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author: Roger Frank <rfrank@pglaf.org> 2025-10-15 01:19:41 -0700
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+<body>
+<h1 class="pg">The Project Gutenberg eBook, The Theory and Practice of Perspective, by
+George Adolphus Storey</h1>
+<pre>
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at <a href = "http://www.gutenberg.org">www.gutenberg.org</a></pre>
+<p>Title: The Theory and Practice of Perspective</p>
+<p>Author: George Adolphus Storey</p>
+<p>Release Date: December 22, 2006  [eBook #20165]</p>
+<p>Language: English</p>
+<p>Character set encoding: ISO-8859-1</p>
+<p>***START OF THE PROJECT GUTENBERG EBOOK THE THEORY AND PRACTICE OF PERSPECTIVE***</p>
+<br><br><center><h3 class="pg">E-text prepared by Louise Hope, Suzanne Lybarger, Jonathan Ingram,<br>
+and the Project Gutenberg Online Distributed Proofreading Team<br>
+(http://www.pgdp.net/c/)</h3></center><br><br>
+<p>&nbsp;</p>
+<div class = "mynote">
+
+<p>Lines in the sample drawings are not always parallel. In some cases
+this may be an artifact of the scanning process, but more often the
+pictures were not positioned evenly in the original book. Page numbers
+shown in brackets [&nbsp;] held illustrations without text. They will
+sometimes be out of sequence with adjoining page numbers.</p>
+
+<p>A few typographical errors have been corrected. They have been
+marked in the text with <ins class = "correction" title =
+"like this">mouse-hover popups</ins>.</p>
+
+</div>
+<p>&nbsp;</p>
+<hr class="full" noshade>
+
+<!--png 001-->
+
+
+<p>&nbsp;<br>&nbsp;</p>
+
+<h5>HENRY FROWDE, M.A.</h5>
+<h6>PUBLISHER TO THE UNIVERSITY OF OXFORD<br>
+LONDON, EDINBURGH, NEW YORK<br>
+TORONTO AND MELBOURNE</h6>
+
+<p>&nbsp;<br>&nbsp;</p>
+
+<!--png 002-->
+<h4>THE</h4>
+
+<h2>THEORY AND PRACTICE<br>
+OF PERSPECTIVE</h2>
+
+<p>&nbsp;</p>
+
+<h6>BY</h6>
+
+<h4>G. A. STOREY, A.R.A.</h4>
+
+<h6>TEACHER OF PERSPECTIVE AT THE ROYAL ACADEMY</h6>
+
+<p>&nbsp;</p>
+
+<p class = "illustration">
+<img src = "images/titlepage.png" width = "128" height = "109"
+alt = "drawing" title = "drawing">
+<br>
+<span class = "smallroman">&lsquo;QU� FIT?&rsquo;</span>
+</p>
+
+<p>&nbsp;</p>
+
+<h5>OXFORD<br>
+AT THE CLARENDON PRESS</h5>
+<h6>1910</h6>
+
+<p>&nbsp;<br>&nbsp;</p>
+
+<!--png 003-->
+<h5>OXFORD</h5>
+<h6>PRINTED AT THE CLARENDON PRESS<br>
+BY HORACE HART, M.A.<br>
+PRINTER TO THE UNIVERSITY</h6>
+
+<p>&nbsp;<br>&nbsp;</p>
+
+
+<hr class = "mid">
+
+<span class = "pagenum">iii</span>
+<a name = "pageiii" id = "pageiii"> </a>
+<!--png 004-->
+
+<div class = "sans">
+<h5>DEDICATED</h5>
+
+<h6>TO</h6>
+
+<h4>SIR EDWARD J. POYNTER</h4>
+
+<h5>BARONET</h5>
+
+<h5>PRESIDENT OF THE ROYAL ACADEMY</h5>
+
+<h5>IN TOKEN OF FRIENDSHIP</h5>
+
+<h5>AND REGARD</h5>
+</div>
+
+<hr class = "mid">
+
+<!--png 005-->
+
+<span class = "pagenum">v</span>
+<a name = "pagev" id = "pagev"> </a>
+<!--png 006-->
+<h4 class = "chapter">PREFACE</h4>
+
+
+<p><span class = "smallcaps">It</span> is much easier to understand and
+remember a thing when a reason is given for it, than when we are merely
+shown how to do it without being told why it is so done; for in the
+latter case, instead of being assisted by reason, our real help in all
+study, we have to rely upon memory or our power of imitation, and to do
+simply as we are told without thinking about it. The consequence is that
+at the very first difficulty we are left to flounder about in the dark,
+or to remain inactive till the master comes to our assistance.</p>
+
+<p>Now in this book it is proposed to enlist the reasoning faculty from
+the very first: to let one problem grow out of another and to be
+dependent on the foregoing, as in geometry, and so to explain each thing
+we do that there shall be no doubt in the mind as to the correctness of
+the proceeding. The student will thus gain the power of finding out any
+new problem for himself, and will therefore acquire a true knowledge of
+perspective.</p>
+
+
+<!--png 007-->
+
+
+<span class = "pagenum">vii</span>
+<a name = "pagevii" id = "pagevii"> </a>
+<!--png 008-->
+<h4 class = "chapter">CONTENTS</h4>
+
+<table class = "toc" summary = "table of contents">
+<tr>
+<td class = "center head" colspan = "3">BOOK I</td>
+</tr>
+<tr>
+<td></td><td></td>
+<td class = "number smallcaps">page</td>
+</tr>
+<tr>
+<td class = "smallcaps" colspan = "2"><p><a href = "#necessity">
+The Necessity of the Study of Perspective To Painters, Sculptors, and
+Architects</a></p></td>
+<td class = "number">1</td>
+</tr>
+<tr>
+<td class = "smallcaps" colspan = "2"><p><a href = "#what_is">
+What Is Perspective?</a></p></td>
+<td class = "number">6</td>
+</tr>
+<tr>
+<td class = "smallcaps" colspan = "2"><p><a href = "#theory">
+The Theory of Perspective:</a></p></td>
+<td></td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapI">
+I.</a></td>
+<td>Definitions</td>
+<td class = "number">13</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapII">
+II.</a></td>
+<td><p>
+The Point of Sight, the Horizon, and the Point of Distance.</p></td>
+<td class = "number">15</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapIII">
+III.</a></td>
+<td>Point of Distance</td>
+<td class = "number">16</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapIV">
+IV.</a></td>
+<td><p>
+Perspective of a Point, Visual Rays, &amp;c.</p></td>
+<td class = "number">20</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapV">
+V.</a></td>
+<td>Trace and Projection</td>
+<td class = "number">21</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapVI">
+VI.</a></td>
+<td>Scientific Definition of Perspective</td>
+<td class = "number">22</td>
+</tr>
+<tr>
+<td class = "item smallcaps"><a href = "#rules">
+Rules:</a></td>
+<td></td><td></td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapVII">
+VII.</a></td>
+<td>The Rules and Conditions of Perspective</td>
+<td class = "number">24</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapVIII">
+VIII.</a></td>
+<td><p>
+A Table or Index of the Rules of Perspective</p></td>
+<td class = "number">40</td>
+</tr>
+<tr>
+<td class = "center head" colspan = "3">BOOK II</td>
+</tr>
+<tr>
+<td class = "smallcaps" colspan = "2"><p><a href = "#practice">
+The Practice of Perspective:</a></p></td>
+<td></td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapIX">
+IX.</a></td>
+<td>The Square in Parallel Perspective</td>
+<td class = "number">42</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapX">
+X.</a></td>
+<td>The Diagonal</td>
+<td class = "number">43</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXI">
+XI.</a></td>
+<td>The Square</td>
+<td class = "number">43</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXII">
+XII.</a></td>
+<td><p>
+Geometrical and Perspective Figures Contrasted</p></td>
+<td class = "number">46</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXIII">
+XIII.</a></td>
+<td><p>
+Of Certain Terms made use of in Perspective</p></td>
+<td class = "number">48</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXIV">
+XIV.</a></td>
+<td><p>
+How to Measure Vanishing or Receding Lines</p></td>
+<td class = "number">49</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXV">
+XV.</a></td>
+<td>How to Place Squares in Given Positions</td>
+<td class = "number">50</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXVI">
+XVI.</a></td>
+<td>How to Draw Pavements, &amp;c.</td>
+<td class = "number">51</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXVII">
+XVII.</a></td>
+<td><p>
+Of Squares placed Vertically and at Different Heights, or the Cube in
+Parallel Perspective</p></td>
+<td class = "number">53</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXVIII">
+XVIII.</a></td>
+<td>The Transposed Distance</td>
+<td class = "number">53</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXIX">
+XIX.</a></td>
+<td><p>
+The Front View of the Square and of the Proportions of Figures at
+Different Heights</p></td>
+<td class = "number">54</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXX">
+XX.</a></td>
+<td><p>
+Of Pictures that are Painted according to the Position they are to
+Occupy</p></td>
+<td class = "number">59</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXI">
+XXI.</a></td>
+<td>Interiors</td>
+<td class = "number">62</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXII">
+XXII.</a></td>
+<td>The Square at an Angle of 45�</td>
+<td class = "number">64</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXIII">
+XXIII.</a></td>
+<td>The Cube at an Angle of 45�</td>
+<td class = "number">65</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXIV">
+XXIV.</a></td>
+<td><p>
+Pavements Drawn by Means of Squares at 45�</p></td>
+<td class = "number">66</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXV">
+XXV.</a></td>
+<td>The Perspective Vanishing Scale</td>
+<td class = "number">68</td>
+</tr>
+<tr>
+<td class = "item">
+<span class = "pagenum">viii</span>
+<a name = "pageviii" id = "pageviii"> </a>
+<!--png 009-->
+<a href = "#chapXXVI">
+XXVI.</a></td>
+<td><p>
+The Vanishing Scale can be Drawn to any Point on the Horizon</p></td>
+<td class = "number">69</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXVII">
+XXVII.</a></td>
+<td><p>
+Application of Vanishing Scales to Drawing Figures</p></td>
+<td class = "number">71</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXVIII">
+XXVIII.</a></td>
+<td><p>
+How to Determine the Heights of Figures on a Level Plane</p></td>
+<td class = "number">71</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXIX">
+XXIX.</a></td>
+<td>The Horizon above the Figures</td>
+<td class = "number">72</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXX">
+XXX.</a></td>
+<td>Landscape Perspective</td>
+<td class = "number">74</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXI">
+XXXI.</a></td>
+<td><p>
+Figures of Different Heights. The Chessboard</p></td>
+<td class = "number">74</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXII">
+XXXII.</a></td>
+<td><p>
+Application of the Vanishing Scale to Drawing Figures at an Angle when
+their Vanishing Points are Inaccessible or Outside the Picture</p></td>
+<td class = "number">77</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXIII">
+XXXIII.</a></td>
+<td><p>
+The Reduced Distance. How to Proceed when the Point of Distance is
+Inaccessible</p></td>
+<td class = "number">77</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXIV">
+XXXIV.</a></td>
+<td><p>
+How to Draw a Long Passage or Cloister by Means of the Reduced
+Distance</p></td>
+<td class = "number">78</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXV">
+XXXV.</a></td>
+<td><p>
+How to Form a Vanishing Scale that shall give the Height, Depth, and
+Distance of any Object in the Picture</p></td>
+<td class = "number">79</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXVI">
+XXXVI.</a></td>
+<td>Measuring Scale on Ground</td>
+<td class = "number">81</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXVII">
+XXXVII.</a></td>
+<td><p>
+Application of the Reduced Distance and the Vanishing Scale to Drawing a
+Lighthouse, &amp;c.</p></td>
+<td class = "number">84</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXVIII">
+XXXVIII.</a></td>
+<td><p>
+How to Measure Long Distances such as a Mile or Upwards</p></td>
+<td class = "number">85</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXXXIX">
+XXXIX.</a></td>
+<td><p>
+Further Illustration of Long Distances and Extended Views.</p></td>
+<td class = "number">87</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXL">
+XL.</a></td>
+<td><p>
+How to Ascertain the Relative Heights of Figures on an Inclined
+Plane</p></td>
+<td class = "number">88</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLI">
+XLI.</a></td>
+<td><p>
+How to Find the Distance of a Given Figure or Point from the Base
+Line</p></td>
+<td class = "number">89</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLII">
+XLII.</a></td>
+<td><p>
+How to Measure the Height of Figures on Uneven Ground</p></td>
+<td class = "number">90</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLIII">
+XLIII.</a></td>
+<td><p>
+Further Illustration of the Size of Figures at Different Distances and
+on Uneven Ground</p></td>
+<td class = "number">91</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLIV">
+XLIV.</a></td>
+<td>Figures on a Descending Plane</td>
+<td class = "number">92</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLV">
+XLV.</a></td>
+<td><p>
+Further Illustration of the Descending Plane</p></td>
+<td class = "number">95</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLVI">
+XLVI.</a></td>
+<td>Further Illustration of Uneven Ground</td>
+<td class = "number">95</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLVII">
+XLVII.</a></td>
+<td>The Picture Standing on the Ground</td>
+<td class = "number">96</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLVIII">
+XLVIII.</a></td>
+<td>The Picture on a Height</td>
+<td class = "number">97</td>
+</tr>
+<tr>
+<td class = "center head" colspan = "3">BOOK III</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXLIX">
+XLIX.</a></td>
+<td>Angular Perspective</td>
+<td class = "number">98</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapL">
+L.</a></td>
+<td><p>
+How to put a Given Point into Perspective</p></td>
+<td class = "number">99</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLI">
+LI.</a></td>
+<td><p>
+A Perspective Point being given, Find its Position on the Geometrical
+Plane</p></td>
+<td class = "number">100</td>
+</tr>
+<tr>
+<td class = "item">
+<span class = "pagenum">ix</span>
+<a name = "pageix" id = "pageix"> </a>
+<!--png 010-->
+<a href = "#chapLII">
+LII.</a></td>
+<td><p>
+How to put a Given Line into Perspective</p></td>
+<td class = "number">101</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLIII">
+LIII.</a></td>
+<td><p>
+To Find the Length of a Given Perspective Line</p></td>
+<td class = "number">102</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLIV">
+LIV.</a></td>
+<td><p>
+To Find these Points when the Distance-Point is Inaccessible</p></td>
+<td class = "number">103</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLV">
+LV.</a></td>
+<td><p>
+How to put a Given Triangle or other Rectilineal Figure into
+Perspective</p></td>
+<td class = "number">104</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLVI">
+LVI.</a></td>
+<td><p>
+How to put a Given Square into Angular Perspective</p></td>
+<td class = "number">105</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLVII">
+LVII.</a></td>
+<td>Of Measuring Points</td>
+<td class = "number">106</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLVIII">
+LVIII.</a></td>
+<td><p>
+How to Divide any Given Straight Line into Equal or Proportionate
+Parts</p></td>
+<td class = "number">107</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLIX">
+LIX.</a></td>
+<td><p>
+How to Divide a Diagonal Vanishing Line into any Number of Equal or
+Proportional Parts</p></td>
+<td class = "number">107</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLX">
+LX.</a></td>
+<td>Further Use of the Measuring Point O</td>
+<td class = "number">110</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXI">
+LXI.</a></td>
+<td>Further Use of the Measuring Point O</td>
+<td class = "number">110</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXII">
+LXII.</a></td>
+<td><p>
+Another Method of Angular Perspective, being that Adopted in our Art
+Schools</p></td>
+<td class = "number">112</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXIII">
+LXIII.</a></td>
+<td><p>
+Two Methods of Angular Perspective in one Figure</p></td>
+<td class = "number">115</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXIV">
+LXIV.</a></td>
+<td>To Draw a Cube, the Points being Given</td>
+<td class = "number">115</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXV">
+LXV.</a></td>
+<td><p>
+Amplification of the Cube Applied to Drawing a Cottage</p></td>
+<td class = "number">116</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXVI">
+LXVI.</a></td>
+<td>How to Draw an Interior at an Angle</td>
+<td class = "number">117</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXVII">
+LXVII.</a></td>
+<td><p>
+How to Correct Distorted Perspective by Doubling the Line of
+Distance</p></td>
+<td class = "number">118</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXVIII">
+LXVIII.</a></td>
+<td><p>
+How to Draw a Cube on a Given Square, using only One Vanishing
+Point</p></td>
+<td class = "number">119</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXIX">
+LXIX.</a></td>
+<td><p>
+A Courtyard or Cloister Drawn with One Vanishing Point</p></td>
+<td class = "number">120</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXX">
+LXX.</a></td>
+<td><p>
+How to Draw Lines which shall Meet at a Distant Point, by Means of
+Diagonals</p></td>
+<td class = "number">121</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXI">
+LXXI.</a></td>
+<td><p>
+How to Divide a Square Placed at an Angle into a Given Number of Small
+Squares</p></td>
+<td class = "number">122</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXII">
+LXXII.</a></td>
+<td><p>
+Further Example of how to Divide a Given Oblique Square into a Given
+Number of Equal Squares, say Twenty-five</p></td>
+<td class = "number">122</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXIII">
+LXXIII.</a></td>
+<td>Of Parallels and Diagonals</td>
+<td class = "number">124</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXIV">
+LXXIV.</a></td>
+<td><p>
+The Square, the Oblong, and their Diagonals</p></td>
+<td class = "number">125</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXV">
+LXXV.</a></td>
+<td><p>
+Showing the Use of the Square and Diagonals in Drawing Doorways,
+Windows, and other Architectural Features</p></td>
+<td class = "number">126</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXVI">
+LXXVI.</a></td>
+<td>How to Measure Depths by Diagonals</td>
+<td class = "number">127</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXVII">
+LXXVII.</a></td>
+<td><p>
+How to Measure Distances by the Square and Diagonal</p></td>
+<td class = "number">128</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXVIII">
+LXXVIII.</a></td>
+<td><p>
+How by Means of the Square and Diagonal we can Determine the Position of
+Points in Space</p></td>
+<td class = "number">129</td>
+</tr>
+<tr>
+<td class = "item">
+<span class = "pagenum">x</span>
+<a name = "pagex" id = "pagex"> </a>
+<!--png 011-->
+<a href = "#chapLXXIX">
+LXXIX.</a></td>
+<td><p>
+Perspective of a Point Placed in any Position within the Square</p></td>
+<td class = "number">131</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXX">
+LXXX.</a></td>
+<td><p>
+Perspective of a Square Placed at an Angle. New Method</p></td>
+<td class = "number">133</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXI">
+LXXXI.</a></td>
+<td><p>
+On a Given Line Placed at an Angle to the Base Draw a Square in Angular
+Perspective, the Point of Sight, and Distance, being given</p></td>
+<td class = "number">134</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXII">
+LXXXII.</a></td>
+<td><p>
+How to Draw Solid Figures at any Angle by the New Method</p></td>
+<td class = "number">135</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXIII">
+LXXXIII.</a></td>
+<td>Points in Space</td>
+<td class = "number">137</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXIV">
+LXXXIV.</a></td>
+<td><p>
+The Square and Diagonal Applied to Cubes and Solids Drawn
+Therein</p></td>
+<td class = "number">138</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXV">
+LXXXV.</a></td>
+<td><p>
+To Draw an Oblique Square in Another Oblique Square without Using
+Vanishing-points</p></td>
+<td class = "number">139</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXVI">
+LXXXVI.</a></td>
+<td><p>
+Showing how a Pedestal can be Drawn by the New Method</p></td>
+<td class = "number">141</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXVII">
+LXXXVII.</a></td>
+<td>Scale on Each Side of the Picture</td>
+<td class = "number">143</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXVIII">
+LXXXVIII.</a></td>
+<td>The Circle</td>
+<td class = "number">145</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapLXXXIX">
+LXXXIX.</a></td>
+<td><p>
+The Circle in Perspective a True Ellipse</p></td>
+<td class = "number">145</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXC">
+XC.</a></td>
+<td>Further Illustration of the Ellipse</td>
+<td class = "number">146</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCI">
+XCI.</a></td>
+<td><p>
+How to Draw a Circle in Perspective Without a Geometrical Plan</p></td>
+<td class = "number">148</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCII">
+XCII.</a></td>
+<td><p>
+How to Draw a Circle in Angular Perspective</p></td>
+<td class = "number">151</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCIII">
+XCIII.</a></td>
+<td><p>
+How to Draw a Circle in Perspective more Correctly, by Using Sixteen
+Guiding Points</p></td>
+<td class = "number">152</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCIV">
+XCIV.</a></td>
+<td><p>
+How to Divide a Perspective Circle into any Number of Equal
+Parts</p></td>
+<td class = "number">153</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCV">
+XCV.</a></td>
+<td>How to Draw Concentric Circles</td>
+<td class = "number">154</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCVI">
+XCVI.</a></td>
+<td><p>
+The Angle of the Diameter of the Circle in Angular and Parallel
+Perspective</p></td>
+<td class = "number">156</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCVII">
+XCVII.</a></td>
+<td><p>
+How to Correct Disproportion in the Width of Columns</p></td>
+<td class = "number">157</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCVIII">
