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<DATE>2012-02-23</DATE>
<RESPSTMT>
<NAME>Simon Charles</NAME>
<RESP>MURP</RESP>
</RESPSTMT>
<ITEM>Proofed and reviewed.</ITEM>
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<EEBO>
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<STC T="W">S5686</STC>
<STC T="T">E1375_3</STC>
<BIBNO T="oclc">50811913</BIBNO>
<VID>169874</VID>
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<TEXT LANG="eng">
<FRONT>
<DIV1 TYPE="title page">
<PB REF="1" MS="y">
<P>THE ARTIFICERS PLAIN SCALE: OR, The <HI>Carpenters</HI> new Rule. In two Parts. The fir&s;t, &s;hewing how to mea&s;ure all Superficies and Solids, as Timber, Stone, Board, Gla&s;&s;e, &amp;c. Geometrically, without the help of <HI>Arithmetick:</HI> it being a new way not heretofore practi&s;ed. The &s;econd &s;hewing how to mea&s;ure Board and Timber In&s;trumentally, upon the Scale it &s;elfe, without <HI>Arithme|tick</HI> or <HI>Geometry,</HI> but what is common to every man. ALSO, How to take Heights and Di&s;tances &s;everall wayes, and to draw the Plot of a Town or City,</P>
<P>By <HI>Thomas Stirrup.</HI> Philomat,</P>
<P><HI>London,</HI> Printed by <HI>R.</HI> &amp; <HI>W. Leybourn,</HI> for <HI>Thomas Pirrepont,</HI> at the <HI>Sun</HI> in <HI>Pauls</HI> Church yard. 1651.</P>
</DIV1>
<DIV1 TYPE="illustration">
<PB REF="2">
<P><FIGURE>
<FIGDESC>depiction of the measurement of altitude or height</FIGDESC>
</FIGURE></P>
<PB REF="2" MS="y">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
</DIV1>
<DIV1 TYPE="to the reader">
<PB REF="3">
<PB REF="3">
<HEAD>To THE READER.</HEAD>
<OPENER><SALUTE>Gentle Reader:</SALUTE></OPENER>
<P>_ALthough ma|ny excellent both in <HI>Arith|metick</HI> &amp; <HI>Geo|metry,</HI> upon infallible grounds have put forth divers mo&s;t certain
<PB REF="4"> and &s;ufficient rules for the mea&s;uring of <HI>board</HI> &amp; <HI>tim|ber,</HI> yet very few of our common <HI>Artificers</HI> have been furthered thereby, be|cau&s;e they have not the art of <HI>Arithmetick,</HI> upon which mo&s;t of their rules depend.</P>
<P>The con&s;ideration of which, with the aptne&s;&s;e which I &s;ee in &s;ome of them for the rai&s;ing of a <HI>Perpen|dicular,</HI> and the drawing of <HI>a Parallel Line,</HI> upon which mo&s;t of this <HI>Book</HI> depends:
<PB REF="4"> this, I &s;ay, hath been the cau&s;e which hath moved me to give them &s;ome rules <HI>Geometrical,</HI> whereby they may mea&s;ure both <HI>board</HI> &amp; <HI>timber,</HI> without the help of <HI>Arithmetick.</HI></P>
<P>Therefore to thy view, <HI>Gentle Reader,</HI> that want|eth the art of <HI>Arithmetick,</HI> doe I prefer this &s;hort and plain <HI>Treati&s;e,</HI> wherein, &amp; in the beginning, is decla|red the infallible grounds upon which the whol <HI>work</HI>
<PB REF="5"> doth depend, &amp; then doth follow the applying of tho&s;e rules to the pre&s;ent purpo&s;e, with the declara|tion of three <HI>tables,</HI> one for <HI>Board,</HI> and one for <HI>&s;quare Timber,</HI> and the third for <HI>round Timber,</HI> very fit for all &s;uch as &s;tand in need thereof, and yet want both <HI>In&s;truments</HI> and <HI>Arithme|tick,</HI> whereby to u&s;e the &s;ame.</P>
<P><HI>In the &s;econd Part of this</HI> Book <HI>is &s;hewed a &s;econd
<PB REF="5"> way whereby you may mea|&s;ure</HI> Board <HI>and</HI> Timber <HI>by</HI> Rule <HI>and</HI> Compa&s;&s;e <HI>only, without drawing of</HI> lines: <HI>&amp; al&s;o how to take</HI> Heights <HI>and</HI> Di&s;tances <HI>&s;everal ways without</HI> In&s;trument, <HI>all which are grounded upon in|fallible principles</HI> Geome|tricall.</P>
<P>Thus de&s;iring thee to accept of this little <HI>Booke</HI> as a ta&s;te of my good will towards thee, which I wi&s;h even &s;o to further thee, as
<PB REF="6"> I know it &s;ufficient for the true mea&s;uring both of <HI>Board</HI> and <HI>Timber.</HI></P>
<CLOSER><SALUTE>Farewell.</SALUTE></CLOSER>
</DIV1>
<DIV1 TYPE="table of contents">
<PB REF="6">
<HEAD>THE CONTENTS.</HEAD>
<DIV2 N="1" TYPE="part">
<LIST>
<ITEM>THe meaning of certain terms of Geometry u&s;ed in this Book. <HI>Page</HI> 1</ITEM>
<ITEM>How to rai&s;e a perpendicular on any part of a right line given. <HI>Page</HI> 5</ITEM>
<ITEM>How to let fall a perpendicular from a point a&s;&s;igned, to a line given. <HI>Page</HI> 7</ITEM>
<ITEM>To a line given to draw a parallel line at any di&s;tance required. <HI>Page</HI> 8</ITEM>
<ITEM>To perform the former propo&s;ition at a di&s;tance required, and by a point limited. <HI>Page</HI> 10</ITEM>
<ITEM>Having two lines given to find a third proportionall line to them. <HI>Page</HI> 12</ITEM>
<ITEM>Having three lines given to finde a fourth proportionall line to them. <HI>Page</HI> 14</ITEM>
<PB REF="7">
<ITEM>The making of a Rule or Scale, for the mea&s;uring of Board and Timber. <HI>Page</HI> 17</ITEM>
<ITEM>How any Board may be mea&s;ured Ge|ometrically. <HI>Page</HI> 20</ITEM>
<ITEM>How Timber may be mea&s;ured Geo|metrically. <HI>Page</HI> 26</ITEM>
<ITEM>Of Round Timber. <HI>Page</HI> 33</ITEM>
<ITEM>How Triangled Timber, or Timber which hath but three &s;ides may be mea|&s;ured. <HI>Page</HI> 42</ITEM>
<ITEM>How Timber who&s;e end is a Rhombus is to be mea&s;ured. <HI>Page</HI> 45</ITEM>
<ITEM>How Timber who&s;e end is a Rhom|boiades is mea&s;ured. <HI>Page</HI> 47</ITEM>
<ITEM>How to mea&s;ure Timber who&s;e end is a Trape&s;iam. <HI>Page</HI> 49</ITEM>
<ITEM>How to mea&s;ure Timber who&s;e &s;ides are many, as 5, 6, 7, 8, 9, 10, or more, &s;o they be all equall. <HI>Page</HI> 51</ITEM>
<ITEM>How to finde the length of a foot of Board, at any breadth given. <HI>Page</HI> 54</ITEM>
<ITEM>The breadth and thickne&s;&s;e of a piece of Timber given, to finde how much in length &s;hall make a foot of &s;quare Tim|ber
<PB REF="7"> at that breadth and thickne&s;&s;e. <HI>Page</HI> 56</ITEM>
<ITEM>How to finde a mean proportionall line between two lines given. <HI>Page</HI> 66</ITEM>
</LIST>
</DIV2>
<DIV2 N="2" TYPE="part">
<HEAD>The &s;econd Part.</HEAD>
<LIST>
<ITEM>OF the Scale, and the graduations or divi&s;ions thereof, and how they are to be u&s;ed. <HI>Page</HI> 71</ITEM>
<ITEM>To divide a line given, into any number of equall parts. <HI>Page</HI> 73</ITEM>
<ITEM>To take any part or parts of a line <HI>Page</HI> 74</ITEM>
<ITEM>A line conteining any part or parts of a line, thereby to finde the whole line. <HI>Page</HI> 75</ITEM>
<ITEM>A line being given, conteining any number of equall parts, to cut off from it &s;o many as &s;hall be required. <HI>Page</HI> 77</ITEM>
<ITEM>To lay down &s;odainly two, three, or more lines in proportion required. <HI>Page</HI> 78</ITEM>
<ITEM>In a Map or Plot, the length of any line being known thereby to find the
<PB REF="8"> length of all or any of the re&s;t <HI>Page</HI> 80</ITEM>
<ITEM>Unto two lines given, to finde a third in proportion. <HI>Page</HI> 82</ITEM>
<ITEM>Unto three Lines given, to finde a fourth in proportion, that is, to perform the <HI>Rule of Three</HI> in Lines. <HI>Page</HI> 84</ITEM>
<ITEM>To divide a line given into two &s;uch parts, bearing proportion one to the other as two numbers given. <HI>Page</HI> 86</ITEM>
<ITEM>To mea&s;ure flat Mea&s;ure. <HI>Page</HI> 87</ITEM>
<ITEM>To mea&s;ure Board that is broader at one end than at the other. <HI>Page</HI> 91</ITEM>
<ITEM>To finde how many &s;quare feet any whole Board conteineth, without find|ing how much in length makes a foot <HI>Page</HI> 93</ITEM>
<ITEM>To mea&s;ure Board that is broader at the one end then at the other, in the &s;ame manner. <HI>Page</HI> 95</ITEM>
<ITEM>To mea&s;ure timber. <HI>Page</HI> 97</ITEM>
<ITEM>To mea&s;ure timber that is broader at one end than at the other. <HI>Page</HI> 100</ITEM>
<ITEM>How Perpendicular heights may be found without either In&s;trument or A|rithmetick. <HI>Page</HI> 101</ITEM>
<PB REF="8">
<ITEM>How to take the altitude or height of a building by a bowl of water. <HI>Page</HI> 105</ITEM>
<ITEM>How to take the altitude of a Buil|ding by a line and plummet the Sun &s;hi|ning. <HI>Page</HI> 107</ITEM>
<ITEM>How to finde the altitude of a Buil|ding by two &s;ticks joyned in a right an|gle. <HI>Page</HI> 109</ITEM>
<ITEM>To finde a Di&s;tance by the two &s;ticks joyned &s;quare. <HI>Page</HI> 112</ITEM>
<ITEM>How to de&s;cribe a <HI>Town</HI> or <HI>City</HI> ac|cording to <HI>Chorographicall</HI> proportion, by the help of a plain gla&s;&s;e. <HI>Page</HI> 116</ITEM>
</LIST>
</DIV2>
</DIV1>
<DIV1 TYPE="notice">
<PB REF="9">
<HEAD>An Adverti&s;ement To the READER.</HEAD>
<P>FOra&s;much as throughout this whole Book, there is mention made of <HI>Rules</HI> and <HI>Scales,</HI> the making whereof is different from tho&s;e which are vulgarly made and &s;old: if any therefore be de&s;irous to have any particular Rule mentioned in this book, or one Rule to performe all the work in generall, he may have them exactly made by Ma&s;ter <HI>Anthony Thomp&s;on</HI> in <HI>Ho&s;ier lane,</HI> neer <HI>Smith|field.</HI></P>
</DIV1>
</FRONT>
<BODY>
<DIV1 N="1" TYPE="part">
<PB N="1" REF="9">
<HEAD>THE ARTIFICERS Plain Scale.</HEAD>
<DIV2 N="1" TYPE="chapter">
<HEAD>CHAP. I. The meaning of certain terms of <HI>Geometry</HI> u&s;ed in this Book.</HEAD>
<P>_BEcau&s;e all Carpenters or other Artificers, in their Trade or Calling, doe in a manuer (and according to their fa|&s;hion) u&s;e &s;ome kind of Geometry, although them&s;elves be
<PB N="2" REF="10"> ignorant thereof, therefore I did con|&s;ider that they might bee &s;ooner brought to mea&s;ure Board and Tim|ber by that art of Geometry (&s;eeing they have their Rule and Compa&s;&s;es by them) then by Arithmetick, being but few of them can write, and there|fore uncapable of that art: and of them few which can write, not one in ten that hath Arithmetick, which is the only cau&s;e (as I &s;uppo&s;e) that mo&s;t of them are &s;o ignorant in this art (which doth &s;o much concern them) notwith|&s;tanding, all tho&s;e excellent Rules which have been formerly delivered by the learned: But now to our in|tended purpo&s;e.</P>
<P>SEeing I &s;hall have occa&s;ion in this Work to u&s;e &s;ome terms of Geometry, by which I may with more ea&s;e deliver, and you with more plainne&s;&s;e perceive my minde in the&s;e things: I have therefore &s;et down the
<PB N="3" REF="10"> meaning, as plainly as I can, of &s;ome Geometricall terms, which mo&s;t &s;erve for our pre&s;ent purpo&s;e.</P>
<LIST>
<ITEM>1 An <HI>Angle</HI> is nothing el&s;e but a corner, made by the meeting of two lines (for I &s;peak not of &s;olid angles.)</ITEM>
<ITEM>2 A <HI>right Angle</HI> (which we call a &s;quare angle) is that who&s;e two lines comprehending or making the angle, &s;tand perpendicular or plumb the one to the other.</ITEM>
