of the Rising line, if it be to find the Sweep, it must be multiplied by itself, or the half length of the Timber must be Multiplied in itself, as 180 by 180, so cometh 32400, which must be divided by 12 the rounding, cometh in the quotient 2700, to which must be added the 12 again, the rounding of the piece, and so it is 2712 the whole Circle, the half of this 2712 is 1356 for the length of the Sweep, and so in all other matters where the Sweep is required: This I read in Mr. Gunter's Book, where he calls it the half Chord, being given, and the Versed fine, to find the Diameter, and Semidiameter of the circle thereto belonging: Example in the Draught foregoing. Where the length of the Rising line is from the point E, to the point i, 32 foot; and half the height thereof is the line D i, 10 foot: turn both Sums into inches, as 32 foot multiplied by 12 produceth, adding the ½ foot 6 inches, 390 inches length for the Rising line: then turn the height of the Rising into inches, as 10 foot multiplied by 12, produceth 120 inches, from which 4 inches must be substracted, because of the dead Rising is 4 inches, so then the height is 116 inches: Now multiply the length 390 inches by itself, 390 maketh 152100. 390 390 000 3510 1170 152100 This Multiplication of the sum 152100, must be divided by 116 inches, the height of the Rising, and so cometh in the quotient of the division 1311 inches; unto this 1311 inches, must be added the 116 inches, the height of the Rising 116/1427, and it maketh 1427, which is the whole 112 3323 46344 152100 (1311 116666 0111 11 Circle: divide it by 2, to find the half of it, so have you in the quotient 713 inches ½ inch for the length of the Sweep, which divided by 12, to bring it into feet, maketh 59 feet, 5 inches and a half, and so for all other Circular lines whatever, when the length is known, and the rounding of them also known; as for the hanging of Waals, the height of them known in the Midships from the Keel, substracted from the height, at the Post, and that will be the hanging of them, which is the same with the height of the Rising line on the Post, in the Arithmetical Work, and is the same with the Versed sine in Geometry; these I think Examples sufficient, to signify the Construction of this way of Working by Sweeps. 1 1427 (713 222 It followeth now that I show the manner of finding the Risings of Timbers by Arithmetic also. To find the Rising of the line F E, in the Figure foregoing. The Sweep being first found to be as before 203 inches, as the side D E signifieth, than there is known the side E G, 108 inches; now these two sides being given, we are to find the third side D G, so here is made a right Angled Triangle, two sides thereof are given to find a third, which to do, proceed thus; Multiply the two sides given by themselves, and subtract the Multiplication of the shortest side, from the Multiplication made of the other sides, and extract the square Root of the remainder, so have you the third side sought for. Example in the following Triangle. Having the side D C, 12 foot, which is 144 inches▪ and the side A C, 10 foot, otherwise 120 inches; to find the side D A, multiply the sides given, in themselves, which is called squaring of them: as, multiply the side D C, 144 inches, by 144 inches, so cometh 20736. Then multiply the other side A C 120 also by itself, so cometh in the quotient 14400, which must be substracted from the other Multiplication, as you see, so cometh in the quotient 6336, from which the greatest square must be extracted, called extraction of the square root, 144 144 576 576 144 20736 120 120 000 240 120 14400 20736 14400 6336 which is 79 inches, and almost another by the Fraction, that is 6 foot, and very near 8 inches. 1 595 147 6●3● (79 14 Note, These Demonstrations, this and the former, are laid down by the first Scale, made to show the Demonstration of a Scale in this Book, at the beginning. Another Example. So in the last Figure foregoing but one, the side D E, 203 inches, which squared, or multiplied in itself, is 41209. 203 203 609 000 406 41209 Then the other side G E, 108, multiplied in itself, which is squaring of it, is 12664, as you see. 108 108 864 000 168 11664 Which substracted from the other multiplication, as 11664 substracted from 41209, resteth 29545, the square Root extracted from it, or the side of the greatest square that can be taken from the substraction being found, is 171, and ¾; which 171 ¾, substracted from 203, the length of the Sweep for one side, is always the length of the Sweep, resteth 31 inches ¼, for the Rising of the line E F, and the like for any other Rising. 41209 11664 29545 (3 156 (08 29549 (●7 .2. 34 Another Example. As at the place K I, the Rising thereof is required, the side D I is as D E, 203 inches. Note, The length of the Sweep being found, always is one of the sides, in the finding the Rising of any Timber, and is always one of the numbers, which when you have squared, note in a piece of Paper by itself, where you may always see what it is, so that in the finding of Risings, after the Sweep is found, all you have to do, is to know how many feet, or inches, the Timber you seek for is removed from the beginning, or foot of the Rising line, which is the second side, and in this third Example it is 11 foot, or 132 inches K I, from the foot of the line A, which squared, is 17424, which must be substracted from the square made of Radius, which in the other example is 41209, and so resteth 23775, from which extract the side of the Square therein contained, and it is 154 inches and ¼, which substracted from the length of the Sweep, leaveth 48 inches for the Rising, and ¾ inches, or 4 foot, and ¾ of an inch, and so much is the Rising of the said Timber. 132 132 264 396 132 17424 41209 17424 23775 10 132 (69 23785 (154 .2.0. 3 One Example in the Draught, The length of that Sweep we found heretofore to be 713 inches, than we will seek to find the Rising for Timber 13, standing aft from the point E, or foot of the Rising line 324 inches, these are the given Sides; then proceed; square the Semidiameter of the Sweep 713, so it maketh squared 508363; then square the distance of the Timber 13, which is 324, and it maketh 104976; these substracted from the former figures, resteth 403387, the square Root thereof is 635 ¼, nearest, which substracted from the Radius 713, resteth 77 inches and ¼, that is 6 foot 5 inches, which with 4 inches Dead Rising, is 6 foot 9 inches ¼; and so much is the Rising of Timber 13 from the Keel. I suppose these Examples are sufficient to illustrate the truth and plainness of this Arithmetical Work, for the truth of it; it hath this to say for itself, that it is the very exact truth itself: The great Objection may be, that many know not the way to Extract the Square Root, and therefore cannot attain to this Work, by reason of that let, or hindrance. To this I Answer, There are many Books that will instruct thee in it, that thou mayest buy, or borrow; but to answer thee better, I will briefly show thee the manner of Extracting the Square, not doubting but thou canst perform Addition, Substraction, Multiplication, and Division already. CHAP. VIII. How to Extract the Square Root. KNow then that a square number hath its sides equal every way, as are the sides of 4, represented by ⸬ pricks; and you see that every way of all the 4 sides it containeth 2, and so 2 times 2, make 4, which is the squaring of a number, so you see ⋮ ⋮ ⋮ 9 pricks is a square, or 9 is a square number, whose side is 3, and 3 times 3 make 9, but 2 times 3 is not a square number, as you see:::, being but 2 one way, and the other way 3, that make but 6; so than all the numbers between 4 and 9, are not square numbers: by the like reason, a square, made of the Next square number 4 is 16, for 4 times 4 is 16, as by the Pricks you may see it represented here, every of the 4 sides containing 4, make a squared number of 16, and all the numbers that are between 9 and 16, as 2 times 4, or 3 times 4, are not squares, but have a fraction annexed to them; so also any number between 16 and 25, are not squares, as 4 times 5, or 2 times 5, or 3 times 5, these are not square numbers, but 5 times 5 is a squared number, and maketh 25, where note, that to square a number, and to extract the square root, is two different things; for when we say, to square a number, is to multiply it in itself, or by it itself; or Thus you may conceive of the Squares of 6, for 6 times 6 make 36; 7 times 7 make 49; 8 times 8 make 64; 9 times 9 make 81; 10 times 10 make 100: there is all the squares made of the 9 Figures, expressed by this little Table annexed, as against each Figure is the square made of them, as 2 times 2 is 4, so is 4 against 2, as you see. 