+XCVIII.</a></td>
+<td><p>
+How to Draw a Circle over a Circle or a Cylinder</p></td>
+<td class = "number">158</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapXCIX">
+XCIX.</a></td>
+<td>To Draw a Circle Below a Given Circle</td>
+<td class = "number">159</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapC">
+C.</a></td>
+<td>Application of Previous Problem</td>
+<td class = "number">160</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCI">
+CI.</a></td>
+<td>Doric Columns</td>
+<td class = "number">161</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCII">
+CII.</a></td>
+<td><p>
+To Draw Semicircles Standing upon a Circle at any Angle</p></td>
+<td class = "number">162</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCIII">
+CIII.</a></td>
+<td>A Dome Standing on a Cylinder</td>
+<td class = "number">163</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCIV">
+CIV.</a></td>
+<td>Section of a Dome or Niche</td>
+<td class = "number">164</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCV">
+CV.</a></td>
+<td>A Dome</td>
+<td class = "number">167</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCVI">
+CVI.</a></td>
+<td><p>
+How to Draw Columns Standing in a Circle</p></td>
+<td class = "number">169</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCVII">
+CVII.</a></td>
+<td>Columns and Capitals</td>
+<td class = "number">170</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCVIII">
+CVIII.</a></td>
+<td><p>
+Method of Perspective Employed by Architects</p></td>
+<td class = "number">170</td>
+</tr>
+<tr>
+<td class = "item">
+<span class = "pagenum">xi</span>
+<a name = "pagexi" id = "pagexi"> </a>
+<!--png 012-->
+<a href = "#chapCIX">
+CIX.</a></td>
+<td>The Octagon</td>
+<td class = "number">172</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCX">
+CX.</a></td>
+<td><p>
+How to Draw the Octagon in Angular Perspective</p></td>
+<td class = "number">173</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXI">
+CXI.</a></td>
+<td><p>
+How to Draw an Octagonal Figure in Angular Perspective</p></td>
+<td class = "number">174</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXII">
+CXII.</a></td>
+<td><p>
+How to Draw Concentric Octagons, with Illustration of a Well</p></td>
+<td class = "number">174</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXIII">
+CXIII.</a></td>
+<td><p>
+A Pavement Composed of Octagons and Small Squares</p></td>
+<td class = "number">176</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXIV">
+CXIV.</a></td>
+<td>The Hexagon</td>
+<td class = "number">177</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXV">
+CXV.</a></td>
+<td>A Pavement Composed of Hexagonal Tiles</td>
+<td class = "number">178</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXVI">
+CXVI.</a></td>
+<td><p>
+A Pavement of Hexagonal Tiles in Angular Perspective</p></td>
+<td class = "number">181</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXVII">
+CXVII.</a></td>
+<td>Further Illustration of the Hexagon</td>
+<td class = "number">182</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXVIII">
+CXVIII.</a></td>
+<td><p>
+Another View of the Hexagon in Angular Perspective</p></td>
+<td class = "number">183</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXIX">
+CXIX.</a></td>
+<td><p>
+Application of the Hexagon to Drawing a Kiosk</p></td>
+<td class = "number">185</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXX">
+CXX.</a></td>
+<td>The Pentagon</td>
+<td class = "number">186</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXI">
+CXXI.</a></td>
+<td>The Pyramid</td>
+<td class = "number">189</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXII">
+CXXII.</a></td>
+<td>The Great Pyramid</td>
+<td class = "number">191</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXIII">
+CXXIII.</a></td>
+<td>The Pyramid in Angular Perspective</td>
+<td class = "number">193</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXIV">
+CXXIV.</a></td>
+<td><p>
+To Divide the Sides of the Pyramid Horizontally</p></td>
+<td class = "number">193</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXV">
+CXXV.</a></td>
+<td>Of Roofs</td>
+<td class = "number">195</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXVI">
+CXXVI.</a></td>
+<td>Of Arches, Arcades, Bridges, &amp;c.</td>
+<td class = "number">198</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXVII">
+CXXVII.</a></td>
+<td><p>
+Outline of an Arcade with Semicircular Arches</p></td>
+<td class = "number">200</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXVIII">
+CXXVIII.</a></td>
+<td><p>
+Semicircular Arches on a Retreating Plane</p></td>
+<td class = "number">201</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXIX">
+CXXIX.</a></td>
+<td>An Arcade in Angular Perspective</td>
+<td class = "number">202</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXX">
+CXXX.</a></td>
+<td>A Vaulted Ceiling</td>
+<td class = "number">203</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXI">
+CXXXI.</a></td>
+<td>A Cloister, from a Photograph</td>
+<td class = "number">206</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXII">
+CXXXII.</a></td>
+<td>The Low or Elliptical Arch</td>
+<td class = "number">207</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXIII">
+CXXXIII.</a></td>
+<td>Opening or Arched Window in a Vault</td>
+<td class = "number">208</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXIV">
+CXXXIV.</a></td>
+<td>Stairs, Steps, &amp;c.</td>
+<td class = "number">209</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXV">
+CXXXV.</a></td>
+<td>Steps, Front View</td>
+<td class = "number">210</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXVI">
+CXXXVI.</a></td>
+<td>Square Steps</td>
+<td class = "number">211</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXVII">
+CXXXVII.</a></td>
+<td><p>
+To Divide an Inclined Plane into Equal Parts&mdash;such as a Ladder
+Placed against a Wall</p></td>
+<td class = "number">212</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXVIII">
+CXXXVIII.</a></td>
+<td>Steps and the Inclined Plane</td>
+<td class = "number">213</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXXXIX">
+CXXXIX.</a></td>
+<td>Steps in Angular Perspective</td>
+<td class = "number">214</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXL">
+CXL.</a></td>
+<td>A Step Ladder at an Angle</td>
+<td class = "number">216</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLI">
+CXLI.</a></td>
+<td>Square Steps Placed over each other</td>
+<td class = "number">217</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLII">
+CXLII.</a></td>
+<td><p>
+Steps and a Double Cross Drawn by Means of Diagonals and one Vanishing
+Point</p></td>
+<td class = "number">218</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLIII">
+CXLIII.</a></td>
+<td>A Staircase Leading to a Gallery</td>
+<td class = "number">221</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLIV">
+CXLIV.</a></td>
+<td>Winding Stairs in a Square Shaft</td>
+<td class = "number">222</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLV">
+CXLV.</a></td>
+<td>Winding Stairs in a Cylindrical Shaft</td>
+<td class = "number">225</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLVI">
+CXLVI.</a></td>
+<td>Of the Cylindrical Picture or Diorama</td>
+<td class = "number">227</td>
+</tr>
+<tr>
+<td>
+<span class = "pagenum">xii</span>
+<a name = "pagexii" id = "pagexii"> </a>
+<!--png 013-->
+</td>
+<td class = "center head">BOOK IV</td>
+<td></td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLVII">
+CXLVII.</a></td>
+<td>The Perspective of Cast Shadows</td>
+<td class = "number">229</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLVIII">
+CXLVIII.</a></td>
+<td>The Two Kinds of Shadows</td>
+<td class = "number">230</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCXLIX">
+CXLIX.</a></td>
+<td>Shadows Cast by the Sun</td>
+<td class = "number">232</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCL">
+CL.</a></td>
+<td><p>
+The Sun in the Same Plane as the Picture</p></td>
+<td class = "number">233</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLI">
+CLI.</a></td>
+<td>The Sun Behind the Picture</td>
+<td class = "number">234</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLII">
+CLII.</a></td>
+<td><p>
+Sun Behind the Picture, Shadows Thrown on a Wall</p></td>
+<td class = "number">238</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLIII">
+CLIII.</a></td>
+<td><p>
+Sun Behind the Picture Throwing Shadow on an Inclined Plane</p></td>
+<td class = "number">240</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLIV">
+CLIV.</a></td>
+<td>The Sun in Front of the Picture</td>
+<td class = "number">241</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLV">
+CLV.</a></td>
+<td>The Shadow of an Inclined Plane</td>
+<td class = "number">244</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLVI">
+CLVI.</a></td>
+<td>Shadow on a Roof or Inclined Plane</td>
+<td class = "number">245</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLVII">
+CLVII.</a></td>
+<td><p>
+To Find the Shadow of a Projection or Balcony on a Wall</p></td>
+<td class = "number">246</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLVIII">
+CLVIII.</a></td>
+<td><p>
+Shadow on a Retreating Wall, Sun in Front</p></td>
+<td class = "number">247</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLIX">
+CLIX.</a></td>
+<td>Shadow of an Arch, Sun in Front</td>
+<td class = "number">249</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLX">
+CLX.</a></td>
+<td>Shadow in a Niche or Recess</td>
+<td class = "number">250</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXI">
+CLXI.</a></td>
+<td>Shadow in an Arched Doorway</td>
+<td class = "number">251</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXII">
+CLXII.</a></td>
+<td>Shadows Produced by Artificial Light</td>
+<td class = "number">252</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXIII">
+CLXIII.</a></td>
+<td><p>
+Some Observations on Real Light and Shade</p></td>
+<td class = "number">253</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXIV">
+CLXIV.</a></td>
+<td>Reflection</td>
+<td class = "number">257</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXV">
+CLXV.</a></td>
+<td>Angles of Reflection</td>
+<td class = "number">259</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXVI">
+CLXVI.</a></td>
+<td><p>
+Reflections of Objects at Different Distances</p></td>
+<td class = "number">260</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXVII">
+CLXVII.</a></td>
+<td>Reflection in a Looking-glass</td>
+<td class = "number">262</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXVIII">
+CLXVIII.</a></td>
+<td>The Mirror at an Angle</td>
+<td class = "number">264</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXIX">
+CLXIX.</a></td>
+<td><p>
+The Upright Mirror at an Angle of 45� to the Wall</p></td>
+<td class = "number">266</td>
+</tr>
+<tr>
+<td class = "item"><a href = "#chapCLXX">
+CLXX.</a></td>
+<td>Mental Perspective</td>
+<td class = "number">269</td>
+</tr>
+<tr>
+<td></td>
+<td><a href = "#index"><i>Index</i></a></td>
+<td class = "number">270</td>
+</tr>
+</table>
+
+
+<span class = "pagenum">1</span>
+<a name = "page1" id = "page1"> </a>
+<!--png 014-->
+<h3 class = "chapter">BOOK FIRST</h3>
+
+<h4><a name = "necessity" id = "necessity">THE NECESSITY OF THE STUDY OF
+PERSPECTIVE TO PAINTERS, SCULPTORS, AND ARCHITECTS</a></h4>
+
+
+<p><span class = "smallcaps">Leonardo da Vinci</span> tells us in his
+celebrated <i>Treatise on Painting</i> that the young artist should
+first of all learn perspective, that is to say, he should first of all
+learn that he has to depict on a flat surface objects which are in
+relief or distant one from the other; for this is the simple art of
+painting. Objects appear smaller at a distance than near to us, so by
+drawing them thus we give depth to our canvas. The outline of a ball is
+a mere flat circle, but with proper shading we make it appear round, and
+this is the perspective of light and shade.</p>
+
+<p>&lsquo;The next thing to be considered is the effect of the
+atmosphere and light. If two figures are in the same coloured dress, and
+are standing one behind the other, then they should be of slightly
+different tone, so as to separate them. And in like manner, according to
+the distance of the mountains in a landscape and the greater or less
+density of the air, so do we depict space between them, not only making
+them smaller in outline, but less distinct.&rsquo;<a class = "tag" name
+= "tag1" id = "tag1" href = "#note1">1</a></p>
+
+<p>Sir Edwin Landseer used to say that in looking at a figure in a
+picture he liked to feel that he could walk round it, and this exactly
+expresses the impression that the true art of painting should make upon
+the spectator.</p>
+
+<p>There is another observation of Leonardo&rsquo;s that it is well I
+should here transcribe; he says: &lsquo;Many are desirous of learning to
+draw, and are very fond of it, who are notwithstanding void of a proper
+disposition for it. This may be known by their want of perseverance;
+like boys who draw everything in a hurry, never finishing or
+shadowing.&rsquo; This shows they do not care for their work, and all
+instruction is thrown away upon them. At the present time there is too
+much of this &lsquo;everything in a hurry&rsquo;,
+<span class = "pagenum">2</span>
+<a name = "page2" id = "page2"> </a>
+<!--png 015-->
+and beginning in this way leads only to failure and disappointment.
+These observations apply equally to perspective as to drawing and
+painting.</p>
+
+<p>Unfortunately, this study is too often neglected by our painters,
+some of them even complacently confessing their ignorance of it; while
+the ordinary student either turns from it with distaste, or only endures
+going through it with a view to passing an examination, little thinking
+of what value it will be to him in working out his pictures. Whether the
+manner of teaching perspective is the cause of this dislike for it,
+I&nbsp;cannot say; but certainly most of our English books on the
+subject are anything but attractive.</p>
+
+<p>All the great masters of painting have also been masters of
+perspective, for they knew that without it, it would be impossible to
+carry out their grand compositions. In many cases they were even
+inspired by it in choosing their subjects. When one looks at those sunny
+interiors, those corridors and courtyards by De Hooghe, with their
+figures far off and near, one feels that their charm consists greatly in
+their perspective, as well as in their light and tone and colour. Or if
+we study those Venetian masterpieces by Paul Veronese, Titian,
+Tintoretto, and others, we become convinced that it was through their
+knowledge of perspective that they gave such space and grandeur to their
+canvases.</p>
+
+<p>I need not name all the great artists who have shown their interest
+and delight in this study, both by writing about it and practising it,
+such as Albert D�rer and others, but I cannot leave out our own Turner,
+who was one of the greatest masters in this respect that ever lived;
+though in his case we can only judge of the results of his knowledge as
+shown in his pictures, for although he was Professor of Perspective at
+the Royal Academy in 1807&mdash;over a hundred years ago&mdash;and took
+great pains with the diagrams he prepared to illustrate his lectures,
+they seemed to the students to be full of confusion and obscurity; nor
+am I aware that any record of them remains, although they must have
+contained some valuable teaching, had their author possessed the art of
+conveying&nbsp;it.</p>
+
+<p>However, we are here chiefly concerned with the necessity of this
+study, and of the necessity of starting our work with&nbsp;it.</p>
+
+<p><span class = "pagenum">3</span>
+<a name = "page3" id = "page3"> </a>
+<!--png 016-->
+Before undertaking a large composition of figures, such as the
+&lsquo;Wedding-feast at Cana&rsquo;, by Paul Veronese, or &lsquo;The
+School of Athens&rsquo;, by Raphael, the artist should set out his
+floors, his walls, his colonnades, his balconies, his steps, &amp;c., so
+that he may know where to place his personages, and to measure their
+different sizes according to their distances; indeed, he must make his
+stage and his scenery before he introduces his actors. He can then
+proceed with his composition, arrange his groups and the accessories
+with ease, and above all with correctness. But I have noticed that some
+of our cleverest painters will arrange their figures to please the eye,
+and when fairly advanced with their work will call in an expert, to (as
+they call it) put in their perspective for them, but as it does not form
+part of their original composition, it involves all sorts of
+difficulties and vexatious alterings and rubbings out, and even then is
+not always satisfactory. For the expert may not be an artist, nor in
+sympathy with the picture, hence there will be a want of unity in it;
+whereas the whole thing, to be in harmony, should be the conception of
+one mind, and the perspective as much a part of the composition as the
+figures.</p>
+
+<p>If a ceiling has to be painted with figures floating or flying in the
+air, or sitting high above us, then our perspective must take a
+different form, and the point of sight will be above our heads instead
+of on the horizon; nor can these difficulties be overcome without an
+adequate knowledge of the science, which will enable us to work out for
+ourselves any new problems of this kind that we may have to solve.</p>
+
+<p>Then again, with a view to giving different effects or impressions in
+this decorative work, we must know where to place the horizon and the
+points of sight, for several of the latter are sometimes required when
+dealing with large surfaces such as the painting of walls, or stage
+scenery, or panoramas depicted on a cylindrical canvas and viewed from
+the centre thereof, where a fresh point of sight is required at every
+twelve or sixteen feet.</p>
+
+<p>Without a true knowledge of perspective, none of these things can be
+done. The artist should study them in the great compositions of the
+masters, by analysing their pictures and seeing
+<span class = "pagenum">4</span>
+<a name = "page4" id = "page4"> </a>
+<!--png 017-->
+how and for what reasons they applied their knowledge. Rubens put low
+horizons to most of his large figure-subjects, as in &lsquo;The Descent
+from the Cross&rsquo;, which not only gave grandeur to his designs, but,
+seeing they were to be placed above the eye, gave a more natural
+appearance to his figures. The Venetians often put the horizon almost on
+a level with the base of the picture or edge of the frame, and sometimes
+even below it; as in &lsquo;The Family of Darius at the Feet of
+Alexander&rsquo;, by Paul Veronese, and &lsquo;The Origin of the
+&ldquo;Via Lactea&rdquo;&rsquo;, by Tintoretto, both in our National
+Gallery. But in order to do all these things, the artist in designing
+his work must have the knowledge of perspective at his fingers' ends,
+and only the details, which are often tedious, should he leave to an
+assistant to work out for him.</p>
+
+<p>We must remember that the line of the horizon should be as nearly as
+possible on a level with the eye, as it is in nature; and yet one of the
+commonest mistakes in our exhibitions is the bad placing of this line.
+We see dozens of examples of it, where in full-length portraits and
+other large pictures intended to be seen from below, the horizon is
+placed high up in the canvas instead of low down; the consequence is
+that compositions so treated not only lose in grandeur and truth, but
+appear to be toppling over, or give the impression of smallness rather
+than bigness. Indeed, they look like small pictures enlarged, which is a
+very different thing from a large design. So that, in order to see them
+properly, we should mount a ladder to get upon a level with their
+horizon line (see <a href = "#fig66">Fig. 66</a>, double-page
+illustration).</p>
+
+<p>We have here spoken in a general way of the importance of this study
+to painters, but we shall see that it is of almost equal importance to
+the sculptor and the architect.</p>
+
+<p>A sculptor student at the Academy, who was making his drawings rather
+carelessly, asked me of what use perspective was to a sculptor.
+&lsquo;In the first place,&rsquo; I&nbsp;said, &lsquo;to reason out
+apparently difficult problems, and to find how easy they become, will
+improve your mind; and in the second, if you have to do monumental work,
+it will teach you the exact size to make your figures according to the
+height they are to be placed, and also the boldness with which they
+should be treated to give them their full effect.&rsquo;
+<span class = "pagenum">5</span>
+<a name = "page5" id = "page5"> </a>
+<!--png 018-->
+He at once acknowledged that I was right, proved himself an efficient
+pupil, and took much interest in his work.</p>
+
+<p>I cannot help thinking that the reason our public monuments so often
+fail to impress us with any sense of grandeur is in a great measure
+owing to the neglect of the scientific study of perspective. As an
+illustration of what I mean, let the student look at a good engraving or
+photograph of the Arch of Constantine at Rome, or the Tombs of the
+Medici, by Michelangelo, in the sacristy of San Lorenzo at Florence. And
+then, for an example of a mistake in the placing of a colossal figure,
+let him turn to the Tomb of Julius II in San Pietro in Vinculis, Rome,
+and he will see that the figure of Moses, so grand in itself, not only
+loses much of its dignity by being placed on the ground instead of in
+the niche above it, but throws all the other figures out of proportion
+or harmony, and was quite contrary to Michelangelo&rsquo;s intention.