<ITEM>3 A <HI>Perpendicular line</HI> is that which &s;tands plumb upright upon another, leaning neither the one way nor the other.</ITEM>
<ITEM>4 A <HI>Superficies</HI> is that which hath only Length and Breadth, and no Thickne&s;&s;e at all.</ITEM>
<ITEM>5 A <HI>Solid,</HI> or a <HI>Body,</HI> is that which hath Length, Breadth, and Thick|ne&s;&s;e.</ITEM>
<ITEM>6 <HI>Parallels</HI> are tho&s;e lines that dif|fer every where alike, or are not nee|rer together in one place then another.</ITEM>
<PB N="4" REF="11">
<ITEM>7 A <HI>Figure</HI> is any kinde of Su|perficies or Solid that is bounded a|bout, as <HI>Triangles, Squares, Circles, Globes, Cones, Pri&s;mes,</HI> and the re&s;t.</ITEM>
<ITEM>8 The <HI>Ba&s;e</HI> of a Figure is any &s;ide thereof upon which it may be &s;uppo|&s;ed to &s;tand; or if you take any &s;ide of a Figure for the Ground or Bottome, or lower part thereof, that &s;ame is the Ba&s;e.</ITEM>
<ITEM>9 The height of a Figure is the length of a Perpendicular or plumb line, falling from the top thereof to the Ba&s;e or bottome thereof.</ITEM>
</LIST>
</DIV2>
<DIV2 N="2" TYPE="chapter">
<PB N="5" REF="11">
<HEAD>CHAP. II. How to rai&s;e a <HI>Perpendi|cular</HI> on any part of a right line given.</HEAD>
<P>LEt AB be a right line given, and let C be a point therein, whereon I would rai&s;e a perpen|dicular, open the Compa&s;&s;es to any convenient di&s;tance, and &s;etting one foot in the point C, with the other mark on either &s;ide thereof, the equall di&s;tances CE, and CF: then opening your Compa&s;&s;es to any convenient wider di&s;tance, with one foot in the points E and F, &s;trike two arch lines, cro&s;&s;ing each other as in D, from whence draw the line DC, which is
<PB REF="12">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB REF="12">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB REF="13">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB REF="13">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB REF="14">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB REF="14">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB REF="15">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB N="1" REF="15">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB N="2" REF="16">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB N="3" REF="16">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB N="4" REF="17">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB N="5" REF="17">
<GAP DESC="duplicate" RESP="OXF" EXTENT="1 page">
<PB N="6" REF="18"> a perpendicular to AB, or as we call it, a &s;quare line to the line AB.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of perpendicular lines (AB and CD)</FIGDESC>
</FIGURE></P>
<P>Or you may from the given point C, prick out any five equal di&s;tances, and opening your Compa&s;&s;es to 4 of them, with one foot in C, &s;trike an arch or piece of a Circle towards N, then opening your Compa&s;&s;es to all 5 divi&s;ions, with one foot in 3, cro&s;s the &s;ame arch line in N, from whence draw the line NC, which is a perpen|dicular to the line AC, as before, for if 3 lines be joyned together, &s;o they be in &s;uch proportion, as 3, 4, and 5; they will make a right angle.</P>
</DIV2>
<DIV2 N="3" TYPE="chapter">
<PB N="7" REF="18">
<HEAD>CHAP. III. How to let fall a <HI>Perpendi|cular</HI> from a point a&s;sign|ed, to a line given.</HEAD>
<P>LEt the point given be D in the former Chapter, and let the line whereon it &s;hould fall be AB, open the Compa&s;&s;es to any conveni|ent di&s;tance, &amp; &s;etting 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 finde the middle between tho&s;e two Inter&s;ections, and from that mid|dle, draw a line to the point D (which is the point given) and that line &s;hal be perpendicular or plumb from the point D to the line AB, as was required.</P>
</DIV2>
<DIV2 N="4" TYPE="chapter">
<PB N="8" REF="19">
<HEAD>CHAP. IV. To a line given, to draw a parallel line, at any di|&s;tance required.</HEAD>
<P>SUppo&s;e the line given to be AB, unto which I mu&s;t draw a paral|lel.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of parallel lines (AB and CD)</FIGDESC>
</FIGURE></P>
<P>Open your Compa&s;&s;es to the di|&s;tance
<PB N="9" REF="19"> required, and &s;etting one foot of your Compa&s;&s;es in the end A, &s;trike an arch on that &s;ide the given line whereon the parallel is to be drawn, as the arch C, then doe the like in the end B, as the arch line D, then draw the line CD, &s;o as it may but touch or be a touch line to the&s;e two arches C and D, and this line &s;o drawn &s;hall be parallel to the line AB, as was re|quired.</P>
</DIV2>
<DIV2 N="5" TYPE="chapter">
<PB N="10" REF="20">
<HEAD>CHAP. V. To perform the former pro|po&s;ition at a di&s;tance re|quired, and by a point li|mited.</HEAD>
<P>ADmit AB in the former Chap|ter, to be a right line given, whereunto it is required to have a parallel line drawn at the di|&s;tance, and by the point C.</P>
<P>Place therefore one foot of your Compa&s;&s;es in C, from whence take the &s;horte&s;t di&s;tance to the line AB, as CA, at which di&s;tance, with one
<PB N="11" REF="20"> foot in the end B, with the other &s;trike the arch line D, by the extream 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.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of parallel lines (AB and CD)</FIGDESC>
</FIGURE></P>
</DIV2>
<DIV2 N="6" TYPE="chapter">
<PB N="12" REF="21">
<HEAD>CHAP. VI. Having two lines given, to finde a third proportionall line to them.</HEAD>
<P>THe two lines given are A and B, and it is required to finde a third line, which &s;hall bee in &s;uch proportion to A, as A is to B. Make any angle what&s;oever, as the angle HEC.</P>
<P>Here note, that an angle is always repre&s;ented by three letters, whereof the middle letter repre&s;ents the angle intended.</P>
<P>Then place the line A, from the angle E to D, and the line B from E to F, and draw the line DF.</P>
<PB N="13" REF="21">
<P>Place al&s;o the line A from E to H, and la&s;tly, by the 4 Chapter, from the point H, draw the line HC parallel to FD.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of proportional lines (A, B, and EC)</FIGDESC>
</FIGURE></P>
<P>So &s;hall EC be a third proportio|nall line to the two given lines, as was required.</P>
</DIV2>
<DIV2 N="7" TYPE="chapter">
<PB N="14" REF="22">
<HEAD>CHAP. VII. Having three lines given to finde a fourth proporti|onall line to them.</HEAD>
<P>THE three lines given are A B and C, and let it be requi|red to finde a fourth line, which &s;hall have &s;uch proportion to A, as B hath to C, make any angle, as DGK, now &s;eeing the line C hath the &s;ame proportion to the linne B, as the line A to the line &s;ought 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
<PB N="15" REF="22"> point I draw the line EI parallel to FH, till it cutteth DG in E, &s;o have you EG the fourth proporti|onall line required, which is 24.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of proportional lines (A, B, C, and EG)</FIGDESC>
</FIGURE></P>
<P>For as the line 12, is to the line 16, &s;o is the line 18, to the line 24, which is the length of the line we &s;ought for.</P>
<P>The&s;e two la&s;t Chapters, would I have you diligently to con&s;ider, and throughly to learne, becau&s;e it is the ground-work of that which I
<PB N="16" REF="23"> intend to deliver in this Booke: which being well under&s;tood, will bring much plea&s;ure and profit to the unlearned Artificer, for who&s;e &s;ake this was written.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of proportional lines (A, B, C, and EG)</FIGDESC>
</FIGURE></P>
</DIV2>
<DIV2 N="8" TYPE="chapter">
<PB N="17" REF="23">
<HEAD>CHAP. VIII. The making of a Rule or Scale, for the mea&s;uring of Board and Timber.</HEAD>
<P>BEfore we can give the true con|tent of any Board or piece of timber, we mu&s;t have &s;ome Rule or Scale whereby we may mea&s;ure the length, bredth, &amp; thickne&s;s thereof, &amp; for this purpo&s;e, the common Rules, commonly divided into inches, half in|ches quarters, and half quarters, will very wel &s;erve for our purpo&s;e; both for to mea&s;ure lengths, bredths, and thickne&s;&s;es withall: and likewi&s;e for to
<PB N="18" REF="24"> give the true content thereof, by the Rules to that purpo&s;e delivered, only herein, I would have you take no|tice, that this Scale thus divided,
<FIGURE>
<FIGDESC>depiction of a rule or scale</FIGDESC>
</FIGURE> I would have drawn upon both edges a|like, but differently figured, only for rea|dine&s;&s;e, after this manner; that &s;idethat you would have the readie&s;t for the mea|&s;uring of lengths, bredths, and thick|ne&s;&s;es, I would have you number them with 1, 2, 3, 4, 5, 6, and &s;o unto 24, dividing the whole two-foot Rule, into 24 equall parts or inches; and the o|ther
<PB N="19" REF="24"> edge, which hath the &s;ame divi|&s;ions, I would have you number with 4, 8, 12, 16, 20, 24. unto 96: dividing the whole two foot into 96 equall parts; that is every inch into 4 parts, only for readine&s;&s;e &s;ake, in the work which doth follow.</P>
<P>This line thus divided, is called a Scale; which is no other thing, but a right line divided into any number of equall parts, be they greater or le&s;&s;er, wider or narrower, &s;o they be equall: every part, or divi&s;ion of which line, may &s;tand for a mile, a rod, a yard, a foot, an inch, or any other kinde of mea&s;ure what you will, or have u&s;e of: and this line I would have you u&s;e, in giving up the Content of either Board, or Timber.</P>
<P>I have de&s;cribed this Scale but to 4 inches; but you may thereby per|ceive what I mean by the whole Rule.</P>
</DIV2>
<DIV2 N="9" TYPE="chapter">
<PB N="20" REF="25">
<HEAD>CHAP. IX. How any Board may be mea&s;ured Geometrically.</HEAD>
<P>IF you do well under&s;tand that which hath been delivered in the &s;eventh Chapter, you may thereby mea&s;ure any Board with ea&s;e and de|light; for as there is three numbers, or three lines given, whereby the fourth proportionall is found: &s;o in every Board, there is three lines, or numbers (which you will) which be given us, whereby, we may by the &s;eventh Chapter finde a fourth proportionall line or number: which is the number of feet, conteined in the whole board.</P>
<PB N="21" REF="25">
<P>The fir&s;t of the three Numbers given, is always 12, which is the &s;ide of a &s;quare foot of Board, or the &s;ide of a cubicall foot of Timber;</P>
<P>The &s;econd number is alwayes the number of feet, conteined in the length of the Board, the third number is alwayes the number of inches con|teined in the breadth of the Board.</P>
<P>And the fourth number which is here &s;ought for, will alwayes be the number of &s;quare feet, conteined in the whole Board: the proportion will be always thus.
