THE ARTIFICERS PLAIN SCALE: OR, The Carpenters new Rule. In two Parts. The first, showing how to measure all Superficies and Solids, as Timber, Stone, Board, Glass, etc. Geometrically, without the help of Arithmetic: it being a new way not heretofore practised. The second showing how to measure Board and Timber Instrumentally, upon the Scale itself, without Arithmetic or Geometry, but what is common to every man. ALSO, How to take Heights and Distances several ways, and to draw the Plot of a Town or City, By Thomas Stirrup. Philomat, London, Printed by R. & W. Leybourn, for Thomas Pirrepont, at the Sun in Paul's Church yard. 1651. depiction of the measurement of altitude or height To THE READER. Gentle Reader: ALthough many excellent both in Arithmetic & Geometry, upon infallible grounds have put forth divers most certain and sufficient rules for the measuring of board & timber, yet very few of our common Artificers have been furthered thereby, because they have not the art of Arithmetic, upon which most of their rules depend. The consideration of which, with the aptness which I see in some of them for the raising of a Perpendicular, and the drawing of a Parallel Line, upon which most of this Book depends: this, I say, hath been the cause which hath moved me to give them some rules Geometrical, whereby they may measure both board & timber, without the help of Arithmetic. Therefore to thy view, Gentle Reader, that wanteth the art of Arithmetic, do I prefer this short and plain Treatise, wherein, & in the beginning, is declared the infallible grounds upon which the whole work doth depend, & then doth follow the applying of those rules to the present purpose, with the declaration of three tables, one for Board, and one for square Timber, and the third for round Timber, very fit for all such as stand in need thereof, and yet want both Instruments and Arithmetic, whereby to use the same. In the second Part of this Book is showed a second way whereby you may measure Board and Timber by Rule and Compass only, without drawing of lines: & also how to take Heights and Distances several ways without Instrument, all which are grounded upon infallible principles Geometrical. Thus desiring thee to accept of this little Book as a taste of my good will towards thee, which I wish even so to further thee, as I know it sufficient for the true measuring both of Board and Timber. Farewell. THE CONTENTS. THe meaning of certain terms of Geometry used in this Book. Page 1 How to raise a perpendicular on any part of a right line given. Page 5 How to let fall a perpendicular from a point assigned, to a line given. Page 7 To a line given to draw a parallel line at any distance required. Page 8 To perform the former proposition at a distance required, and by a point limited. Page 10 Having two lines given to find a third proportional line to them. Page 12 Having three lines given to find a fourth proportional line to them. Page 14 The making of a Rule or Scale, for the measuring of Board and Timber. Page 17 How any Board may be measured Geometrically. Page 20 How Timber may be measured Geometrically. Page 26 Of Round Timber. Page 33 How Triangled Timber, or Timber which hath but three sides may be measured. Page 42 How Timber whose end is a Rhombus is to be measured. Page 45 How Timber whose end is a Rhomboiades is measured. Page 47 How to measure Timber whose end is a Trapesiam. Page 49 How to measure Timber whose sides are many, as 5, 6, 7, 8, 9, 10, or more, so they be all equal. Page 51 How to find the length of a foot of Board, at any breadth given. Page 54 The breadth and thickness of a piece of Timber given, to find how much in length shall make a foot of square Timber at that breadth and thickness. Page 56 How to find a mean proportional line between two lines given. Page 66 The second Part. OF the Scale, and the graduations or divisions thereof, and how they are to be used. Page 71 To divide a line given, into any number of equal parts. Page 73 To take any part or parts of a line Page 74 A line containing any part or parts of a line, thereby to find the whole line. Page 75 A line being given, containing any number of equal parts, to cut off from it so many as shall be required. Page 77 To lay down suddenly two, three, or more lines in proportion required. Page 78 In a Map or Plot, the length of any line being known thereby to find the length of all or any of the rest Page 80 Unto two lines given, to find a third in proportion. Page 82 Unto three Lines given, to find a fourth in proportion, that is, to perform the Rule of Three in Lines. Page 84 To divide a line given into two such parts, bearing proportion one to the other as two numbers given. Page 86 To measure flat Measure. Page 87 To measure Board that is broader at one end than at the other. Page 91 To find how many square feet any whole Board containeth, without finding how much in length makes a foot Page 93 To measure Board that is broader at the one end then at the other, in the same manner. Page 95 To measure timber. Page 97 To measure timber that is broader at one end than at the other. Page 100 How Perpendicular heights may be found without either Instrument or Arithmetic. Page 101 How to take the altitude or height of a building by a bowl of water. Page 105 How to take the altitude of a Building by a line and plummet the Sun shining. Page 107 How to find the altitude of a Building by two sticks joined in a right angle. Page 109 To find a Distance by the two sticks joined square. Page 112 How to describe a Town or City according to chorographical proportion, by the help of a plain glass. Page 116 An Advertisement To the READER. FOrasmuch as throughout this whole Book, there is mention made of Rules and Scales, the making whereof is different from those which are vulgarly made and sold: if any therefore be desirous to have any particular Rule mentioned in this book, or one Rule to perform all the work in general, he may have them exactly made by Master Anthony Thompson in Hosier lane, near Smithfield. THE ARTIFICERS Plain Scale. CHAP. I. The meaning of certain terms of Geometry used in this Book. BEcause all Carpenters or other Artificers, in their Trade or Calling, do in a manuer (and according to their fashion) use some kind of Geometry, although themselves be ignorant thereof, therefore I did consider that they might be sooner brought to measure Board and Timber by that art of Geometry (seeing they have their Rule and Compasses by them) then by Arithmetic, being but few of them can write, and therefore uncapable of that art: and of them few which can write, not one in ten that hath Arithmetic, which is the only cause (as I suppose) that most of them are so ignorant in this art (which doth so much concern them) notwithstanding, all those excellent Rules which have been formerly delivered by the learned: But now to our intended purpose. SEeing I shall have occasion in this Work to use some terms of Geometry, by which I may with more ease deliver, and you with more plainness perceive my mind in these things: I have therefore set down the meaning, as plainly as I can, of some Geometrical terms, which most serve for our present purpose. 1 An Angle is nothing else but a corner, made by the meeting of two lines (for I speak not of solid angles.) 