+Indeed, this tomb, which was to have been the finest thing of its kind
+ever done, was really the tragedy of the great sculptor&rsquo;s
+life.</p>
+
+<p>The same remarks apply in a great measure to the architect as to the
+sculptor. The old builders knew the value of a knowledge of perspective,
+and, as in the case of Serlio, Vignola, and others, prefaced their
+treatises on architecture with chapters on geometry and perspective. For
+it showed them how to give proper proportions to their buildings and the
+details thereof; how to give height and importance both to the interior
+and exterior; also to give the right sizes of windows, doorways,
+columns, vaults, and other parts, and the various heights they should
+make their towers, walls, arches, roofs, and so forth. One of the most
+beautiful examples of the application of this knowledge to architecture
+is the Campanile of the Cathedral, at Florence, built by Giotto and
+Taddeo Gaddi, who were painters as well as architects. Here it will be
+seen that the height of the windows is increased as they are placed
+higher up in the building, and the top windows or openings into the
+belfry are about six times the size of those in the lower story.</p>
+
+
+
+
+<span class = "pagenum">6</span>
+<a name = "page6" id = "page6"> </a>
+<!--png 019-->
+<h4 class = "chapter"><a name = "what_is" id = "what_is">
+WHAT IS PERSPECTIVE?</a></h4>
+
+<table class = "float left" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig1" id = "fig1"> </a>
+<img src = "images/fig1.png" width = "342" height = "300"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 1.</td>
+</tr>
+</table>
+
+<p><span class = "smallcaps">Perspective</span> is a subtle form of
+geometry; it represents figures and objects not as they are but as we
+see them in space, whereas geometry represents figures not as we see
+them but as they are. When we have a front view of a figure such as a
+square, its perspective and geometrical appearance is the same, and we
+see it as it really is, that is, with all its sides equal and all its
+angles right angles, the perspective only varying in size according to
+the distance we are from it; but if we place that square flat on the
+table and look at it sideways or at an angle, then we become conscious
+of certain changes in its form&mdash;the side farthest from us appears
+shorter than that near to us,
+<span class = "pagenum">7</span>
+<a name = "page7" id = "page7"> </a>
+<!--png 020-->
+and all the angles are different. Thus <span class =
+"smallroman">A</span> (Fig.&nbsp;2) is a geometrical square and <span
+class = "smallroman">B</span> is the same square seen in
+perspective.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig2" id = "fig2"> </a>
+<img src = "images/fig2a.png" width = "83" height = "68"
+alt = "figure" title = "figure">
+</td>
+<td>
+<img src = "images/fig2b.png" width = "76" height = "41"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps" colspan = "2">
+Fig. 2.</td>
+</tr>
+</table>
+
+<p>The science of perspective gives the dimensions of objects seen in
+space as they appear to the eye of the spectator, just as a perfect
+tracing of those objects on a sheet of glass placed vertically between
+him and them would do; indeed its very name is derived from
+<i>perspicere</i>, to see through. But as no tracing done by hand could
+possibly be mathematically correct, the mathematician teaches us how by
+certain points and measurements we may yet give a perfect image of them.
+These images are called projections, but the artist calls them pictures.
+In this sketch <span class = "smallroman">K</span> is the vertical
+transparent plane or picture, <span class = "smallroman">O</span> is a
+cube placed on one side of it. The young student is the spectator on the
+other side of it, the dotted lines drawn from the corners of the cube to
+the eye of the spectator are the visual rays, and the points on the
+transparent picture plane where these visual rays pass through it
+indicate the perspective position
+<span class = "pagenum">8</span>
+<a name = "page8" id = "page8"> </a>
+<!--png 021-->
+of those points on the picture. To find these points is the main object
+or duty of linear perspective.</p>
+
+<p class = "illustration">
+<a name = "fig3" id = "fig3"> </a>
+<img src = "images/fig3.png" width = "336" height = "159"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 3.</p>
+
+<p>Perspective up to a certain point is a pure science, not depending
+upon the accidents of vision, but upon the exact laws of reasoning. Nor
+is it to be considered as only pertaining to the craft of the painter
+and draughtsman. It has an intimate connexion with our mental
+perceptions and with the ideas that are impressed upon the brain by the
+appearance of all that surrounds us. If we saw everything as depicted by
+plane geometry, that is, as a map, we should have no difference of view,
+no variety of ideas, and we should live in a world of unbearable
+monotony; but as we see everything in perspective, which is infinite in
+its variety of aspect, our minds are subjected to countless phases of
+thought, making the world around us constantly interesting, so it is
+devised that we shall see the infinite wherever we turn, and marvel at
+it, and delight in it, although perhaps in many cases unconsciously.</p>
+
+<p>In perspective, as in geometry, we deal with parallels, squares,
+triangles, cubes, circles, &amp;c.; but in perspective the same figure
+takes an endless variety of forms, whereas in geometry it has but one.
+Here are three equal geometrical squares: they are all alike. Here are
+three equal perspective squares, but all varied
+<span class = "pagenum">9</span>
+<a name = "page9" id = "page9"> </a>
+<!--png 022-->
+in form; and the same figure changes in aspect as often as we view it
+from a different position. A&nbsp;walk round the dining-room table will
+exemplify this.</p>
+
+<p class = "illustration">
+<a name = "fig4" id = "fig4"> </a>
+<img src = "images/fig4.png" width = "229" height = "55"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 4.</p>
+
+<p class = "illustration">
+<a name = "fig5" id = "fig5"> </a>
+<img src = "images/fig5.png" width = "225" height = "87"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 5.</p>
+
+<p>It is in proving that, notwithstanding this difference of appearance,
+the figures do represent the same form, that much of our work consists;
+and for those who care to exercise their reasoning powers it becomes not
+only a sure means of knowledge, but a study of the greatest
+interest.</p>
+
+<p>Perspective is said to have been formed into a science about the
+fifteenth century. Among the names mentioned by the unknown but pleasant
+author of <i>The Practice of Perspective</i>, written by a Jesuit of
+Paris in the eighteenth century, we find Albert D�rer, who has left us
+some rules and principles in the fourth book of his <i>Geometry</i>;
+Jean Cousin, who has an express treatise on the art wherein are many
+valuable things; also Vignola, who altered the plans of St.
+Peter&rsquo;s left by Michelangelo; Serlio, whose treatise is one of the
+best I have seen of these early writers; Du Cerceau, Serigati, Solomon
+de Cause, Marolois, Vredemont; Guidus Ubaldus, who first introduced
+foreshortening; the Sieur de Vaulizard, the Sieur Dufarges, Joshua
+Kirby, for whose <i>Method of Perspective made Easy</i> <ins class =
+"correction" title = "question mark in original">(?)</ins> Hogarth drew
+the well-known frontispiece; and lastly, the above-named <i>Practice of
+Perspective</i> by a Jesuit of Paris, which is very clear and excellent
+as far as it goes, and was the book used by Sir Joshua Reynolds.<a class
+= "tag" name = "tag2" id = "tag2" href = "#note2">2</a> But nearly all
+these authors treat chiefly of parallel perspective, which they do with
+clearness and simplicity, and also mathematically, as shown in the short
+treatise in Latin by Christian Wolff, but they scarcely touch upon the
+more difficult problems of angular and oblique perspective. Of modern
+books, those to which I am most indebted are the <i>Trait� Pratique de
+Perspective</i> of M.&nbsp;A. Cassagne (Paris, 1873), which is
+thoroughly artistic, and full of pictorial examples admirably done; and
+to M.&nbsp;Henriet&rsquo;s <i>Cours Rational de Dessin</i>. There are
+many other foreign books of excellence, notably M.&nbsp;Thibault's
+<i>Perspective</i>, and some German and Swiss books, and yet,
+notwithstanding this imposing array of authors, I&nbsp;venture to say
+that many new features and original
+<span class = "pagenum">10</span>
+<a name = "page10" id = "page10"> </a>
+<!--png 023-->
+problems are presented in this book, whilst the old ones are not
+neglected. As, for instance, How to draw figures at an angle without
+vanishing points (see p.&nbsp;141, <a href = "#fig162">Fig. 162</a>,
+&amp;c.), a&nbsp;new method of angular perspective which dispenses with
+the cumbersome setting out usually adopted, and enables us to draw
+figures at any angle without vanishing lines, &amp;c., and is almost, if
+not quite, as simple as parallel perspective (see p.&nbsp;133, <a href =
+"#fig150">Fig. 150</a>, &amp;c.). How to measure distances by the square
+and diagonal, and to draw interiors thereby (p.&nbsp;128, <a href =
+"#fig144">Fig. 144</a>). How to explain the theory of perspective by
+ocular demonstration, using a vertical sheet of glass with strings,
+placed on a drawing-board, which I have found of the greatest use (see
+p.&nbsp;29, <a href = "#fig29">Fig. 29</a>). Then again, I&nbsp;show how
+all our perspective can be done inside the picture; that we can measure
+any distance into the picture from a foot to a mile or twenty miles (see
+p.&nbsp;86, <a href = "#fig94">Fig. 94</a>); how we can draw the Great
+Pyramid, which stands on thirteen acres of ground, by putting it 1,600
+feet off (<a href = "#fig224">Fig. 224</a>), &amp;c., &amp;c. And while
+preserving the mathematical science, so that all our operations can be
+proved to be correct, my chief aim has been to make it easy of
+application to our work and consequently useful to the artist.</p>
+
+<p>The Egyptians do not appear to have made any use of linear
+perspective. Perhaps it was considered out of character with their
+particular kind of decoration, which is to be looked upon as picture
+writing rather than pictorial art; a&nbsp;table, for instance, would be
+represented like a ground-plan and the objects upon it in elevation or
+standing up. A&nbsp;row of chariots with their horses and drivers side
+by side were placed one over the other, and although the Egyptians had
+no doubt a reason for this kind of representation, for they were grand
+artists, it seems to us very primitive; and indeed quite young beginners
+who have never drawn from real objects have a tendency to do very much
+the same thing as this ancient people did, or even to emulate the
+mathematician and represent things not as they appear but as they are,
+and will make the top of a table an almost upright square and the
+objects upon it as if they would fall off.</p>
+
+<p>No doubt the Greeks had correct notions of perspective, for the
+paintings on vases, and at Pompeii and Herculaneum, which were either by
+Greek artists or copied from Greek pictures,
+<span class = "pagenum">11</span>
+<a name = "page11" id = "page11"> </a>
+<!--png 024-->
+show some knowledge, though not complete knowledge, of this science.
+Indeed, it is difficult to conceive of any great artist making his
+perspective very wrong, for if he can draw the human figure as the
+Greeks did, surely he can draw an angle.</p>
+
+<p>The Japanese, who are great observers of nature, seem to have got at
+their perspective by copying what they saw, and, although they are not
+quite correct in a few things, they convey the idea of distance and make
+their horizontal planes look level, which are two important things in
+perspective. Some of their landscapes are beautiful; their trees,
+flowers, and foliage exquisitely drawn and arranged with the greatest
+taste; whilst there is a character and go about their figures and birds,
+&amp;c., that can hardly be surpassed. All their pictures are lively and
+intelligent and appear to be executed with ease, which shows their
+authors to be complete masters of their craft.</p>
+
+<p>The same may be said of the Chinese, although their perspective is
+more decorative than true, and whilst their taste is exquisite their
+whole art is much more conventional and traditional, and does not remind
+us of nature like that of the Japanese.</p>
+
+<p>We may see defects in the perspective of the ancients, in the
+mediaeval painters, in the Japanese and Chinese, but are we always right
+ourselves? Even in celebrated pictures by old and modern masters there
+are occasionally errors that might easily have been avoided, if a ready
+means of settling the difficulty were at hand. We should endeavour then
+to make this study as simple, as easy, and as complete as possible, to
+show clear evidence of its correctness (according to its conditions),
+and at the same time to serve as a guide on any and all occasions that
+we may require&nbsp;it.</p>
+
+<p>To illustrate what is perspective, and as an experiment that any one
+can make, whether artist or not, let us stand at a window that looks out
+on to a courtyard or a street or a garden, &amp;c., and trace with a
+paint-brush charged with Indian ink or water-colour the outline of
+whatever view there happens to be outside, being careful to keep the eye
+always in the same place by means of a rest; when this is dry, place a
+piece of drawing-paper over it and trace through with a pencil. Now we
+will rub out the tracing on the glass, which is sure to be rather
+clumsy, and, fixing
+<span class = "pagenum">12</span>
+<a name = "page12" id = "page12"> </a>
+<!--png 025-->
+our paper down on a board, proceed to draw the scene before us, using
+the main lines of our tracing as our guiding lines.</p>
+
+<p>If we take pains over our work, we shall find that, without troubling
+ourselves much about rules, we have produced a perfect perspective of
+perhaps a very difficult subject. After practising for some little time
+in this way we shall get accustomed to what are called perspective
+deformations, and soon be able to dispense with the glass and the
+tracing altogether and to sketch straight from nature, taking little
+note of perspective beyond fixing the point of sight and the
+horizontal-line; in fact, doing what every artist does when he goes out
+sketching.</p>
+
+<p class = "illustration">
+<a name = "fig6" id = "fig6"> </a>
+<img src = "images/fig6.png" width = "340" height = "213"
+alt = "picture described in caption"
+title = "picture described in caption">
+</p>
+
+<p class = "caption left">
+<span class = "smallcaps">Fig. 6.</span> This is a much reduced
+reproduction of a drawing made on my studio window in this way some
+twenty years ago, when the builder started covering the fields at the
+back with rows and rows of houses.</p>
+
+
+
+
+<span class = "pagenum">13</span>
+<a name = "page13" id = "page13"> </a>
+<!--png 026-->
+<h4 class = "chapter"><a name = "theory" id = "theory">
+THE THEORY OF PERSPECTIVE</a></h4>
+
+<h5 class = "smallcaps">Definitions</h5>
+
+<h5><a name = "chapI" id = "chapI">I</a></h5>
+
+
+<p>Fig. 7. In this figure, <span class = "smallroman">AKB</span>
+represents the picture or transparent vertical plane through which the
+objects to be represented can be seen, or on which they can be traced,
+such as the cube <span class = "smallroman">C</span>.</p>
+
+<p class = "illustration">
+<a name = "fig7" id = "fig7"> </a>
+<img src = "images/fig7.png" width = "312" height = "208"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 7.</p>
+
+<p>The line <span class = "smallroman">HD</span> is the
+<b>Horizontal-line</b> or <b>Horizon</b>, the chief line in perspective,
+as upon it are placed the principal points to which our perspective
+lines are drawn. First, the <b>Point of Sight</b> and next <span class =
+"smallroman">D</span>, the <b>Point of Distance</b>. The chief vanishing
+points and measuring points are also placed on this line.</p>
+
+<p>Another important line is <span class = "smallroman">AB</span>, the
+<b>Base</b> or <b>Ground line</b>, as it is on this that we measure the
+width of any object to be represented, such as <i>ef</i>, the base of
+the square <i>efgh</i>, on which the cube <span class =
+"smallroman">C</span> is raised. <span class = "smallroman">E</span> is
+the position of the eye of the spectator, being drawn in perspective,
+and is called the <b>Station-point</b>.</p>
+
+<p>Note that the perspective of the board, and the line <span class =
+"smallroman">SE</span>, is not
+<span class = "pagenum">14</span>
+<a name = "page14" id = "page14"> </a>
+<!--png 027-->
+the same as that of the cube in the picture <span class =
+"smallroman">AKB</span>, and also that so much of the board which is
+behind the picture plane partially represents the
+<b>Perspective-plane</b>, supposed to be perfectly level and to extend
+from the base line to the horizon. Of this we shall speak further on. In
+nature it is not really level, but partakes in extended views of the
+rotundity of the earth, though in small areas such as ponds the
+roundness is infinitesimal.</p>
+
+<p class = "illustration">
+<a name = "fig8" id = "fig8"> </a>
+<img src = "images/fig8.png" width = "335" height = "206"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 8.</p>
+
+<p>Fig. 8. This is a side view of the previous figure, the picture plane
+<span class = "smallroman">K</span> being represented edgeways, and the
+line <span class = "smallroman">SE</span> its full length. It also shows
+the position of the eye in front of the point of sight <span class =
+"smallroman">S</span>. The horizontal-line <span class =
+"smallroman">HD</span> and the base or ground-line <span class =
+"smallroman">AB</span> are represented as receding from us, and in that
+case are called vanishing lines, a&nbsp;not quite satisfactory term.</p>
+
+<p>It is to be noted that the cube <span class = "smallroman">C</span>
+is placed close to the transparent picture plane, indeed touches it, and
+that the square <i>fj</i> faces the spectator <span class =
+"smallroman">E</span>, and although here drawn in perspective it appears
+to him as in the other figure. Also, it is at the same time a
+perspective and a geometrical figure, and can therefore be measured with
+the compasses. Or in other words, we can
+<span class = "pagenum">15</span>
+<a name = "page15" id = "page15"> </a>
+<!--png 028-->
+touch the square <i>fj</i>, because it is on the surface of the picture,
+but we cannot touch the square <i>ghmb</i> at the other end of the cube
+and can only measure it by the rules of perspective.</p>
+
+
+<h5 class = "section"><a name = "chapII" id = "chapII">
+II</a></h5>
+
+<h5 class = "smallcaps">The Point of Sight, the Horizon, and the Point
+of Distance</h5>
+
+
+<p>There are three things to be considered and understood before we can
+begin a perspective drawing. First, the position of the eye in front of
+the picture, which is called the <b>Station-point</b>, and of course is
+not in the picture itself, but its position is indicated by a point on
+the picture which is exactly opposite the eye of the spectator, and is
+called the <b>Point of Sight</b>, or <b>Principal Point</b>, or
+<b>Centre of Vision</b>, but we will keep to the first of these.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig9" id = "fig9"> </a>
+<img src = "images/fig9.png" width = "134" height = "85"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig10" id = "fig10"> </a>
+<img src = "images/fig10.png" width = "133" height = "72"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 9.</td>
+<td class = "caption smallcaps">
+Fig. 10.</td>
+</tr>
+</table>
+
+<p>If our picture plane is a sheet of glass, and is so placed that we
+can see the landscape behind it or a sea-view, we shall find that the
+distant line of the horizon passes through that point of sight, and we
+therefore draw a line on our picture which exactly corresponds with it,
+and which we call the <b>Horizontal-line</b> or <b>Horizon</b>.<a class
+= "tag" name = "tag3" id = "tag3" href = "#note3">3</a> The height of
+the horizon then depends entirely upon the position of the eye of the
+spectator: if he rises, so does the horizon; if he stoops or descends to
+lower ground, so does the horizon follow his movements. You may sit in a
+boat on a calm sea, and the horizon will be as low down as you are, or
+you may go to the top of a high cliff, and still the horizon will be on
+the same level as your eye.</p>
+
+<p><span class = "pagenum">16</span>
+<a name = "page16" id = "page16"> </a>
+<!--png 029-->
+This is an important line for the draughtsman to consider, for the
+effect of his picture greatly depends upon the position of the horizon.