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 Now to extract the square Root from greater numbers, as from 144 proceed thus, write down the sum given, as followeth, and make a quotient on the right hand, as you see, then set pricks under every other figure beginning at the right hand, and set pricks towards the left hand, under every other figure, so in this number 144, consisting of 3 figures, there is 2 pricks, and so many figures must the quotient consist of; then begin at the left hand of the sum, and say, or inquire for the greatest square in the figure, or figures, over the first prick, at the left hand, which here is but 1, therefore you can take but 1, for 1 is always the Square, or Cube of 1, therefore write 1 in the quotient, and subtract that 1 from the 1 over the left hand prick, 144 (.. 0 144 (1 .. and Cancel it, nothing remainining, write a Cipher over it, as you see, so have you one figure of the quotient, then double your figure found in the quotient, as 2 times 1 is 2; write that 2 under the figure between the next pricks, which is the Divisor for the second figure, then say, how many times 2 can I have in 4, over the Divisor, I say 2, therefore I write 2 in the quotient, saying, 2 times 2 is 4, which subtract from the 4 over head, Cancel the Divisor, and the 4 over head, and write a Cipher over it, then square the last figure found in the quotient, saying, 2 times 2 is 4, which subtract from the 4 over the prick, and so resteth 0, therefore cancel the 4, and write cyphers over head, signifying, that the number given to find the root of, is a just square number, the root or side is 12, the proof hereof is by Multiplication of the quotient in itself, as 12 by 12 make 144, which, if it be the same with the sum given to be Extracted, it is rightly done▪ if it do not agree, it is not true. 000 144 (12 .2. Example of another Summ. Let 625 be given to find the Square Root of it, write down the sum, make a quotient, and set pricks under every other figure; then inquire for the greatest square in the figure, over the prick, at the left hand; I say, 2 is the greatest square can be taken: for 3 times 3 is 9, and here the figure is but 6; so I write 2 in the quotient, and square it, saying, 2 times 2 is 4, taken from 6, so resteth 2; I cancel the 6, and write 2 2 625 (2 .. over it, as you see, then double the figure in the quotient, saying, 2 times 2 is 4; this 4 is the second Divisor, I write it between the two next pricks, and say, how many times 4, can I have in 22, and I find 5 times; for 5 times 4 is 20, taken from 22, the figures over 4, so resteth 2; therefore I write 5 in the quotient, and saying, 5 times 4 make 20▪ therefore I cancel the 4 Divisor, and the 22, and write 2 overhead, then square the last figure found, 5 by 5 make 25, taken from 25 over head▪ resteth nothing, so the number given is a square number. 2 625 (2 .4. 22 625 (25 .4. 22 625 (25 .4. A Sum of 5476, given to find the nearest Square Root in it, write down the Sum, and make a quotient and prick underneath, as afore showed; say, What is the greatest Square in the figures over the left hand prick? and I find it to be 7, for 7 times 7 make 49, but 8 times 8 make 64, 10 too much, therefore I write 7 in the quotient, and take 7 times 7, that is 49 from 54, so resteth 5, which I write over the prick, and Cancel the 5 and the 4; then I double the figure in the quotient, which maketh 14, for the Divisor, I write the first figure of the Divisor, if there be more than 1 under the figure, between the two next pricks, and all the other figures, in their places, toward the left hand; then inquire how many times can 1 be taken from 5, overhead, and I find it may be taken 4 times; I write therefore 4▪ 5 5476 (7 .. 1 51 5476 (74 .. 14 in the quotient, and say, 4 times 1 is 4, from 5; so resteth 1: I Cancel the 1 and the 5, and write 1 over the 5, than I say, 4 times 4 make 16, from 17 resteth 1: I Cancel the 4 Divisor, and write 1 over 7, and Cancel the other 1 and the 7; then I square the last figure found, for so it must be at every prick, 4 times 4 make 16, which I subtract from the 16 over the last prick, and so I see nothing remaineth, that showeth the sum given, to be a just square sum. 10 510 5476 (74 .. 14 Example of another Summ. As if 528563 be given to find the greatest side of the Square therein, I write down the Sum, as followeth, and make the quotient, and set the pricks under every other figure, as you see; and seeing there is 3 pricks, it telleth, that there must be 3 figures in the quotient, then beginning at the figures over the left hand prick, I take the greatest square in 52, and I find it 7, for 7 times 7 make 49; therefore I write down 7 in the quotient, and Subtract 49 from 52, so resteth 3, therefore Cancel the 52, and write 3 over the 2, as you see; then double the quotient 7, it maketh 14, for a new Divisor, which write down, the first figure thereof, under the figure between the two next pricks, namely 4 under 8, and the other figure of the Divisor one place farther to the left hand, under the 3, as you see; then take the Divisor 1 as many times as you can, from the figure 3 528563 (7 ... 3 528563 (7 ... 14 3 over head, so as that after the Division be made, there may be the square of the last figure of the quotient, taken from the figures over the next prick, as I can take 1 but 2 times from 2, therefore I write 2 in the quotient, and Cancel the Divisor 1, saying, 2 times 1 is 2, from 3; so resteth 1: I Cancel the figure 3 also, and write 1 overhead, as you see: then 2 times 4 is 8, from 8 over head resteth nothing; therefore I Cancel the second figure of the Divisor, 4 and 8, and write a Cipher over 8, as you see; then the next place being a prick, I must square the last figure found, saying, 2 times 2 make 4, from 5; the figure over the prick resteth 1, as you see; therefore I Cancel the 5, and write 1 over it, as you see, and here is a fraction of 101. 