2 A right Angle (which we call a square angle) is that whose two lines comprehending or making the angle, stand perpendicular or plumb the one to the other. 3 A Perpendicular line is that which stands plumb upright upon another, leaning neither the one way nor the other. 4 A Superficies is that which hath only Length and Breadth, and no Thickness at all. 5 A Solid, or a Body, is that which hath Length, Breadth, and Thickness. 6 Parallels are those lines that differ every where alike, or are not nearer together in one place then another. 7 A Figure is any kind of Superficies or Solid that is bounded about, as Triangles, Squares, Circles, Globes, Cones, Prismes, and the rest. 8 The Base of a Figure is any side thereof upon which it may be supposed to stand; or if you take any side of a Figure for the Ground or Bottom, or lower part thereof, that same is the Base. 9 The height of a Figure is the length of a Perpendicular or plumb line, falling from the top thereof to the Base or bottom thereof. CHAP. II. How to raise a Perpendicular on any part of a right line given. LEt AB be a right line given, and let C be a point therein, whereon I would raise a perpendicular, open the Compasses to any convenient distance, and setting one foot in the point C, with the other mark on either side thereof, the equal distances CE, and CF: then opening your Compasses to any convenient wider distance, with one foot in the points E and F, strike two arch lines, crossing each other as in D, from whence draw the line DC, which is a perpendicular to AB, or as we call it, a square line to the line AB. diagram of the measurement of perpendicular lines (AB and CD) Or you may from the given point C, prick out any five equal distances, and opening your Compasses to 4 of them, with one foot in C, strike an arch or piece of a Circle towards N, then opening your Compasses to all 5 divisions, with one foot in 3, across the same arch line in N, from whence draw the line NC, which is a perpendicular to the line AC, as before, for if 3 lines be joined together, so they be in such proportion, as 3, 4, and 5; they will make a right angle. CHAP. III. How to let fall a Perpendicular from a point assigned, to a line given. LEt the point given be D in the former Chapter, and let the line whereon it should fall be AB, open the Compasses to any convenient distance, & setting one foot in the point D, make an arch or piece of a Circle with the other foot, till it cut the line AB twice, that is at E and F, then find the middle between those two Intersections, and from that middle, draw a line to the point D (which is the point given) and that line shall be perpendicular or plumb from the point D to the line AB, as was required. CHAP. IU. To a line given, to draw a parallel line, at any distance required. SUppose the line given to be AB, unto which I must draw a parallel. diagram of the measurement of parallel lines (AB and CD) Open your Compasses to the distance required, and setting one foot of your Compasses in the end A, strike an arch on that side the given line whereon the parallel is to be drawn, as the arch C, then do the like in the end B, as the arch line D, then draw the line CD, so as it may but touch or be a touch line to these two arches C and D, and this line so drawn shall be parallel to the line AB, as was required. CHAP. V To perform the former proposition at a distance required, and by a point limited. ADmit AB in the former Chapter, to be a right line given, whereunto it is required to have a parallel line drawn at the distance, and by the point C. Place therefore one foot of your Compasses in C, from whence take the shortest distance to the line AB, as CA, at which distance, with one foot in the end B, with the other strike the arch line D, by the extreme part of which arch line D, and the point C, draw the line CD, which is parallel to the given line AB, which was required. diagram of the measurement of parallel lines (AB and CD) CHAP. VI Having two lines given, to find a third proportional line to them. THe two lines given are A and B, and it is required to find a third line, which shall be in such proportion to A, as A is to B. Make any angle whatsoever, as the angle HEC. Here note, that an angle is always represented by three letters, whereof the middle letter represents the angle intended. Then place the line A, from the angle E to D, and the line B from E to F, and draw the line DF. Place also the line A from E to H, and lastly, by the 4 Chapter, from the point H, draw the line HC parallel to FD. diagram of the measurement of proportional lines (A, B, and EC) So shall EC be a third proportional line to the two given lines, as was required. CHAP. VII. Having three lines given to find a fourth proportional line to them. THE three lines given are A B and C, and let it be required to find a fourth line, which shall have such proportion to A, as B hath to C, make any angle, as DGK, now seeing the line C hath the same proportion to the linne B, as the line A to the line sought for, therefore place the line C from G to H, and the line B from G to F, then draw the line FH, now place the line A from G to I, by which point I draw the line EI parallel to FH, till it cutteth DG in E, so have you EGLANTINE the fourth proportional line required, which is 24. diagram of the measurement of proportional lines (A, B, C, and EG) For as the line 12, is to the line 16, so is the line 18, to the line 24, which is the length of the line we sought for. These two last Chapters, would I have you diligently to consider, and throughly to learn, because it is the groundwork of that which I intent to deliver in this Book: which being well understood, will bring much pleasure and profit to the unlearned Artificer, for whose sake this was written. diagram of the measurement of proportional lines (A, B, C, and EG) CHAP. VIII. The making of a Rule or Scale, for the measuring of Board and Timber. depiction of a rule or scale This line thus divided, is called a Scale; which is no other thing, but a right line divided into any number of equal parts, be they greater or lesser, wider or narrower, so they be equal: every part, or division of which line, may stand for a mile, a rod, a yard, a foot, an inch, or any other kind of measure what you will, or have use of: and this line I would have you use, in giving up the Content of either Board, or Timber. I have described this Scale but to 4 inches; but you may thereby perceive what I mean by the whole Rule. CHAP. IX. How any Board may be measured Geometrically. IF you do well understand that which hath been delivered in the seventh Chapter, you may thereby measure any Board with ease and delight; for as there is three numbers, or three lines given, whereby the fourth proportional is found: so in every Board, there is three lines, or numbers (which you will) which be given us, whereby, we may by the seventh Chapter find a fourth proportional line or number: which is the number of feet, contained in the whole board. The first of the three Numbers given, is always 12, which is the side of a square foot of Board, or the side of a cubical foot of Timber; The second number is always the number of feet, contained