+If you wish to give height and dignity to a mountain or a building, the
+horizon should be low down, so that these things may appear to tower
+above you. If you wish to show a wide expanse of landscape, then you
+must survey it from a height. In a composition of figures, you select
+your horizon according to the subject, and with a view to help the
+grouping. Again, in portraits and decorative work to be placed high up,
+a&nbsp;low horizon is desirable, but I have already spoken of this
+subject in the chapter on the necessity of the study of perspective.</p>
+
+
+<h5 class = "section"><a name = "chapIII" id = "chapIII">
+III</a></h5>
+
+<h5 class = "smallcaps">Point of Distance</h5>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig11" id = "fig11"> </a>
+<img src = "images/fig11.png" width = "312" height = "244"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 11.</td>
+</tr>
+</table>
+
+<p>Fig. 11. The distance of the spectator from the picture is of great
+importance; as the distortions and disproportions arising from too near
+a view are to be avoided, the object of drawing being to make things
+look natural; thus, the floor should look level, and not as if it were
+running up hill&mdash;the top of a table flat, and not on a slant, as if
+cups and what not, placed upon it, would fall off.</p>
+
+<p>In this figure we have a geometrical or ground plan of two squares at
+different distances from the picture, which is represented by the line
+<span class = "smallroman">KK</span>. The spectator is first at <span
+class = "smallroman">A</span>, the corner of the near square <span class
+= "smallroman">A</span><i>cd</i>. If from <span class =
+"smallroman">A</span> we draw a diagonal of that square and produce it
+to the line <span class = "smallroman">KK</span> (which may represent
+the horizontal-line in the picture), where it intersects that line at
+<span class = "smallroman">A�</span> marks the distance that the
+spectator is from the point of sight <span class =
+"smallroman">S</span>. For it will be seen that line <span class =
+"smallroman">SA</span> equals line <span class =
+"smallroman">SA�</span>. In like manner, if the spectator is at <span
+class = "smallroman">B</span>, his distance from the point <span class =
+"smallroman">S</span> is also found on the horizon by means of the
+diagonal <span class = "smallroman">BB�</span>, so that all lines or
+diagonals at 45� are drawn to the point of distance (see
+Rule&nbsp;6).</p>
+
+<p>Figs. 12 and 13. In these two figures the difference is shown between
+the effect of the short-distance point <span class =
+"smallroman">A�</span> and the long-distance point <span class =
+"smallroman">B�</span>; the first, <span class =
+"smallroman">A</span><i>cd</i>, does not appear to lie so flat on the
+ground as the second square, <span class =
+"smallroman">B</span><i>ef</i>.</p>
+
+<p>From this it will be seen how important it is to choose the
+<span class = "pagenum">17</span>
+<a name = "page17" id = "page17"> </a>
+<!--png 030-->
+right point of distance: if we take it too near the point of sight, as
+in Fig. 12, the square looks unnatural and distorted. This, I&nbsp;may
+note, is a common fault with photographs taken with a wide-angle lens,
+which throws everything out of proportion, and will make the east end of
+a church or a cathedral appear higher than the steeple or tower; but as
+soon as we make our
+<span class = "pagenum">18</span>
+<a name = "page18" id = "page18"> </a>
+<!--png 031-->
+line of distance sufficiently long, as at Fig. 13, objects take their
+right proportions and no distortion is noticeable.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig12" id = "fig12"> </a>
+<img src = "images/fig12.png" width = "150" height = "96"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig13" id = "fig13"> </a>
+<img src = "images/fig13.png" width = "255" height = "96"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 12.</td>
+<td class = "caption smallcaps">
+Fig. 13.</td>
+</tr>
+</table>
+
+<p>In some books on perspective we are told to make the angle of vision
+60�, so that the distance <span class = "smallroman">SD</span> (Fig. 14)
+is to be rather less than the length or height of the picture, as at
+<span class = "smallroman">A</span>. The French recommend an angle of
+28�, and to make the distance about double the length of the picture, as
+at <span class = "smallroman">B</span> (Fig. 15), which is far more
+agreeable. For we must remember that the distance-point is not only the
+point from which we are supposed to make our tracing on the vertical
+transparent plane, or a point transferred to the horizon to make our
+measurements by, but it is also the point in front of the canvas that we
+view the picture from, called the station-point. It is ridiculous, then,
+to have it so close that we must almost touch the canvas with our noses
+before we can see its perspective properly.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig14" id = "fig14"> </a>
+<img src = "images/fig14.png" width = "172" height = "110"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig15" id = "fig15"> </a>
+<img src = "images/fig15.png" width = "288" height = "98"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 14.</td>
+<td class = "caption smallcaps">
+Fig. 15.</td>
+</tr>
+</table>
+
+<p>Now a picture should look right from whatever distance we
+<span class = "pagenum">19</span>
+<a name = "page19" id = "page19"> </a>
+<!--png 032-->
+view it, even across the room or gallery, and of course in decorative
+work and in scene-painting a long distance is necessary.</p>
+
+<p>We need not, however, tie ourselves down to any hard and fast rule,
+but should choose our distance according to the impression of space we
+wish to convey: if we have to represent a domestic scene in a small
+room, as in many Dutch pictures, we must not make our distance-point too
+far off, as it would exaggerate the size of the room.</p>
+
+<p class = "illustration">
+<a name = "fig16" id = "fig16"> </a>
+<img src = "images/fig16.png" width = "337" height = "236"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption"><span class = "smallcaps">
+Fig. 16.</span> Cattle. By Paul Potter.</p>
+
+<p><span class = "pagenum">20</span>
+<a name = "page20" id = "page20"> </a>
+<!--png 033-->
+The height of the horizon is also an important consideration in the
+composition of a picture, and so also is the position of the point of
+sight, as we shall see farther&nbsp;on.</p>
+
+<p>In landscape and cattle pictures a low horizon often gives space and
+air, as in this sketch from a picture by Paul Potter&mdash;where the
+horizontal-line is placed at one quarter the height of the canvas.
+Indeed, a&nbsp;judicious use of the laws of perspective is a great aid
+to composition, and no picture ever looks right unless these laws are
+attended to. At the present time too little attention is paid to them;
+the consequence is that much of the art of the day reflects in a great
+measure the monotony of the snap-shot camera, with its everyday and
+wearisome commonplace.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapIV" id = "chapIV">
+IV</a></h5>
+
+<h5 class = "smallcaps">Perspective of a Point, Visual Rays,
+&amp;c.</h5>
+
+
+<p>We perceive objects by means of the visual rays, which are imaginary
+straight lines drawn from the eye to the various points of the thing we
+are looking at. As those rays proceed from the pupil of the eye, which
+is a circular opening, they form themselves into a cone called the
+<b>Optic Cone</b>, the base of which increases in proportion to its
+distance from the eye, so that the larger the view which we wish to take
+in, the farther must we be removed from it. The diameter of the base of
+this cone, with the visual rays drawn from each of its extremities to
+the eye, form the angle of vision, which is wider or narrower according
+to the distance of this diameter.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig17" id = "fig17"> </a>
+<img src = "images/fig17.png" width = "260" height = "139"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 17.</td>
+</tr>
+</table>
+<p>Now let us suppose a visual ray <span class = "smallroman">EA</span>
+to be directed to some small object on the floor, say the head of a
+nail, <span class = "smallroman">A</span> (Fig. 17). If we interpose
+between this nail and our eye a sheet of glass, <span class =
+"smallroman">K</span>, placed vertically on the floor, we continue to
+see the nail through the glass, and it is easily understood that its
+perspective appearance thereon is the point <i>a</i>, where the visual
+ray passes through it. If now we trace on the floor a line <span class =
+"smallroman">AB</span> from the nail to the spot <span class =
+"smallroman">B</span>, just under the eye, and from the point <i>o</i>,
+where this line passes through or under the glass, we raise a
+perpendicular <i>o</i><span class = "smallroman">S</span>, that
+perpendicular passes through the precise point that the visual ray
+<span class = "pagenum">21</span>
+<a name = "page21" id = "page21"> </a>
+<!--png 034-->
+passes through. The line <span class = "smallroman">AB</span> traced on
+the floor is the horizontal trace of the visual ray, and it will be seen
+that the point <i>a</i> is situated on the vertical raised from this
+horizontal trace.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapV" id = "chapV">
+V</a></h5>
+
+<h5 class = "smallcaps">Trace and Projection</h5>
+
+
+<p>If from any line <span class = "smallroman">A</span> or <span class =
+"smallroman">B</span> or <span class = "smallroman">C</span> (Fig. 18),
+&amp;c., we drop perpendiculars from different points of those lines on
+to a horizontal plane, the intersections of those verticals with the
+plane will be on a line called the horizontal trace or projection of the
+original line. We may liken these projections to sun-shadows when the
+sun is in the meridian, for it will be remarked that the trace does not
+represent the length of the original line, but only so much of it as
+would be embraced by the verticals dropped from each end of it, and
+although line <span class = "smallroman">A</span> is the same length as
+line <span class = "smallroman">B</span> its horizontal
+<span class = "pagenum">22</span>
+<a name = "page22" id = "page22"> </a>
+<!--png 035-->
+trace is longer than that of the other; that the projection of a curve
+(<span class = "smallroman">C</span>) in this upright position is a
+straight line, that of a horizontal line (<span class =
+"smallroman">D</span>) is equal to it, and the projection of a
+perpendicular or vertical (<span class = "smallroman">E</span>) is a
+point only. The projections of lines or points can likewise be shown on
+a vertical plane, but in that case we draw lines parallel to the
+horizontal plane, and by this means we can get the position of a point
+in space; and by the assistance of perspective, as will be shown farther
+on, we can carry out the most difficult propositions of descriptive
+geometry and of the geometry of planes and solids.</p>
+
+<p class = "illustration">
+<a name = "fig18" id = "fig18"> </a>
+<img src = "images/fig18.png" width = "326" height = "72"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 18.</p>
+
+<p>The position of a point in space is given by its projection on a
+vertical and a horizontal plane&mdash;</p>
+
+<p class = "illustration">
+<a name = "fig19" id = "fig19"> </a>
+<img src = "images/fig19.png" width = "250" height = "125"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 19.</p>
+
+<p>Thus <i>e�</i> is the projection of <span class =
+"smallroman">E</span> on the vertical plane <span class =
+"smallroman">K</span>, and <i>e��</i> is the projection of <span class =
+"smallroman">E</span> on the horizontal plane; <i>fe��</i> is the
+horizontal trace of the plane <i>f</i><span class =
+"smallroman">E</span>, and <i>e�f</i> is the trace of the same plane on
+the vertical plane <span class = "smallroman">K</span>.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapVI" id = "chapVI">
+VI</a></h5>
+
+<h5 class = "smallcaps">Scientific Definition of Perspective</h5>
+
+
+<p>The projections of the extremities of a right line which passes
+through a vertical plane being given, one on either side of it, to find
+the intersection of that line with the vertical plane. <span class =
+"smallroman">AE</span> (Fig. 20) is the right line. The projection of
+its extremity <span class = "smallroman">A</span> on the vertical plane
+is <i>a�</i>, the projection of <span class = "smallroman">E</span>, the
+other extremity, is <i>e�</i>. <span class = "smallroman">AS</span> is
+the horizontal trace of <span class = "smallroman">AE</span>, and
+<i>a�e�</i> is its trace
+<span class = "pagenum">23</span>
+<a name = "page23" id = "page23"> </a>
+<!--png 036-->
+on the vertical plane. At point <i>f</i>, where the horizontal trace
+intersects the base <span class = "smallroman">B</span><i>c</i> of the
+vertical plane, raise perpendicular <i>f</i><span class =
+"smallroman">P</span> till it cuts <i>a�e�</i> at point <span class =
+"smallroman">P</span>, which is the point required. For it is at the
+same time on the given line <span class = "smallroman">AE</span> and the
+vertical plane <span class = "smallroman">K</span>.</p>
+
+<p class = "illustration">
+<a name = "fig20" id = "fig20"> </a>
+<img src = "images/fig20.png" width = "335" height = "136"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 20.</p>
+
+<p>This figure is similar to the previous one, except that the extremity
+<span class = "smallroman">A</span> of the given line is raised from the
+ground, but the same demonstration applies to&nbsp;it.</p>
+
+<p class = "illustration">
+<a name = "fig21" id = "fig21"> </a>
+<img src = "images/fig21.png" width = "269" height = "163"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 21.</p>
+
+<p>And now let us suppose the vertical plane <span class =
+"smallroman">K</span> to be a sheet of glass, and the given line <span
+class = "smallroman">AE</span> to be the visual ray passing from
+<span class = "pagenum">24</span>
+<a name = "page24" id = "page24"> </a>
+<!--png 037-->
+the eye to the object <span class = "smallroman">A</span> on the other
+side of the glass. Then if <span class = "smallroman">E</span> is the
+eye of the spectator, its projection on the picture is <span class =
+"smallroman">S</span>, the point of sight.</p>
+
+<p>If I draw a dotted line from <span class = "smallroman">E</span> to
+little <i>a</i>, this represents another visual ray, and <i>o</i>, the
+point where it passes through the picture, is the perspective of little
+<i>a</i>. I&nbsp;now draw another line from <i>g</i> to <span class =
+"smallroman">S</span>, and thus form the shaded figure <i>ga�</i><span
+class = "smallroman">P</span><i>o</i>, which is the perspective of
+<i>a</i><span class = "smallroman">A</span><i>a�g</i>. <!-- gah! --></p>
+
+<p>Let it be remarked that in the shaded perspective figure the lines
+<i>a�</i><span class = "smallroman">P</span> and <i>go</i> are both
+drawn towards <span class = "smallroman">S</span>, the point of sight,
+and that they represent parallel lines <span class =
+"smallroman">A</span><i>a�</i> and <i>ag</i>, which are at right angles
+to the picture plane. This is the most important fact in perspective,
+and will be more fully explained farther on, when we speak of retreating
+or so-called vanishing lines.</p>
+
+
+
+
+<h5 class = "section"><a name = "rules" id = "rules">RULES</a></h5>
+
+<h5><a name = "chapVII" id = "chapVII">
+VII</a></h5>
+
+<h5 class = "smallcaps">The Rules and Conditions of Perspective</h5>
+
+
+<p>The conditions of linear perspective are somewhat rigid. In the first
+place, we are supposed to look at objects with one eye only; that is,
+the visual rays are drawn from a single point, and not from two. Of this
+we shall speak later on. Then again, the eye must be placed in a certain
+position, as at <span class = "smallroman">E</span> (Fig. 22), at a
+given height from the ground, <span class = "smallroman">S�E</span>, and
+at a given distance from the picture, as <span class =
+"smallroman">SE</span>. In the next place, the picture or picture plane
+itself must be vertical and perpendicular to the ground or horizontal
+plane, which plane is supposed to be as level as a billiard-table, and
+to extend from the base line, <i>ef</i>, of the picture to the horizon,
+that is, to infinity, for it does not partake of the rotundity of the
+earth.</p>
+
+<p class = "illustration">
+<a name = "fig22" id = "fig22"> </a>
+<img src = "images/fig22.png" width = "340" height = "214"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 22.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig23" id = "fig23"> </a>
+<img src = "images/fig23.png" width = "194" height = "108"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 23.</span>
+Front view of above figure.</td>
+</tr>
+</table>
+
+<p>We can only work out our propositions and figures in space with
+mathematical precision by adopting such conditions as the above. But
+afterwards the artist or draughtsman may modify and suit them to a more
+elastic view of things; that is, he can make his figures separate from
+one another, instead of their outlines coming close together as they do
+when we look at them
+<span class = "pagenum">25</span>
+<a name = "page25" id = "page25"> </a>
+<!--png 038-->
+with only one eye. Also he will allow for the unevenness of the ground
+and the roundness of our globe; he may even move his head and his eyes,
+and use both of them, and in fact make himself quite at his ease when he
+is out sketching, for Nature does all his perspective for him. At the
+same time, a&nbsp;knowledge of this rigid perspective is the sure and
+unerring basis of his freehand drawing.</p>
+
+
+<span class = "pagenum">26</span>
+<a name = "page26" id = "page26"> </a>
+<!--png 039-->
+<h5 class = "smallcaps"><a name = "rule1" id = "rule1">Rule 1</a></h5>
+
+<p>All straight lines remain straight in their perspective appearance.<a
+class = "tag" name = "tag4" id = "tag4" href = "#note4">4</a></p>
+
+
+<h5 class = "smallcaps"><a name = "rule2" id = "rule2">Rule 2</a></h5>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig24" id = "fig24"> </a>
+<img src = "images/fig24.png" width = "247" height = "127"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 24.</td>
+</tr>
+</table>
+
+<p>Vertical lines remain vertical in perspective, and are divided in the
+same proportion as <span class = "smallroman">AB</span> (Fig. 24), the
+original line, and <i>a�b�</i>, the perspective line, and if the one is
+divided at <span class = "smallroman">O</span> the other is divided at
+<i>o�</i> in the same way.</p>
+
+<p>It is not an uncommon error to suppose that the vertical lines of a
+high building should converge towards the top; so they would if we stood
+at the foot of that building and looked up, for then we should alter the
+conditions of our perspective, and our point of sight, instead of being
+on the horizon, would be up in the sky. But if we stood sufficiently far
+away, so as to bring the whole of the building within our angle of
+vision, and the point of sight down to the horizon, then these same
+lines would appear perfectly parallel, and the different stories in
+their true proportion.</p>
+
+
+<h5 class = "smallcaps"><a name = "rule3" id = "rule3">Rule 3</a></h5>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig25" id = "fig25"> </a>
+<img src = "images/fig25.png" width = "242" height = "124"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 25.</td>
+</tr>
+</table>
+
+<p>Horizontals parallel to the base of the picture are also parallel to
+that base in the picture. Thus <i>a�b�</i> (Fig. 25) is parallel to
+<span class = "smallroman">AB</span>,
+<span class = "pagenum">27</span>
+<a name = "page27" id = "page27"> </a>
+<!--png 040-->
+and to <span class = "smallroman">GL</span>, the base of the picture.