1 30 528563 (72 14 1 301 528563 (72 14 Then for a new Divisor, Double the quotient 72, and it makes 144, which is a new Divisor, the first figure thereof write under the figure between the next pricks, as the first 4 under 6, in the sum; and the other figures towards the left hand, in the order as you see: then, how many times 1 in 10 over head, and I see I cannot take 8 times, for that there will not be left to take out the other figures from, nor for the square of the last figure, which if it were 8, would be 64 from the figure over the prick, therefore I take but 7, for by a light examination I see that will do, therefore I write down 7 in the quotient, and proceed to the 1 301 528563 (727 14 4 14 Division, thus, 7 times 1 is 7, from 10 over head remaineth 3, which I write down, and Cancel the 10, as you see; then 7 times 4 is 28, from 31 over head, so remaineth 3, which I also write down, and Cancel the 31; then again, 7 times 4, the other figure of the Divisor, is also 28, which taken from 36 over head, resteth 8, which I write down over 6, and so Cancel the 36, and then the Sum standeth as you see. 13 301 528563 (727 14 4 14 133 301 528563 (727 14 4 14 133 3018 528563 (727 14 4 14 133 (3 530●8 (4 528563 (727 14 4 14 Then last, square the last figure of the quotient, 7 times 7 make 49, taken from 83, the figures over the prick, resteth 34, as a fraction, and the Sum is finished: But in regard here is a fraction, by that it tells you, that the Sum given was no square number; and the greatest square therein is 727, the proof is by Multiplication adding in the fraction thus, 727 Multiplied by 727, make 528529, then adding in the fraction of 34, maketh it 528563, the just Sum given. But some may Object, and say, That this is a very tedious way of Work, and will take up a great deal of time; It is true, it is more labour than demonstration, but the truth of it might very well plead for patience to Work it, but it is not necessary you perform all the parts by it, that is, in every particular: as the exact hanging of the Waal at every Timber, but it may suffice at every third or fourth Timber, to find the hanging of the Waals, only the Risings allow, afoare and abaft, I would work to every Timber there. But to make it more brief, here followeth a Table that the numbers are therein contrived to the same purpose, to avoid the tedious Extraction of the Root, and only use Addition and Substraction, only being but a very little difference between the finding the Risings by this Table, and by the Draught, for in this kind of Arithmetical Work, it mattereth not, whether or no there be any Draught drawn at all, or no, if the builder only note in his Book the length by the Keel, and the breadth at the Beam, the Rack of the Stem, Rack of the Post, depth of the Water, to Sail in depth of the Hold, height of the Waals abaft, afoare, at the Midships, and all the remarkable things to be noted, he may be able to Build a Vessel, and never draw a Draught at all, and yet affirm his Work to be absolutely true, according to Art, and a great deal more exact than by Draught: I shall in few words show you the use of the Table, and so conclude. CHAP. IX. A Description of the Table of Squares. TO save the Practitioner a labour of Extracting of Roots, for here they are ready done to thy hand of purpose, and all the use of Arithmetic required is only Substraction, as Example in the Figure of the Sweep foregoing, being found to be 203 inches, as you saw it found before, which is, I say, always one side of the Triangle, made of the side DI, then knowing the length of ●I, 132 inches, which is the distance of the point, of which the Rising is sought at; seek in the Tables, under the Title of inches, at the head of the Tables, for 132, you will find it in the second Page, and the twelfth line; and right against it, in the same line, under the next Title of Squares, you have 17424, the square made of 132, which Subtract from the Square made of 203, which is 41209, which is found in the second Page of the Tables, and the third line: Now the other number 17424, Substracted from 41209, so resteth 23775; seek the number nearest to it in the Table, under the Title of Squares, which you will find in the second Page, 34 line, you find not just the same number, for in stead of 23775, you find 23716 too little by 59, and the Root answering thereto, is in the same line, under the Title of inches, towards the left hand, which is 154; now if you take the next square lower to the left hand 35 line, it is 24025, 250 too much, so you may see it is nearer to the 34th line, because there it was too little but by 59, so that you may see it will be ¾ of an inch less than the number of inches, belonging to the 35th line, and about ¼ of an inch more than the numbers in the 34th line; so that you see it is answered, the third side D 0 is 154, and ¼ of an inch, which Substracted from the whole Sweep 203, leaveth 48¾ inches for the Rising, so you have no need of extraction of the Roots by these Tables, it is already done to your hand; the Column that is between the inches and the squares, and written feet inches in the head, is to show you, how may feet, and inches of the foot any number of inches is; as here, the number 203 inches sought, and found in the Tables, in the second page, and third line, just against it, in the same line, between that and the squares, is 16— 11, showing that it is 16 feet and 11 inches; or if the square were given, as 41209, found at the second page, and third line, next toward the left hand, you have 16 foot, 11 inches; and if you seek for it in inches, in the third Column toward the left hand, and the same line, you have 203 inches: Thus is it very ready to reduce inches into foot measure, or feet into inches. Another Example. In the same figure, to find the Rising at the point F, the sweep being 203 inches, as before is said, is always one side, throughout the whole Work of the same Rising line is 41209, as is found in the second page, the third line; the other fide from the point A to F, is 9 foot, or 108 inches, whose square is 11664, found in the first page, and the 28th line; now subtract the square made of the side A F, 11664, from the square of the side D E, so remaineth 29545 41209 11664 29545 Seek in the Table of squares for that number, and I find in the second page, and 12 line, and the sixth Column, 29584, the nearest number to it, yet it is a little too much near the ¼ of an inch; and toward the left hand in the same line, the next Column under the title feet inch, you find 14; 4 signifying that to be 14 foot, 4 inches: and in one Column more to the left hand, and the same line, you see under the Title of inches 172 over the head you tituled inches, which must be subracted from 203 inch, so remaineth 3 inches for the Rising of F E, which is 2 foot, 7 inches, as in the first page of the Table, and the 31 line. 203 172 031 These few Examples I think may be sufficient to show the use of the following Tables of squares, the benefit where of may be very great, for such as shall make use of the same: If any desire the finding of the Fractions of these squares, when he findeth not his just figures in the squares, let him do thus, subtract the Figures under his number, from the Figures above his number, which shall be the denominator, than these Figures given, substracted, from which the next squares less, shall be the denominator to that Fraction. As for Example, In the foregoing figures, after substraction, should have been 29553; the nearest agreeing in the Tables, is 29584, the next lesser square number in the Table is 29241, which is more a great deal too little, than the other is too great; then subtract the lesser square number 29241, from 29584, and so resteth 343, which must be the denominator, then again subtract the true number given, 29553, the next lesser square number in the Table is 29241, which must be substracted, I say, from the true number given, 29553, and so resteth after substraction 312, which is the Numerator to the Fraction, and must be thus written, ● so than the number belonging to 29584, is 171 inches, and 312/343 parts of an inch, which being abreviated, is something more than ¼ of one inch, and not full ⅞ of one inch. 29584 29241 343 Thus he that pleaseth may find the rising of any Timber, or narrowing of any place by these Tables and the help of Substraction, exactly to any Circle whatsoever, but it may suffice, that a Man, going to his Tables, may see which square his figures have greatest affinity with, and may