in the length of the Board, the third number is always the number of inches contained in the breadth of the Board. And the fourth number which is here sought for, will always be the number of square feet, contained in the whole Board: the proportion will be always thus. As 12 to the length in Feet: So the breadth in inches, To the Content in Feet. And seeing examples teacheth better than many words: therefore let us suppose the three lines given us in the seventh Chapter, to be three such numbers as here we have spoken of. And therefore let the first line C, be a number of 12, taken from some Scale with your Compasses, and placed from G to H, which 12 doth signify 12 inches, which is the side of a square foot of Board. And let the second line B, be a number of 16, taken from the same Scale, and placed from the angle at G unto F, and draw theline FH, and this number of 16 doth signify 16 feet, the length of a supposed Board. And so let the third line A, be a number of 18, taken from the same Scale and placed from G to I, and this number of 18 doth signify 18 inches; the breadth of the supposed board: now from the point I, by the fifth Chapter, draw the line IE parallel to FH, till it cutteth EGLANTINE in E: So have you EGLANTINE the fourth proportional line required; which being taken between your Compasses, and aplyed to your Scale, will show it to be 24, and so many square feet are in that Board, whose length is 16 foot, and breadth is 18 inches. diagram of the measurement of a board (lines AB and CB) And here note, that when you have any odd parts of an inch in the breadth of your board you must take the like parts of one division from your Scale, more than your even parts was: and so must you do when you have odd parts of a foot in the length of your Board: as for example, suppose a Board to be eight foot, and three quarters long; now for to set down this length you must take from your Scale eight whole divisions, and three quarters of one; and so apply them to your use: and this must be noted throughout this Book. And here note also, that if your Board be taper grown, that is, wider at one end then at the other; then measure the breadth thereof in the middle, and with that wideness proceed according to your Rules given: And this may very well suffice for Timber that doth taper also. CHAP. X. How Timber may be measured geometrically. THE measuring of Timber doth little differ from measuring of Board, by the last Chapter, but only in measuring of Timber we have a double work; but the last Chapter well understood, will give light sufficient hereunto. Therefore by the last Chapter, first, measure how many square feet of flat measure there is in one of the sides of your Timber, as if it was a board by itself, which being done, you have three numbers given you, whereby you may by the seventh Chapter find a fourth in proportion unto them, which fourth number, is the number of cubical feet contained in that piece of Timber. The First number is always 12. The Second is always the number of square feet contained in one of the sides, I mean, of flat measure. The Third, is always the number of inches contained in the thickness of the Timber: and this will be always the proportion for this work. First, As 12 is to the length in feet, so is the breadth in inches, To the superficial content, of that same side. Secondly, As 12 to this superficial content, So is the thickness in inches, to the solid content in feet, As for example, suppose the figure A to be a piece of Timber to be measured, whose length is 8 foot, and breadth 18 inches, and thickness 14 inches. diagram of the measurement of a block (lines BE and HE) Now draw two lines, so as they may make any angle as the lines BE, and HE, meeting in the angle E, this being done, first, place 12 (which is the side of a cubical foot, of Timber) from E to F, than place 8 (the length of your piece in feet) from E to D, and draw the line DF, and then place 18 (the breadth of your piece in inches) from E to H, and then by the 5 Chapter, draw the line HC, parallel to DF, till it cutteth BE, in C: So shall CE be the number of feet of flat measure, contained in the broadest side of the piece of Timber. Thus far we have proceeded according to the last Chapter; and now we have three numbers more given us, whereby we may find a fourth proportional unto them. Wherefore, first, we have 12 already placed, from E to F, secondly, we have the superficial content of the broadest side, already placed from E to C: there fore draw the line FC, and thirdly, we have 14 the number of inches contained in the thickness, which we must place from E to G: and lastly, from the point G, by the fifth Chap. draw the line GB parallel to FC, till it cutteth BE at B: So shall BE be the number of cubical feet contained in that piece of Timber noted with the Letter A, which being taken between your Compasses, and applied unto your Scale, will reach unto 14, and so many feet is in that piece. Now here I will give you one example, of a piece of Timber hewed just square: Let the figure B be a piece of Timber so hewed, whose length is 9 foot. And let it be 8 inches square, now having made any angle, as the angle ADC, first place 12 from D to C. And 9 foot the length, from D to A, and draw the line CA, then place 8, the thickness of one of the sides, from D to F, and by the 5 Chapter, draw the line FE parallel to CA, till it cuteth DA, at E, so shall ED, be the superficial content, of one of the sides, thus far according to the 9th. Chapter, as if it had been a board. diagram of the measurement of a block (lines AE and CE) This Chapter would I have you well to consider, because I do not intent to repeat, what I have hear delivered, but only describe unto you, the end of some pieces, according to their forms, and so give you some Rules, for to measure them by this Chapter. CHAP. XI. Of round Timber HEre first I would have you to understand, what the Circumference, the Centre and Diameter of a Circle is, the Circumference is the line encompassing the Circle, the Centre is the point in the midst thereof, the Diameter is a right line passing by the Centre through the whole Circle, and divideth the same into two equal parts, either half of which Diameter is called the Semidiameter. Now having found the Circumference of a round piece of Timbe, by girding it about with some line, I think it is hear needful, to give you a Rule, for the finding of the Diameter of the same piece. diagram of the measurement of a circle Now the Circumference and Diameter being found, you may find the solid content, after this manner. First, take one half of the Circumference, for the breadth of your piece, and one half of the Diameter for the thickness thereof, according to which breadth, and thickness, you may proceed in all things, (by the former part of the tenth Chapter) as if it were an unequal squared piece of Timber, as in the figure A, take 22 inches, (the Circumference of the piece) for the breadth thereof. Or take a quarter of 44 that is 11 for the one side, and the whole 14 for the other. And take 7, which is the half of 14 the Diameter, for the thickness thereof, and so with this breadth, and thickness, proceed in all things according to the former