<LIST>
<ITEM>As 12 to the length in Feet:</ITEM>
<ITEM>So the bredth in inches,</ITEM>
<ITEM>To the Content in Feet.</ITEM>
</LIST></P>
<P>And &s;eeing examples teacheth better then many words: therefore let us &s;uppo&s;e the three lines given us in the &s;eventh Chapter, to be three &s;uch numbers as here we have &s;poken of.</P>
<PB N="22" REF="26">
<P>And therefore let the fir&s;t line C, be a number of 12, taken from &s;ome Scale with your Compa&s;&s;es, and placed from G to H, which 12 doth &s;ignifie 12 inches, which is the &s;ide of a &s;quare foot of Board.</P>
<P>And let the &s;econd line B, be a number of 16, taken from the &s;ame Scale, and placed from the angle at G unto F, and draw theline FH, and this number of 16 doth &s;ignifie 16 feet, the length of a &s;uppo&s;ed Board.</P>
<P>And &s;o let the third line A, be a number of 18, taken from the &s;ame Scale and placed from G to I, and this number of 18 doth &s;ignifie 18 inches; the breadth of the &s;up|po&s;ed board: now from the point I, by the fifth Chapter, draw the line IE parallel to FH, till it cutteth EG in E: So have you EG the fourth proportionall line required; which being taken between your Compa&s;&s;es, and aplyed to your
<PB N="23" REF="26"> Scale, will &s;hew it to be 24, and &s;o many &s;quare feet are in that Board, who&s;e length is 16 foot, and breadth is 18 inches.</P>
<P>Let us take one Example more, for plainne&s;&s;e &s;ake, let the figure A, be a Board to be mea&s;ured, who&s;e length is 9 foot, and breadth 16 inches:
<FIGURE>
<FIGDESC>diagram of the measurement of a board (lines AB and CB)</FIGDESC>
</FIGURE> now draw two lines, &s;o as they make any angle, as the angle ABC, which being done, fir&s;t, take
<PB N="24" REF="27"> 12 from your Scale, and place it from your Scale, and place it from B to D, then take 9 of the &s;ame divi&s;ions, which doth &s;ignifie 9 foot, the length of the Board, and place them from B to E, and draw the line ED, then take 16 of the &s;ame divi&s;ions, (which doth &s;ignifie 16 inches, which is the breadth of the Board,) and place them from B to C: and la&s;tly, by the fifth Chapter, draw the line C, A, parallel to DE, till it cutteth AB in A, &s;o &s;hall AB be the fourth proportionall number; which being applyed unto your Scale, will reach unto 12; and &s;o many &s;quare feet are in that Board.
<LIST>
<ITEM>For, as 12 is to 9 foot, the length:</ITEM>
<ITEM>So is 16 the bredth,</ITEM>
<ITEM>To 12 foot the Content.</ITEM>
</LIST></P>
<PB N="25" REF="27">
<P>And here note, that when you have any odde parts of an inch in the breadth of your board you mu&s;t take the like parts of one divi&s;i|on from your Scale, more then your even parts was: and &s;o mu&s;t you doe when you have odde parts of a foot in the length of your Board: as for example, &s;uppo&s;e a Board to be eight foot, and three quarters long; now for to &s;et down this length you mu&s;t take from your Scale eight whole divi&s;ions, and three quarters of one; and &s;o apply them to your u&s;e: and this mu&s;t be noted throughout this Booke.</P>
<P>And here note al&s;o, that if your Board be taper grown, that is, wider at one end then at the other; then mea&s;ure the breadth thereof in the middle, and with that widene&s;s pro|ceed according to your Rules given:</P>
<P>And this may very well &s;uffice for Timber that doth taper al&s;o.</P>
</DIV2>
<DIV2 N="10" TYPE="chapter">
<PB N="26" REF="28">
<HEAD>CHAP. X. How Timber may be mea|&s;ured geometrically.</HEAD>
<P>THE mea&s;uring of Timber doth little differ from mea|&s;uring of Board, by the la&s;t Chapter, but only in mea&s;uring of Timber we have a double work; but the la&s;t Chapter well under&s;tood, will give light &s;ufficient hereunto.</P>
<P>Therefore by the la&s;t Chapter, fir&s;t, mea&s;ure how many &s;quare feet of flat mea&s;ure there is in one of the &s;ides of your Timber, as if it was a board by it &s;elf, which being done, you have three numbers given you, whereby you may by the &s;eventh Chapter finde a fourth in propor|tion unto them, which fourth num|ber,
<PB N="27" REF="28"> is the number of Cubicall feet conteined in that piece of Timber.</P>
<P>The Fir&s;t number is always 12.</P>
<P>The Second is always the number of &s;quare feet conteined in one of the &s;ides, I mean, of flat mea&s;ure.</P>
<P>The Third, is always the number of inches conteined in the thickne&s;s of the Timber: and this will be always the proportion for this work.</P>
<P><HI>Fir&s;t,</HI>
<LIST>
<ITEM>As 12 is to the length in feet, &s;o is the breadth in inches,</ITEM>
<ITEM>To the &s;uperficiall content, of that &s;ame &s;ide.</ITEM>
</LIST></P>
<P><HI>Secondly,</HI>
<LIST>
<ITEM>As 12 to this &s;uperficiall content,</ITEM>
<ITEM>So is the thickne&s;s in inches, to the &s;olid content in feet,</ITEM>
</LIST></P>
<P>As for example, &s;uppo&s;e the fi|gure A to be a piece of Timber to be mea&s;ured, who&s;e length is 8 foot, and breadth 18 inches, and thickne&s;s 14 inches.</P>
<PB N="28" REF="29">
<P><FIGURE>
<FIGDESC>diagram of the measurement of a block (lines BE and HE)</FIGDESC>
</FIGURE></P>
<P>Now draw two lines, &s;o as they may make any angle as the lines BE, and HE, meeting in the angle E, this being done, fir&s;t, place 12 (which is the &s;ide of a Cubicall foot, of Timber) from E to F, then place 8 (the length of your piece in feet) from E to D, and draw the line DF,
<PB N="29" REF="29"> and then place 18 (the bredth of your piece in inches) from E to H, and then by the 5 Chapter, draw the line HC, parallel to DF, till it cut|teth BE, in C: So &s;hall CE be the number of feet of flat mea&s;ure, con|teined in the broade&s;t &s;ide of the piece of Timber.</P>
<P>Thus far we have proceeded ac|cording to the la&s;t Chapter; and now we have three numbers more given us, whereby we may finde a fourth proportionall unto them.</P>
<P>Wherefore, fir&s;t, we have 12 already placed, from E to F, &s;econdly, we have the &s;uperficial content of the broade&s;t &s;ide, already placed from E to C: there fore draw the line FC, and thirdly, we have 14 the number of inches con|teined in the thickne&s;s, which we mu&s;t place from E to G: and la&s;tly, from the point G, by the fifth Chap. draw the line GB parallel to FC, till it cutteth BE at B: So &s;hal BE be the number of
<PB N="30" REF="30"> cubicall feet conteined in that piece of Timber noted with the Letter A, which being taken between your Compa&s;&s;es, and applyed unto your Scale, will reach unto 14, and &s;o many feet is in that piece.</P>
<P>Now here I will give you one ex|ample, of a piece of Timber hewed ju&s;t &s;quare: Let the figure B be a piece of Timber &s;o hewed, who&s;e length is 9 foot. And let it be 8 inches &s;quare, now having made any angle, as the angle ADC, fir&s;t place 12 from D to C.</P>
<P>And 9 foot the length, from D to A, and draw the line CA, then place 8, the thickne&s;&s;e of one of the &s;ides, from D to F, and by the 5 Chapter, draw the line FE parallel to CA, till it cuteth DA, at E, &s;o &s;hall ED, be the &s;uperficiall content, of one of the &s;ides, thus far according to the 9th. Chapter, as if it had been a board.</P>
<P>And now hear we have three num|bers,
<PB N="31" REF="30">
<FIGURE>
<FIGDESC>diagram of the measurement of a block (lines AE and CE)</FIGDESC>
</FIGURE> already placed, whereby we may finde a fourth, after this manner. Fir&s;t, we have 12 here placed from D to C, and we have the &s;uperficiall content of one of the &s;ides, placed from D to E, and therefore draw the line CE, and hear we have al&s;o, 8 the thickne&s;&s;e of one of the &s;ides, already placed from
<PB N="32" REF="31"> D to F, from which point F, draw the line FG, parallel to CE, till it cutteth AD, in G, &s;o &s;hall GD, &s;hew you the &s;olid content of that piece, which being applyed unto your Scale, will reach unto foure and &s;o many Cubi|call feet is there in that piece, marked with the letter B.</P>
<P>This Chapter would I have you well to con&s;ider, becau&s;e I do not in|tend to repeat, what I have heare de|livered, but only de&s;cribe unto you, the end of &s;ome pieces, according to their formes, and &s;o give you &s;ome Rules, for to mea&s;ure them by this Chapter.</P>
</DIV2>
<DIV2 N="11" TYPE="chapter">
<PB N="33" REF="31">
<HEAD>CHAP. XI. Of round Timber</HEAD>
<DIV3 TYPE="section">
<P>HEre fir&s;t I would have you to under&s;tand, what the Cir|cumference, the Center and Diameter of a Circle is, the Circum|ference is the line incompa&s;&s;ing the Circle, the Center is the point in the midde&s;t thereof, the Diameter is a right line pa&s;&s;ing by the Center through the whole Circle, and divi|deth the &s;ame into two equal parts, ei|ther halfe of which Diameter is cal|led the Semidiameter.</P>
<P>Now having found the Circum|ference of a round piece of Timbe, by girding it about with &s;ome line, I thinke it is heare needfull, to give you
<PB N="34" REF="32"> a Rule, for the finding of the Diame|ter of the &s;ame piece.</P>
<P>Therefore con&s;ider that every Cir|cumference, is in &s;uch proportion to
<FIGURE>
<FIGDESC>diagram of the measurement of a circle</FIGDESC>
</FIGURE> his Diameter, as 22 is to 7, therefore having the&s;e two numbers given you, and the Circumference of your piece of Timber, you may by the 7 Chap|ter, finde a fourth proportionall unto them, which will be the Diameters &s;ought for. As for example, let the
<PB N="35" REF="32"> figure A be the end of a round piece of Timber, who&s;e Circumference is found by girding it about, for to be 44 inches, now working by the rule given, you &s;hall finde the Diameter to be 14 inches.