+Indeed, the same argument may be used with regard to horizontal lines as
+with verticals. If we look at a straight wall in front of us, its top
+and its rows of bricks, &amp;c., are parallel and horizontal; but if we
+look along it sideways, then we alter the conditions, and the parallel
+lines converge to whichever point we direct the eye.</p>
+
+<p>This rule is important, as we shall see when we come to the
+consideration of the perspective vanishing scale. Its use may be
+illustrated by this sketch, where the houses, walls, &amp;c., are
+parallel to the base of the picture. When that is the case, then objects
+<span class = "pagenum">28</span>
+<a name = "page28" id = "page28"> </a>
+<!--png 041-->
+exactly facing us, such as windows, doors, rows of boards, or of bricks
+or palings, &amp;c., are drawn with their horizontal lines parallel to
+the base; hence it is called parallel perspective.</p>
+
+<p class = "illustration">
+<a name = "fig26" id = "fig26"> </a>
+<img src = "images/fig26.png" width = "261" height = "162"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 26.</p>
+
+
+<h5 class = "smallcaps"><a name = "rule4" id = "rule4">Rule 4</a></h5>
+
+<p>All lines situated in a plane that is parallel to the picture plane
+diminish in proportion as they become more distant, but do not undergo
+any perspective deformation; and remain in the same relation and
+proportion each to each as the original lines. This is called the front
+view.</p>
+
+<p class = "illustration">
+<a name = "fig27" id = "fig27"> </a>
+<img src = "images/fig27.png" width = "308" height = "105"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 27.</p>
+
+
+<h5 class = "smallcaps"><a name = "rule5" id = "rule5">Rule 5</a></h5>
+
+<p>All horizontals which are at right angles to the picture plane are
+drawn to the point of sight.</p>
+
+<p>Thus the lines <span class = "smallroman">AB</span> and <span class =
+"smallroman">CD</span> (Fig. 28) are horizontal or parallel to the
+ground plane, and are also at right angles to the picture plane <span
+class = "smallroman">K</span>. It will be seen that the perspective
+lines <span class = "smallroman">B</span><i>a�</i>, <span class =
+"smallroman">D</span><i>c�</i>, must, according to the laws of
+projection, be drawn to the point of sight.</p>
+
+<p class = "illustration">
+<a name = "fig28" id = "fig28"> </a>
+<img src = "images/fig28.png" width = "319" height = "155"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 28.</p>
+
+<p>This is the most important rule in perspective (see <a href =
+"#fig7">Fig.&nbsp;7</a> at beginning of Definitions).</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig29" id = "fig29"> </a>
+<img src = "images/fig29.png" width = "195" height = "158"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 29.</td>
+</tr>
+<tr>
+<td class = "picture">
+<a name = "fig30" id = "fig30"> </a>
+<img src = "images/fig30.png" width = "199" height = "225"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 30.</td>
+</tr>
+</table>
+
+<p>An arrangement such as there indicated is the best means of
+illustrating this rule. But instead of tracing the outline of the square
+or cube on the glass, as there shown, I&nbsp;have a hole drilled through
+at the point <span class = "smallroman">S</span> (Fig. 29), which I
+select for the point of sight, and through which I pass two loose
+strings <span class = "smallroman">A</span> and <span class =
+"smallroman">B</span>, fixing their ends at <span class =
+"smallroman">S</span>.</p>
+
+<span class = "pagenum">29</span>
+<a name = "page29" id = "page29"> </a>
+<!--png 042-->
+
+<p>As <span class = "smallroman">SD</span> represents the distance the
+spectator is from the glass or picture, I&nbsp;make string <span class =
+"smallroman">SA</span> equal in length to <span class =
+"smallroman">SD</span>. Now if the pupil takes this string in one hand
+and holds it at right angles to the glass, that is, exactly in front of
+<span class = "smallroman">S</span>, and then places one eye at the end
+<span class = "smallroman">A</span> (of course with the string
+extended), he will be at the proper distance from the picture. Let him
+then take the other string, <span class = "smallroman">SB</span>, in the
+other hand, and apply it to point <i>b�</i> where the square touches the
+glass, and he will find that it exactly tallies with the side <i>b�f</i>
+<span class = "pagenum">30</span>
+<a name = "page30" id = "page30"> </a>
+<!--png 043-->
+of the square <i>a�b�fe</i>. If he applies the same string to <i>a�</i>,
+the other corner of the square, his string will exactly tally or cover
+the side <i>a�e</i>, and he will thus have ocular demonstration of this
+important rule.</p>
+
+<p>In this little picture (Fig. 30) in parallel perspective it will be
+seen that the lines which retreat from us at right angles to the picture
+plane are directed to the point of sight <span class =
+"smallroman">S</span>.</p>
+
+
+<h5 class = "smallcaps"><a name = "rule6" id = "rule6">Rule 6</a></h5>
+
+<p>All horizontals which are at 45�, or half a right angle to the
+picture plane, are drawn to the point of distance.</p>
+
+<p>We have already seen that the diagonal of the perspective square, if
+produced to meet the horizon on the picture, will mark on that horizon
+the distance that the spectator is from the point of sight (see <a href
+= "#chapIII">definition</a>, p.&nbsp;16). This point of distance becomes
+then the measuring point for all horizontals at right angles to the
+picture plane.</p>
+
+<p class = "illustration">
+<a name = "fig31" id = "fig31"> </a>
+<img src = "images/fig31.png" width = "339" height = "222"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 31.</p>
+
+<p><span class = "pagenum">31</span>
+<a name = "page31" id = "page31"> </a>
+<!--png 044-->
+Thus in Fig. 31 lines <span class = "smallroman">AS</span> and <span
+class = "smallroman">BS</span> are drawn to the point of sight <span
+class = "smallroman">S</span>, and are therefore at right angles to the
+base <span class = "smallroman">AB</span>. <span class =
+"smallroman">AD</span> being drawn to <span class =
+"smallroman">D</span> (the distance-point), is at an angle of 45� to the
+base <span class = "smallroman">AB</span>, and <span class =
+"smallroman">AC</span> is therefore the diagonal of a square. The line
+1<span class = "smallroman">C</span> is made parallel to <span class =
+"smallroman">AB</span>, consequently <span class =
+"smallroman">A</span>1<span class = "smallroman">CB</span> is a square
+in perspective. The line <span class = "smallroman">BC</span>,
+therefore, being one side of that square, is equal to <span class =
+"smallroman">AB</span>, another side of it. So that to measure a length
+on a line drawn to the point of sight, such as <span class =
+"smallroman">BS</span>, we set out the length required, say <span class
+= "smallroman">BA</span>, on the base-line, then from <span class =
+"smallroman">A</span> draw a line to the point of distance, and where it
+cuts <span class = "smallroman">BS</span> at <span class =
+"smallroman">C</span> is the length required. This can be repeated any
+number of times, say five, so that in this figure <span class =
+"smallroman">BE</span> is five times the length of <span class =
+"smallroman">AB</span>.</p>
+
+
+<h5 class = "smallcaps"><a name = "rule7" id = "rule7">Rule 7</a></h5>
+
+<p>All horizontals forming any other angles but the above are drawn to
+some other points on the horizontal line. If the angle is greater than
+half a right angle (Fig. 32), as <span class = "smallroman">EBG</span>,
+the point is within the point of distance, as at <span class =
+"smallroman">V�</span>. If it is less, as <span class =
+"smallroman">ABV��</span>, then
+<span class = "pagenum">32</span>
+<a name = "page32" id = "page32"> </a>
+<!--png 045-->
+it is beyond the point of distance, and consequently farther from the
+point of sight.</p>
+
+<p class = "illustration">
+<a name = "fig32" id = "fig32"> </a>
+<img src = "images/fig32.png" width = "337" height = "71"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 32.</p>
+
+<p>In Fig. 32, the dotted line <span class = "smallroman">BD</span>,
+drawn to the point of distance <span class = "smallroman">D</span>, is
+at an angle of 45� to the base <span class = "smallroman">AG</span>. It
+will be seen that the line <span class = "smallroman">BV�</span> is at a
+greater angle to the base than <span class = "smallroman">BD</span>; it
+is therefore drawn to a point <span class = "smallroman">V�</span>,
+within the point of distance and nearer to the point of sight <span
+class = "smallroman">S</span>. On the other hand, the line <span class =
+"smallroman">BV��</span> is at a more acute angle, and is therefore
+drawn to a point some way beyond the other distance point.</p>
+
+<p><i>Note.</i>&mdash;When this vanishing point is a long way outside
+the picture, the architects make use of a centrolinead, and the painters
+fix a long string at the required point, and get their perspective lines
+by that means, which is very inconvenient. But I will show you later on
+how you can dispense with this trouble by a very simple means, with
+equally correct results.</p>
+
+
+<h5 class = "smallcaps"><a name = "rule8" id = "rule8">Rule 8</a></h5>
+
+<p>Lines which incline upwards have their vanishing points above the
+horizontal line, and those which incline downwards, below it. In both
+cases they are on the vertical which passes through the vanishing point
+(<span class = "smallroman">S</span>) of their horizontal
+projections.</p>
+
+<span class = "pagenum">33</span>
+<a name = "page33" id = "page33"> </a>
+<!--png 046-->
+<p class = "illustration">
+<a name = "fig33" id = "fig33"> </a>
+<img src = "images/fig33.png" width = "219" height = "172"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 33.</p>
+
+<p>This rule is useful in drawing steps, or roads going uphill and
+downhill.</p>
+
+<p class = "illustration">
+<a name = "fig34" id = "fig34"> </a>
+<img src = "images/fig34.png" width = "338" height = "229"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 34.</p>
+
+
+<span class = "pagenum">34</span>
+<a name = "page34" id = "page34"> </a>
+<!--png 047-->
+<h5 class = "smallcaps"><a name = "rule9" id = "rule9">Rule 9</a></h5>
+
+<p>The farther a point is removed from the picture plane the nearer does
+its perspective appearance approach the horizontal line so long as it is
+viewed from the same position. On the contrary, if the spectator
+retreats from the picture plane <span class = "smallroman">K</span>
+(which we suppose to be transparent), the point remaining at the same
+place, the perspective appearance of this point will approach the
+ground-line in proportion to the distance of the spectator.</p>
+
+<p class = "illustration">
+<a name = "fig35" id = "fig35"> </a>
+<img src = "images/fig35.png" width = "342" height = "175"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 35.</p>
+
+<p class = "illustration">
+<a name = "fig36" id = "fig36"> </a>
+<img src = "images/fig36.png" width = "342" height = "136"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 36.</p>
+
+<p class = "caption">The spectator at two different distances from the
+picture.</p>
+
+<p><span class = "pagenum">35</span>
+<a name = "page35" id = "page35"> </a>
+<!--png 048-->
+Therefore the position of a given point in perspective above the
+ground-line or below the horizon is in proportion to the distance of the
+spectator from the picture, or the picture from the point.</p>
+
+<p class = "illustration">
+<a name = "fig37" id = "fig37"> </a>
+<img src = "images/fig37.png" width = "272" height = "124"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 37.</p>
+
+<p>Figures 38 and 39 are two views of the same gallery from different
+distances. In Fig. 38, where the distance is too short, there is a want
+of proportion between the near and far objects, which is corrected in
+Fig. 39 by taking a much longer distance.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "caption" colspan = "2">
+The picture at two different distances from the point.</td>
+</tr>
+<tr>
+<td class = "picture">
+<a name = "fig38" id = "fig38"> </a>
+<img src = "images/fig38.png" width = "157" height = "178"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig39" id = "fig39"> </a>
+<img src = "images/fig39.png" width = "132" height = "178"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 38.</td>
+<td class = "caption smallcaps">
+Fig. 39.</td>
+</tr>
+</table>
+
+
+<span class = "pagenum">36</span>
+<a name = "page36" id = "page36"> </a>
+<!--png 049-->
+
+<h5 class = "smallcaps"><a name = "rule10" id = "rule10">Rule 10</a></h5>
+
+<p>Horizontals in the same plane which are drawn to the same point on
+the horizon are parallel to each other.</p>
+
+<p class = "illustration">
+<a name = "fig40" id = "fig40"> </a>
+<img src = "images/fig40.png" width = "453" height = "103"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 40.</p>
+
+<p>This is a very important rule, for all our perspective drawing
+depends upon it. When we say that parallels are drawn to the same point
+on the horizon it does not imply that they meet at that point, which
+would be a contradiction; perspective parallels never reach that point,
+although they appear to do so. Fig. 40 will explain this.</p>
+
+<p>Suppose <span class = "smallroman">S</span> to be the spectator,
+<span class = "smallroman">AB</span> a transparent vertical plane which
+represents the picture seen edgeways, and <span class =
+"smallroman">HS</span> and <span class = "smallroman">DC</span> two
+parallel lines, mark off spaces between these parallels equal to <span
+class = "smallroman">SC</span>, the height of the eye of the spectator,
+and raise verticals 2, 3, 4, 5, &amp;c., forming so many squares.
+Vertical line&nbsp;2 viewed from <span class = "smallroman">S</span>
+will appear on <span class = "smallroman">AB</span> but half its length,
+vertical&nbsp;3 will be only a third, vertical&nbsp;4 a fourth, and so
+on, and if we multiplied these spaces <i>ad infinitum</i> we must keep
+on dividing the line <span class = "smallroman">AB</span> by the same
+number. So if we suppose <span class = "smallroman">AB</span> to be a
+yard high and the distance from one vertical to another to be also a
+yard, then if one of these were a thousand yards away its representation
+at <span class = "smallroman">AB</span> would be the thousandth part of
+a yard, or ten thousand yards away, its representation at <span class =
+"smallroman">AB</span> would be the ten-thousandth part, and whatever
+the distance it must always be something; and therefore <span class =
+"smallroman">HS</span> and <span class = "smallroman">DC</span>, however
+far they may be produced
+<span class = "pagenum">37</span>
+<a name = "page37" id = "page37"> </a>
+<!--png 050-->
+and however close they may appear to get, can never meet.</p>
+
+<p class = "illustration">
+<a name = "fig41" id = "fig41"> </a>
+<img src = "images/fig41.png" width = "344" height = "187"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 41.</p>
+
+<p>Fig. 41 is a perspective view of the same figure&mdash;but more
+extended. It will be seen that a line drawn from the tenth upright <span
+class = "smallroman">K</span> to <span class = "smallroman">S</span>
+cuts off a tenth of <span class = "smallroman">AB</span>. We look then
+upon these two lines <span class = "smallroman">SP, OP</span>, as the
+sides of a long parallelogram of which <span class =
+"smallroman">SK</span> is the diagonal, as <i>cefd</i>, the figure on
+the ground, is also a parallelogram.</p>
+
+<p>The student can obtain for himself a further illustration of this
+rule by placing a looking-glass on one of the walls of his studio and
+then sketching himself and his surroundings as seen therein.
+<span class = "pagenum">38</span>
+<a name = "page38" id = "page38"> </a>
+<!--png 051-->
+He will find that all the horizontals at right angles to the glass will
+converge to his own eye. This rule applies equally to lines which are at
+an angle to the picture plane as to those that are at right angles or
+perpendicular to it, as in Rule&nbsp;7. It also applies to those on an
+inclined plane, as in Rule&nbsp;8.</p>
+
+<p class = "illustration">
+<a name = "fig42" id = "fig42"> </a>
+<img src = "images/fig42.png" width = "281" height = "309"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 42.</span> Sketch of artist in
+studio.</p>
+
+<p><span class = "pagenum">39</span>
+<a name = "page39" id = "page39"> </a>
+<!--png 052-->
+With the above rules and a clear notion of the definitions and
+conditions of perspective, we should be able to work out any proposition
+or any new figure that may present itself. At any rate, a&nbsp;thorough
+understanding of these few pages will make the labour now before us
+simple and easy. I&nbsp;hope, too, it may be found interesting. There is
+always a certain pleasure in deceiving and being deceived by the senses,
+and in optical and other illusions, such as making things appear far off
+that are quite near, in making a picture of an object on a flat surface
+to look as if it stood out and in relief by a kind of magic. But there
+is, I&nbsp;think, a&nbsp;still greater pleasure than this, namely, in
+invention and in overcoming difficulties&mdash;in finding out how to do
+things for ourselves by our reasoning faculties, in originating or being
+original, as it were. Let us now see how far we can go in this
+respect.</p>
+
+
+<span class = "pagenum">40</span>
+<a name = "page40" id = "page40"> </a>
+<!--png 053-->
+<h5 class = "section"><a name = "chapVIII" id = "chapVIII">
+VIII</a></h5>
+
+<h5 class = "smallcaps">A Table or Index of the Rules of
+Perspective</h5>
+
+<p>The rules here set down have been fully explained in the previous
+pages, and this table is simply for the student's ready reference.</p>
+
+
+<h5 class = "smallcaps">Rule 1</h5>
+
+<p>All straight lines remain straight in their perspective
+appearance.</p>
+
+
+<h5 class = "smallcaps">Rule 2</h5>
+
+<p>Vertical lines remain vertical in perspective.</p>
+
+
+<h5 class = "smallcaps">Rule 3</h5>
+
+<p>Horizontals parallel to the base of the picture are also parallel to
+that base in the picture.</p>
+
+
+<h5 class = "smallcaps">Rule 4</h5>
+
+<p>All lines situated in a plane that is parallel to the picture plane
+diminish in proportion as they become more distant, but do not undergo
+any perspective deformation. This is called the front view.</p>
+
+
+<h5 class = "smallcaps">Rule 5</h5>
+
+<p>All horizontal lines which are at right angles to the picture plane
+are drawn to the point of sight.</p>
+
+
+<h5 class = "smallcaps">Rule 6</h5>
+
+<p>All horizontals which are at 45� to the picture plane are drawn to
+the point of distance.</p>
+
+
+<h5 class = "smallcaps">Rule 7</h5>
+
+<p>All horizontals forming any other angles but the above are drawn to
+some other points on the horizontal line.</p>
+
+
+<h5 class = "smallcaps">Rule 8</h5>
+
+<p>Lines which incline upwards have their vanishing points above the
+horizon, and those which incline downwards, below it. In both cases they
+are on the vertical which passes through the vanishing point of their
+ground-plan or horizontal projections.</p>
+
+
+<span class = "pagenum">41</span>
+<a name = "page41" id = "page41"> </a>
+<!--png 054-->
+
+<h5 class = "smallcaps">Rule 9</h5>
+
+<p>The farther a point is removed from the picture plane the nearer does
+it appear to approach the horizon, so long as it is viewed from the same
+position.</p>
+
+
+<h5 class = "smallcaps">Rule 10</h5>
+
+<p>Horizontals in the same plane which are drawn to the same point on
+the horizon are perspectively parallel to each other.</p>
+
+
+
+<span class = "pagenum">42</span>
+<a name = "page42" id = "page42"> </a>
+<!--png 055-->
+
+<h3 class = "chapter">BOOK SECOND</h3>
+
+<h5><a name = "practice" id = "practice">THE PRACTICE OF
+PERSPECTIVE</a></h5>
+
+
+<p>In the foregoing book we have explained the theory or science of
+perspective; we now have to make use of our knowledge and to apply it to
+the drawing of figures and the various objects that we wish to
+depict.</p>
+
+<p>The first of these will be a square with two of its sides parallel to
+the picture plane and the other two at right angles to it, and which we
+call</p>
+
+
+<h5 class = "section"><a name = "chapIX" id = "chapIX">
+IX</a></h5>
+
+<h5 class = "smallcaps">The Square in Parallel Perspective</h5>
+
+<p>From a given point on the base line of the picture draw a line at
+right angles to that base. Let <span class = "smallroman">P</span> be
+the given point on the base line <span class = "smallroman">AB</span>,
+and <span class = "smallroman">S</span> the point of sight. We simply
+draw a line along the ground to the point of sight <span class =
+"smallroman">S</span>, and this line will be at right angles to the
+base, as explained in Rule&nbsp;5, and consequently angle <span class =
+"smallroman">APS</span> will be equal to angle <span class =
+"smallroman">SPB</span>, although it does not look so here. This is our
+first difficulty, but one that we shall soon get over.</p>
+
+<p class = "illustration">
+<a name = "fig43" id = "fig43"> </a>
+<img src = "images/fig43.png" width = "244" height = "98"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 43.</p>
+
+<p><span class = "pagenum">43</span>
+<a name = "page43" id = "page43"> </a>
+<!--png 056-->
+In like manner we can draw any number of lines at right angles to the
+base, or we may suppose the point <span class = "smallroman">P</span> to
+be placed at so many different positions, our only difficulty being to
+conceive these lines to be parallel to each other. See Rule&nbsp;10.</p>
+
+<p class = "illustration">
+<a name = "fig44" id = "fig44"> </a>
+<img src = "images/fig44.png" width = "223" height = "94"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 44.</p>
+
+
+<h5 class = "section"><a name = "chapX" id = "chapX">
+X</a></h5>
+
+<h5 class = "smallcaps">The Diagonal</h5>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig45" id = "fig45"> </a>
+<img src = "images/fig45.png" width = "230" height = "97"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 45.</td>
+</tr>
+</table>
+
+<p>From a given point on the base line draw a line at 45�, or half a
+right angle, to that base. Let <span class = "smallroman">P</span> be
+the given point. Draw a line from <span class = "smallroman">P</span> to
+the point of distance <span class = "smallroman">D</span> and this line
+<span class = "smallroman">PD</span> will be at an angle of 45�, or at
+the same angle as the diagonal of a square. See definitions.</p>
+
+
+<h5 class = "section"><a name = "chapXI" id = "chapXI">
+XI</a></h5>
+
+<h5 class = "smallcaps">The Square</h5>
+
+<p>Draw a square in parallel perspective on a given length on the base
+line. Let <i>ab</i> be the given length. From its two
+<span class = "pagenum">44</span>
+<a name = "page44" id = "page44"> </a>
+<!--png 057-->
+extremities <i>a</i> and <i>b</i> draw <i>a</i><span class =
+"smallroman">S</span> and <i>b</i><span class = "smallroman">S</span> to
+the point of sight <span class = "smallroman">S</span>. These two lines
+will be at right angles to the base (see <a href = "#fig43">Fig.