part of the tenth Chapter. Of the halfround, or quarter or any other portion or part of a Circle. FOr this half Circle, take half the arch line CDB, which is 11, for the breadth of your piece. And one half the Diameter, which is 7 for the thickness thereof, and proceeding with this breadth and thickness, by the tenth Chapter, you shall find the content. diagram of the measurement of a semi-circle (CDB) Now having a piece of Timber, whose end shall be like unto this portion of a Circle, noted with these letters ABCD, before we can give the content thereof, it will be needful to to find out the Centre, which for to do work as followeth. A Segment of a Circle being given, to find out the Centre, and consequently the Diameter, and so if need be, the whole Circle. diagram of the measurement of a circular segment (ABC) The centre being found, draw the lines EA and EC, and cast up the whole figure ABCE, as before is showed, and then by the next Chapter, find the content of the Triangle ACE, and take it from the content of the whole figure ABCE, and that which is lift, shall be the content, of the figure ABCD, as was required. By this Rule, (observed with discretion,) may all manner of Segments, or parts of a Circle, whether greater or lesser than a Semicircle, be easily measured, without further instruction. Hithereto have we shown the measuring of such Timber, as is most in use, that is to say, of equal squared, and also of unequal squared Timber, so likewise have we shown, how round Timber, and its parts, may be measured, by the former Rules. so So now will I show how some pieces of extraordinary forms, may be brought to be measured, by the former Rules. diagram of the measurement of a circular segment (ABC) CHAP. XII. How triangled Timber, or Timber which hath but three sides, may be measured TRiangles, are made of strait lines, o● crooked, or of both together, but I speak only of Right lined Triangles, which is nothing else, but a figure made of three right lines, as the figure ABC. Triangles are divers, both in respect of their sides and angles, and may be measured divers ways, but let this one way serve for all: take half of the base, and suppose it to be one side of a squared piece of Timber; & take the whole hieght, or perpendicular, for the other side of the same piece, and so measure it by the former part of the tenth Chapter, in all respects, as there is showed. diagram of the measurement of a triangle (ABC) Let the Triangle ABC, be the end of a piece of Timber to be measured, which hath but only three sides. diagram of the measurement of a triangle (ABC) CHAP. XIII. How Timber, whose end is a Rhombus, (or Diamond form) is to be measured. A Rhombus (or Diamond) is a figure of four equal sides) but no right Angles, such as is the figure ABCD, for the measuring whereof, observe this example. Let the said figure ABCD, be the end of a piece of Timber to be measured: now taking the length of the side or base AB, which is 14 inches for one of the sides of a squared piece of timber, and the length of the perpendicular DE, which will be found to be 12, and something better than the eighth part of one more, for the other side diagram of the measurement of a rhombus (ABCD) of the same piece, with which two sides, as if it were an unequal squared piece of timber, proceed in all things, according to the former part of the tenth Chapter. CHAP. XV. How Timber whose end is a Rhomboides (or Diamond-like) is measured. A Rhomboides (or Diamond-like) is a figure, whose opposite sides, and opposite Angles, are only equal, and it hath no right Angles. Such as is the figure FGHI, and may diagram of the measurement of a rhomboid (FGHI) be measured after this manner: take the length of the side HI, or FG, which is 16 inches for one side of a squared piece of timber, and take the perpendicular FL, which is 10 inches, for the other side of the same piece, & so you may measure it by the former part of the tenth Chapter, as if it were an unequal squared piece of 16 inches broad, and 10 inches thick. All other four sided figures besides the true Square, and the unequal Square in the tenth Chapter, and the Rhombus in the last Chapter, and the Rhomboides in this, are called Trapezias or Tables. diagram of the measurement of a rhomboid (FGHI) CHAP. XV. How to measure Timber, whose end is a Trapeziam. A Trapeziam is any irregular four sided figure of what fashion soever, as the figure ABCD is a Trapeziam, and may be cast into two diagram of the measurement of a trapezium (ABCD) Triangles, by drawing the diagonal line AC, and so each Triangle measured as is before showed, which being done, add the contents of them both together, and you shall have the content of the whole Trapeziam ABCD. Or you may more readily measure it thus: Take one half the diagonal line AC, which in this example will be 8 inches, for one side of your piece, and take the two perpendiculars BF and DE, and join them both together in one sum, so shall you have in this example 10 inches for the other side of your piece, with which two sides, (as if it were an unequal squared piece of Timber) proceed as before, in the former part of the tenth Chapter. CHAP. XVI. How to measure Timber, whose sides are many, as 5, 6, 7, 8, 9, 10, or more, so they be all equal. MAny sided figures are those which have more sides than four, and are generally called Pollygons. A piece of timber whose end shall have more sides than four, may be measured after this manner, add all their sides together, and take half that number for one side of an unequal squared piece of Timber, then let fall a perpendicular from the centre or midst of the figure, to the midst of some one side, and take that length for the other side of the same piece, with which two sides proceed as before is showed. diagram of the measurement of a polygon (regular pentagon) Suppose the figure A to be the end of a piece of Timber of five sides, being all equal, and each side containing 12 inches, which being added together into one sum will make 60, the half whereof will be 30 for the breadth of your piece, then take the length of the perpendicular (falling from the centre A to the midst of one of the sides,) which here is 8 inches, for the thickness of the same piece, with which breadth and thickness proceed in all things according to the former part of the tenth Chapter. This rule is general in all kind of regular Polygons, how many sides soever they have. Here I might have proceeded to have showed by what means Pyramidal or picked Timber, or Steeples may be measured: but considering how little this appertaineth to Carpenters, and how sufficiently they be handled by Master Diggs in his Geometrical works, I forbear here to write of them. CHAP. XVII. How to find the length of a Foot of Board, at any breadth given. THe breadth of a Board being given, with the number of 12, (the side of a square foot of Board,) you may by the sixth Chapter find how much in length will make a foot at any given breadth, by finding a third proportional number, which shall be to 12, as 12 is to the given breadth. As suppose a Board to be 16 inches broad, and I would know how much in length will make a foot thereof. diagram of the measurement of length (angle ABC) CHAP. XVIII. The