<LIST>
<ITEM>For, as 22 is to 7.</ITEM>
<ITEM>So is 44 to 14.</ITEM>
</LIST></P>
<P>Now the Circumference and Di|ameter being found, you may finde the &s;olid content, after this manner. Fir&s;t, take one halfe of the Circumfe|rence, for the breadth of your piece, and one halfe of the Diameter for the thickne&s;&s;e thereof, according to which breadth, and thickne&s;&s;e, you may pro|ceed in all things, (by the former part of the tenth Chapter) as if it were an unequall &s;quared piece of Timber, as in the figure A, take 22 inches, (the Circumference of the piece) for the breadth thereof.</P>
<PB N="36" REF="33">
<P><HI>Or take a quarter of 44 that is 11 for the one &s;ide, and the whole 14 for the other.</HI></P>
<P>And take 7, which is the halfe of 14 the Diameter, for the thickne&s;&s;e thereof, and &s;o with this breadth, and thickne&s;&s;e, proceed in all things accor|ding to the former part of the tenth Chapter.</P>
</DIV3>
<DIV3 TYPE="section">
<HEAD>Of the half-round, or quar|ter or any other portion or part of a Circle.</HEAD>
<P>FOr this halfe Circle, take halfe the arch line CDB, which is 11, for the breadth of your piece.</P>
<P>And one halfe the Diameter, which is 7 for the thickne&s;&s;e thereof, and proceeding with this breadth and thickne&s;&s;e, by the tenth Chapter, you &s;hall finde the content.</P>
<P>And &s;o for a quarter of a Circle, or
<PB N="37" REF="33"> any other portion, (which goeth to the center,) take one halfe of the arch, belonging to that part of the Circle, for the thickne&s;&s;e of one of the &s;ides,
<FIGURE>
<FIGDESC>diagram of the measurement of a semi-circle (CDB)</FIGDESC>
</FIGURE> and take the Semidiameter, for the thickne&s;s of the other &s;ide. As in the quarter ACD, take one half the arch line CD, which will be five and a halfe, for the thickne&s;&s;e of one of the &s;ides, of that piece of Timber, and take the Diameter AC, which is 7 for the other &s;ide of the &s;ame piece, with which two &s;ides, as if it were an unequall &s;quared piece, of Timber, ca&s;t up the content, by the former part
<PB N="38" REF="34"> of the tenth Chapter, performing all things, as before in that Chapter.</P>
<P>Now having a piece of Timber, who&s;e end &s;hall be like unto this por|tion of a Circle, noted with the&s;e let|ters ABCD, before we can give the content thereof, it will be needfull to to finde out the Center, which for to doe work as followeth.</P>
</DIV3>
<DIV3 TYPE="section">
<HEAD>A Segment of a Circle being given, to finde out the Center, and con&s;equently the Diameter, and &s;o if need be, the whole Circle.</HEAD>
<P>LEt ABC, be part of a Circle gi|ven, to finde out the center there|of, fir&s;t take a point at plea&s;ure, with mo&s;t convenience, in the arch ABC, as at B, now on the point B, at any
<PB N="39" REF="34"> meet di&s;tance, de&s;cribe the Circle FGHITK, which being done, re|move the Compa&s;&s;es to the point H, (where the Circle cro&s;&s;eth the arch line given,) now one foot being in the point H, and at the &s;ame di&s;tance, as
<FIGURE>
<FIGDESC>diagram of the measurement of a circular segment (ABC)</FIGDESC>
</FIGURE> before, cro&s;&s;e the Circle twice, as at G and I and with the &s;ame di&s;tance, on the point K cro&s;&s;e the &s;aid Circle
<PB N="40" REF="35"> twice more, as in T and F, and la&s;tly, by the&s;e inter&s;ections or cro&s;&s;ings, draw the lines FE and GE, till they meet, or cro&s;&s;e the one the other, in the point E, which &s;hall be the center required.</P>
<P>The center being found, draw the lines EA and EC, and ca&s;t up the whole figure ABCE, as before is &s;hew'd, and then by the next Chapter, finde the content of the Triangle ACE, and take it from the content of the whole figure ABCE, and that which is lift, &s;hall be the content, of the figure ABCD, as was required.</P>
<P>By this Rule, (ob&s;erved with di&s;|cretion,) may all manner of Segments, or parts of a Circle, whether greater or le&s;&s;er then a Semicircle, be ea&s;ily mea&s;ured, without further in&s;truction.</P>
<P>Hithereto have wee &s;hewed the mea&s;uring of &s;uch Timber, as is mo&s;t in u&s;e, that is to &s;ay, of equal &s;quared, and al&s;o of unequall &s;quared Timber,
<PB N="41" REF="35"> &s;o likewi&s;e have we &s;hewed, how round Timber, and its parts, may be mea&s;ured, by the former Rules. &s;o So now will I &s;hew how &s;ome pieces of extraordinary formes, may be brought to be mea&s;ured, by the former Rules.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of a circular segment (ABC)</FIGDESC>
</FIGURE></P>
</DIV3>
</DIV2>
<DIV2 N="12" TYPE="chapter">
<PB N="42" REF="36">
<HEAD>CHAP. XII. How triangled Timber, or Timber which hath but three &s;ides, may be mea|&s;ured</HEAD>
<P>TRiangles, are made of &s;traight lines, o<GAP DESC="illegible" RESP="OXF" EXTENT="1 letter"> crooked, or of both to|gether, but I &s;peake only of Right lined Triangles, which is no|thing el&s;e, but a figure made of three right lines, as the figure ABC.</P>
<P>Triangles are divers, both in re|&s;pect of their &s;ides and angles, and may be mea&s;ured divers wayes, but let this one way &s;erve for all: take half of the ba&s;e, and &s;uppo&s;e it to be one &s;ide of a &s;quared piece of Timber; &amp; take
<PB N="43" REF="36"> the whole hieght, or perpendicular, for the other &s;ide of the &s;ame piece, and &s;o mea&s;ure it by the former part of the tenth Chapter, in all re&s;pects, as there is &s;hewed.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of a triangle (ABC)</FIGDESC>
</FIGURE></P>
<P>Let the Triangle ABC, be the end of a piece of Timber to be mea|&s;ured, which hath but only three &s;ides.</P>
<PB N="44" REF="37">
<P>Now &s;eeing the Ba&s;e BC is 16, I take one halfe thereof, which is eight inches, for on &s;ide of a &s;quared piece of Timber, &amp; I take 10, the whole length of the perpendicular, (which is the
<FIGURE>
<FIGDESC>diagram of the measurement of a triangle (ABC)</FIGDESC>
</FIGURE> pricked line AD) for the other &s;ide of the &s;ame piece: and &s;o as if it were a piece of 10 inches broad, and eight inches thick, you may ca&s;t up the con|tent, by the former Rules.</P>
</DIV2>
<DIV2 N="13" TYPE="chapter">
<PB N="45" REF="37">
<HEAD>CHAP. XIII. How Timber, who&s;e end is a <HI>Rhombus,</HI> (or Diamond form) is to be mea&s;ured.</HEAD>
<P>A <HI>Rhombus</HI> (or Diamond) is a fi|gure of foure equall &s;ides) but no right Angles, &s;uch as is the figure ABCD, for the mea&s;uring whereof, ob&s;erve this example. Let the &s;aid figure ABCD, be the end of a piece of Timber to be mea&s;ured: now taking the length of the &s;ide or ba&s;e AB, which is 14 inches for one of the &s;ides of a &s;quared piece of tim|ber, and the length of the perpendicu|lar DE, which wil be found to be 12,
<PB N="46" REF="38"> and &s;omething better then the eighth part of one more, for the other &s;ide
<FIGURE>
<FIGDESC>diagram of the measurement of a rhombus (ABCD)</FIGDESC>
</FIGURE> of the &s;ame piece, with which two &s;ides, as if it were an unequall &s;quared piece of timber, proceed in all things, according to the former part of the tenth Chapter.</P>
</DIV2>
<DIV2 N="15" TYPE="chapter">
<PB N="47" REF="38">
<HEAD>CHAP. XV. How Timber who&s;e end is a <HI>Rhomboides</HI> (or Dia|mond-like) is mea&s;ured.</HEAD>
<P>A <HI>Rhomboides</HI> (or Diamond-like) is a figure, who&s;e oppo&s;ite &s;ides, and oppo&s;ite Angles, are only equall, and it hath no right Angles. Such as is the figure FGHI, and may
<FIGURE>
<FIGDESC>diagram of the measurement of a rhomboid (FGHI)</FIGDESC>
</FIGURE>
<PB N="48" REF="39"> be mea&s;ured after this manner: take the length of the &s;ide HI, or FG, which is 16 inches for one &s;ide of a &s;quared piece of timber, and take the perpendicular FL, which is 10 inches, for the other &s;ide of the &s;ame piece, &amp; &s;o you may mea&s;ure it by the former part of the tenth Chapter, as if it were an unequall &s;quared piece of 16 inches broad, and 10 inches thick.</P>
<P>All other four &s;ided figures be&s;ides the true Square, and the unequall Square in the tenth Chapter, and the Rhombus in the la&s;t Chapter, and the Rhomboides in this, are called <HI>Trape|zias</HI> or Tables.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of a rhomboid (FGHI)</FIGDESC>
</FIGURE></P>
</DIV2>
<DIV2 N="14" TYPE="chapter">
<PB N="49" REF="39">
<HEAD>CHAP. XV. How to mea&s;ure Timber, who&s;e end is a <HI>Trapezi|am.</HI></HEAD>
<P>A <HI>Trapeziam</HI> is any irregular four &s;ided figure of what fa&s;hion &s;o|ever, as the figure ABCD is a Trapeziam, and may be ca&s;t into two
<FIGURE>
<FIGDESC>diagram of the measurement of a trapezium (ABCD)</FIGDESC>
</FIGURE> Triangles, by drawing the Diagonall
<PB N="50" REF="40"> line AC, and &s;o each Triangle mea|&s;ured as is before &s;hewed, which being done, adde the contents of them both together, and you &s;hall have the con|tent of the whole Trapeziam ABCD. Or you may more readily mea&s;ure it thus: Take one half the Diagonall line AC, which in this example will be 8 inches, for one &s;ide of your piece, and take the two perpendiculars BF and DE, and joyn them both toge|ther in one &s;um, &s;o &s;hall you have in this example 10 inches for the other &s;ide of your piece, with which two &s;ides, (as if it were an unequall &s;quared piece of Timber) proceed as before, in the former part of the tenth Chapter.</P>
</DIV2>
<DIV2 N="16" TYPE="chapter">
<PB N="51" REF="40">
<HEAD>CHAP. XVI. How to mea&s;ure Timber, who&s;e &s;ides are many, as 5, 6, 7, 8, 9, 10, or more, &s;o they be all equall.</HEAD>
<P>MAny &s;ided figures are tho&s;e which have more &s;ides than foure, and are generally cal|led <HI>Pollygons.</HI></P>
<P>A piece of timber who&s;e end &s;hall have more &s;ides than foure, may be mea&s;ured after this manner, adde all their &s;ides together, and take halfe that number for one &s;ide of an unequal &s;quared piece of Timber, then let fall a perpendicular from the centre or mid&s;t of the figure, to the mid&s;t of &s;ome one
<PB N="52" REF="41"> &s;ide, and take that length for the other &s;ide of the &s;ame piece, with which two &s;ides proceed as before is &s;hewed.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of a polygon (regular pentagon)</FIGDESC>
</FIGURE></P>
<P>Suppo&s;e the figure A to be the end of a piece of Timber of five &s;ides, be|ing all equall, and each &s;ide conteining 12 inches, which being added toge|ther into one &s;um will make 60, the half whereof wil be 30 for the breadth of your piece, then take the length of the perpendicular (falling from the
<PB N="53" REF="41"> center A to the mid&s;t of one of the &s;ides,) which here is 8 inches, for the thickne&s;&s;e of the &s;ame piece, with which breadth and thickne&s;&s;e proceed in all things according to the former part of the tenth Chapter.</P>
<P>This rule is generall in all kinde of regular <HI>Polygons,</HI> how many &s;ides &s;oever they have.</P>
<P><HI>Here I might have proceeded to have &s;hewed by what means Pyramidall or picked Timber, or Steeples may be mea|&s;ured: but con&s;idering how little this appertaineth to Carpenters, and how &s;ufficiently they be handled by Ma&s;ter</HI> Diggs <HI>in his Geometricall works, I for|beare here to write of them.</HI></P>
</DIV2>
<DIV2 N="17" TYPE="chapter">
<PB N="54" REF="42">
<HEAD>CHAP. XVII. How to finde the length of a Foot of Board, at any breadth given.</HEAD>