estimate the difference near enough, without seeking for the fraction, which will be easily known by much practice herein. HEre followeth a Table of Square Roots, ready Extracted, from one Inch to 1300 Inches, which is to 108 foot▪ and 4 Inches, and it is thus contraved, That from one Inch, to 840 Inches, all the Inches are reduced into Feet and Inches▪ for the ease and help of Workmen, who always take their Measures by Feet and Inches; but from thence to the end of the table you have the Inches only, and the Squares thereof against them as the Titles over every Page do make appear. A Table of Square Roots. Inch Feet Inches Squares 1 1 1 2 2 4 3 3 9 4 4 16 5 5 25 6 6 36 7 7 49 8 8 64 9 9 81 10 10 100 11 11 121 12 1 00 144 13 1 1 169 14 1 2 196 15 1 3 225 16 1 4 256 17 1 5 289 18 1 6 324 19 1 7 361 20 1 8 400 21 1 9 441 22 1 10 484 23 1 11 529 24 2 00 576 25 2 01 625 26 2 2 676 27 2 3 729 28 2 4 784 29 2 5 841 30 2 6 900 31 2 7 961 32 2 8 1024 33 2 9 1089 34 2 10 1156 35 2 11 1225 36 3 00 1296 37 3 1 1369 38 3 2 1444 39 3 3 1521 40 3 4 1600 41 3 5 16●1 42 3 6 1764 43 3 7 1849 44 3 8 1936 45 3 9 2025 46 3 10 2116 47 3 11 2209 48 4 00 2304 49 4 1 2401 50 4 2 2500 51 4 3 2601 52 4 4 2704 53 4 5 2809 54 4 6 2916 55 4 7 3025 56 4 8 3136 57 4 9 3249 58 4 10 3364 59 4 11 3481 60 5 00 3600 61 5 1 3721 62 5 2 3844 63 5 3 3964 64 5 4 4096 65 5 5 4225 66 5 6 4356 67 5 7 4489 68 5 8 4624 69 5 9 4761 70 5 10 4900 71 5 11 5041 72 6 00 5184 73 6 1 5329 74 6 2 5476 75 6 3 5625 76 6 4 5776 77 6 5 5929 78 6 6 6084 79 6 7 6241 80 6 8 6400 81 6 9 6561 82 6 10 6724 83 6 11 6889 84 7 00 7056 85 7 1 7225 86 7 2 7396 87 7 3 7569 88 7 4 7744 89 7 5 7921 90 7 6 8●00 91 7 7 8●81 92 7 8 8464 93 7 9 8649 94 7 10 8836 95 7 11 9025 96 8 0 9226 97 8 1 9409 98 8 2 9604 99 8 3 9801 100 8 4 10000 101 8 5 10201 102 8 6 10404 103 8 7 10609 104 8 8 10816 105 8 9 11025 106 8 10 11236 107 8 11 11449 108 9 0 11664 109 9 1 11881 110 9 2 12100 111 9 3 12321 112 9 4 12544 113 9 5 12769 114 9 6 12996 115 9 7 13225 116 9 8 13456 117 9 9 13689 118 9 10 13924 219 9 11 14162 120 10 0 14400 121 10 1 14641 122 10 2 14884 123 10 3 15229 124 10 4 15376 125 10 5 15625 126 10 6 15876 127 10 7 16029 128 10 8 16384 129 10 9 16641 130 10 10 16900 131 10 11 17161 132 11 00 17424 133 11 1 17689 134 11 2 17956 135 11 3 18225 136 11 4 18496 137 11 5 18769 138 11 6 19044 139 11 7 19321 140 11 8 19600 141 11 9 19881 142 11 10 20164 143 11 11 20449 144 12 00 20736 145 12 01 21025 146 12 2 22416 147 12 3 21609 148 12 4 21904 149 12 5 22201 150 12 6 22500 151 12 7 22801 152 12 8 23104 153 12 9 23409 154 12 10 23716 155 12 11 24025 156 13 00 24336 157 13 1 24649 158 13 2 24964 159 13 3 25381 160 13 4 25600 161 13 5 25921 162 13 6 26244 163 13 7 26569 164 13 8 26956 165 13 9 27225 166 13 10 27556 167 13 11 27889 168 14 00 28224 169 14 1 28561 170 14 2 28900 171 14 3 29241 172 14 4 29584 173 14 5 29929 174 14 6 30276 175 14 7 30625 176 14 8 31076 177 14 9 31329 178 14 10 31684 179 14 11 32041 180 15 00 32400 181 15 1 32761 182 15 2 33124 183 15 3 33489 184 15 4 33856 185 15 5 34025 186 15 6 34596 187 15 7 34969 188 15 8 35344 189 15 9 35721 190 15 10 36100 191 15 11 36481 192 16 00 36864 193 16 1 37249 194 16 2 37636 195 16 3 38025 196 16 4 38416 197 16 5 38809 198 16 6 39204 199 16 7 39601 200 16 8 40000 201 16 9 40401 202 16 10 40844 203 16 11 41209 204 17 00 41616 205 17 1 42025 206 17 2 42436 207 17 3 42849 208 17 4 43264 209 17 5 43681 210 17 6 44100 211 17 7 44521 212 17 8 44944 213 17 9 45369 214 17 10 45796 215 17 11 46224 216 18 0 46656 217 18 1 47089 218 18 2 47524 219 18 3 47961 220 18 4 48400 221 18 5 48841 222 18 6 49284 223 18 7 49729 224 18 8 50176 225 18 9 50625 226 18 10 51076 227 18 11 51529 228 19 0 51984 229 19 1 52441 230 19 2 52900 231 19 3 53361 232 19 4 53824 233 19 5 54289 234 19 6 54656 235 19 7 55225 236 19 8 55696 237 19 9 56069 238 19 10 56644 239 19 11 57121 240 20 0 57600 241 20 1 58081 242 20 2 58564 243 20 3 59049 244 20 4 59536 245 20 5 60025 246 20 6 60516 247 20 7 61009 248 20 8 61504 249 20 9 62001 250 20 10 62500 251 20 11 63001 252 21 0 63504 253 21 1 64009 254 21 2 64516 255 21 3 65025 256 21 4 65536 257 21 5 66049 258 21 6 66564 259 21 7 67081 260 21 8 67600 261 21 9 68121 262 21 10 68644 263 21 11 69169 264 22 0 69596 265 22 1 70425 266 22 2 70756 267 22 3 71289 268 22 4 71824 269 22 5 72361 270 22 6 72900 271 22 7 73441 272 22 8 73984 273 22 9 74529 274 32 10 75076 275 22 11 75575 276 23 0 76176 277 23 1 76729 278 23 2 77284 279 23 3 77841 280 23 4 78400 281 23 5 78961 282 23 6 79524 283 23 7 80089 284 23 8 80656 285 23 9 81225 286 23 10 81796 287 23 11 82369 288 24 0 82944 289 24 1 83521 290 24 2 84100 291 24 3 84681 292 24 4 85264 293 24 5 85849 294 24 6 86436 295 24 7 87025 296 24 8 87616 297 24 9 88209 298 24 10 88804 299 24 11 89401 200 25 0 90000 301 25 1 90601 302 25 2 91204 303 25 3 91809 304 25 4 92416 305 25 5 93025 306 25 6 93636 307 25 7 94241 308 25 8 94864 309 25 9 95481 310 25 10 96100 311 25 11 96721 312 26 0 97344 313 26 1 97969 314 26 2 98596 315 26 3 99225 316 26 4 99856 317 26 5 100489 318 26 6 101124 319 26 7 101761 320 26 8 102400 321 26 9 103041 322 26 10 103684 323 26 11 104329 324 27 0 104976 325 27 1 105625 326 27 2 106276 327 27 3 106929 328 27 4 107584 329 27 5 108241 330 27 6 108900 331 27 7 109561 332 27 8 110224 333 27 9 110889 334 27 10 111556 335 27 11 112225 336 28 0 112896 337 28 1 113569 338 28 2 114244 339 28 3 114921 340 28 4 115600 341 28 5 116281 342 28 6 116964 343 28 7 117349 344 28 8 118336 345 28 9 119025 346 28 10 119716 347 28 11 120409 348 29 0 121104 349 29 1 121801 350 29 2 122505 351 29 3 123206 352 29 4 123909 353 29 5 124604 354 29 6 125311 355 29 7 126025 356 29 8 126736 357 29 9 127449 358 29 10 128164 359 29 11 128881 360 30 0 129600 361 30 1 130321 362 30 2 131044 363 30 3 131779 364 30 4 132496 365 30 5 133225 366 30 6 133956 367 30 7 134689 368 30 8 135424 369 30 9 136161 370 30 10 136900 371 30 11 137640 372 31 0 138384 373 31 1 139129 374 31 2 139876 375 31 3 140625 376 41 4 141676 377 31 5 142129 378 31 6 142984 379 31 7 143641 380 31 8 144400 381 31 9 145161 382 31 10 145924 383 31 11 146689 384 32 0 147456 385 32 1 148225 386 32 2 149006 387 32 3 149769 388 32 4 150544 389 32 5 151321 390 32 6 152210 391 32 7 152831 392 32 8 153664 393 32 9 15444● 394 32 10 155236 395 32 11 156025 396 33 0 156816 397 33 1 157609 398 33 2 158104 399 33 3 159201 400 33 4 160000 401 33 5 160801 402 33 6 161604 403 33 7 162409 404 33 8 163216 405 33 9 164025 406 33 10 164836 407 33 11 165649 408 34 0 166464 409 34 1 167281 410 34 2 168100 411 34 3 168921 412 34 4 169744 413 34 5 170569 414 34 6 171396 415 34 7 172225 416 34 8 173056 417 34 9 173889 418 34 10 1747●4 419 34 11 175561 420 35 0 176400 421 35 1 177241 422 35 2 178084 423 35 3 178959 424 35 4 17977● 425 35 5 180625 426 35 6 181476 427 35 7 182329 428 35 8 183184 429 35 9 184041 430 35 10 184900 431 35 11 185761 432 36 0 186624 433 36 1 187789 434 36 2 188●56 435 36 3 1898●5 436 36 4 190096 437 36 5 190960 438 36 6 191044 439 36 7 192721 440 36 8 193600 441 36 9 194481 442 36 10 195364 443 36 11 196249 444 37 0 197136 445 37 1 198025 446 37 2 198916 447 37 3 199809 448 37 4 200704 449 37 5 201601 450 37 6 202509 451 37 7 203401 452 37 8 204304 453 37 9 205●90 454 37 10 206116 455 37 11 2070●5 456 38 0 207936 457 38 1 208849 458 38 2 209●64 459 38 3 210681 460 38 4 2116●0 461 38 5 212521 462 38 6 213444 463 38 7 214369 464 38 8 215296 465 38 9 2162●5 466 38 10 217156 467 38 11 218089 468 39 0 219024 469 39 1 219961 470 39 2 220●●0 471 39 3 221841 472 39 4 222784 473 39 5 223729 474 39 6 224676 475 39 7 225625 476 39 8 226576 477 39 9 227429 478 39 10 228484 479 39 11 229141 480 40 0 2304●● 481 40 1 231361 482 40 2 232324 483 40 3 233289 484 40 4 234216 