+43</a>). From <i>a</i> draw diagonal <i>a</i><span class =
+"smallroman">D</span> to point of distance <span class =
+"smallroman">D</span>; this line will be 45� to base. At point <i>c</i>,
+where it cuts <i>b</i><span class = "smallroman">S</span>, draw
+<i>dc</i> parallel to <i>ab</i> and <i>abcd</i> is the square
+required.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig46" id = "fig46"> </a>
+<img src = "images/fig46.png" width = "287" height = "106"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig47" id = "fig47"> </a>
+<img src = "images/fig47.png" width = "109" height = "108"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 46.</td>
+<td class = "caption smallcaps">
+Fig. 47.</td>
+</tr>
+</table>
+
+<p>We have here proceeded in much the same way as in drawing a
+geometrical square (Fig. 47), by drawing two lines <span class =
+"smallroman">AE</span> and <span class = "smallroman">BC</span> at right
+angles to a given line, <span class = "smallroman">AB</span>, and from
+<span class = "smallroman">A</span>, drawing the diagonal <span class =
+"smallroman">AC</span> at 45� till it cuts <span class =
+"smallroman">BC</span> at <span class = "smallroman">C</span>, and then
+through <span class = "smallroman">C</span> drawing <span class =
+"smallroman">EC</span> parallel to <span class = "smallroman">AB</span>.
+Let it be remarked that because the two perspective lines (Fig. 48)
+<span class = "smallroman">AS</span> and <span class =
+"smallroman">BS</span> are at right angles to the base, they must
+consequently be parallel to each other, and therefore are perspectively
+equidistant, so that all lines parallel to <span class =
+"smallroman">AB</span> and lying between them, such as <i>ad</i>,
+<i>cf</i>, &amp;c., must be equal.</p>
+
+<p class = "illustration">
+<a name = "fig48" id = "fig48"> </a>
+<img src = "images/fig48.png" width = "321" height = "127"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 48.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig49" id = "fig49"> </a>
+<img src = "images/fig49.png" width = "122" height = "150"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 49.</td>
+</tr>
+</table>
+
+<p>So likewise all diagonals drawn to the point of distance, which
+<span class = "pagenum">45</span>
+<a name = "page45" id = "page45"> </a>
+<!--png 058-->
+are contained between these parallels, such as <span class =
+"smallroman">A</span><i>d</i>, <i>af</i>, &amp;c., must be equal. For
+all straight lines which meet at any point on the horizon are
+perspectively parallel to each other, just as two geometrical parallels
+crossing two others at any angle, as at Fig. 49. Note also (Fig. 48)
+that all squares formed between the two vanishing lines <span class =
+"smallroman">AS</span>, <span class = "smallroman">BS</span>, and by the
+aid of these diagonals, are also equal, and further, that any number of
+squares such as are shown in this figure (Fig. 50), formed in the same
+way and having equal bases, are also equal; and the nine squares
+contained in the square <i>abcd</i> being equal, they divide each side
+of the larger square into three equal parts.</p>
+
+<p>From this we learn how we can measure any number of given
+<span class = "pagenum">46</span>
+<a name = "page46" id = "page46"> </a>
+<!--png 059-->
+lengths, either equal or unequal, on a vanishing or retreating line
+which is at right angles to the base; and also how we can measure any
+width or number of widths on a line such as <i>dc</i>, that is, parallel
+to the base of the picture, however remote it may be from that base.</p>
+
+<p class = "illustration">
+<a name = "fig50" id = "fig50"> </a>
+<img src = "images/fig50.png" width = "350" height = "120"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 50.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXII" id = "chapXII">
+XII</a></h5>
+
+<h5 class = "smallcaps">Geometrical and Perspective Figures
+Contrasted</h5>
+
+
+<p>As at first there may be a little difficulty in realizing the
+resemblance between geometrical and perspective figures, and also about
+certain expressions we make use of, such as horizontals, perpendiculars,
+parallels, &amp;c., which look quite different in perspective,
+I&nbsp;will here make a note of them and also place side by side the two
+views of the same figures.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig51a" id = "fig51a"> </a>
+<img src = "images/fig51a.png" width = "141" height = "96"
+alt = "figure" title = "figure">
+<td class = "picture">
+<a name = "fig51b" id = "fig51b"> </a>
+<img src = "images/fig51b.png" width = "172" height = "85"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 A.</span> The geometrical view.</td>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 B.</span> The perspective view.</td>
+</tr>
+<tr class = "space">
+<td class = "picture">
+<span class = "pagenum">47</span>
+<a name = "page47" id = "page47"> </a>
+<!--png 060-->
+<a name = "fig51c" id = "fig51c"> </a>
+<img src = "images/fig51c.png" width = "54" height = "55"
+alt = "figure" title = "figure">
+<td class = "picture">
+<a name = "fig51d" id = "fig51d"> </a>
+<img src = "images/fig51d.png" width = "95" height = "64"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 C.</span> A geometrical square.</td>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 D.</span> A perspective square.</td>
+</tr>
+<tr class = "space">
+<td class = "picture">
+<a name = "fig51e" id = "fig51e"> </a>
+<img src = "images/fig51e.png" width = "88" height = "73"
+alt = "figure" title = "figure">
+<td class = "picture">
+<a name = "fig51f" id = "fig51f"> </a>
+<img src = "images/fig51f.png" width = "140" height = "58"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 E.</span> Geometrical parallels.</td>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 F.</span> Perspective parallels.</td>
+</tr>
+<tr class = "space">
+<td class = "picture">
+<a name = "fig51g" id = "fig51g"> </a>
+<img src = "images/fig51g.png" width = "98" height = "96"
+alt = "figure" title = "figure">
+<td class = "picture">
+<a name = "fig51h" id = "fig51h"> </a>
+<img src = "images/fig51h.png" width = "118" height = "105"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 G.</span> Geometrical
+perpendicular.</td>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 H.</span> Perspective
+perpendicular.</td>
+</tr>
+<tr class = "space">
+<td class = "picture">
+<a name = "fig51i" id = "fig51i"> </a>
+<img src = "images/fig51i.png" width = "85" height = "71"
+alt = "figure" title = "figure">
+<td class = "picture">
+<a name = "fig51j" id = "fig51j"> </a>
+<img src = "images/fig51j.png" width = "112" height = "92"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 I.</span> Geometrical equal
+lines.</td>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 J.</span> Perspective equal
+lines.</td>
+</tr>
+<tr class = "space">
+<td class = "picture">
+<span class = "pagenum">48</span>
+<a name = "page48" id = "page48"> </a>
+<!--png 061-->
+<a name = "fig51k" id = "fig51k"> </a>
+<img src = "images/fig51k.png" width = "100" height = "100"
+alt = "figure" title = "figure">
+<td class = "picture">
+<a name = "fig51l" id = "fig51l"> </a>
+<img src = "images/fig51l.png" width = "117" height = "82"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 K.</span> A geometrical circle.</td>
+<td class = "caption">
+<span class = "smallcaps">Fig. 51 L.</span> A perspective circle.</td>
+</tr>
+</table>
+
+
+
+<h5 class = "section"><a name = "chapXIII" id = "chapXIII">
+XIII</a></h5>
+
+<h5 class = "smallcaps">Of Certain Terms made use of in Perspective</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig52" id = "fig52"> </a>
+<img src = "images/fig52.png" width = "169" height = "80"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 52.</span> Horizontals.</td>
+</tr>
+</table>
+
+<p>Of course when we speak of <b>Perpendiculars</b> we do not mean
+verticals only, but straight lines at right angles to other lines in any
+position. Also in speaking of <b>lines</b> a right or <b>straight
+line</b> is to be understood; or when we speak of <b>horizontals</b> we
+mean all straight lines that are parallel to the perspective plane, such
+as those on Fig. 52, no matter what direction they take so long as they
+are level. They are not to be confused with the horizon or
+horizontal-line.</p>
+
+<p>There are one or two other terms used in perspective which are not
+satisfactory because they are confusing, such as vanishing lines and
+vanishing points. The French term, <i>fuyante</i> or <i>lignes
+fuyantes</i>, or going-away lines, is more expressive; and <i>point de
+fuite</i>, instead of vanishing point, is much better. I&nbsp;have
+occasionally called the former retreating lines, but the simple meaning
+is, lines that are not parallel to the picture plane; but a vanishing
+line implies a line that disappears, and a vanishing point implies
+<span class = "pagenum">49</span>
+<a name = "page49" id = "page49"> </a>
+<!--png 062-->
+a point that gradually goes out of sight. Still, it is difficult to
+alter terms that custom has endorsed. All we can do is to use as few of
+them as possible.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXIV" id = "chapXIV">
+XIV</a></h5>
+
+<h5 class = "smallcaps">How to Measure Vanishing or Receding Lines</h5>
+
+
+<p>Divide a vanishing line which is at right angles to the picture plane
+into any number of given measurements. Let <span class =
+"smallroman">SA</span> be the given line. From <span class =
+"smallroman">A</span> measure off on the base line the divisions
+required, say five of 1&nbsp;foot each; from each division draw
+diagonals to point of distance <span class = "smallroman">D</span>, and
+where these intersect the line <span class = "smallroman">AC</span> the
+corresponding divisions will be found. Note that as lines <span class =
+"smallroman">AB</span> and <span class = "smallroman">AC</span> are two
+sides of the same square they are necessarily equal, and so also are the
+divisions on <span class = "smallroman">AC</span> equal to those on
+<span class = "smallroman">AB</span>.</p>
+
+<p class = "illustration">
+<a name = "fig53" id = "fig53"> </a>
+<img src = "images/fig53.png" width = "333" height = "102"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 53.</p>
+
+<table class = "float left" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig54" id = "fig54"> </a>
+<img src = "images/fig54.png" width = "96" height = "143"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 54.</td>
+</tr>
+</table>
+
+<p>The line <span class = "smallroman">AB</span> being the base of the
+picture, it is at the same time a perspective line and a geometrical
+one, so that we can use it as a scale for measuring given lengths
+thereon, but should there not be enough room on it to measure the
+required number we draw a second line, <span class =
+"smallroman">DC</span>, which we divide in the same proportion and
+proceed to divide <i>cf</i>. This geometrical figure gives, as it were,
+a&nbsp;bird's-eye view or ground-plan of the above.</p>
+
+
+
+
+<span class = "pagenum">50</span>
+<a name = "page50" id = "page50"> </a>
+<!--png 063-->
+<h5 class = "section"><a name = "chapXV" id = "chapXV">
+XV</a></h5>
+
+<h5 class = "smallcaps">How to Place Squares in Given Positions</h5>
+
+
+<p>Draw squares of given dimensions at given distances from the base
+line to the right or left of the vertical line, which passes through the
+point of sight.</p>
+
+<p class = "illustration">
+<a name = "fig55" id = "fig55"> </a>
+<img src = "images/fig55.png" width = "336" height = "146"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 55.</p>
+
+<p>Let <i>ab</i> (Fig. 55) represent the base line of the picture
+divided into a certain number of feet; <span class =
+"smallroman">HD</span> the horizon, <span class = "smallroman">VO</span>
+the vertical. It is required to draw a square 3&nbsp;feet wide,
+2&nbsp;feet to the right of the vertical, and 1&nbsp;foot from the
+base.</p>
+
+<p>First measure from <span class = "smallroman">V</span>, 2&nbsp;feet
+to <i>e</i>, which gives the distance from the vertical. Second, from
+<i>e</i> measure 3&nbsp;feet to <i>b</i>, which gives the width of the
+square; from <i>e</i> and <i>b</i> draw <i>e</i><span class =
+"smallroman">S</span>, <i>b</i><span class = "smallroman">S</span>, to
+point of sight. From either <i>e</i> or <i>b</i> measure 1&nbsp;foot to
+the left, to <i>f</i> or <i>f�</i>. Draw <i>f</i><span class =
+"smallroman">D</span> to point of distance, which intersects
+<i>e</i><span class = "smallroman">S</span> at <span class =
+"smallroman">P</span>, and gives the required distance from base. Draw
+<span class = "smallroman">P</span><i>g</i> and <span class =
+"smallroman">B</span> parallel to the base, and we have the required
+square.</p>
+
+<p>Square <span class = "smallroman">A</span> to the left of the
+vertical is 2� feet wide, 1&nbsp;foot from the vertical and 2&nbsp;feet
+from the base, and is worked out in the same way.</p>
+
+<p><i>Note.</i>&mdash;It is necessary to know how to work to scale,
+especially in architectural drawing, where it is indispensable, but in
+working
+<span class = "pagenum">51</span>
+<a name = "page51" id = "page51"> </a>
+<!--png 064-->
+out our propositions and figures it is not always desirable.
+A&nbsp;given length indicated by a line is generally sufficient for our
+requirements. To work out every problem to scale is not only tedious and
+mechanical, but wastes time, and also takes the mind of the student away
+from the reasoning out of the subject.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXVI" id = "chapXVI">
+XVI</a></h5>
+
+<h5 class = "smallcaps">How To Draw Pavements, &amp;c.</h5>
+
+
+<p>Divide a vanishing line into parts varying in length. Let <span class
+= "smallroman">BS�</span> be the vanishing line: divide it into
+4&nbsp;long and 3&nbsp;short spaces; then proceed as in the previous
+figure. If we draw horizontals through the points thus obtained and from
+these raise verticals, we form, as it were, the interior of a building
+in which we can place pillars and other objects.</p>
+
+<p class = "illustration">
+<a name = "fig56" id = "fig56"> </a>
+<img src = "images/fig56.png" width = "320" height = "202"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 56.</p>
+
+<p><span class = "pagenum">52</span>
+<a name = "page52" id = "page52"> </a>
+<!--png 065-->
+Or we can simply draw the plan of the pavement as in this figure.</p>
+
+<p class = "illustration">
+<a name = "fig57" id = "fig57"> </a>
+<img src = "images/fig57.png" width = "222" height = "221"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 57.</p>
+
+<p>And then put it into perspective.</p>
+
+<p class = "illustration">
+<a name = "fig58" id = "fig58"> </a>
+<img src = "images/fig58.png" width = "338" height = "170"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 58.</p>
+
+
+
+
+<span class = "pagenum">53</span>
+<a name = "page53" id = "page53"> </a>
+<!--png 066-->
+<h5 class = "section"><a name = "chapXVII" id = "chapXVII">
+XVII</a></h5>
+
+<h5 class = "smallcaps">Of Squares placed Vertically and at Different
+Heights, or the Cube in Parallel Perspective</h5>
+
+
+<p>On a given square raise a cube.</p>
+
+<p class = "illustration">
+<a name = "fig59" id = "fig59"> </a>
+<img src = "images/fig59.png" width = "344" height = "137"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 59.</p>
+
+<p><span class = "smallroman">ABCD</span> is the given square; from
+<span class = "smallroman">A</span> and <span class =
+"smallroman">B</span> raise verticals <span class =
+"smallroman">AE</span>, <span class = "smallroman">BF</span>, equal to
+<span class = "smallroman">AB</span>; join <span class =
+"smallroman">EF</span>. Draw <span class = "smallroman">ES</span>, <span
+class = "smallroman">FS</span>, to point of sight; from <span class =
+"smallroman">C</span> and <span class = "smallroman">D</span> raise
+verticals <span class = "smallroman">CG</span>, <span class =
+"smallroman">DH</span>, till they meet vanishing lines <span class =
+"smallroman">ES</span>, <span class = "smallroman">FS</span>, in <span
+class = "smallroman">G</span> and <span class = "smallroman">H</span>,
+and the cube is complete.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXVIII" id = "chapXVIII">
+XVIII</a></h5>
+
+<h5 class = "smallcaps">The Transposed Distance</h5>
+
+
+<p>The transposed distance is a point <span class =
+"smallroman">D�</span> on the vertical <span class =
+"smallroman">VD�</span>, at exactly the same distance from the point of
+sight as is the point of distance on the horizontal line.</p>
+
+<p>It will be seen by examining this figure that the diagonals of the
+squares in a vertical position are drawn to this vertical
+distance-point, thus saving the necessity of taking the measurements
+first on the base line, as at <span class = "smallroman">CB</span>,
+which in the case of distant objects, such as the farthest window, would
+be very inconvenient. Note that the windows at <span class =
+"smallroman">K</span> are twice as high as they are wide.
+<span class = "pagenum">54</span>
+<a name = "page54" id = "page54"> </a>
+<!--png 067-->
+Of course these or any other objects could be made of any
+proportion.</p>
+
+<p class = "illustration">
+<a name = "fig60" id = "fig60"> </a>
+<img src = "images/fig60.png" width = "351" height = "318"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 60.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXIX" id = "chapXIX">
+XIX</a></h5>
+
+<h5 class = "smallcaps">The Front View of the Square and of the
+Proportions of Figures at Different Heights</h5>
+
+
+<p>According to Rule&nbsp;4, all lines situated in a plane parallel to
+the picture plane diminish in length as they become more distant, but
+remain in the same proportions each to each as the original lines; as
+squares or any other figures retain the same form. Take the two squares
+<span class = "smallroman">ABCD</span>, <i>abcd</i> (Fig. 61), one
+inside the other; although moved back from square <span class =
+"smallroman">EFGH</span> they retain the same form. So
+<span class = "pagenum">55</span>
+<a name = "page55" id = "page55"> </a>
+<!--png 068-->
+in dealing with figures of different heights, such as statuary or
+ornament in a building, if actually equal in size, so must we represent
+them.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig61" id = "fig61"> </a>
+<img src = "images/fig61.png" width = "222" height = "202"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture middle">
+<a name = "fig62" id = "fig62"> </a>
+<img src = "images/fig62.png" width = "148" height = "162"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 61.</td>
+<td class = "caption smallcaps">
+Fig. 62.</td>
+</tr>
+</table>
+
+<p>In this square <span class = "smallroman">K</span>, with the checker
+pattern, we should not think of making the top squares smaller than the
+bottom ones; so it is with figures.</p>
+
+<p><span class = "pagenum">56</span>
+<a name = "page56" id = "page56"> </a>
+<!--png 069-->
+This subject requires careful study, for, as pointed out in our opening
+chapter, there are certain conditions under which we have to modify and
+greatly alter this rule in large decorative work.</p>
+
+<p class = "illustration">
+<a name = "fig63" id = "fig63"> </a>
+<img src = "images/fig63.png" width = "326" height = "321"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 63.</p>
+
+<p>In Fig. 63 the two statues <span class = "smallroman">A</span> and
+<span class = "smallroman">B</span> are the same size. So if traced
+through a vertical sheet of glass, <span class = "smallroman">K</span>,
+as at <i>c</i> and <i>d</i>, they would also be equal; but as the angle
+<i>b</i> at which the upper one is seen is smaller than angle <i>a</i>,
+at which the lower figure or statue is seen, it will appear smaller to
+the spectator (<span class = "smallroman">S</span>) both in reality and
+in the picture.</p>
+
+<span class = "pagenum">57</span>
+<a name = "page57" id = "page57"> </a>
+<!--png 070-->
+<p class = "illustration">
+<a name = "fig64" id = "fig64"> </a>
+<img src = "images/fig64.png" width = "337" height = "290"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 64.</p>
+
+<p>But if we wish them to appear the same size to the spectator who is
+viewing them from below, we must make the angles <i>a</i> and <i>b</i>
+(Fig. 64), at which they are viewed, both equal. Then draw lines through
+equal arcs, as at <i>c</i> and <i>d</i>, till they cut the vertical
+<span class = "smallroman">NO</span> (representing the side of the
+building where the figures are to be placed). We shall then obtain the
+exact size of the figure at that height, which will make it look the
+same size as the lower one, <span class = "smallroman">N</span>. The
+same rule applies to the picture <span class = "smallroman">K</span>,
+when it is of large proportions. As an example in painting, take
+Michelangelo&rsquo;s large altar-piece in the Sistine Chapel, &lsquo;The
+Last Judgement&rsquo;; here the figures forming the upper group, with
+our Lord in judgement surrounded by saints, are about four times the
+size, that is, about twice the height, of those at the lower part of the
+fresco. The
+<span class = "pagenum">58</span>
+<a name = "page58" id = "page58"> </a>
+<!--png 071-->
+figures on the ceiling of the same chapel are studied not only according
+to their height from the pavement, which is 60 ft., but to suit the
+arched form of it. For instance, the head of the figure of Jonah at the
+end over the altar is thrown back in the design, but owing to the
+curvature in the architecture is actually more forward than the feet.
+Then again, the prophets and sybils seated round the ceiling, which are
+perhaps the grandest figures in the whole range of art, would be 18 ft.