<P>THe breadth of a Board being given, with the number of 12, (the &s;ide of a &s;quare foot of Board,) you may by the &s;ixt Chapter finde how much in length will make a foot at any given breadth, by finding a third proportionall number, which &s;hall be to 12, as 12 is to the given breadth.</P>
<P>As &s;uppo&s;e a Board to be 16 inches broad, and I would know how much in length will make a foot thereof.</P>
<P>Fir&s;t, make any angle, as ABC,
<PB N="55" REF="42"> then place 16 inches, the breadth of your Board, from B to D, &amp; 12 inches from B to E, and draw the line DE:<FIGURE>
<FIGDESC>diagram of the measurement of length (angle ABC)</FIGDESC>
</FIGURE> Then again place 12 from B to G, fro~ which point G (by the fifth Chapter) draw the line GF parallel to DE, till it cutteth AB in F, &s;o &s;hall FB be the length of a foot of Board at 16 inches broad, which being applyed to your Scale, will reach unto 9, which doth &s;hew, that at 16 inches breadth, 9 inches in length doth make a foot of &s;quare Board: For,
<PB N="56" REF="43">
<LIST>
<ITEM>As 16 is to 12;</ITEM>
<ITEM>So is 12 to 9.</ITEM>
</LIST></P>
</DIV2>
<DIV2 N="18" TYPE="chapter">
<HEAD>CHAP. XVIII. The breadth and thickne&s;s of a piece of Timber given, to find how much in length &s;hall make a foot of &s;quare Timber at that breadth and thickne&s;&s;e.</HEAD>
<P>SUppo&s;e a piece of Timber to be 18 inches broad, and 14 inches thick. Fir&s;t, make any angle, as DFB, and place 18 inches from F to A, this is the &s;uppo&s;ed breadth of your piece; then place 12, the &s;ide of a Cubicall foot of Timber from F to E,
<PB N="57" REF="43"> and draw the line AE: So likewi&s;e place 12 from F to G, from which point G, draw the line GH parallel to AE, till it cutteth FD in H, &s;o &s;hall HF be the length of a&s;quare foot of flat mea&s;ure at the former breadth gi|ven: thus far according to the la&s;t Chapter.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of length (angle DFB)</FIGDESC>
</FIGURE></P>
<P>Now to proceed, place 14 the thick|ne&s;&s;e of your piece from F to C, and draw the line CG; and la&s;tly, from the point H, draw the line HI parallel
<PB N="58" REF="44"> to CG, till it cutteth FB in I, &s;o &s;hall IF be the length of a foot required, which being applyed to your Scale, will reach almo&s;t unto 7 inches, it wanteth but one &s;eventh part of an inch, and &s;uch is the length of a foot of Timber who&s;e breadth is 18 inch|es, and thickne&s;&s;e 14 inches.</P>
</DIV2>
<DIV2 N="19" TYPE="chapter">
<HEAD>CHAP. XIX. Of the Table for Board and Square Timber, and al&s;o for round Timber.</HEAD>
<DIV3 TYPE="section">
<P>COncerning the u&s;e of the&s;e Ta|bles, I would have you to un|der&s;tand that I have &s;uppo&s;ed the Inch to be divided into 10 equall parts, and each part divided into 10 e|quall parts, and &s;o the whole inch will contain 100 equall parts.</P>
<PB N="59" REF="44">
<P><TABLE>
<HEAD>A Table for Board mea&s;ure.</HEAD>
<ROW>
<CELL ROLE="label">Inches.</CELL>
<CELL ROLE="label">Feet.</CELL>
<CELL ROLE="label">Inches.</CELL>
<CELL ROLE="label">10 part of In</CELL>
<CELL ROLE="label">10 part of a 10 part.</CELL>
</ROW>
<ROW>
<CELL>1</CELL>
<CELL>12</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>2</CELL>
<CELL>06</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>3</CELL>
<CELL>04</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>4</CELL>
<CELL>03</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>5</CELL>
<CELL>02</CELL>
<CELL>04</CELL>
<CELL>8</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>6</CELL>
<CELL>02</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>7</CELL>
<CELL>01</CELL>
<CELL>08</CELL>
<CELL>5</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>8</CELL>
<CELL>01</CELL>
<CELL>06</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>9</CELL>
<CELL>01</CELL>
<CELL>04</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>10</CELL>
<CELL>01</CELL>
<CELL>02</CELL>
<CELL>4</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>11</CELL>
<CELL>01</CELL>
<CELL>01</CELL>
<CELL>0</CELL>
<CELL>9</CELL>
</ROW>
<ROW>
<CELL>12</CELL>
<CELL>01</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>13</CELL>
<CELL>&nbsp;</CELL>
<CELL>11</CELL>
<CELL>0</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>14</CELL>
<CELL>&nbsp;</CELL>
<CELL>10</CELL>
<CELL>2</CELL>
<CELL>8</CELL>
</ROW>
<ROW>
<CELL>15</CELL>
<CELL>&nbsp;</CELL>
<CELL>09</CELL>
<CELL>6</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>16</CELL>
<CELL>&nbsp;</CELL>
<CELL>9</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>17</CELL>
<CELL>&nbsp;</CELL>
<CELL>8</CELL>
<CELL>4</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>18</CELL>
<CELL>&nbsp;</CELL>
<CELL>8</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>19</CELL>
<CELL>&nbsp;</CELL>
<CELL>7</CELL>
<CELL>5</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>20</CELL>
<CELL>&nbsp;</CELL>
<CELL>7</CELL>
<CELL>2</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>21</CELL>
<CELL>&nbsp;</CELL>
<CELL>6</CELL>
<CELL>8</CELL>
<CELL>5</CELL>
</ROW>
<ROW>
<CELL>22</CELL>
<CELL>&nbsp;</CELL>
<CELL>6</CELL>
<CELL>5</CELL>
<CELL>4</CELL>
</ROW>
<ROW>
<CELL>23</CELL>
<CELL>&nbsp;</CELL>
<CELL>6</CELL>
<CELL>2</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>24</CELL>
<CELL>&nbsp;</CELL>
<CELL>6</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>25</CELL>
<CELL>&nbsp;</CELL>
<CELL>5</CELL>
<CELL>7</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>16</CELL>
<CELL>&nbsp;</CELL>
<CELL>5</CELL>
<CELL>5</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>27</CELL>
<CELL>&nbsp;</CELL>
<CELL>5</CELL>
<CELL>3</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>28</CELL>
<CELL>&nbsp;</CELL>
<CELL>5</CELL>
<CELL>1</CELL>
<CELL>4</CELL>
</ROW>
<ROW>
<CELL>29</CELL>
<CELL>&nbsp;</CELL>
<CELL>4</CELL>
<CELL>9</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>30</CELL>
<CELL>&nbsp;</CELL>
<CELL>4</CELL>
<CELL>8</CELL>
<CELL>0</CELL>
</ROW>
</TABLE></P>
<PB N="60" REF="45">
<P><TABLE>
<HEAD>A Table of &s;quare Timber mea&s;ure.</HEAD>
<ROW>
<CELL ROLE="label">Inches.</CELL>
<CELL ROLE="label">Feet.</CELL>
<CELL ROLE="label">Inches.</CELL>
<CELL ROLE="label">10 part of In</CELL>
<CELL ROLE="label">10 part of a 10 part.</CELL>
</ROW>
<ROW>
<CELL>1</CELL>
<CELL>144</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>2</CELL>
<CELL>36</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>3</CELL>
<CELL>16</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>4</CELL>
<CELL>9</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>5</CELL>
<CELL>5</CELL>
<CELL>09</CELL>
<CELL>1</CELL>
<CELL>2</CELL>
</ROW>
<ROW>
<CELL>6</CELL>
<CELL>4</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>7</CELL>
<CELL>2</CELL>
<CELL>11</CELL>
<CELL>2</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>8</CELL>
<CELL>2</CELL>
<CELL>03</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>9</CELL>
<CELL>1</CELL>
<CELL>09</CELL>
<CELL>3</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>10</CELL>
<CELL>1</CELL>
<CELL>05</CELL>
<CELL>2</CELL>
<CELL>8</CELL>
</ROW>
<ROW>
<CELL>11</CELL>
<CELL>1</CELL>
<CELL>02</CELL>
<CELL>2</CELL>
<CELL>8</CELL>
</ROW>
<ROW>
<CELL>12</CELL>
<CELL>1</CELL>
<CELL>00</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>13</CELL>
<CELL>&nbsp;</CELL>
<CELL>10</CELL>
<CELL>2</CELL>
<CELL>2</CELL>
</ROW>
<ROW>
<CELL>14</CELL>
<CELL>&nbsp;</CELL>
<CELL>08</CELL>
<CELL>8</CELL>
<CELL>1</CELL>
</ROW>
<ROW>
<CELL>15</CELL>
<CELL>&nbsp;</CELL>
<CELL>07</CELL>
<CELL>6</CELL>
<CELL>8</CELL>
</ROW>
<ROW>
<CELL>16</CELL>
<CELL>&nbsp;</CELL>
<CELL>6</CELL>
<CELL>7</CELL>
<CELL>5</CELL>
</ROW>
<ROW>
<CELL>17</CELL>
<CELL>&nbsp;</CELL>
<CELL>5</CELL>
<CELL>9</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>18</CELL>
<CELL>&nbsp;</CELL>
<CELL>5</CELL>
<CELL>3</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>19</CELL>
<CELL>&nbsp;</CELL>
<CELL>4</CELL>
<CELL>7</CELL>
<CELL>8</CELL>
</ROW>
<ROW>
<CELL>20</CELL>
<CELL>&nbsp;</CELL>
<CELL>4</CELL>
<CELL>3</CELL>
<CELL>2</CELL>
</ROW>
<ROW>
<CELL>21</CELL>
<CELL>&nbsp;</CELL>
<CELL>3</CELL>
<CELL>9</CELL>
<CELL>1</CELL>
</ROW>
<ROW>
<CELL>22</CELL>
<CELL>&nbsp;</CELL>
<CELL>3</CELL>
<CELL>5</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>23</CELL>
<CELL>&nbsp;</CELL>
<CELL>3</CELL>
<CELL>2</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>24</CELL>
<CELL>&nbsp;</CELL>
<CELL>3</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>25</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>7</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>26</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>5</CELL>
<CELL>5</CELL>
</ROW>
<ROW>
<CELL>27</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>3</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>28</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>2</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>29</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>0</CELL>
<CELL>5</CELL>
</ROW>
<ROW>
<CELL>30</CELL>
<CELL>&nbsp;</CELL>
<CELL>1</CELL>
<CELL>9</CELL>
<CELL>2</CELL>
</ROW>
</TABLE></P>
<PB N="61" REF="45">
<P><TABLE>
<HEAD>A Table of round Timber mea&s;ure.</HEAD>
<ROW>
<CELL ROLE="label">Inches.</CELL>
<CELL ROLE="label">Feet.</CELL>
<CELL ROLE="label">Inches.</CELL>
<CELL ROLE="label">10 part of In</CELL>
<CELL ROLE="label">10 part of a 10 part.</CELL>
</ROW>
<ROW>
<CELL>1</CELL>
<CELL>113</CELL>
<CELL>01</CELL>
<CELL>7</CELL>
<CELL>1</CELL>
</ROW>
<ROW>
<CELL>2</CELL>
<CELL>28</CELL>
<CELL>03</CELL>
<CELL>4</CELL>
<CELL>2</CELL>
</ROW>
<ROW>
<CELL>3</CELL>
<CELL>12</CELL>
<CELL>06</CELL>
<CELL>8</CELL>
<CELL>5</CELL>
</ROW>
<ROW>
<CELL>4</CELL>
<CELL>7</CELL>
<CELL>00</CELL>
<CELL>8</CELL>
<CELL>5</CELL>
</ROW>
<ROW>
<CELL>5</CELL>
<CELL>4</CELL>
<CELL>06</CELL>
<CELL>3</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>6</CELL>
<CELL>3</CELL>
<CELL>01</CELL>
<CELL>7</CELL>
<CELL>1</CELL>
</ROW>
<ROW>
<CELL>7</CELL>
<CELL>2</CELL>
<CELL>03</CELL>
<CELL>7</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>8</CELL>
<CELL>1</CELL>
<CELL>09</CELL>
<CELL>2</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>9</CELL>
<CELL>1</CELL>
<CELL>04</CELL>
<CELL>7</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>10</CELL>
<CELL>1</CELL>
<CELL>01</CELL>
<CELL>5</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>11</CELL>
<CELL>&nbsp;</CELL>
<CELL>11</CELL>
<CELL>2</CELL>
<CELL>2</CELL>
</ROW>
<ROW>
<CELL>12</CELL>
<CELL>&nbsp;</CELL>
<CELL>09</CELL>
<CELL>4</CELL>
<CELL>2</CELL>
</ROW>
<ROW>
<CELL>13</CELL>
<CELL>&nbsp;</CELL>
<CELL>08</CELL>
<CELL>0</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>14</CELL>
<CELL>&nbsp;</CELL>
<CELL>06</CELL>
<CELL>9</CELL>
<CELL>2</CELL>
</ROW>
<ROW>
<CELL>15</CELL>
<CELL>&nbsp;</CELL>
<CELL>06</CELL>
<CELL>0</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>16</CELL>
<CELL>&nbsp;</CELL>
<CELL>5</CELL>
<CELL>3</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>17</CELL>
<CELL>&nbsp;</CELL>
<CELL>4</CELL>
<CELL>6</CELL>
<CELL>9</CELL>
</ROW>
<ROW>
<CELL>18</CELL>
<CELL>&nbsp;</CELL>
<CELL>4</CELL>
<CELL>1</CELL>
<CELL>9</CELL>
</ROW>
<ROW>
<CELL>19</CELL>
<CELL>&nbsp;</CELL>
<CELL>3</CELL>
<CELL>7</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>20</CELL>
<CELL>&nbsp;</CELL>
<CELL>3</CELL>
<CELL>3</CELL>
<CELL>9</CELL>
</ROW>
<ROW>
<CELL>21</CELL>
<CELL>&nbsp;</CELL>
<CELL>3</CELL>
<CELL>1</CELL>
<CELL>1</CELL>
</ROW>
<ROW>
<CELL>22</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>8</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>23</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>5</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>24</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>3</CELL>