485 40 5 235225 486 40 6 236196 487 40 7 237●69 488 40 8 238144 489 40 9 239121 490 40 10 240100 491 40 11 240981 492 41 0 2420●4 493 41 1 243049 494 41 2 244036 495 41 3 245025 496 41 4 246016 497 41 5 246509 498 41 6 247004 499 41 7 249001 500 41 8 250000 501 41 9 251001 502 41 10 252004 503 41 11 253009 504 42 0 254016 505 42 1 255025 506 42 2 256036 507 42 3 257049 508 42 4 258064 509 42 5 269081 510 42 6 260100 511 42 7 261121 512 42 8 262144 513 42 9 363169 514 42 10 264196 515 42 11 265225 516 43 0 266256 517 43 1 267289 518 43 2 268324 519 43 3 269361 520 43 4 270400 521 43 5 271441 522 43 6 272448 523 43 7 273529 524 43 8 274576 525 43 9 275625 526 43 10 276676 527 43 11 2777●9 528 44 0 278784 529 44 1 280●41 530 44 2 280900 531 44 3 281961 532 44 4 284●24 533 44 5 2870●9 534 44 6 285156 535 44 7 286225 536 44 8 287296 537 44 9 288369 538 44 10 290444 539 44 11 290521 540 45 0 291600 541 45 1 292681 542 45 2 293764 543 45 3 294849 544 45 4 295936 545 45 5 297025 546 45 6 298016 547 45 7 299209 548 45 8 300304 549 45 9 301401 550 45 10 302500 551 45 11 303601 552 46 0 304704 553 46 1 305809 554 46 2 306916 555 46 3 308025 556 46 4 309136 557 46 5 310●49 558 46 6 311364 559 46 7 312481 560 46 8 313600 561 46 9 314721 562 46 10 315844 563 46 11 316969 564 47 0 318096 565 47 1 319225 566 47 2 320356 567 47 3 321489 568 47 4 322624 569 47 5 323761 570 47 6 324900 571 47 7 326041 572 47 8 327184 573 47 9 328329 574 47 10 330276 575 47 11 330625 576 48 0 331776 577 48 1 332929 578 48 2 384048 579 48 3 335241 580 48 4 336400 581 48 5 337561 582 48 6 338724 583 48 7 340089 584 48 8 341056 585 48 9 3422●5 586 48 10 343396 587 48 11 344669 588 49 0 345744 589 49 1 346921 590 49 2 348100 591 49 3 349281 592 49 4 350464 593 49 5 351649 594 49 6 352836 595 49 7 353925 596 49 8 354216 597 49 9 355409 598 49 10 356●04 599 49 11 358801 600 50 0 360000 601 50 1 361201 602 50 2 362404 603 50 3 363609 604 50 4 364816 605 50 5 366025 606 50 6 367236 607 50 7 368449 608 50 8 369664 609 50 9 370881 610 50 10 372100 611 50 11 373321 612 51 0 374544 613 51 1 375769 614 51 2 376996 615 51 3 378225 616 51 4 379456 617 51 5 380689 618 51 6 381924 619 51 7 383161 620 51 8 384400 621 51 9 385641 622 51 10 386884 623 51 11 388129 624 52 0 389376 625 52 1 390625 626 52 2 391876 627 52 3 393129 628 52 4 394384 629 52 5 395641 630 52 6 396900 631 52 7 398161 632 52 8 399424 633 52 9 400489 634 52 10 401956 635 52 11 403225 636 53 0 404496 637 53 1 405769 638 53 2 407044 639 53 3 408321 640 53 4 409600 641 53 5 410881 642 53 6 412164 643 53 7 413449 644 53 8 414736 645 53 9 416025 646 53 10 417316 647 53 11 418609 648 54 0 429904 649 54 1 421201 650 54 2 422500 651 54 3 423801 652 54 4 425104 653 54 5 426403 654 54 6 427716 655 54 7 429025 656 54 8 430336 657 54 9 431449 658 54 10 432969 659 54 11 434181 660 55 0 435600 661 55 1 436921 662 55 2 438244 663 55 3 439569 664 55 4 440896 665 55 5 442225 666 55 6 443556 667 55 7 444889 668 55 8 446224 669 55 9 447561 670 55 10 448900 671 55 11 450241 672 56 0 451544 673 56 1 452829 674 56 2 454276 675 56 3 455625 676 56 4 456976 677 56 5 458329 678 56 6 459684 679 56 7 461041 680 56 8 462400 681 56 9 463761 682 56 10 465124 683 56 11 466489 684 57 0 467856 685 57 1 469225 686 57 2 470596 687 57 3 471939 688 57 4 473344 689 57 5 475721 690 57 6 476700 691 57 7 477841 692 57 8 478864 693 57 9 480269 694 57 10 481636 695 57 11 482825 696 58 0 484416 697 58 1 485809 698 58 2 487204 699 58 3 488601 700 58 4 490000 701 58 5 491401 702 58 6 492804 703 58 7 494209 704 58 8 495616 705 58 9 497025 706 58 10 498436 707 58 11 498849 708 59 0 501264 709 59 1 502681 710 59 2 504100 711 59 3 505521 712 59 4 506944 713 59 5 508669 714 59 6 509796 715 59 7 511225 716 59 8 512656 717 59 9 514089 718 59 10 515824 719 59 11 516961 720 60 0 518400 721 60 1 519841 722 60 2 521284 723 60 3 522729 724 60 4 524176 725 60 5 525625 726 60 6 526976 727 60 7 528529 728 60 8 529984 729 60 9 521421 730 60 10 522900 731 60 11 524361 732 61 0 535844 733 61 1 537289 734 61 2 538656 735 61 3 540225 736 61 4 541696 737 61 5 543169 738 61 6 544644 739 61 7 546031 740 61 8 547600 741 61 9 549081 742 61 10 550564 743 61 11 552049 744 62 0 553436 745 62 1 555025 746 62 2 556516 747 62 3 558009 748 62 4 559504 749 62 5 561001 750 62 6 562500 751 62 7 564001 752 62 8 565504 753 62 9 567009 754 62 10 568516 755 62 11 570025 756 63 0 571536 757 63 1 573049 758 63 2 574564 759 63 3 576081 760 63 4 577600 761 63 5 579121 762 63 6 580644 763 63 7 582169 764 63 8 583696 765 63 9 585225 766 63 10 586756 767 63 11 588289 768 64 0 589824 769 64 1 591361 770 64 2 592900 771 64 3 594441 772 64 4 595984 773 64 5 597529 774 64 6 599076 775 64 7 600625 776 64 8 602176 777 64 9 604729 778 64 10 606284 779 64 11 607841 780 65 0 608400 781 65 1 609961 782 65 2 611524 783 95 3 613099 784 65 4 614656 785 65 5 616225 786 65 6 617796 787 65 7 619369 788 65 8 620944 789 65 9 622521 790 65 10 624100 791 65 11 625681 792 66 0 627964 793 66 1 628849 794 66 2 630466 795 66 3 632125 796 66 4 633616 797 66 5 635209 798 66 6 637404 799 66 7 638401 800 66 8 〈◊〉 801 66 9 641601 802 66 10 642204 803 66 11 644809 804 67 0 646416 805 67 1 648025 806 67 2 649836 807 67 3 651249 808 67 4 652864 809 67 5 654481 810 67 6 656100 811 67 7 657721 812 67 8 659344 813 67 9 660969 814 67 10 662596 815 67 11 664225 816 68 0 665856 817 68 1 667429 818 68 2 669124 819 68 3 671771 820 68 4 672400 821 68 5 674041 822 68 6 675684 823 68 7 677329 824 68 8 678976 825 68 9 680625 826 68 10 682276 827 68 11 684129 828 69 0 685584 829 69 1 688241 830 69 2 688900 831 69 3 689661 832 69 4 692224 833 69 5 693889 834 69 6 695556 835 69 7 697225 836 69 1 698896 837 69 9 700569 838 69 10 702244 839 69 11 703921 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Inch Squares 841 707281 842 708964 843 710649 844 711336 845 714025 846 715716 847 717309 848 719004 849 720801 850 722500 851 724●01 852 725904 853 727609 854 729216 855 721025 856 732736 857 734449 858 736164 859 737681 860 739600 861 741321 862 743044 863 744769 864 746396 865 748225 866 749956 867 753689 868 753824 869 755161 870 756900 871 758641 872 760384 873 762129 874 763776 875 765625 876 767376 877 769129 878 770884 879 772641 880 774400 881 777161 882 777924 883 779589 884 781456 885 783225 886 784996 887 786769 888 788544 889 790321 890 792100 891 793081 892 795664 893 797449 894 799236 895 801025 896 802816 897 804609 898 805904 899 808201 900 810000 901 811801 902 813604 903 815400 904 817216 905 819025 906 820836 907 822649 908 824464 909 826281 910 828100 911 829921 912 831741 913 833569 914 835369 915 837225 916 839056 917 840789 918 842724 919 844561 920 846400 921 847241 922 850084 923 851929 924 853746 925 855625 926 857476 927 859329 928 861●84 929 863041 930 864900 931 866761 932 868624 933 870489 934 872356 935 874225 936 876096 937 877869 938 879844 939 881721 940 883600 941 885481 942 886364 943 889249 944 881136 945 893025 946 894916 947 896809 948 898704 949 900601 950 902500 951 904401 952 906304 953 908209 954 910016 955 912025 956 913936 957 915849 958 917764 959 919681 960 921600 961 923521 962 926444 963 928369 964 929296 965 931225 966 933256 967 935089 968 937024 969 939961 970 940900 971 942741 972 944784 973 946729 974 948676 975 950625 976 952576 977 954529 978 956484 979 958441 980 960400 981 962361 982 964324 983 966●89 984 968256 985 970225 986 972196 987 974169 988 976144 989 978121 990 980100 991 982081 992 984064 993 986049 994 988036 995 990025 996 992016 997 994009 998 996004 999 998001 1000 1000000 1001 1002001 1002 1004004 1003 1006009 1004 1008016 1005 1010025 1006 1012036 1007 1014049 1008 1016064 1009 1018081 1010 1020100 1011 1022121 1012 1024104 1013 1026196 1014 1028196 1015 1030225 1016 1032256 1017 