breadth and thickness of a piece of Timber given, to find how much in length shall make a foot of square Timber at that breadth and thickness. SUppose a piece of Timber to be 18 inches broad, and 14 inches thick. First, make any angle, as DFB, and place 18 inches from F to A, this is the supposed breadth of your piece; then place 12, the side of a cubical foot of Timber from F to E, and draw the line A So likewise place 12 from F to G, from which point G, draw the line GH parallel to A, till it cutteth FD in H, so shall HF be the length of asquare foot of flat measure at the former breadth given: thus far according to the last Chapter. diagram of the measurement of length (angle DFB) Now to proceed, place 14 the thickness of your piece from F to C, and draw the line CG; and lastly, from the point H, draw the line HI parallel to CG, till it cutteth FB in I, so shall IF be the length of a foot required, which being applied to your Scale, will reach almost unto 7 inches, it wanteth but one seventh part of an inch, and such is the length of a foot of Timber whose breadth is 18 inches, and thickness 14 inches. CHAP. XIX. Of the Table for Board and Square Timber, and also for round Timber. COncerning the use of these Tables, I would have you to understand that I have supposed the Inch to be divided into 10 equal parts, and each part divided into 10 equal parts, and so the whole inch will contain 100 equal parts. A Table for Board measure. Inches. Feet. Inches. 10 part of In 10 part of a 10 part. 1 12 00 0 0 2 06 00 0 0 3 04 00 0 0 4 03 00 0 0 5 02 04 8 0 6 02 00 0 0 7 01 08 5 7 8 01 06 0 0 9 01 04 0 0 10 01 02 4 0 11 01 01 0 9 12 01 00 0 0 13 11 0 7 14 10 2 8 15 09 6 0 16 9 0 0 17 8 4 7 18 8 0 0 19 7 5 7 20 7 2 0 21 6 8 5 22 6 5 4 23 6 2 6 24 6 0 0 25 5 7 6 16 5 5 3 27 5 3 3 28 5 1 4 29 4 9 6 30 4 8 0 A Table of square Timber measure. Inches. Feet. Inches. 10 part of In 10 part of a 10 part. 1 144 00 0 0 2 36 00 0 0 3 16 00 0 0 4 9 00 0 0 5 5 09 1 2 6 4 00 0 0 7 2 11 2 6 8 2 03 0 0 9 1 09 3 3 10 1 05 2 8 11 1 02 2 8 12 1 00 0 0 13 10 2 2 14 08 8 1 15 07 6 8 16 6 7 5 17 5 9 7 18 5 3 3 19 4 7 8 20 4 3 2 21 3 9 1 22 3 5 7 23 3 2 6 24 3 0 0 25 2 7 6 26 2 5 5 27 2 3 7 28 2 2 0 29 2 0 5 30 1 9 2 A Table of round Timber measure. Inches. Feet. Inches. 10 part of In 10 part of a 10 part. 1 113 01 7 1 2 28 03 4 2 3 12 06 8 5 4 7 00 8 5 5 4 06 3 0 6 3 01 7 1 7 2 03 7 0 8 1 09 2 3 9 1 04 7 6 10 1 01 5 7 11 11 2 2 12 09 4 2 13 08 0 3 14 06 9 2 15 06 0 3 16 5 3 0 17 4 6 9 18 4 1 9 19 3 7 6 20 3 3 9 21 3 1 1 22 2 8 0 23 2 5 6 24 2 3 5 25 2 1 7 26 2 0 0 27 1 8 6 28 1 7 3 29 1 6 1 30 1 5 1 The first column towards the left hand, doth contain any number of inches, from one to 30. In each of these Tables, is set down the length of a foot in feet & inches, & the tenth part of an inch, and so to the tenth part of one tenth part of an inch, that is to the hundreth part of an inch. Of Board Measure. An example upon each Table, will give more light than many words, and therefore, first, of Board: suppose a Board to be 7 inches broad: then find 7 in the first column towards the left hand, and over against it, under the title of Board Measure; you shall find one foot 8 inches, 5 tenths of an inch, and 7 tenths of one tenth part of an inch, and such is the length of a foot of Board at that breadth. And so if a Board be 14 inches broad, look 14 in the column towards the left hand, and against it under the title of Board Measure, you shall find 10 inches, two tenths of an inch, and eight parts of one tenth part of an inch; for the length of a foot at that breadth: and the like is to be observed for Timber. Of square Timber. Suppose a piece of Timber to be 10 inches square, look 10 to the left hand, and over against it, under the title of square Timber, you shall find one foot, five inches, two tenths of an inch, and eight parts of one tenth, for the length of a foot. If the square given be 16 inches, than over against 16, under the title of square Timber you shall find 6 inches, 7 tenths of an inch, and 5 parts of one tenth. Of Round Timber. For Round Timber, gird the piece about with some line, and with a quarter thereof enter your Table, and over against it under the title of Round Timber, you shall find the length of a foot. As suppose a stick to be 44 inches about, the quarter whereof is eleven inches, with which I enter the Table, in the column towards the left hand, and over against it, under the title of Round Timber, you shall find 11 inches, two tenths of an inch, & two parts of one tenth part of an inch, which is the length of a foot, at that thickness, and if that piece had been but 28 inches about, than the quarter thereof would have been but 7 inches, which being found to the left hand of the Table, over against it under the title of Round Timber, you shall find two feet three inches, and seven tenth parts of an inch, which is the length of a foot at that thickness. Note. And hereby will appear, that gross error which Carpenters use in taking a quarter of the Circumference, for the true square of that piece, which indeed it is not: for here against 7, in Timber which is square, there standeth two foot, 11 inches, two tenths of an inch, and six parts of one tenth, whereby it doth plainly appear, that at this thickness they do make their foot too long by 7 inches, 5 tenths, and 6 parts of one tenth part of an inch. These three Tables may be placed upon your Rule, according to the ordinary manner. CHAP. XX. How to find a mean proportional line between two lines given. THe Tables of Timber measure servs for such timber as is just square or round, it will not be unnecessary to show you how to find a mean proportional line, between two lines given, or to bring an unequal squared piece of Timber, to a true square, and so to apply it to your Table. Let a piece of Timber to be measured, be 9 inches broad, and 4 inches thick. Now: because it is not just square, it cannot be measured by the Table, therefore we must find a mean between the two given sides, after this manner. diagram of the measurement of a semi-circle (ABC) So that it doth appear, that a piece of Timber that is 9 inches broad, and 4 inches thick, is equal to a piece of 6 inches square. And hereby doth another of our Carpenters errors appear, which is this, they do put both sides together, and then they take half of that number, for the true square of that piece, which is merely false. For in this example, join 9 and 4 together, and they will make 13, the half whereof is 6 and a half, which indeed, according to our Rule should be but six. diagram of the measurement of a semi-circle (ABC) Having found the mean proportional number, you may enter the Table of Timber therewith, as hath been formerly showed, concerning square Timber. By this Rule the ingenious practitioner may bring any of the former pieces, of what fashion soever, to be measured by the Table of square Timber. And he that hath Arithmetic, may apply the proportions given in this Book, to the Rule of Three, and thereby he shall find the contents as before. The end of the first Part. THE ARTIFICERS Plain Scale. The second Part. CHAP. I. Of the Scale, and the graduations or divisions thereof, and how they are to be used. THe Scale here mentioned