<CELL>5</CELL>
</ROW>
<ROW>
<CELL>25</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>1</CELL>
<CELL>7</CELL>
</ROW>
<ROW>
<CELL>26</CELL>
<CELL>&nbsp;</CELL>
<CELL>2</CELL>
<CELL>0</CELL>
<CELL>0</CELL>
</ROW>
<ROW>
<CELL>27</CELL>
<CELL>&nbsp;</CELL>
<CELL>1</CELL>
<CELL>8</CELL>
<CELL>6</CELL>
</ROW>
<ROW>
<CELL>28</CELL>
<CELL>&nbsp;</CELL>
<CELL>1</CELL>
<CELL>7</CELL>
<CELL>3</CELL>
</ROW>
<ROW>
<CELL>29</CELL>
<CELL>&nbsp;</CELL>
<CELL>1</CELL>
<CELL>6</CELL>
<CELL>1</CELL>
</ROW>
<ROW>
<CELL>30</CELL>
<CELL>&nbsp;</CELL>
<CELL>1</CELL>
<CELL>5</CELL>
<CELL>1</CELL>
</ROW>
</TABLE></P>
<PB N="62" REF="46">
<P>The fir&s;t columne towards the left hand, doth contein any number of inches, from one to 30.</P>
<P>In each of the&s;e Tables, is &s;et down the length of a foot in feet &amp; inches, &amp; the tenth part of an inch, and &s;o to the tenth part of one tenth part of an inch, that is to the hundreth part of an inch.</P>
</DIV3>
<DIV3 TYPE="section">
<HEAD>Of Board Mea&s;ure.</HEAD>
<P>An example upon each Table, will give more light than many words, and therefore, fir&s;t, of Board: &s;uppo&s;e a Board to be 7 inches broad: then find 7 in the fir&s;t columne towards the left hand, and over again&s;t it, under the title of Board Mea&s;ure; you &s;hall find one foot 8 inches, 5 tenths of an inch, and 7 tenths of one tenth part of an inch, and &s;uch is the length of a foot of Board at that breadth.</P>
<P>And &s;o if a Board be 14 inches broad, look 14 in the column towards
<PB N="63" REF="46"> the left hand, and again&s;t it under the title of Board Mea&s;ure, you &s;hall 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 ob&s;erved for Timber.</P>
</DIV3>
<DIV3 TYPE="section">
<HEAD>Of &s;quare Timber.</HEAD>
<P>Suppo&s;e a piece of Timber to be 10 inches &s;quare, look 10 to the left hand, and over again&s;t it, under the ti|tle of &s;quare Timber, you &s;hall finde one foot, five inches, two tenths of an inch, and eight parts of one tenth, for the length of a foot.</P>
<P>If the &s;quare given be 16 inches, then over again&s;t 16, under the title of &s;quare Timber you &s;hall find 6 inches, 7 tenths of an inch, and 5 parts of one tenth.</P>
</DIV3>
<DIV3 TYPE="section">
<PB N="64" REF="47">
<HEAD>Of Round Timber.</HEAD>
<P>For Round Timber, gird the piece about with &s;ome line, and with a quar|ter thereof enter your Table, and o|ver again&s;t it under the title of Round Timber, you &s;hall finde the length of a foot.</P>
<P>As &s;uppo&s;e a &s;tick to be 44 inches a|bout, the quarter whereof is eleven inches, with which I enter the Table, in the columne towards the left hand, and over again&s;t it, under the title of Round Timber, you &s;hall finde 11 inches, two tenths of an inch, &amp; two parts of one tenth part of an inch, which is the length of a foot, at that thickne&s;s, and if that piece had been but 28 inches about, then the quarter thereof would have been but 7 inches, which being found to the left hand of the Table, over again&s;t it under the title of Round Timber, you &s;hall find
<PB N="65" REF="47"> two feet three inches, and &s;even tenth parts of an inch, which is the length of a foot at that thickne&s;s.</P>
</DIV3>
<DIV3 TYPE="note">
<HEAD>Note.</HEAD>
<P>And hereby will appear, that gro&s;s errour which Carpenters u&s;e in taking a quarter of the Circumference, for the true &s;quare of that piece, which in|deed it is not: for here again&s;t 7, in Timber which is &s;quare, there &s;tandeth two foot, 11 inches, two tenths of an inch, and &s;ix parts of one tenth, where|by it doth plainly appear, that at this thickne&s;s they do make their foot too long by 7 inches, 5 tenths, and 6 parts of one tenth part of an inch. The&s;e three Tables may be placed upon your Rule, according to the ordinary manner.</P>
</DIV3>
</DIV2>
<DIV2 N="20" TYPE="chapter">
<PB N="66" REF="48">
<HEAD>CHAP. XX. How to finde a mean propor|tionall line between two lines given.</HEAD>
<P>THe Tables of Timber mea&s;ure &s;ervs for &s;uch timber as is ju&s;t &s;quare or round, it will not be unnece&s;&s;ary to &s;hew you how to finde a mean proportional line, between two lines given, or to bring an unequall &s;quared piece of Timber, to a true &s;quare, and &s;o to apply it to your Ta|ble.</P>
<P>Let a piece of Timber to be mea|&s;ured, be 9 inches broad, and 4 inches thick. Now: becau&s;e it is not ju&s;t &s;quare, it cannot be mea&s;ured by the Table, therefore we mu&s;t finde a mean between the two given &s;ides, after this manner.</P>
<PB N="67" REF="48">
<P>Fir&s;t, take 9 from your Scale, and place it on the line AC fro~ C to D, &amp; place 4 of the &s;ame divi&s;ions, from D to A, upon which line AC de&s;cribe the Semicircle ABC, and one the
<FIGURE>
<FIGDESC>diagram of the measurement of a semi-circle (ABC)</FIGDESC>
</FIGURE> point D (where the two lines are joyn|ed) by the &s;econd Chapter, rai&s;e a perpendicular to cut the Circumfe|rence in B, &s;o &s;hall BD be the mean proportionall line de&s;ired, which be|ing applyed to your Scale, will reach unto 6.</P>
<P>So that it doth appear, that a piece of Timber that is 9 inches broad, and 4 inches thick, is equall to a piece of 6 inches &s;quare.</P>
<PB N="68" REF="49">
<P>And hereby doth another of our Carpenters errours appear, which is this, they do put both &s;ides together, and then they take half of that num|ber, for the true &s;quare of that piece, which is meerly fal&s;e. For in this ex|ample, joyn 9 and 4 together, and they will make 13, the halfe whereof is 6 and a halfe, which indeed, according to our Rule &s;hould be but &s;ix.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of a semi-circle (ABC)</FIGDESC>
</FIGURE></P>
<P>Having found the mean proportio|nall number, you may enter the Ta|ble of Timber therewith, as hath bin
<PB N="69" REF="49"> formerly &s;hewed, concerning &s;quare Timber. By this Rule the ingenious Practicioner may bring any of the for|mer pieces, of what fa&s;hion &s;oever, to be mea&s;ured by the Table of &s;quare Timber.</P>
<P>And hee that hath Arithmetick, may apply the proportions given in this Book, to the <HI>Rule of Three,</HI> and thereby he &s;hall finde the contents as before.</P>
<TRAILER>The end of the fir&s;t Part.</TRAILER>
</DIV2>
</DIV1>
<DIV1 N="2" TYPE="part">
<PB REF="50">
<PB N="71" REF="50">
<HEAD>THE ARTIFICERS Plain Scale.</HEAD>
<HEAD>The &s;econd Part.</HEAD>
<DIV2 N="1" TYPE="chapter">
<HEAD>CHAP. I. Of the Scale, and the gradu|ations or divi&s;ions there|of, and how they are to be u&s;ed.</HEAD>
<P>_THe Scale here mentioned in this &s;econd Part, is a two foot Rule made with a joynt, having a line of e|qual
<PB N="72" REF="51"> parts i&s;&s;uing fro~ the center there|of, and divided into 100 equall parts, upon the flat or edge thereof, may be made the other Rule according to the directions and figures in the preceding Part, &s;o that any Artificer having his Rule and a pair of Compa&s;&s;es about him, may mea&s;ure his Board, Timber, or Stone, two &s;everall ways.</P>
<P>And before I come to any perticu|lar Propo&s;itions of the mea&s;uring of any Timber, or Board, I wil fir&s;t &s;hew the u&s;e of this opening Scale, in find|ing of proportionall Lines, which is the ground of the way of mea&s;uring in the former Book, and al&s;o apply the u&s;e of them to &s;ome other Conclu&s;i|ons following.</P>
</DIV2>
<DIV2 N="2" TYPE="chapter">
<PB N="73" REF="51">
<HEAD>CHAP. II. To divide a line gi|ven, into any num|ber of equall parts.</HEAD>
<P>THe line given is AB, and it is required to di|vide the &s;ame into five equall parts.</P>
<P><FIGURE>
<FIGDESC>diagram of the division of a line (AB)</FIGDESC>
</FIGURE></P>
<P>Take with your Compa&s;|&s;es the given line AB, and fit that in any number that may be equally parted into five without any remainer, as fit it in 100, there let the Scale re&s;t, then take it over in one fifth part thereof, <HI>viz.</HI> in 20, and that di&s;tance &s;et from A to 1,
<PB N="74" REF="52"> &s;o is A 1 one fifth part of the given line AB, which was required.</P>
</DIV2>
<DIV2 N="3" TYPE="chapter">
<HEAD>CHAP. III. To take any part or parts of a line.</HEAD>
<P>THe line given is AB, now of the &s;ame line it is re|quired to cut off three eight parts thereof.</P>
<P><FIGURE>
<FIGDESC>diagram of the division of a line (AB)</FIGDESC>
</FIGURE></P>
<P>Take the given line AB, and apply it to &s;ome number that may be parted in 8, which 8 is the Denominator, and take it o|ver in the Numerator, or fit it in &s;o many times the Numera|tor.</P>
<P>As fit AB in 80, which is 10
<PB N="75" REF="52"> times 8 the Denominator, there let re&s;t the Scale, then take it over in ten times the Numerator, <HI>viz.</HI> in 30, and that di&s;tance &s;et from A to C, &s;o is AC three eight parts of AB given: then mu&s;t CB be the re&s;t, which is five eights. For if you take it over in 50, which is five eights of 80, (as the Scale &s;tood wherein the line AB was fitted) and CB will appear to be the re&s;t.</P>
</DIV2>
<DIV2 N="4" TYPE="chapter">
<HEAD>CHAP. IV. A line containing any part or parts of a line, thereby to finde the whole line.</HEAD>
<P>SUppo&s;e that AB be three eight parts of &s;ome line, and let it be required to find the whole line.</P>
<P>Take both Numerator and Deno|minator
<PB N="76" REF="53">
<FIGURE>
<FIGDESC>diagram of the measurement of parts of a line (AB and CD)</FIGDESC>
</FIGURE> &s;o many times as you plea&s;e, as take each ten times, makes 30 and 50, then take the line AB and fit that in 30, and there let re&s;t the Scale, then take it over in 50, &amp; that di&s;tance lay down for the line CD, and &s;o &s;hall CD be the whole line, whereof AB was three eights.</P>
</DIV2>
<DIV2 N="5" TYPE="chapter">
<PB N="77" REF="53">
<HEAD>CHAP. V. A line being given, conteining any number of equall parts, to cut off from it &s;o many as &s;hall be required.</HEAD>
<P>AS let AB be a line given, conteining 52 parts upon &s;ome Scale, and let it be required to cut off from it 24 of the &s;ame parts.</P>
<P><FIGURE>
<FIGDESC>diagram of the division of a line (AB)</FIGDESC>
</FIGURE></P>
<P>Take with your Com|pa&s;&s;es the given line AB, and &s;et that in 52 of the e|quall parts, there let re&s;t the
<PB N="78" REF="54"> Scale, then take it over in 24, the parts required, and that di&s;tance &s;et from A to C, &s;o is AC 24 of the &s;ame parts whereof AB is 52, which was required to be done.</P>
</DIV2>
<DIV2 N="6" TYPE="chapter">
<HEAD>CHAP. VI. To lay down &s;odainly 2, 3, or more lines in proportion required.</HEAD>
<P>IT is required to lay down foure lines in proportion one to another, as the&s;e foure numbers following. The numbers given
<LIST>
<ITEM>60 A</ITEM>
<ITEM>50 B</ITEM>
<ITEM>32 C</ITEM>
<ITEM>23 D</ITEM>
</LIST></P>
<P>Open by chance your Scale, and
<PB N="79" REF="54"> there let it re&s;t, then take it over in 60 and in 50, and lay them both down, al&s;o take it over in 32 and 23, and lay them down, and &s;o have you four lines A, B, C, D, in proportion, ac|cording to the four numbers given.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of proportional lines (A, B, C, and D)</FIGDESC>
</FIGURE></P>
</DIV2>
<DIV2 N="7" TYPE="chapter">
<PB N="80" REF="55">
<HEAD>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 re&s;t.</HEAD>
<P>AS in the Plot ABCDEF, let the length of the line AB be known to be 47 parts, on &s;ome Scale, now it is required to finde the length of the line CD.</P>