1034289 1018 1036324 1019 1038361 1020 1040400 1021 1042441 1022 1044484 1023 1046529 1024 1048576 1025 1050625 1026 1052676 1027 1054729 1028 1056784 1029 1058841 1030 1060900 1031 1060961 1032 1065024 1033 1067089 1034 1069156 1035 1071225 1036 1073296 1037 1075369 1038 1077444 1039 1079521 1040 1081600 1041 1082681 1042 1085764 1043 1087●49 1044 1089936 1045 1092025 1046 1094116 1047 1096209 1048 1098304 1049 1100401 1050 1102550 1051 1104601 1052 1106704 1053 1108809 1054 1110916 1055 1113025 1056 1115136 1057 1117249 1058 1119364 1059 1120489 1060 1123600 1061 1125721 1062 1127844 1063 1129969 1064 1132096 1065 1134225 1066 1136358 1067 1138489 1068 1140624 1069 1142761 1070 1144900 1071 1147041 1072 1149184 1073 1151329 1074 1153476 1075 1155625 1076 1157976 1077 1159929 1078 1162074 1079 1164241 1080 1166400 1081 1168561 1082 1170724 1083 1172889 1084 1175056 1085 1177225 1086 1179396 1087 1181569 1088 1183744 1089 1185921 1090 1188100 1091 1190281 1092 1192464 1093 1194649 1094 1196836 1095 1199025 1096 1201216 1097 1203409 1098 1205604 1099 1207801 1100 1210000 1101 1212201 1102 1214404 1103 1216609 1104 1218816 1105 1221025 1106 1223396 1107 1225449 1108 1227664 1109 1229881 1110 1232100 1111 1234321 1112 1236544 1113 1238769 1114 1240969 1115 1242625 1116 1245459 1117 1247689 1118 1249924 1119 1252161 1120 1254400 1121 1256●41 1122 1258884 1123 1261029 1124 1263376 1125 1265625 1126 1267876 1127 1270029 1128 1272384 1129 1274641 1130 1276900 1131 1279161 1132 1281426 1133 1283689 1134 1285956 1135 1288225 1136 1287496 1137 1292769 1138 1294094 1139 1297321 1140 1299640 1141 1301881 1142 1304164 1143 1306449 1144 1308736 1145 1311025 1146 1313316 1147 1315509 1148 1317904 1149 1320201 1150 1322500 1151 1324801 1152 1327104 1153 1329409 1154 1331716 1155 1334025 1156 1336336 1157 1338649 1158 1340964 1159 1343381 1160 1345600 1161 1347921 1162 1350244 1163 1352569 1164 1354396 1165 1357225 1166 1358556 1167 1361689 1168 1364124 1169 1366921 1170 1368900 1171 1371240 1172 1373584 1173 1375929 1174 1378276 1175 1380625 1176 1382979 1177 1383329 1178 1387284 1179 1390041 1180 1392400 1181 1394761 1182 1397124 1183 1399489 1184 1401856 1185 1404225 1186 1406606 1187 1408904 1188 1411124 1189 1413711 1190 1416100 1191 1418481 1192 1420864 1193 1423249 1194 1425639 1195 1428025 1196 1430416 1197 1432809 1198 1435204 1199 1437601 1200 1440000 1201 1442401 1202 1444804 1203 1447209 1204 1449616 1205 1452025 1206 1454436 1207 1456849 1208 1459264 1209 1461681 1210 1464100 1211 1466521 1212 1468944 1213 1471369 1214 1473796 1215 1476225 1216 1478656 1217 1480989 1218 1483924 1219 1485961 1220 1488400 1221 1490841 1222 1493244 1223 1495729 1224 1498246 1225 1500125 1226 1503076 1227 1505529 1228 1507984 1229 1510441 1230 1512900 1231 1515361 1232 1517824 1233 1520289 1234 1522656 1235 1525225 1236 1527696 1237 1530169 1238 1334244 1239 1535121 1240 1537600 1241 15400●1 1242 1542564 1243 1545049 1244 1547536 1245 1550025 1246 1552516 1247 1555009 1248 1557504 1249 1560001 1250 1562500 1251 1565001 1252 1567504 1253 1570009 1254 1572416 1255 1575025 1256 1577536 1257 1580049 1258 1582564 1259 1585081 1260 1587600 1261 1590121 1262 1592644 1263 1595169 1264 1597706 1265 1600225 1266 1602756 1267 1605289 1268 1607824 1269 1609361 1270 1612900 1271 1615441 1272 1617984 1273 1620529 1274 1622076 1275 1625625 1276 1628176 1277 1530729 1278 1633464 1279 1635841 1280 1638400 1281 1640961 1282 1643524 1283 1645989 1284 1645656 1285 1651225 1286 1653796 1287 1656369 1288 1658944 1289 1661521 1290 1664100 1291 1666681 1292 1669264 1293 1671849 1294 1674336 1295 1677025 1296 1679616 1297 1682209 1298 1683804 1299 1687401 1300 1690000 CHAP. XI. Showing, how to Hang a Rising line by several Sweeps, to make it rounder aftward, then at the beginning of the same. IF any be desirous to have a Rising line rounder aftward than it is at the foar part of it, they must proceed thus; first Work by the Sweep that they would have first, and then begin again, and find the other Sweep, that they would have the roundest; An Example of this will make it more plain, as in the following Figure will appear. Let D E represent the length of a Rising line E I, the height thereof 8 foot, on the after end thereof; first I find the Sweep that Sweepeth it, by Multiplying of 20 foot the length, which is 240 inches: for if you look in the Tables, under the Title of Feet-Inches, for 20 feet, you will see in the next Column, toward the left hand, 240, over head is written Inches, signifying, that in 20 feet is 240 inches; and just against it, and in the same line, toward the right hand, under the Title of Squares, you will see written 57600, signifying, that the square of 240 is 57600, these numbers you will find in the second Page of the Tables, and the last line, the seventh, eighth, and ninth Columns. This squared number 57600, made by the Multiplication of D E, 240 inches, must be divided by the height of the Rising line assigned E ay, 8 foot, or 96 inches, so remaineth in the quotient 600, to which must be added the height of the Rising, as is afore taught, and they make 696, which is the Diameter of the whole Circle: the half thereof is 348 inches, which is 29 foot, as you may see by dividing it by 12; or else, if you turn to the Tables, and seek under the Title of Inches for 348, you will see in the same line, toward the left hand, 29 feet, which you will find in the third Page, and the 28th line, the seventh and eighth Column; then I Work by that Sweep to 3/5 of the length of the Rising line, or 12 foot of the same, at the point C it is represented, at which point I seek the Rising C B, I seek in the Table for the Square made of 144, and I find it in the second Page, 24 line, at the first Column; and toward the right hand, under the Title of Squares, I find 20736, which is the Square made of 144: then I seek for the Square made of the Sweep, or side A B, 348 inches, and I find it in the Tables to be 121104, from this 121104 I Subtract the other Square, made of the side D C, 144 being 20736, and there remaineth 100368, whose Root I find in the Tables, in the third Page, and the 37th line, and the sixth Column, 100489, which is too much by near 121; but the other number afore it being much more too little, the number answering hereunto is 316 inches, and near ¼, Substracted from 348, the whole side leaveth 31 inches ¼, or two foot 7 inches ¼ for the Rising, 0 30 57600 (600 9666 89 121104 20736 100368 at the point C: Now to make a rounder Sweep aftward on, or at the other end of the line, as from B to F, which runeth higher up, or Roundeth more, as from I to F: Here will be something more of trouble to find the Sweep that shall exactly touch the two points assigned, as from B to F▪ then to find the former Sweep. Now the Demonstration will show it to be thus. Let B and F be the two points to which the Sweep is confined to touch; draw a straight line from B to F, as you see, and so you have a Right lined Triangle, made of the sides B H, the length of the line to be swept by the second Sweep, and the side H F, the height of the same, together with the Subtending side B F; then a straight line drawn from the middle of the side B F, and perpendicular, or square, to the same line B F, and extended, till it touch the side D A, the place where it toucheth shall be the Centre of the same Sweep, as is the line G H, passing through the middle of the side B F, at the point O, which to find Arithmetically, proceed thus; find first the length of the side B F, as before is taught, of two sides of a Right Angled Triangle given, to find the third side, which will be found to be 134 ½ inches, the half whereof is 67 inches, ¼ from B to O, then if a perpendicular be let fall from O to the