in this second Part, is a two foot Rule made with a joint, having a line of equal parts issuing from the centre thereof, and divided into 100 equal parts, upon the flat or edge thereof, may be made the other Rule according to the directions and figures in the preceding Part, so that any Artificer having his Rule and a pair of Compasses about him, may measure his Board, Timber, or Stone, two several ways. And before I come to any particular Propositions of the measuring of any Timber, or Board, I will first show the use of this opening Scale, in finding of proportional Lines, which is the ground of the way of measuring in the former Book, and also apply the use of them to some other Conclusions following. CHAP. II. To divide a line given, into any number of equal parts. THe line given is AB, and it is required to divide the same into five equal parts. diagram of the division of a line (AB) Take with your Compasses the given line AB, and fit that in any number that may be equally parted into five without any remainder, as fit it in 100, there let the Scale rest, then take it over in one fifth part thereof, viz. in 20, and that distance set from A to 1, so is A 1 one fifth part of the given line AB, which was required. CHAP. III. To take any part or parts of a line. THe line given is AB, now of the same line it is required to cut off three eight parts thereof. diagram of the division of a line (AB) Take the given line AB, and apply it to some number that may be parted in 8, which 8 is the Denominator, and take it over in the Numerator, or fit it in so many times the Numerator. As fit AB in 80, which is 10 times 8 the Denominator, there let rest the Scale, then take it over in ten times the Numerator, viz. in 30, and that distance set from A to C, so is AC three eight parts of AB given: then must CB be the rest, which is five eights. For if you take it over in 50, which is five eights of 80, (as the Scale stood wherein the line AB was fitted) and CB will appear to be the rest. CHAP. IU. A line containing any part or parts of a line, thereby to find the whole line. SUppose that AB be three eight parts of some line, and let it be required to find the whole line. Take both Numerator and Denominator diagram of the measurement of parts of a line (AB and CD) so many times as you please, as take each ten times, makes 30 and 50, then take the line AB and fit that in 30, and there let rest the Scale, then take it over in 50, & that distance lay down for the line CD, and so shall CD be the whole line, whereof AB was three eights. CHAP. V A line being given, containing any number of equal parts, to cut off from it so many as shall be required. AS let AB be a line given, containing 52 parts upon some Scale, and let it be required to cut off from it 24 of the same parts. diagram of the division of a line (AB) Take with your Compasses the given line AB, and set that in 52 of the equal parts, there let rest the Scale, then take it over in 24, the parts required, and that distance set from A to C, so is AC 24 of the same parts whereof AB is 52, which was required to be done. CHAP. VI To lay down suddenly 2, 3, or more lines in proportion required. IT is required to lay down four lines in proportion one to another, as these four numbers following. The numbers given 60 A 50 B 32 C 23 D Open by chance your Scale, and there let it rest, then take it over in 60 and in 50, and lay them both down, also take it over in 32 and 23, and lay them down, and so have you four lines A, B, C, D, in proportion, according to the four numbers given. diagram of the measurement of proportional lines (A, B, C, and D) CHAP. VII. In a Map or Plot, the length of any line being known, thereby to find the length of all or any of the rest. AS in the Plot ABCDEF, let the length of the line AB be known to be 47 parts, on some Scale, now it is required to find the length of the line CD. Take the known line AB, and fit that in 47, and let the Scale rest, then take CD, and bring it along the equal parts, till it be equally fitted on each side, which is 73 parts, so is CD 73 of the same as AB is 47, the like of all the rest. diagram of the measurement of a plot (ABCDEF) CHAP. VIII. Unto two lines given to find a third in proportion. THe two lines given are A and B, and it is required to find a third in proportion. diagram of the measurement of proportional lines (A, B, and C) Take the two lines given, and apply them to any Scale of equal parts, and see how many parts they contain, and let A contain 24 parts, and B 36 of the same parts, then take 36 the length of the line B on some Scale of equal parts, and fit that on 24 the line A, then let the Scale rest, then take it over in the line B, viz. 36, and that distance lay down for the line C, which shall be 54, a third line in proportion required. The Reason. For as 36 is 24, one time half, so is 54 once, 36 and a half, and so consequently 54 is the third proportional required. CHAP. IX. Unto 3 Lines given, to find a fourth in proportion, that is to perform the Rule of Three in Lines. AS let A B C be the 3 lines, unto the which it is required to find a fourth in proportion, that is, as the first is to the second, so is the third to the fourth. Take the three lines one after another with your Compasses, and apply them to any Scale of equal parts to know their length, and suppose you find them as the numbers which stand by them. diagram of the measurement of proportional lines (A, B, C, and D) CHAP. X. To divide a line given into two such parts, bearing proportion one to the other, as two numbers given. AS let it be required to divide the given line AB into two such parts, bearing proportion one to the other, as 28 to 21, viz. that AC may be to CB, as 28 to 21. diagram of the division of a line (AB) Add your two given numbers together, viz. 28 and 21, make 49, then take with your Compasses the given line, AB and fit it in 49, there let the Scale rest, then take it over in 28, which set from A to C, so is AC to BC, as 28 to 21, which was required to be done. CHAP. XI. To measure flat Measure. A Board being 16 inches broad, now it is required to find how much in length makes one foot. Take on any Scale of equal parts 12, the number of inches in one foot, and fit that in the breadth of the board which is 16, there let the Scale rest, then take it over in 12 always, and that apply to the same Scale of equal parts where the 12 was taken, and it showeth 9, and so many inches in length make a foot of board required, for if a board have 16 inches in length and 9 in breadth, these two numbers multiplied together make 144 inches, the number of square inches contained in a foot of square board, or glass, etc. Let a board be seven inches & three quarters broad, now it is required to find how much in length makes one foot. Take (as before) 12 of some Scale of equal parts, and fit it on seven three quarters, the breadth thereof, and then take it over in 12, as before, but to fit it in seven three quarters, would open the Scale too wide, therefore take four times seven three quarters, which is 31, & fit 12 in that, and take it over in four times 12, which is 48, and that distance applied to the same Scale where the 12 was taken, showeth 18 three fifths, and