<P>Take the known line AB, and fit that in 47, and let the Scale re&s;t, then take CD, and bring it along the e|quall parts, till it be equally fitted on each &s;ide, which is 73 parts, &s;o is CD 73 of the &s;ame as AB is 47, the like of all the re&s;t.</P>
<PB N="81" REF="55">
<P>But if you de&s;ire the di&s;tance from C to F, take it, and bring it along
<FIGURE>
<FIGDESC>diagram of the measurement of a plot (ABCDEF)</FIGDESC>
</FIGURE> your Scale as it &s;tandeth, &s;o finde you 90.</P>
</DIV2>
<DIV2 N="8" TYPE="chapter">
<PB N="82" REF="56">
<HEAD>CHAP. VIII. &V;nto two lines given to find a third in proportion.</HEAD>
<DIV3 TYPE="section">
<P>THe two lines given are A and B, and it is required to finde a third in proportion.</P>
<P><FIGURE>
<FIGDESC>diagram of the measurement of proportional lines (A, B, and C)</FIGDESC>
</FIGURE></P>
<P>Take the two lines given, and apply them to any Scale of equall parts, and &s;ee how ma|ny parts they contein, and let A contain 24 parts, and B 36 of the &s;ame parts, then take 36 the length of the line B on &s;ome Scale of equall parts, and fit that on 24 the line A,
<PB N="83" REF="56"> then let the Scale re&s;t, then take it over in the line B, <HI>viz.</HI> 36, and that di&s;tance lay downe for the line C, which &s;hall be 54, a third line in pro|portion required.</P>
</DIV3>
<DIV3 TYPE="section">
<HEAD>The Rea&s;on.</HEAD>
<P>For as 36 is 24, one time half, &s;o is 54 once, 36 and a half, and &s;o con&s;e|quently 54 is the third proportionall required.</P>
</DIV3>
</DIV2>
<DIV2 N="9" TYPE="chapter">
<PB N="84" REF="57">
<HEAD>CHAP. IX. &V;nto 3 Lines given, to find a fourth in proportion, that is to perform the <HI>Rule of Three</HI> in Lines.</HEAD>
<P>AS let A B C be the 3 lines, unto the which it is required to finde a fourth in proportion, that is, as the fir&s;t is to the &s;econd, &s;o is the third to the fourth.</P>
<P>Take the three lines one after ano|ther with your Compa&s;&s;es, and apply them to any Scale of equall parts to know their length, and &s;uppo&s;e you find them as the numbers which &s;tand by them.</P>
<PB N="85" REF="57">
<P>Then take the &s;econd line B, and apply it to &s;ome Scale of equall parts,
<FIGURE>
<FIGDESC>diagram of the measurement of proportional lines (A, B, C, and D)</FIGDESC>
</FIGURE> and fit that over in the length of the fir&s;t line 48, there let the Scale re&s;t, then take it over in the third line C, <HI>viz.</HI> in 40, and that di&s;tance lay down for D, which is 30 parts of the &s;ame Scale, where the line A is 48, and is the fourth proportionall required: for as A is to B, &s;o is C to D, &amp;c.</P>
</DIV2>
<DIV2 N="10" TYPE="chapter">
<PB N="86" REF="58">
<HEAD>CHAP. X. To divide a line given into two &s;uch parts, bearing proportion one to the other, as two numbers given.</HEAD>
<P>AS let it be required to di|vide the given line AB into two &s;uch parts, bear|ing proportion one to the other, as 28 to 21, <HI>viz.</HI> that AC may be to CB, as 28 to 21.</P>
<P><FIGURE>
<FIGDESC>diagram of the division of a line (AB)</FIGDESC>
</FIGURE></P>
<P>Adde your two given num|bers together, <HI>viz.</HI> 28 and 21, make 49, then take with your Compa&s;&s;es the given line, AB
<PB N="87" REF="58"> and fit it in 49, there let the Scale re&s;t, then take it over in 28, which &s;et from A to C, &s;o is AC to BC, as 28 to 21, which was required to be done.</P>
</DIV2>
<DIV2 N="11" TYPE="chapter">
<HEAD>CHAP. XI. To mea&s;ure flat Mea&s;ure.</HEAD>
<P>A Board being 16 inches broad, now it is required to finde how much in length makes one foot.</P>
<P>Take on any Scale of equall 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 re&s;t, then take it over in 12 alwayes, and that apply to the &s;ame Scale of equall parts where the 12 was taken, and it &s;heweth 9, and &s;o many inches in length make a foot of board required,
<PB N="88" REF="59"> for if a board have 16 inches in length and 9 in breadth, the&s;e two numbers multiplyed together make 144 inches, the number of &s;quare inches contein|ed in a foot of &s;quare board, or gla&s;s, &amp;c.</P>
<P>Let a board be &s;even inches &amp; three quarters broad, now it is required to finde how much in length makes one foot.</P>
<P>Take (as before) 12 of &s;ome Scale of equall parts, and fit it on &s;even three quarters, the breadth thereof, and then take it over in 12, as before, but to fit it in &s;even three quarters, would open the Scale too wide, there|fore take four times &s;even three quar|ters, which is 31, &amp; fit 12 in that, and take it over in four times 12, which is 48, and that di&s;tance applyed to the &s;ame Scale where the 12 was taken, &s;heweth 18 three fifths, and &s;o many in length &s;hall make one foot.</P>
<P>If a board be two inches broad,
<PB N="89" REF="59"> how much in length &s;hal make a foot?</P>
<P>Multiply two the breadth of the Board, and 12 the inches in the foot by 10 makes 20, and 120, then take of &s;ome &s;mall Scale 120 (which may be done upon &s;ome 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 &s;o many times 12, and that apply to the &s;ame Scale, where the 120 was ta|ken, and it &s;heweth 72 inches, and &s;o many in length is a foot of Board, the Board being two inches broad.</P>
<P>Let a Board be three inches and three quarters broad, now you de&s;ire to know how much in length maketh a foot.</P>
<P>Bring three inches and a quarter into quarters, and it maketh 13 quar|ters, then multiply 12, the inches in a foot, into quarters, and it maketh 48: take then 48 parts of &s;ome &s;mall Scale and fit that in 13, then let the Scale
<PB N="90" REF="60"> re&s;t, and take it over in 12, and apply that to the &s;ame Scale where the 48 was taken, &s;heweth 44 and one third part, and &s;o many inches in length is required to make a foot.</P>
<P>But having taken your 48 on &s;ome &s;mall 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 &s;o 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 &s;heweth, 44 and one third, as before, being apply|ed to the &s;ame Scale where the 48 was taken.</P>
<P>Again, let a Board be 5 inches, and three eight parts of an inch broad, and it is required to finde how many inches in length make a foot.</P>
<P>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 re&s;t,
<PB N="91" REF="60"> and take it over in 12, and al&s;o apply it to the &s;ame Scale where 96 was ta|ken, and it &s;heweth 26 and three quarters, and &s;o much in length makes a foot of Board, the breadth being 5 inches, and three eight parts of an inch, which is the thing de&s;i|red.</P>
</DIV2>
<DIV2 N="12" TYPE="chapter">
<HEAD>CHAP. XII. To mea&s;ure Board that is broader at one end then at the other.</HEAD>
<P>SUppo&s;e a Board be broad at one end 20 inches, and at the other 16: now it is required to finde how much in length makes one foot throughout the whole Board.</P>
<P>Adde the breadth at both ends to|gether,
<PB N="92" REF="61"> and take halfe thereof for a mean breadth, &s;o finde 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.</P>
<P>Take 12 and fit it in 18, and take it over in 12, and &s;o much makes a foot.</P>
<P>Let a board be broad at one end ten inches and a quarter, and at the o|ther &s;even and a halfe, now the de&s;ire is to know how much in length makes a foot.</P>
<P>Adde both the numbers together and take halfe, which maketh 8 inch|es and &s;even eight parts of an inch for the common breadth; then bring 8 inches and &s;even eights of an inch &amp; 12 inches into eights, and it maketh 71 eights, and 96 eights. Take then 96 in &s;ome Scale, and fit that in 71, then let the Scale re&s;t, then take it o|ver in 12, and that apply to the &s;ame Scale where the 96 was taken, and it &s;heweth 16 and a quarter, and &s;o ma|ny
<PB N="93" REF="61"> inches in length make one foot of Board.</P>
</DIV2>
<DIV2 N="13" TYPE="chapter">
<HEAD>CHAP. XIII. To finde how many &s;quare feet any whole Board con|taineth, without finding how much in length makes a foot.</HEAD>
<P>IMagine a Board be 15 foot long, and 16 inches broad, and it is re|quired to finde how many &s;quare foot of Board it containeth.</P>
<P>Take the length of 15 on &s;ome Scale of equall parts, and fit that in 12 the inches in a foot (alwayes) there let the Scale re&s;t, then take it over in 16 the breadth, and apply it to the &s;ame Scale where the length was taken, it
<PB N="94" REF="62"> &s;heweth 20, and &s;o many &s;quare foot is found to be therein contained.</P>
<P>Let a Board be 17 foot and a quar|ter long, and 16 inches and a halfe broad, and the de&s;ire is to know how many foot it containeth.</P>
<P>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 &s;ame Scale whe 17 and a quarter the length was taken, it &s;heweth 23 and two thirds, and &s;o many foot it contain|eth.</P>
<P>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 halfe the breadth, &s;o finde you 23 and two thirds as before.</P>
</DIV2>
<DIV2 N="14" TYPE="chapter">
<PB N="95" REF="62">
<HEAD>CHAP. XIV. To mea&s;ure Board that is broader at the one end then at the other, in the &s;ame manner.</HEAD>
<P>SUppo&s;e a Board be broad at the one end 18 inches, and at the o|ther end 14, and long 21 foot, I demand how many &s;quare foot it con|taineth.</P>
<P>Adde the breadth at both ends to|gether, makes 32 inches, who&s;e halfe 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, &s;o finde you 28 for the area required.</P>
<PB N="96" REF="63">
<P>Again, let a Board be broad at the one end 11 inches and a halfe, and at she other 7 and three quarters, and 15 foot and three quarters long, now the Area is required.</P>
<P>Fir&s;t, adde them both together, and take half, makes 9 five eight parts, for the mean breadth.</P>
<P>Then take 15 three quarters, the length on any Scale, and fit in 12, and take it over in 9 five eights, and that applyed to the &s;ame Scale where the length was taken, and it &s;heweth how many foot it containeth.</P>
<P>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 &s;heweth on the &s;ame Scale where the 15 three quarters was taken, twelve two thirds, the Area de&s;ired.</P>
</DIV2>
<DIV2 N="15" TYPE="chapter">
<PB N="97" REF="63">
<HEAD>CHAP. XV. To mea&s;ure Timber.</HEAD>
<P>SUppo&s;e a piece of Timber be 18 inches broad, and deep 16 inches, it is required to finde how much in length doth make a foot.</P>
<P>Take twelve the inches in a foot on any Scale of equal parts, &amp; fit that in the breadth eighteen, and take that over in twelve, alwayes. Again, &s;et that di&s;tance in &s;ixteen the depth, and take it over in twelve &s;till, and that ap|ply to the &s;ame Scale where the twelve was taken &s;hew &s;ix, and &s;o many inches in length make a foot, the thing required.</P>
<P>Again, let a piece of Timber be broad &s;ixteen inches, and deep thir|teen
<PB N="98" REF="64"> and a halfe, and it is required to finde how much in length make one foot.</P>
<P>As before, fit twelve in &s;ixteen, and take it over in 12 &s;till, &amp; that apply to the &s;ame Scale where the twelve was taken, &s;heweth eight inches, and &s;o many inches in length make one foot.</P>
<P>Again, let a piece of Timber be fifteen three quarters broad, and ele|ven three quarters deep: I demand how much &s;hall make a foot?</P>
<P>Bring fifteen three quarters, and twelve into quarters, makes &s;ixty three, and forty eight, then take twelve on &s;ome Scale of equall parts, and fit it in &s;ixty three, and take it over in 48, and that di&s;tance fit in eleven one quarter, and take it over in 12: Or as before, bring eleven one quarter, and twelve both into quarters, makes for|ty eight and forty five, then fit it in forty five, and take it over in forty