line B H, it will cut that Base line also in halves, as at the point P, being 48 inches: then again, find the side O H, and that will be, in this Example, equal to the side B O, but in other cases it may not so fall out: So then, those two sides being known, as the side O H, 67, ½ inches, and the side P H, 48 inches, and the whole length of the side K H, 240 inches, you may then Work by the Rule of Three, saying, if 48, the side P H, give 67 ½ inch, for the side O H, what will 240 give, for the side K H, as thus; If 48 give 67 ½, what will 240 240 2 67 144 1680 4640 1440 16080 (335 48888 16880 44 If you Multiply the two first numbers together, and divide by the first number, you will beget in the quotient 335, for the length of the whole side G H. I here neglected the ½ inch in this Multiplication, for the ½ inch should have been Multiplied into the 240, by adding to the Sum 16080, 120, the half of 240, and it maketh 16200, which divided by 48, maketh 337 ⅓ inches for the whole side G H; So then, these two sides being found, find the side G K, thus, as before is taught, look in the Table of Squares for the Square made of the side 337, and it will be 113569, from which Subtract the Square made of 240, the other side, being 57600, there resteth 55969, as you may see, for that number sought for in the Tables, and you find the nearest number to it, to be 56069, and the root of it to be 237, for the side G K, to which must be added the Rising of the point C B, or K D, which is all one, and is as we found it before to be, 31 ¼ inches, added to 237, maketh 268 ¼ inches, or 22 foot 4 inches; showing, that at 22 foot 4 inches, from the point D, towards G, will be the point where the Centre of the Rounder Circle ought to stand: Then again, you have the side G K, found as before, to be 237, and the side K B 144, and if you work as is taught before, but remember, that if the longest side be sought for, as is now in the last side sought for, G B being the longest side, you must add the squares made of the other two sides together, and the square of those two Sums shall be the longest side G B, 277 inches, that is 23 feet, 1 inch, which is the length of the second Sweep: and so have you the length of the Sweep. The same order you may observe to round your Sweep as often as you please. 113569 57600 55969 237 31 ¼ 268 ¼ If any have knowledge of the Doctrine of Triangles, it may be found more readier, that I leave to those that know the use thereof. Note also, that when you seek for any number in the Tables, take heed that you mind the number of Figures you seek for, to agree in number with those that directeth you to seek for them. As for Example, In the other figures abovementioned, 55969, they are in number 5, by their places, as you see; then repairing to the Table, I find 559504, but telling the Figures, I see that they are in number 6, but should be but 5: therefore this number, represented in the seventh Page; and the 28th line, and third Column, is not the place I seek for, than I turn toward the beginning of the Table, till I see that the Columns of Squares contain but 5 figures, and there seek the nearest number agreeing to 55969, and in the second Page, 37th line, last Column, I find 56069, the nearest agreeing to it, which is the place answering to the other directory figures. Note also, That the Example of finding the Sweep aforegoing, is laid down by the small Scale of the Draught, by which you may try it for your better directions. And in that Table you may see that any farther than 70 foot, being the end of the seventh Page, I have not mentioned the Feet and Inches belonging to the number of Inches, but have left it out because they are of little use any further, because that will reach far enough for the length of any Rising line of any Ship whatever: If any be desirous to convert any of the following numbers into inches, he may do it by Dividing by 12. Thus I think I have spoken enough to the Ingenuous, concerning the singular use of the Tables, or of this way of Working by Segments o Circles. CHAP. XII. Concerning Measuring of Ships. 60 20 1200 10 120100 I Shall say something concerning it; the Shipwrights have to themselves a custom of measuring at London, or on the River of Thames thus, they multiply the length of the Keel into the breadth of the Ship, at the broadest place, taken from outside to outside, and the product of that by the half breadth, this second product of the multiplication they divide by 94 or sometimes 100, and according to that division, the quotient thereof; they are paid for so many Tuns; as suppose in the former draught being in length 60 foot and 20 foot broad, 60, being multiplied by 20, the breadth, produce 1200, that 1200 being again multiplied by 10, the half breadth produce 12000, if you divide by 100, you need do no more than cut off the two last figures toward the right hand, which shall be the answer and rendereth the Ship to be 120 Tuns, but if you divide the sum 12000 by 94, you will have 127 2/3 of a Tun very near, but this cannot be the true ability of the ship to carry or lift, because two ships by this rule of equal breadth and length shall be of equal burden, notwithstanding the fullness or sharpness of those Vessels, which may differ them very much, or the one ship may have more timber than the other in her building, & so shall carry less than the other: But the true way of measure must be by measure of the body and bulk of the ship underwater, for if one ship be longer in the floor than another of the same breadth and length, she shall be more in burden than the other; as a Flemish ship shall carry more than a French or Italian Vessel of the same length and breadth; Therefore I say the measure of the ship being known by measuring her, as a piece of timber may be measured of the same form, to the draught of water assigned her, the weight of the same body of the same water that the ship swimmeth in shall be the exact weight of the ship; and all things therein; loading, rigging, victuals included therein: then if the ship be measured to her light mark as she will swim at being launched, the weight of so much water being taken or substracted from the weight of the water when she is laden, the residue shall be the weight that must load her, or her ability of carrying, called her burden, by this means you may know the weight of the ship light, and what she will carry to every foot of water assigned to her, which cannot be done by no general rules in Arithmetic because of their great irregularity, according to the differing minds of Shipwrights; you may if you please first measure the content of the Keel and Post and Stem-rudder, all of it that is without the Plank, and under the water line, and note it by itself, then measure the body of the ship in the Midships, made by the square made of the multiplying of the depth of the water line, and the breadth, than you may find; the content of the want by the circular part of the ship under water, being narrower downward, and subtract this from the whole content of the squared body of the depth of the Water-line and breadth of the ship, and this shall be the solid content of that part of the ship, I mean in solid foot measure of 1728 inches to the foot, then proceed to the fore part or the after part of the ship, and to 3 or 4 Timbers more, find the mean breadth at the narrowing aloft at the water-line, and allow at the floor and the mean depth, and measure that piece of the ship, as I told you of the middle part of the Ship, and so measure the whole Ship by pieces and add them together, and so many feet as it maketh, so many feet of water shall be the weight of the said ship; and the reason may be considered thus; there is a ponderosity in