+high if they stood up; these, too, are not on a flat surface, so that it
+required great knowledge to give them their right effect.</p>
+
+<table class = "float left" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig65" id = "fig65"> </a>
+<img src = "images/fig65.png" width = "219" height = "199"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 65.</td>
+</tr>
+</table>
+
+<p>Of course, much depends upon the distance we view these statues or
+paintings from. In interiors, such as churches, halls, galleries,
+&amp;c., we can make a fair calculation, such as the length of the nave,
+if the picture is an altar-piece&mdash;or say, half the length; so also
+with statuary in niches, friezes, and other architectural ornaments. The
+nearer we are to them, and the more we have to look up, the larger will
+the upper figures have to be; but if these are on the outside of a
+building that can be looked at from a long distance, then it is better
+not to have too great a difference.</p>
+
+<p><span class = "pagenum">59</span>
+<a name = "page59" id = "page59"> </a>
+<!--png 075-->
+For the farther we recede the more equal are the angles at which we view
+the objects at their different stages, so that in each case we may have
+to deal with, we must consider the conditions attending&nbsp;it.</p>
+
+<p>These remarks apply also to architecture in a great measure.
+Buildings that can only be seen from the street below, as pictures in a
+narrow gallery, require a different treatment from those out in the
+open, that are to be looked at from a distance. In the former case the
+same treatment as the Campanile at Florence is in some cases desirable,
+but all must depend upon the taste and judgement of the architect in
+such matters. All I venture to do here is to call attention to the
+subject, which seems as a rule to be ignored, or not to be considered of
+importance. Hence the many mistakes in our buildings, and the
+unsatisfactory and mean look of some of our public monuments.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXX" id = "chapXX">
+XX</a></h5>
+
+<h5 class = "smallcaps">Of Pictures that are Painted according to the
+Position they are to Occupy</h5>
+
+
+<p>In this double-page illustration of the wall of a picture-gallery,
+I&nbsp;have, as it were, hung the pictures in accordance with the style
+in which they are painted and the perspective adopted by their painters.
+It will be seen that those placed on the line level with the eye have
+their horizon lines fairly high up, and are not suited to be placed any
+higher. The Giorgione in the centre, the Monna Lisa to the right, and
+the Velasquez and Watteau to the left, are all pictures that fit that
+position; whereas the grander compositions above them are so designed,
+and are so large in conception, that we gain in looking up to them.</p>
+
+<!--png 072-->
+<!--png 073-->
+<p class = "illustration">
+<a name = "fig66" id = "fig66"> </a>
+<img src = "images/fig66thumb.png" width = "317" height = "208"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption"><span class = "smallcaps">
+Fig. 66.</span><br>
+<br>
+<a href = "images/fig66large.png"><i>Larger View</i></a></p>
+<!--png 074-->
+
+<p>Note how grandly the young prince on his pony, by Velasquez, tells
+out against the sky, with its low horizon and strong contrast of light
+and dark; nor does it lose a bit by being placed where it is, over the
+smaller pictures.</p>
+
+<p>The Rembrandt, on the opposite side, with its burgomasters in black
+hats and coats and white collars, is evidently intended and painted for
+a raised position, and to be looked up to, which is evident from the
+perspective of the table. The grand Titian in
+<span class = "pagenum">60</span>
+<a name = "page60" id = "page60"> </a>
+<!--png 076-->
+the centre, an altar-piece in one of the churches in Venice (here
+reversed), is also painted to suit its elevated position, with low
+horizon and figures telling boldly against the sky. Those placed low
+down are modern French pictures, with the horizon high up and almost
+above their frames, but placed on the ground they fit into the general
+harmony of the arrangement.</p>
+
+<p>It seems to me it is well, both for those who paint and for those who
+hang pictures, that this subject should be taken into consideration. For
+it must be seen by this illustration that a bigger style is adopted by
+the artists who paint for high places in palaces or churches than by
+those who produce smaller easel-pictures intended to be seen close.
+Unfortunately, at our picture exhibitions, we see too often that nearly
+all the works, whether on large or small canvases, are painted for the
+line, and that those which happen to get high up look as if they were
+toppling over, because they have such a high horizontal line; and
+instead of the figures telling against the sky, as in this picture of
+the &lsquo;Infant&rsquo; by Velasquez, the Reynolds, and the fat man
+treading on a flag, we have fields or sea or distant landscape almost to
+the top of the frame, and all, so methinks, because the perspective is
+not sufficiently considered.</p>
+
+
+<p><i>Note.</i>&mdash;Whilst on this subject, I&nbsp;may note that the
+painter in his large decorative work often had difficulties to contend
+with, which arose from the form of the building or the shape of the wall
+on which he had to place his frescoes. Painting on the ceiling was no
+easy task, and Michelangelo, in a humorous sonnet addressed to Giovanni
+da Pistoya, gives a burlesque portrait of himself while he was painting
+the Sistine Chapel:&mdash;</p>
+
+<h5><i>&ldquo;I&rsquo;ho gi� fatto un gozzo in questo
+stento.&rdquo;</i></h5>
+
+<p class = "verse">
+Now have I such a goitre &rsquo;neath my chin</p>
+<p class = "verse">
+That I am like to some Lombardic cat,</p>
+<p class = "verse">
+My beard is in the air, my head i&rsquo; my back,</p>
+<p class = "verse">
+My chest like any harpy&rsquo;s, and my face</p>
+<p class = "verse">
+Patched like a carpet by my dripping brush.</p>
+<p class = "verse">
+Nor can I see, nor can I budge a step;</p>
+<p class = "verse">
+My skin though loose in front is tight behind,</p>
+<p class = "verse">
+And I am even as a Syrian bow.</p>
+<p class = "verse">
+Alas! methinks a bent tube shoots not well;</p>
+<p class = "verse">
+So give me now thine aid, my Giovanni.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig67" id = "fig67"> </a>
+<img src = "images/fig67.png" width = "153" height = "181"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 67.</td>
+</tr>
+<tr>
+<td class = "caption left" width = "180px">
+&lsquo;Draw upon part of wall <span class = "smallroman">MN</span> half
+the figure you mean to represent, and the other half upon the cove above
+(<span class = "smallroman">MR</span>).&rsquo; Leonardo da Vinci&rsquo;s
+<i>Treatise on Painting</i>.</td>
+</tr>
+</table>
+
+<p><span class = "pagenum">61</span>
+<a name = "page61" id = "page61"> </a>
+<!--png 077-->
+At present that difficulty is got over by using large strong canvas, on
+which the picture can be painted in the studio and afterwards placed on
+the wall.</p>
+
+<p>However, the other difficulty of form has to be got over also.
+A&nbsp;great portion of the ceiling of the Sistine Chapel, and notably
+the prophets and sibyls, are painted on a curved surface, in which case
+a similar method to that explained by Leonardo da Vinci has to be
+adopted.</p>
+
+<p>In Chapter CCCI he shows us how to draw a figure twenty-four braccia
+high upon a wall twelve braccia high. (The braccia is 1&nbsp;ft.
+10&#x215E; in.). He first draws the figure upright, then from the
+various points draws lines to a point <span class =
+"smallroman">F</span> on the floor of the building, marking their
+intersections on the profile of the wall somewhat in the manner we have
+indicated, which serve as guides in making the outline to be traced.</p>
+
+
+
+
+<span class = "pagenum">62</span>
+<a name = "page62" id = "page62"> </a>
+<!--png 078-->
+<h5 class = "section"><a name = "chapXXI" id = "chapXXI">
+XXI</a></h5>
+
+<h5 class = "smallcaps">Interiors</h5>
+
+
+<p class = "illustration">
+<a name = "fig68" id = "fig68"> </a>
+<img src = "images/fig68.png" width = "303" height = "312"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 68.</span> Interior by de Hoogh.</p>
+
+<p>To draw the interior of a cube we must suppose the side facing us to
+be removed or transparent. Indeed, in all our figures which represent
+solids we suppose that we can see through them,
+<span class = "pagenum">63</span>
+<a name = "page63" id = "page63"> </a>
+<!--png 079-->
+and in most cases we mark the hidden portions with dotted lines. So also
+with all those imaginary lines which conduct the eye to the various
+vanishing points, and which the old writers called
+&lsquo;occult&rsquo;.</p>
+
+<p class = "illustration">
+<a name = "fig69" id = "fig69"> </a>
+<img src = "images/fig69.png" width = "321" height = "130"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 69.</p>
+
+<p>When the cube is placed below the horizon (as in <a href =
+"#fig59">Fig. 59</a>), we see the top of it; when on the horizon, as in
+the above (Fig. 69), if the side facing us is removed we see both top
+and bottom of it, or if a room, we see floor and ceiling, but otherwise
+we should see but one side (that facing us), or at most two sides. When
+the cube is above the horizon we see underneath&nbsp;it.</p>
+
+<p>We shall find this simple cube of great use to us in architectural
+subjects, such as towers, houses, roofs, interiors of rooms, &amp;c.</p>
+
+<p>In this little picture by de Hoogh we have the application of the
+perspective of the cube and other foregoing problems.</p>
+
+
+
+
+<span class = "pagenum">64</span>
+<a name = "page64" id = "page64"> </a>
+<!--png 080-->
+<h5 class = "section"><a name = "chapXXII" id = "chapXXII">
+XXII</a></h5>
+
+<h5 class = "smallcaps">The Square at an Angle of 45�</h5>
+
+
+<p>When the square is at an angle of 45� to the base line, then its
+sides are drawn respectively to the points of distance, <span class =
+"smallroman">DD</span>, and one of its diagonals which is at right
+angles to the base is drawn to the point of sight <span class =
+"smallroman">S</span>, and the other <i>ab</i>, is parallel to that base
+or ground line.</p>
+
+<p class = "illustration">
+<a name = "fig70" id = "fig70"> </a>
+<img src = "images/fig70.png" width = "340" height = "89"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 70.</p>
+
+<p>To draw a pavement with its squares at this angle is but an
+amplification of the above figure. Mark off on base equal distances, 1,
+2, 3, &amp;c., representing the diagonals of required squares, and from
+each of these points draw lines to points of distance <span class =
+"smallroman">DD�</span>. These lines will intersect each other, and so
+form the squares of the pavement; to ensure correctness, lines should
+also be drawn from these points 1, 2, 3, to the point of sight <span
+class = "smallroman">S</span>, and also horizontals parallel to the
+base, as <i>ab</i>.</p>
+
+<p class = "illustration">
+<a name = "fig71" id = "fig71"> </a>
+<img src = "images/fig71.png" width = "307" height = "72"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 71.</p>
+
+
+
+
+<span class = "pagenum">65</span>
+<a name = "page65" id = "page65"> </a>
+<!--png 081-->
+<h5 class = "section"><a name = "chapXXIII" id = "chapXXIII">
+XXIII</a></h5>
+
+<h5 class = "smallcaps">The Cube at an Angle of 45�</h5>
+
+
+<p>Having drawn the square at an angle of 45�, as shown in the previous
+figure, we find the length of one of its sides, <i>dh</i>, by drawing a
+line, <span class = "smallroman">SK</span>, through <i>h</i>, one of its
+extremities, till it cuts the base line at <span class =
+"smallroman">K</span>. Then, with the other extremity <i>d</i> for
+centre and <i>d</i><span class = "smallroman">K</span> for radius,
+describe a quarter of a circle <span class =
+"smallroman">K</span><i>m</i>; the chord thereof <i>m</i><span class =
+"smallroman">K</span> will be the geometrical length of <i>dh</i>. At
+<i>d</i> raise vertical <i>d</i><span class = "smallroman">C</span>
+equal to <i>m</i><span class = "smallroman">K</span>, which gives us the
+height of the cube, then raise verticals at <i>a</i>, <i>h</i>, &amp;c.,
+their height being found by drawing <span class = "smallroman">CD</span>
+and <span class = "smallroman">CD�</span> to the two points of distance,
+and so completing the figure.</p>
+
+<p class = "illustration">
+<a name = "fig72" id = "fig72"> </a>
+<img src = "images/fig72.png" width = "334" height = "142"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 72.</p>
+
+
+
+
+<span class = "pagenum">66</span>
+<a name = "page66" id = "page66"> </a>
+<!--png 082-->
+<h5 class = "section"><a name = "chapXXIV" id = "chapXXIV">
+XXIV</a></h5>
+
+<h5 class = "smallcaps">Pavements Drawn by Means of Squares at 45�</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig75" id = "fig75"> </a>
+<img src = "images/fig75.png" width = "147" height = "180"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 75.</td>
+</tr>
+</table>
+
+<p>The square at 45� will be found of great use in drawing pavements,
+roofs, ceilings, &amp;c. In Figs. 73, 74 it is shown how
+<span class = "pagenum">67</span>
+<a name = "page67" id = "page67"> </a>
+<!--png 083-->
+having set out one square it can be divided into four or more equal
+squares, and any figure or tile drawn therein. Begin by making a
+geometrical or ground plan of the required design, as at Figs. <ins
+class = "correction" title= "text reads '74 and 75'">73 and 74</ins>,
+where we have bricks placed at right angles to each other in rows,
+a&nbsp;common arrangement in brick floors, or tiles of an octagonal form
+as at Fig.&nbsp;75.</p>
+
+<p class = "illustration">
+<a name = "fig73" id = "fig73"> </a>
+<img src = "images/fig73.png" width = "337" height = "116"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 73.</p>
+
+<p class = "illustration">
+<a name = "fig74" id = "fig74"> </a>
+<img src = "images/fig74.png" width = "300" height = "233"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 74.</p>
+
+
+
+
+<span class = "pagenum">68</span>
+<a name = "page68" id = "page68"> </a>
+<!--png 084-->
+<h5 class = "section"><a name = "chapXXV" id = "chapXXV">
+XXV</a></h5>
+
+<h5 class = "smallcaps">The Perspective Vanishing Scale</h5>
+
+
+<p>The vanishing scale, which we shall find of infinite use in our
+perspective, is founded on the facts explained in Rule 10. We there find
+that all horizontals in the same plane, which are drawn to the same
+point on the horizon, are perspectively parallel to each other, so that
+if we measure a certain height or width on the picture plane, and then
+from each extremity draw lines to any convenient point on the horizon,
+then all the perpendiculars drawn between these lines will be
+perspectively equal, however much they may appear to vary in length.</p>
+
+<p class = "illustration">
+<a name = "fig76" id = "fig76"> </a>
+<img src = "images/fig76.png" width = "259" height = "112"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 76.</p>
+
+<p>Let us suppose that in this figure (76) <span class =
+"smallroman">AB</span> and <span class = "smallroman">A�B�</span> each
+represent 5&nbsp;feet. Then in the first case all the verticals, as
+<i>e</i>, <i>f</i>, <i>g</i>, <i>h</i>, drawn between AO and BO
+represent 5&nbsp;feet, and in the second case all the horizontals
+<i>e</i>, <i>f</i>, <i>g</i>, <i>h</i>, drawn between A�O and B�O also
+represent 5&nbsp;feet each. So that by the aid of this scale we can give
+the exact perspective height and width of any object in the picture,
+however far it may be from the base line, for of course we can increase
+or diminish our measurements at <span class = "smallroman">AB</span> and
+<span class = "smallroman">A�B�</span> to whatever length we
+require.</p>
+
+<p>As it may not be quite evident at first that the points O may be
+taken at random, the following figure will prove&nbsp;it.</p>
+
+
+
+
+<span class = "pagenum">69</span>
+<a name = "page69" id = "page69"> </a>
+<!--png 085-->
+<h5 class = "section"><a name = "chapXXVI" id = "chapXXVI">
+XXVI</a></h5>
+
+<h5 class = "smallcaps">The Vanishing Scale can be Drawn to any Point on
+the Horizon</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig77" id = "fig77"> </a>
+<img src = "images/fig77.png" width = "204" height = "131"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 77.</td>
+</tr>
+</table>
+
+<p>From <span class = "smallroman">AB</span> (Fig. 77) draw <span class
+= "smallroman">AO</span>, <span class = "smallroman">BO</span>, thus
+forming the scale, raise vertical <span class = "smallroman">C</span>.