so many in length shall make one foot. If a board be two inches broad, how much in length shall make a foot? Multiply two the breadth of the Board, and 12 the inches in the foot by 10 makes 20, and 120, then take of some small Scale 120 (which may be done upon some Scale placed upon your Rule) and fit that on 20 on the Scale, and take it over in 12, and fit it in 2, 3, or 4 times 20, and take it over in so many times 12, and that apply to the same Scale, where the 120 was taken, and it showeth 72 inches, and so many in length is a foot of Board, the Board being two inches broad. Let a Board be three inches and three quarters broad, now you desire to know how much in length maketh a foot. Bring three inches and a quarter into quarters, and it maketh 13 quarters, then multiply 12, the inches in a foot, into quarters, and it maketh 48: take then 48 parts of some small Scale and fit that in 13, then let the Scale rest, and take it over in 12, and apply that to the same Scale where the 48 was taken, showeth 44 and one third part, and so many inches in length is required to make a foot. But having taken your 48 on some small Scale, and are to fit it on 13, now if it open your Scale too wide, you may fit it over in two or three times 13, and take it over in so many times 12, as fit 48 in four times 13, that is in 52, and take it over in four times 12, that is, in 48, and it showeth, 44 and one third, as before, being applied to the same Scale where the 48 was taken. Again, let a Board be 5 inches, and three eight parts of an inch broad, and it is required to find how many inches in length make a foot. Bring five and three eights into eights, makes 43, and 12 into eights make 96, then take 96 and fit it in 43, or in twice 43, there let the Scale rest, and take it over in 12, and also apply it to the same Scale where 96 was taken, and it showeth 26 and three quarters, and so much in length makes a foot of Board, the breadth being 5 inches, and three eight parts of an inch, which is the thing desired. CHAP. XII. To measure Board that is broader at one end then at the other. SUppose a Board be broad at one end 20 inches, and at the other 16: now it is required to find how much in length makes one foot throughout the whole Board. Add the breadth at both ends together, and take half thereof for a mean breadth, so find you 18, then is it all one as if your Board were 18 inches, and you would know how much in length makes a foot. Take 12 and fit it in 18, and take it over in 12, and so much makes a foot. Let a board be broad at one end ten inches and a quarter, and at the other seven and a half, now the desire is to know how much in length makes a foot. Add both the numbers together and take half, which maketh 8 inches and seven eight parts of an inch for the common breadth; then bring 8 inches and seven eights of an inch & 12 inches into eights, and it maketh 71 eights, and 96 eights. Take then 96 in some Scale, and fit that in 71, then let the Scale rest, then take it over in 12, and that apply to the same Scale where the 96 was taken, and it showeth 16 and a quarter, and so many inches in length make one foot of Board. CHAP. XIII. To find how many square feet any whole Board containeth, without finding how much in length makes a foot. IMagine a Board be 15 foot long, and 16 inches broad, and it is required to find how many square foot of Board it containeth. Take the length of 15 on some Scale of equal parts, and fit that in 12 the inches in a foot (always) there let the Scale rest, then take it over in 16 the breadth, and apply it to the same Scale where the length was taken, it showeth 20, and so many square foot is found to be therein contained. Let a Board be 17 foot and a quarter long, and 16 inches and a half broad, and the desire is to know how many foot it containeth. Take 17 and a quarter the length, and fit it in 12, and take it over in 16 and a half, and that apply to the same Scale when 17 and a quarter the length was taken, it showeth 23 and two thirds, and so many foot it containeth. Or you may bring 17 and a quarter into quarters, makes 69, and in like manner 12 into quarters, makes 48, and take it over in 16 and a half the breadth, so find you 23 and two thirds as before. CHAP. XIV. To measure Board that is broader at the one end then at the other, in the same manner. SUppose a Board be broad at the one end 18 inches, and at the other end 14, and long 21 foot, I demand how many square foot it containeth. Add the breadth at both ends together, makes 32 inches, whose half is 16 inches for a mean breadth, then proceed as before, take 21 and fit it in 12, and take it over in 16, or fit it in five times 12, and take it over in five times 16, so find you 28 for the area required. Again, let a Board be broad at the one end 11 inches and a half, and at she other 7 and three quarters, and 15 foot and three quarters long, now the Area is required. First, add them both together, and take half, makes 9 five eight parts, for the mean breadth. Then take 15 three quarters, the length on any Scale, and fit in 12, and take it over in 9 five eights, and that applied to the same Scale where the length was taken, and it showeth how many foot it containeth. Or bring 12, and 9 five eights into eights, make 96 and 77, then fit fifteen three quarters the length, in 96, and take it over in 77, and that showeth on the same Scale where the 15 three quarters was taken, twelve two thirds, the Area desired. CHAP. XV. To measure Timber. SUppose a piece of Timber be 18 inches broad, and deep 16 inches, it is required to find how much in length doth make a foot. Take twelve the inches in a foot on any Scale of equal parts, & fit that in the breadth eighteen, and take that over in twelve, always. Again, set that distance in sixteen the depth, and take it over in twelve still, and that apply to the same Scale where the twelve was taken show six, and so many inches in length make a foot, the thing required. Again, let a piece of Timber be broad sixteen inches, and deep thirteen and a half, and it is required to find how much in length make one foot. As before, fit twelve in sixteen, and take it over in 12 still, & that apply to the same Scale where the twelve was taken, showeth eight inches, and so many inches in length make one foot. Again, let a piece of Timber be fifteen three quarters broad, and eleven three quarters deep: I demand how much shall make a foot? Bring fifteen three quarters, and twelve into quarters, makes sixty three, and forty eight, then take twelve on some Scale of equal parts, and fit it in sixty three, and take it over in 48, and that distance fit in eleven one quarter, and take it over in 12: Or as before, bring eleven one quarter, and twelve both into quarters, makes forty eight and forty five, then fit it in forty five, and take it over in forty eight, and that applied to the same Scale where the first twelve was taken, showeth nine four fifths, and so many inches in length will make one foot. If a piece of Timber be seven one quarter broad, and five & a half deep, it is required to find how much in length shall make a foot. Bring seven one quarter, and five an a half into quarters, makes twenty nine, eight hundred twenty two, likewise