warer, but there is a greater in the air, only to the heaviest of things; and there is a ponderosity in water itself, but not so much as in other things more solid as in Iron: Suppose a Gun or an Anchor of Iron, it sinketh in the water, but yet it is not so heavy in the water as in the air, by the weight of so much water as shall make a body of the same water equal to the body of the Gun or Anchor in magnitude; which weight substracted from the weight of the Iron body weighed in the air, and so much must be the weight of it in the water. Again, if a body be lighter in weight, than water of the same bigness, it hath an ability of lifting in the water, and can lift or carry so much as is that difference, as a piece of cork or wood of firr-trees, being lighter than water, it swimmeth on the face of the water, and refuseth to be depressed without more weight added to it. Thus a ship being a concave body, is made capable of lifting according to the greatness or littleness of this concavity, respect being had to the greatness of the Timber put into it, or the nature of it, all which maketh a ship swim deeper or lighter in the water. I have proved by the Thames water, that fresh Water is lighter than salt water, so then salt water being heavier than fresh, causeth that a ship swimmeth deeper in the fresh water than in salt. I shall not need to say any thing more concerning the mesauring, for it will be understood by those that have any Judgement in the mesuring of triangles, the matter itself being but a nicety rather than usefll: I only touched it to show those that are so curious minded, which way they may accomplish their desires; I shall forbear to give examples, because it will much increase my Treatise, and augment the Price, which might prove more prejudicial to young men than advantageous. CHAP. XIII. Concerning the Masts of Ships. FRom the length and breadth is gained the Mainmasts length, and all the other Masts as well as yards, is derived from thence, and there is different proceedings in this case, according to the largeness of the Ships, thus, the main Masts of small Ships to be three times as long as the Ship is in breadth; as a ship of 20 foot broad, by the same rule must have a Mast of 60 foot long. Others for greater Ships, add the breadth to the length, and to that the half breadth, which some they divide by 5, and the quotient is the number of yards, as a ship 114 foot long and 34 foot in breadth, as the breadth added to the length, and the half breadth added together, make 165, that divided by 5, yields 33, and so many yards is the length to be of that Mast, the foremast must be a yard shorter at the head, that is to say besides the height of the step, which commonly in most ships the step of the foremast standeth higher from the bottom of the ship than the step of the Mainmast; the foremast must be shorter by that difference, and one yard more, or the bigness of the ship considered, 4 foot shorter at the head, or besides the difference below, 114 34 17 165 10 165 (33 55 The Top-masts two thirds of the length of the lower Masts. The Main-yard to be 2/● and ●/22 of the Mainmast, as in the Mast aforementioned of 60 foot long, two thirds of 60 is 40, and the 2/12 of 60 is 5, added to 40 make 45, for the length of the main yard. The foreyard to be 6/7 of the Main-Yard, as the Main-yard being 45 foot, divide 45 by 7, so cometh 6 in the quotient, and a fraction remaining of 3, signifying 3/7, so that the 1/7 of 45 will be 6 and 3/7, you must take 6 times so much, a● 6 times 6 makes 36, and if you take 6 times 3/7 make 18/7, that is, two whole numbers, and 4/7 remaining, which added to 36, make 38, and 4/7 of a foot for the length of the fore-yard. The Topsail Yards must be half the length of the lower Yards, the Mizne Yard usualy is made of equal length with the fore-yard, the Crosjack yard, of equal length with the Main topsail yard, and the mizzen Topsail yard to be half the length of the Crosjack yard. The mizzen Mast to be of the length of the Maintop mast from the upper Decks, and so much longer as is the height of the ship between Deck, the Boltspreete to be of length equal to the foremast from the upper Deck of the Forecastle upwards. For the bigness of these Masts, to a yard in length, ¾ of an inch, or else ¼ of an inch to the foot, and so of yards likewise, only the Boltspreet something bigger, would be the better if he be made as big as the foremast. CHAP. XIV. Concerning Rowing of Ships, when they are becalmed. I Have here invented a means of Rowing of a Ship, by the heaving at the Capstane, where will be many benefits; First, of a greater purchase of strength, for it is evident, that 10 Men at a Capstane shall heave a Ship a Head, when 30 Men shall not Haall her a Head by hand, nay 50; neither shall they be so soon tired, for that Owers are a great weariness to the Arms, beside a double motion of the body, as when the stroke is fetched, to way down the Ower, that the blade may be elevated out of the Water, where it must be kept so, until another Let the two lines, C D, and C D, represent the sides of the Ship, 16 foot broad, as is the line C C, 16 foot long, by the Scale of the Draught; and let the two long squares, d d, and d d, represent the two Bitt-pins, with the Cross-piece of, let the Black, between the two Bit-pins, represent a Roll, or Windless, with a Surdge in the middle, as is the Surdge of a Crab, or Capstane; in the two ends of this Roll let there be placed two Winches, as you may see represented by Cranks, a and a; let there be made a hollow place in the Head of the Bit-pins for these two Winches to rest in, that they may turn round in them, and bide in them: then let there be two pieces of Timber, equal in length, to the space of the Ship you would have filled with Oars, represented by the two black lists, marked b b, and b b; then let there be fitted two small pieces, made of good Ash, or some good strong Wood, of equal length as is the two black Lists, n L, m L, fastened into the pieces of the frame, as at the points L and L, by a Bolt, but so, that they may play on that Bolt, and the other ends must be with a Hole made in the ends, put over the handles of the Cranks, at n and m, then in the two long pieces for the frame let the Oars be fixed, as at the points 1, 2, 3, 4, 5, 6, 7, of each side of the Ship representing 7 Oars of a side, they may be fastened in the Frame, by a Mortis made therein, and a Tenent on the Oar, made to go slack, in that the Oar may play, and have liberty to fetch a Stroke, in the middle whereof must be placed a Iron bolt, to fasten him, and keep the Oar from launching in and out, and on the Roughtre, or side of the Ship▪ as in C D and C D, must be placed Thoule pins to each Oar, as in Boats that Row; then must you have a Halser splised together, in manner of a Vial, that must take two or three turns about the Roll, in the Surdge, as you may imagine, at the middle of the Black Roll, or notch therein, and pass from thence to the Capstane, with two or three turns there also; then this Vial also, reeved through some other Blocks, as in manner of Snach block, and these blocks placed between the Capstane and the Roll, you may thereby increase that purchase so, as that it may heave very easy, and with great strength, so as to be able to Row a Ship a Head in a calm, or in little wind, two, three or four leagues, a watch or more, according as the Engine shall be better or worse fitted; for if you mark in the Figure of the Work, and suppose the Vial Reeved, and by heaving of the Capstane, shall turn the Roll, as the Cranks goeth round, it shall carry with it about, and then the small pieces shall cause the frame to pass forward to and again, to fetch a stroke with the Oars. FINIS.