<PB N="99" REF="64"> eight, and that applyed to the &s;ame Scale where the fir&s;t twelve was ta|ken, &s;heweth nine foure fifths, and &s;o many inches in length will make one foot.</P>
<P>If a piece of Timber be &s;even one quarter broad, and five &amp; a half deep, it is required to finde how much in length &s;hall make a foot.</P>
<P>Bring &s;even one quarter, and five an a half into quarters, makes twenty nine, eight hundred twenty two, like|wi&s;e twelve makes 48, then take twelve on any Scale of equall parts, and fit it on twenty nine, and take it over in forty eight, which di&s;tance fit again in twenty two, and take it over in forty eight, and that applyed to the &s;ame Scale where the twelve was taken, &s;heweth forty three one third part, and &s;o many inches in length make a foot, which was required to be done.</P>
</DIV2>
<DIV2 N="16" TYPE="chapter">
<PB N="100" REF="65">
<HEAD>CHAP. XVI. To mea&s;ure Timber that is broader at one end than at the other.</HEAD>
<P>SUppo&s;e a piece being broader at the one end than at the other, be given to be mea&s;ured.</P>
<P>Fir&s;t, take &s;ome place neer the big|ger end for a meane part, then take the breadth and depth thereabout, which &s;uppo&s;e to be twenty and fif|teen, then proceed as before, &s;o finde you 5 three quarters, and &s;o many inches in length make a foot.</P>
</DIV2>
<DIV2 N="17" TYPE="chapter">
<PB N="101" REF="65">
<HEAD>CHAP. XVII. How Perpendicular heights may be found without ei|ther In&s;trument or Arith|metick.</HEAD>
<P>TAke a trencher, or any &s;imple boards end, of what fa&s;hion &s;oever, &s;uch as you can get, &amp; draw thereon a line towards one of the &s;ides, as the line AB, and on the point A, rai&s;e 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 thrid with a plummet, then lift up this board with the end A, towards
<PB N="102" REF="66"> the height required, till you bring the two pins into one &s;traight line, with your eye, and the top of the height required, and directly where the thrid falleth, there mark it with a prick of your Compa&s;&s;e, as at E, and draw the line AE, now mea&s;ure the di&s;tance between your &s;tanding, and the ba&s;e of your altitude, which here wee will &s;uppo&s;e to be 36 foot, as from F to G, and take 36 from your Scale, &amp; &s;et it down from A to D, from which point D, rai&s;e a Perpendicular, to cut the plumbe-line AE in E, &s;o &s;hall DE be the height required, which being applyed unto your Scale, will reach unto 32, and &s;o many foot is the alti|tude GH.</P>
<P>But &s;uppo&s;e you cannot come unto the ba&s;e G, and therefore cannot mea|&s;ure the di&s;tance GF, and yet it is re|quired to find the altitude GH, there|fore to perform this, fir&s;t take your &s;tanding at F, as before, and lift up
<PB N="105" REF="66"> your board, till you bring the two pins A and B into one &s;traight line, between your eye, and the top at H, and directly where the thred falleth,
<FIGURE>
<FIGDESC>depiction of the measurement of altitude or height</FIGDESC>
</FIGURE>
<PB N="106" REF="67"> there make a mark, as at E, and draw the line AE, then mea&s;ure out &s;o ma|ny foot as you think good, in a &s;traight line towards the ba&s;e G, and there take your other &s;tanding, as here &s;up|po&s;e 16, this 16 take from your Scale, and place them from A to C, and in the point C hang your thrid and plummet, and lifting up your Board, till you take your &s;ight as before, make a mark directly where your thrid cro&s;&s;eth the former plumb line AE, as at E, from which point E let fall a perpendicular to the line AD, as DE, &s;o &s;hall DE be 32 for the al|titude of GH as before.</P>
<P>Here note, that the altitude thus found, is from the levell of the eye upwards, and therefore the height fro~ the eye downwards is to be added thereto to make it compleat.</P>
</DIV2>
<DIV2 N="18" TYPE="chapter">
<PB N="107" REF="67">
<HEAD>CHAP. XVIII. How to take the altitude or height of a building by a bowl of water.</HEAD>
<P>PLace on the ground a Bowle of water, which done, erect your body &s;traight up, and goe back in a right line from the building, till you e&s;pie in the center or middle of the water the very top of the altitude, which done, ob&s;erve the place of your &s;tanding, and mea&s;ure the height of your eye from the ground, together with the di&s;tance from your &s;tanding to the water, and the di&s;tance from the water to the ba&s;e or foot of the al|titude, which being all exactly taken, will help you to the altitude required by the Rule of proportion.</P>
<PB N="106" REF="68">
<P><HI>Example,</HI> Let the altitude required be AB, the Bowle of water placed on the ground at C, then go back|ward from C (your body erected as
<FIGURE>
<FIGDESC>depiction of the measurement of altitude or height</FIGDESC>
</FIGURE> &s;treight as may be) till your eye at E e&s;pie the top of the altitude AB in the water, which found, ob&s;erve the place of your &s;tanding at D, and mea|&s;ure the altitude from your eye to the
<PB N="107" REF="68"> ground, which is five foot, then mea|&s;ure the di&s;tance from D to C, which is 6 foot, and likewi&s;e the di&s;tance from C to B which is 80 foot: the&s;e three di&s;tances being known, work by the Rule of proportion thus.
<LIST>
<ITEM>As the di&s;tance CD, to the altitude ED;</ITEM>
<ITEM>So is the di&s;tance CB, to the altitude AB:</ITEM>
</LIST></P>
<P>Which will be found to be 66 foot and 8 inches.</P>
</DIV2>
<DIV2 N="19" TYPE="chapter">
<HEAD>CHAP. XIX. How to take the altitude of a Building by a line and plummet the Sun &s;hining.</HEAD>
<P>LEt the Building who&s;e altitude you de&s;ire to know be AB, ca&s;ting his &s;hadow in a right line to C, at C
<PB N="110" REF="69"> let fall a line and plummet (who&s;e length before you know in feet and inches) ob&s;erving where the end of that &s;hadow lighteth, which &s;uppo&s;e
<FIGURE>
<FIGDESC>depiction of the measurement of altitude or height</FIGDESC>
</FIGURE> at D, then mea&s;ure the length of the &s;hadow of the &s;tring, and con&s;equent|ly the &s;hadow of the Building, both which being exactly taken, work thus by the Rule of Proportion.</P>
<P>If CD the &s;hadow of the line and
<PB N="111" REF="69"> plummet 4 foot 5/11, give EC 7 foot in altitude; what altitude doth 14 foot give, which is the &s;hadow of the altitude required.</P>
<P>Multiply and divide according to the Rule, and you &s;hall finde in your Quotient 22 foot, which is the true altitude of the building required.</P>
</DIV2>
<DIV2 N="20" TYPE="chapter">
<HEAD>CHAP. XX. How to find the altitude of a Building by two &s;ticks of one length joyned in a right angle.</HEAD>
<P>CAu&s;e two &s;ticks to be joyned in a right angle, as is in the figure MN and OP, having at O a hole made wherein to hang a thrid &amp; plummet.</P>
<P>The two &s;ticks being thus prepared,
<PB N="110" REF="70"> come to the building who&s;e altitude you require (which building let be AB) then apply the end of your cro&s;&s;e &s;taffe (noted with D) to your eye, &amp; hold it up and down till the third and
<FIGURE>
<FIGDESC>depiction of the measurement of altitude or height</FIGDESC>
</FIGURE> plummet hang ju&s;t upon the perpendi|cular, then goe backward or forward till your eye at D looking over E, e&s;py the top of the building at A, which
<PB N="111" REF="70"> found, marke well the place of your &s;tanding, which is at F, and mea&s;ure the di&s;tance from your eye to the ground, which is DF, and &s;et that &s;ame di&s;tance off from F to C, then mea&s;ure the di&s;tance from C to B, for that is the true height of the building AB.</P>
</DIV2>
<DIV2 N="21" TYPE="chapter">
<PB N="114" REF="71">
<HEAD>CHAP. XXI. To finde a Di&s;tance by the two &s;ticks joyned &s;quare.</HEAD>
<P>THis experiment is grounded upon the fourth propo&s;ition of the 6 Booke of <HI>Euclid.</HI></P>
<P>Let the di&s;tance which you de&s;ire to know be AB, &s;et up a &s;taffe at A of four foot long, (or more or le&s;&s;e at your plea&s;ure) as the &s;taffe AC, at the end of the &s;taffe C place a thrid as CD.</P>
<P>Then hanging the angle of the &s;quare on the top of the &s;taffe at C, move it up or down till you &s;ee the farthe&s;t part of your longitude: the &s;quare &s;o remaining, and the &s;taffe not
<PB N="113" REF="71"> removed, draw the &s;tring that is fa&s;te|ned at C, clo&s;e by the &s;ide of the &s;quare, till it touch the ground at D, then mea&s;ure how many times the di|&s;tance DA is conteined in the Staffe, for &s;o many times is the Staffe con|teined in the longitude.</P>
<P><HI>Example.</HI></P>
<P>The Staff &s;uppo&s;ed four foot high placed at A, and the Square being
<GAP DESC="missing" RESP="OXF" EXTENT="1+ pages"></P>
</DIV2>
<DIV2 N="22" TYPE="chapter">
<PB N="116" REF="72">
<HEAD>CHAP. XXII. How to de&s;cribe a <HI>Town</HI> or <HI>City</HI> according to <HI>Cho|rographicall</HI> proportion, by the helpe of a plain gla&s;&s;e.</HEAD>
<P>TO performe this conclu&s;ion, you mu&s;t re&s;ort to &s;ome high place in the Town or Coun|trey you would de&s;cribe, from whence you may behold all the Ca&s;tles, Ports Harbours, Bays, Gates, Forts, and &s;uch other notable places as you in|tend to de&s;cribe: which place being cho&s;en, provide a plain gla&s;&s;e, which in the mid&s;t of the Platforme hang
<PB N="117" REF="72"> parallel to the Horizon, (in the doing of which you mu&s;t be very carefull) &s;o that moving up and down the plat|forme, you may in the Center of the Gla&s;&s;e, &s;ee all tho&s;e notable places. The foundation being laid, let us now proceed to the worke; and fir&s;t of all on your platforme, you mu&s;t draw a Meridian line, which mu&s;t pa&s;&s;e ju&s;t under the Gla&s;&s;e, &s;o that if a perpendicular line were let fall from the Center of the Gla&s;&s;e to the plat|forme it might cut the Meridian line at right Angles, and by having this line drawn, you may draw the line of Ea&s;t and We&s;t at right Angles to the Meridian; and in like manner, the two and thirty points of the Com|pa&s;&s;e, with Circles and Parallels, as is u&s;uall in the projecting of Sea-charts; &s;o that thereby you may know how all the chief places in the Town are &s;ituate, and how they bear from you: This done, move Circularly about
<PB N="118" REF="73"> the Gla&s;&s;e, ob&s;erving always when you e&s;pie any marke in the Center of your Gla&s;&s;e to &s;et up a &s;taffe, writing thereupon the name of the place, whether it be Village, Port, Road, or &s;uch like, you &s;hall in the end &s;ituate, as it were, the whole Countrey, in due proportion upon your platform, &s;o that mea&s;uring the di&s;tance of eve|ry &s;taffe &s;et up from the Center of your platforme, and the di&s;tance like|wi&s;e of every &s;taffe from other, you may by the Rule of Proportion, finde out the di&s;tance of every Town, Vil|lage, Fort, Haven, and the like, from your platform; and al&s;o the di&s;tance between any two places there de&s;cri|bed. This Experiment is marvei|lous plea&s;ant to practi&s;e, and mo&s;t exactly &s;erving for the de&s;cription of a plaid <HI>Champion</HI> Countrey, which when you have thus traced out upon the platform, you may, by the help of Scale and Compa&s;&s;es, project in pa|per
<PB N="119" REF="73"> or parchment with a Scale of Leagues, Miles, Furlongs, Pa|ces, or other mea&s;ures, as liketh you be&s;t.</P>
</DIV2>
<TRAILER>FINIS.</TRAILER>
</DIV1>
</BODY>
</TEXT>
</EEBO>
</ETS>