twelve makes 48, then take twelve on any Scale of equal parts, and fit it on twenty nine, and take it over in forty eight, which distance fit again in twenty two, and take it over in forty eight, and that applied to the same Scale where the twelve was taken, showeth forty three one third part, and so many inches in length make a foot, which was required to be done. CHAP. XVI. To measure Timber that is broader at one end than at the other. SUppose a piece being broader at the one end than at the other, be given to be measured. First, take some place near the bigger end for a mean part, then take the breadth and depth thereabout, which suppose to be twenty and fifteen, then proceed as before, so find you 5 three quarters, and so many inches in length make a foot. CHAP. XVII. How Perpendicular heights may be found without either Instrument or Arithmetic. TAke a trencher, or any simple board's end, of what fashion soever, such as you can get, & draw thereon a line towards one of the sides, as the line AB, and on the point A, raise a perpendicular, as AD, then in the line AB, knock in two pins, one at A, and the other at B, then on the point or pin at A, hang a third with a plummet, then lift up this board with the end A, towards the height required, till you bring the two pins into one strait line, with your eye, and the top of the height required, and directly where the third falleth, there mark it with a prick of your Compass, as at E, and draw the line A, now measure the distance between your standing, and the base of your altitude, which here we will suppose to be 36 foot, as from F to G, and take 36 from your Scale, & set it down from A to D, from which point D, raise a Perpendicular, to cut the plumbe-line A in E, so shall DE be the height required, which being applied unto your Scale, will reach unto 32, and so many foot is the altitude GH. depiction of the measurement of altitude or height Here note, that the altitude thus found, is from the level of the eye upwards, and therefore the height from the eye downwards is to be added thereto to make it complete. CHAP. XVIII. How to take the altitude or height of a building by a bowl of water. PLace on the ground a Bowl of water, which done, erect your body strait up, and go back in a right line from the building, till you espy in the centre or middle of the water the very top of the altitude, which done, observe the place of your standing, and measure the height of your eye from the ground, together with the distance from your standing to the water, and the distance from the water to the base or foot of the altitude, which being all exactly taken, will help you to the altitude required by the Rule of proportion. depiction of the measurement of altitude or height Which will be found to be 66 foot and 8 inches. CHAP. XIX. How to take the altitude of a Building by a line and plummet the Sun shining. depiction of the measurement of altitude or height If CD the shadow of the line and plummet 4 foot 5/11, give EC 7 foot in altitude; what altitude doth 14 foot give, which is the shadow of the altitude required. Multiply and divide according to the Rule, and you shall find in your Quotient 22 foot, which is the true altitude of the building required. CHAP. XX. How to find the altitude of a Building by two sticks of one length joined in a right angle. 'Cause two sticks to be joined in a right angle, as is in the figure MN and OPEN, having at O a hole made wherein to hang a third & plummet. The two sticks being thus prepared, come to the building whose altitude you require (which building let be AB) then apply the end of your cross staff (noted with D) to your eye, & hold it up and down till the third and depiction of the measurement of altitude or height plummet hang just upon the perpendicular, then go backward or forward till your eye at D looking over E, espy the top of the building at A, which found, mark well the place of your standing, which is at F, and measure the distance from your eye to the ground, which is DF, and set that same distance off from F to C, then measure the distance from C to B, for that is the true height of the building AB. CHAP. XXI. To find a Distance by the two sticks joined square. THis experiment is grounded upon the fourth proposition of the 6 Book of Euclid. Let the distance which you desire to know be AB, set up a staff at A of four foot long, (or more or less at your pleasure) as the staff AC, at the end of the staff C place a third as CD. Then hanging the angle of the square on the top of the staff at C, move it up or down till you see the farthest part of your longitude: the square so remaining, and the staff not removed, draw the string that is fastened at C, close by the side of the square, till it touch the ground at D, then measure how many times the distance DA is contained in the Staff, for so many times is the Staff contained in the longitude. Example. The Staff supposed four foot high placed at A, and the Square being CHAP. XXII. How to describe a Town or City according to chorographical proportion, by the help of a plain glass. TO perform this conclusion, you must resort to some high place in the Town or Country you would describe, from whence you may behold all the Castles, Ports Harbours, Bays, Gates, Forts, and such other notable places as you intent to describe: which place being chosen, provide a plain glass, which in the midst of the Platform hang parallel to the Horizon, (in the doing of which you must be very careful) so that moving up and down the platform, you may in the Centre of the Glass, see all those notable places. The foundation being laid, let us now proceed to the work; and first of all on your platform, you must draw a Meridian line, which must pass just under the Glass, so that if a perpendicular line were let fall from the Centre of the Glass to the platform it might cut the Meridian line at right Angles, and by having this line drawn, you may draw the line of East and West at right Angles to the Meridian; and in like manner, the two and thirty points of the Compass, with Circles and Parallels, as is usual in the projecting of Sea-charts; so that thereby you may know how all the chief places in the Town are situate, and how they bear from you: This done, move Circularly about the Glass, observing always when you espy any mark in the Centre of your Glass to set up a staff, writing thereupon the name of the place, whether it be Village, Port, Road, or such like, you shall in the end situate, as it were, the whole Country, in due proportion upon your platform, so that measuring the distance of every staff set up from the Centre of your platform, and the distance likewise of every staff from other, you may by the Rule of Proportion, find out the distance of every Town, Village, Fort, Haven, and the like, from your platform; and also the distance between any two places there described. This Experiment is marvellous pleasant to practise, and most exactly serving for the description of a played Champion Country, which when you have thus traced out upon the platform, you may, by the help of Scale and Compasses, project in paper or parchment with a Scale of Leagues, Miles, Furlongs, Paces, or other measures, as liketh you best. FINIS.