+Now form a second scale from <span class = "smallroman">AB</span> by
+drawing <span class = "smallroman">AO�</span> <span class =
+"smallroman">BO�</span>, and therein raise vertical <span class =
+"smallroman">D</span> at an equal distance from the base. First, then,
+vertical <span class = "smallroman">C</span> equals <span class =
+"smallroman">AB</span>, and secondly vertical <span class =
+"smallroman">D</span> equals <span class = "smallroman">AB</span>,
+therefore <span class = "smallroman">C</span> equals <span class =
+"smallroman">D</span>, so that either of these scales will measure a
+given height at a given distance.</p>
+
+<p>(See axioms of geometry.)</p>
+
+<span class = "pagenum">71</span>
+<a name = "page71" id = "page71"> </a>
+<!--png 087-->
+
+
+<h5 class = "section"><a name = "chapXXVII" id = "chapXXVII">
+XXVII</a></h5>
+
+<h5 class = "smallcaps">Application of Vanishing Scales to Drawing
+Figures</h5>
+
+
+<p>In this figure we have marked off on a level plain three or four
+points <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, to indicate the places
+where we wish to stand our figures. <span class = "smallroman">AB</span>
+represents their average height, so we have made our scale <span class =
+"smallroman">AO, BO</span>, accordingly. From each point marked we draw
+a line parallel to the base till it reaches the scale. From the point
+where it touches the line <span class = "smallroman">AO</span>, raise
+perpendicular as <i>a</i>, which gives the height required at that
+distance, and must be referred back to the figure itself.</p>
+
+<p class = "illustration">
+<a name = "fig78" id = "fig78"> </a>
+<img src = "images/fig78.png" width = "327" height = "132"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 78.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXVIII" id = "chapXXVIII">
+XXVIII</a></h5>
+
+<h5 class = "smallcaps">How to Determine the Heights of Figures on a
+Level Plane</h5>
+
+<h5><i>First Case.</i></h5>
+
+
+<p>This is but a repetition of the previous figure, excepting that we
+have substituted these schoolgirls for the vertical lines. If we wish to
+make some taller than the others, and some shorter, we can easily do so,
+as must be evident (see Fig.&nbsp;79).</p>
+
+<p class = "illustration">
+<span class = "pagenum">[70a]</span>
+<a name = "page70" id = "page70"> </a>
+<!--png 086-->
+<a name = "fig79" id = "fig79"> </a>
+<img src = "images/fig79.png" width = "338" height = "214"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 79.</span> Schoolgirls.</p>
+
+<p>Note that in this first case the scale is below the horizon, so that
+we see over the heads of the figures, those nearest to us being the
+lowest down. That is to say, we are looking on this scene from a
+slightly raised platform.</p>
+
+
+<span class = "pagenum">72</span>
+<a name = "page72" id = "page72"> </a>
+<!--png 088-->
+
+<h5><i>Second Case.</i></h5>
+
+<p>To draw figures at different distances when their heads are above the
+horizon, or as they would appear to a person sitting on a low seat. The
+height of the heads varies according to the distance of the figures
+(Fig.&nbsp;80).</p>
+
+<p class = "illustration">
+<span class = "pagenum">[70b]</span>
+<!--png 086-->
+<a name = "fig80" id = "fig80"> </a>
+<img src = "images/fig80.png" width = "341" height = "232"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 80.</span> Cavaliers.</p>
+
+
+<h5><i>Third Case.</i></h5>
+
+<p>How to draw figures when their heads are about the height of the
+horizon, or as they appear to a person standing on the same level or
+walking among them.</p>
+
+<p class = "illustration">
+<a name = "fig81" id = "fig81"> </a>
+<img src = "images/fig81.png" width = "333" height = "178"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 81.</p>
+
+<p>In this case the heads or the eyes are on a level with the horizon,
+and we have little necessity for a scale at the side unless it is for
+the purpose of ascertaining or marking their distances from the base
+line, and their respective heights, which of course vary; so in all
+cases allowance must be made for some being taller and some shorter than
+the scale measurement.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXIX" id = "chapXXIX">
+XXIX</a></h5>
+
+<h5 class = "smallcaps">The Horizon above the Figures</h5>
+
+
+<p>In this example from De Hoogh the doorway to the left is higher up
+than the figure of the lady, and the effect seems to me
+<span class = "pagenum">73</span>
+<a name = "page73" id = "page73"> </a>
+<!--png 089-->
+more pleasing and natural for this kind of domestic subject. This
+delightful painter was not only a master of colour, of sunlight effect,
+and perfect composition, but also of perspective, and thoroughly
+understood the charm it gives to a picture, when cunningly introduced,
+for he makes the spectator feel that he
+<span class = "pagenum">74</span>
+<a name = "page74" id = "page74"> </a>
+<!--png 090-->
+can walk along his passages and courtyards. Note that he frequently puts
+the point of sight quite at the side of his canvas, as at S, which gives
+almost the effect of angular perspective whilst it preserves the
+flatness and simplicity of parallel or horizontal perspective.</p>
+
+<p class = "illustration">
+<a name = "fig82" id = "fig82"> </a>
+<img src = "images/fig82.png" width = "336" height = "386"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 82.</span> Courtyard by De Hoogh.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXX" id = "chapXXX">
+XXX</a></h5>
+
+<h5 class = "smallcaps">Landscape Perspective</h5>
+
+
+<p>In an extended view or landscape seen from a height, we have to
+consider the perspective plane as in a great measure lying above it,
+reaching from the base of the picture to the horizon; but of course
+pierced here and there by trees, mountains, buildings, &amp;c. As a rule
+in such cases, we copy our perspective from nature, and do not trouble
+ourselves much about mathematical rules. It is as well, however, to know
+them, so that we may feel sure we are right, as this gives certainty to
+our touch and enables us to work with freedom. Nor must we, when
+painting from nature, forget to take into account the effects of
+atmosphere and the various tones of the different planes of distance,
+for this makes much of the difference between a good picture and a bad
+one; being a more subtle quality, it requires a keener artistic sense to
+discover and depict it. (See <a href = "#fig95">Figs. 95</a> and <a href
+= "#fig103">103</a>.)</p>
+
+<p>If the landscape painter wishes to test his knowledge of perspective,
+let him dissect and work out one of Turner's pictures, or better still,
+put his own sketch from nature to the same test.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXXI" id = "chapXXXI">
+XXXI</a></h5>
+
+<h5 class = "smallcaps">Figures of Different Heights</h5>
+
+<h5 class = "smallcaps">The Chessboard</h5>
+
+
+<p>In this figure the same principle is applied as in the previous one,
+but the chessmen being of different heights we have to arrange the scale
+accordingly. First ascertain the exact height of each piece, as <span
+class = "smallroman">Q, K, B</span>, which represent the queen, king,
+bishop, &amp;c. Refer these dimensions to the scale, as shown at QKB,
+which will give us the perspective measurement of each piece according
+to the square on which it is placed.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[75]</span>
+<a name = "page75" id = "page75"> </a>
+<!--png 091-->
+<a name = "fig83" id = "fig83"> </a>
+<img src = "images/fig83.png" width = "508" height = "188"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 83.</span> Chessboard and Men.</p>
+
+<p><span class = "pagenum">76</span>
+<a name = "page76" id = "page76"> </a>
+<!--png 092-->
+This is shown in the above drawing (Fig. 83) in the case of the white
+queen and the black queen, &amp;c. The castle, the knight, and the pawn
+being about the same height are measured from the fourth line of the
+scale marked <span class = "smallroman">C</span>.</p>
+
+<p class = "illustration">
+<a name = "fig84" id = "fig84"> </a>
+<img src = "images/fig84.png" width = "337" height = "393"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 84.</p>
+
+
+
+
+<span class = "pagenum">77</span>
+<a name = "page77" id = "page77"> </a>
+<!--png 093-->
+<h5 class = "section"><a name = "chapXXXII" id = "chapXXXII">
+XXXII</a></h5>
+
+<h5 class = "smallcaps">Application of the Vanishing Scale to Drawing
+Figures at an Angle when their Vanishing Points are Inaccessible or
+Outside the Picture</h5>
+
+
+<p>This is exemplified in the drawing of a fence (Fig. 84). Form scale
+<i>a</i><span class = "smallroman">S</span>, <i>b</i><span class =
+"smallroman">S</span>, in accordance with the height of the fence or
+wall to be depicted. Let <i>ao</i> represent the direction or angle at
+which it is placed, draw <i>od</i> to meet the scale at <i>d</i>, at
+<i>d</i> raise vertical <i>dc</i>, which gives the height of the fence
+at <i>oo�</i>. Draw lines <i>bo�</i>, <i>eo</i>, <i>ao</i>, &amp;c., and
+it will be found that all these lines if produced will meet at the same
+point on the horizon. To divide the fence into spaces, divide base line
+<i>af</i> as required and proceed as already shown.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXXIII" id = "chapXXXIII">
+XXXIII</a></h5>
+
+<h5 class = "smallcaps">The Reduced Distance. How to Proceed when the
+Point of Distance is Inaccessible</h5>
+
+
+<p>It has already been shown that too near a point of distance is
+objectionable on account of the distortion and disproportion resulting
+from it. At the same time, the long distance-point must be some way out
+of the picture and therefore inconvenient. The object of the reduced
+distance is to bring that point within the picture.</p>
+
+<p class = "illustration">
+<a name = "fig85" id = "fig85"> </a>
+<img src = "images/fig85.png" width = "333" height = "101"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 85.</p>
+
+<p>In Fig. 85 we have made the distance nearly twice the length of the
+base of the picture, and consequently a long way out of it. Draw <span
+class = "smallroman">S</span><i>a</i>, <span class =
+"smallroman">S</span><i>b</i>, and from <i>a</i> draw <i>a</i><span
+class = "smallroman">D</span> to point of distance, which cuts <span
+class = "smallroman">S</span><i>b</i> at <i>o</i>, and determines the
+depth of the square <i>acob</i>. But
+<span class = "pagenum">78</span>
+<a name = "page78" id = "page78"> </a>
+<!--png 094-->
+we can find that same point if we take half the base and draw a line
+from � base to � distance. But even this � distance-point does not come
+inside the picture, so we take a fourth of the base and a fourth of the
+distance and draw a line from � base to � distance. We shall find that
+it passes precisely through the same point <i>o</i> as the other lines
+<i>a</i><span class = "smallroman">D</span>, &amp;c. We are thus able to
+find the required point <i>o</i> without going outside the picture.</p>
+
+<p>Of course we could in the same way take an 8th or even a 16th
+distance, but the great use of this reduced distance, in addition to the
+above, is that it enables us to measure any depth into the picture with
+the greatest ease.</p>
+
+<p>It will be seen in the next figure that without having to extend the
+base, as is usually done, we can multiply that base to any amount by
+making use of these reduced distances on the horizontal line. This is
+quite a new method of proceeding, and it will be seen is mathematically
+correct.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXXIV" id = "chapXXXIV">
+XXXIV</a></h5>
+
+<h5 class = "smallcaps">How to Draw a Long Passage or Cloister by means
+of the Reduced Distance</h5>
+
+
+<p class = "illustration">
+<a name = "fig86" id = "fig86"> </a>
+<img src = "images/fig86.png" width = "337" height = "118"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 86.</p>
+
+<p>In Fig. 86 we have divided the base of the first square into four
+equal parts, which may represent so many feet, so that <span class =
+"smallroman">A</span>4 and <span class = "smallroman">B</span><i>d</i>
+being the retreating sides of the square each represents 4&nbsp;feet.
+But we found point � <span class = "smallroman">D</span> by drawing 3D
+from � base to � distance, and by proceeding in the same way from each
+division,
+<span class = "pagenum">79</span>
+<a name = "page79" id = "page79"> </a>
+<!--png 095-->
+<span class = "smallroman">A</span>,&nbsp;1, 2,&nbsp;3, we mark off on
+<span class = "smallroman">SB</span> four spaces each equal to
+4&nbsp;feet, in all 16 feet, so that by taking the whole base and the �
+distance we find point <span class = "smallroman">O</span>, which is
+distant four times the length of the base <span class =
+"smallroman">AB</span>. We can multiply this distance to any amount by
+drawing other diagonals to 8th distance, &amp;c. The same rule applies
+to this corridor (Fig. 87 and Fig.&nbsp;88).</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig87" id = "fig87"> </a>
+<img src = "images/fig87.png" width = "157" height = "213"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig88" id = "fig88"> </a>
+<img src = "images/fig88.png" width = "178" height = "235"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 87.</td>
+<td class = "caption smallcaps">
+Fig. 88.</td>
+</tr>
+</table>
+
+
+
+<h5 class = "section"><a name = "chapXXXV" id = "chapXXXV">
+XXXV</a></h5>
+
+<h5 class = "smallcaps">How to Form a Vanishing Scale that shall give
+the Height, Depth, and Distance of any Object in the Picture</h5>
+
+
+<p>If we make our scale to vanish to the point of sight, as in Fig. 89,
+we can make <span class = "smallroman">SB</span>, the lower line
+thereof, a&nbsp;measuring line for distances. Let us first of all divide
+the base <span class = "smallroman">AB</span> into eight parts, each
+part representing 5&nbsp;feet. From each division draw lines to 8th
+distance; by their intersections with <span class =
+"smallroman">SB</span> we obtain
+<span class = "pagenum">80</span>
+<a name = "page80" id = "page80"> </a>
+<!--png 096-->
+measurements of 40, 80, 120, 160, &amp;c., feet. Now divide the side of
+the picture <span class = "smallroman">BE</span> in the same manner as
+the base, which gives us the height of 40 feet. From the side <span
+class = "smallroman">BE</span> draw lines 5<span class =
+"smallroman">S</span>, 15<span class = "smallroman">S</span>, &amp;c.,
+to point of sight, and from each division on the base line also draw
+lines 5<span class = "smallroman">S</span>, 10<span class =
+"smallroman">S</span>, 15<span class = "smallroman">S</span>, &amp;c.,
+to point of sight, and from each division on <span class =
+"smallroman">SB</span>, such as 40, 80, &amp;c., draw horizontals
+parallel to base. We thus obtain squares 40 feet wide, beginning at base
+<span class = "smallroman">AB</span> and reaching as far as required.
+Note how the height of the flagstaff, which is 140 feet high and 280
+feet distant, is obtained. So also any buildings or other objects can be
+measured, such as those shown on the left of the picture.</p>
+
+<p class = "illustration">
+<a name = "fig89" id = "fig89"> </a>
+<img src = "images/fig89.png" width = "336" height = "327"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 89.</p>
+
+
+
+
+<span class = "pagenum">81</span>
+<a name = "page81" id = "page81"> </a>
+<!--png 097-->
+<h5 class = "section"><a name = "chapXXXVI" id = "chapXXXVI">
+XXXVI</a></h5>
+
+<h5 class = "smallcaps">Measuring Scale on Ground</h5>
+
+
+<p>A simple and very old method of drawing buildings, &amp;c., and
+giving them their right width and height is by means of squares of a
+given size, drawn on the ground.</p>
+
+<p class = "illustration">
+<a name = "fig90" id = "fig90"> </a>
+<img src = "images/fig90.png" width = "335" height = "334"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 90.</p>
+
+<p>In the above sketch (Fig. 90) the squares on the ground
+<span class = "pagenum">84</span>
+<a name = "page84" id = "page84"> </a>
+<!--png 100-->
+represent 3&nbsp;feet each way, or one square yard. Taking this as our
+standard measure, we find the door on the left is 10 feet high, that the
+archway at the end is 21 feet high and 12 feet wide, and so&nbsp;on.</p>
+
+<p>Fig. 91 is a sketch made at Sandwich, Kent, and shows a somewhat
+similar subject to <a href = "#fig84">Fig.&nbsp;84</a>, but the
+irregularity and freedom of the perspective gives it a charm far beyond
+the rigid precision of the other, while it conforms to its main laws.
+This sketch, however, is the real artist's perspective, or what we might
+term natural perspective.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[82]</span>
+<a name = "page82" id = "page82"> </a>
+<!--png 098-->
+<a name = "fig91" id = "fig91"> </a>
+<img src = "images/fig91.png" width = "335" height = "450"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 91.</span> Natural Perspective.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXXVII" id = "chapXXXVII">
+XXXVII</a></h5>
+
+<h5 class = "smallcaps">Application of the Reduced Distance and the
+Vanishing Scale to Drawing a Lighthouse, &amp;c.</h5>
+
+
+<p>In the drawing of Honfleur (Fig. 92) we divide the base <span class =
+"smallroman">AB</span> as
+<span class = "pagenum">85</span>
+<a name = "page85" id = "page85"> </a>
+<!--png 101-->
+in the previous figure, but the spaces measure 5&nbsp;feet instead of
+3&nbsp;feet: so that taking the 8th distance, the divisions on the
+vanishing line <span class = "smallroman">BS</span> measure 40 feet
+each, and at point <span class = "smallroman">O</span> we have 400 feet
+of distance, but we require 800. So we again reduce the distance to a
+16th. We thus multiply the base by 16. Now let us take a base of 50 feet
+at <i>f</i> and draw line <i>f</i><span class = "smallroman">D</span> to
+16th distance; if we multiply 50 feet by 16 we obtain the 800 feet
+required.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[83]</span>
+<a name = "page83" id = "page83"> </a>
+<!--png 099-->
+<a name = "fig92" id = "fig92"> </a>
+<img src = "images/fig92.png" width = "445" height = "314"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 92.</span> Honfleur.</p>
+
+<p>The height of the lighthouse is found by means of the vanishing
+scale, which is 15 feet below and 15 feet above the horizon, or 30 feet
+from the sea-level. At <span class = "smallroman">L</span> we raise a
+vertical <span class = "smallroman">LM</span>, which shows the position
+of the lighthouse. Then on that vertical measure the height required as
+shown in the figure.</p>
+
+<p class = "caption">Perspective of a lighthouse 135 feet high at 800
+feet distance.</p>
+
+<p class = "illustration">
+<a name = "fig93" id = "fig93"> </a>
+<img src = "images/fig93.png" width = "334" height = "172"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 93.</span> Key to Fig. 92, Honfleur.</p>
+
+<p>The 800 feet could be obtained at once by drawing line <i>f</i><span
+class = "smallroman">D</span>, or 50 feet, to 16th distance. The other
+measurements obtained by 8th distance serve for nearer buildings.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXXXVIII" id = "chapXXXVIII">
+XXXVIII</a></h5>
+
+<h5 class = "smallcaps">How to Measure Long Distances such as a Mile or
+Upwards</h5>
+
+
+<p>The wonderful effect of distance in Turner's pictures is not to be
+achieved by mere measurement, and indeed can only be properly done by
+studying Nature and drawing her perspective as she presents it to us. At
+the same time it is useful to be able to test and to set out distances
+in arranging a composition. This latter, if neglected, often leads to
+great difficulties and sometimes to repainting.</p>
+
+<p>To show the method of measuring very long distances we have to work
+with a very small scale to the foot, and in Fig. 94 I have divided the
+base <span class = "smallroman">AB</span> into eleven parts, each part
+representing 10 feet. First draw <span class = "smallroman">AS</span>
+and <span class = "smallroman">BS</span> to point of sight.
+<span class = "pagenum">86</span>
+<a name = "page86" id = "page86"> </a>
+<!--png 102-->
+From <span class = "smallroman">A</span> draw <span class =
+"smallroman">AD</span> to � distance, and we obtain at 440 on line <span
+class = "smallroman">BS</span> four times the length of <span class =
+"smallroman">AB</span>, or 110 feet �&nbsp;4 = 440 feet. Again, taking
+the whole base and drawing a line from S to 8th distance we obtain eight
+times 110 feet or 880 feet. If now we use the 16th distance we get
+sixteen times 110 feet, or 1,760 feet, one-third of a mile; by repeating
+this process, but by using the base at 1,760, which is the same length
+in perspective as <span class = "smallroman">AB</span>, we obtain 3,520
+feet, and then again using the base at 3,520 and proceeding in the same
+way we obtain 5,280 feet, or one mile to the archway. The flags show
+their heights at their respective distances from the base. By the scale
+at the side of the picture, <span class = "smallroman">BO</span>, we can
+measure any height above or any depth below the perspective plane.</p>
+
+<p class = "illustration">
+<a name = "fig94" id = "fig94"> </a>
+<img src = "images/fig94.png" width = "356" height = "187"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption"><span class = "smallcaps">
+Fig. 94.</span><br>
+<a href = "images/fig94large.png">larger view</a></p>
+
+<p><i>Note</i>.&mdash;This figure (here much reduced) should be drawn
+large by the student, so that the numbering, &amp;c., may be made more
+distinct. Indeed, many of the other figures should be copied large, and
+worked out with care, as lessons in perspective<ins class = "correction"
+title = "period missing">.</ins></p>
+
+
+
+
+<span class = "pagenum">87</span>
+<a name = "page87" id = "page87"> </a>
+<!--png 103-->
+<h5 class = "section"><a name = "chapXXXIX" id = "chapXXXIX">
+XXXIX</a></h5>
+
+<h5 class = "smallcaps">Further Illustration of Long Distances and
+Extended Views</h5>
+
+
+<p>An extended view is generally taken from an elevated position, so
+that the principal part of the landscape lies beneath the perspective
+plane, as already noted, and we shall presently treat of objects and
+figures on uneven ground. In the previous figure is shown how we can
+measure heights and depths to any extent. But when we turn to a drawing
+by Turner, such as the &lsquo;View from Richmond Hill&rsquo;, we feel
+that the only way to accomplish such perspective as this, is to go and
+draw it from nature, and even then to use our judgement, as he did, as
+to how much we may emphasize or even exaggerate certain features.</p>
+
+<p class = "illustration">
+<a name = "fig95" id = "fig95"> </a>
+<img src = "images/fig95.png" width = "344" height = "211"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 95.</span> Turner's View from Richmond
+Hill.</p>
+
+<p>Note in this view the foreground on which the principal figures stand
+is on a level with the perspective plane, while the river and
+surrounding park and woods are hundreds of feet below us
+<span class = "pagenum">88</span>
+<a name = "page88" id = "page88"> </a>
+<!--png 104-->
+and stretch away for miles into the distance. The contrasts obtained by
+this arrangement increase the illusion of space, and the figures in the
+foreground give as it were a standard of measurement, and by their
+contrast to the size of the trees show us how far away those trees
+are.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXL" id = "chapXL">
+XL</a></h5>
+
+<h5 class = "smallcaps">How to Ascertain the Relative Heights of Figures
+on an Inclined Plane</h5>
+
+
+<p>The three figures to the right marked <i>f</i>, <i>g</i>, <i>b</i>
+(Fig. 96) are on level ground, and we measure them by the vanishing
+scale <i>a</i><span class = "smallroman">S</span>, <i>b</i><span class =
+"smallroman">S</span>. Those to the left, which are repetitions of them,
+are on an inclined plane, the vanishing point of which is <span class =
+"smallroman">S�</span>; by the side of this plane we have placed another
+vanishing scale <i>a�</i><span class = "smallroman">S�</span>,
+<i>b�</i><span class = "smallroman">S�</span>, by which we measure the
+figures on that incline in the same way as on the level plane. It will
+be seen that if a horizontal line is drawn from the foot of one of these
+figures, say <span class = "smallroman">G</span>, to point <span class =
+"smallroman">O</span> on the edge of the incline, then dropped
+vertically to <i>o�</i>, then again carried on to <i>o��</i> where the
+other figure <i>g</i> is, we find it is the same height and also that
+the other vanishing scale is the same width at that distance, so that we
+can work from either one or the other. In the event of the rising ground
+being uneven we can make use of the scale on the level plane.</p>
+
+<p class = "illustration">
+<a name = "fig96" id = "fig96"> </a>
+<img src = "images/fig96.png" width = "336" height = "159"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 96.</p>
+
+
+
+
+<span class = "pagenum">89</span>
+<a name = "page89" id = "page89"> </a>
+<!--png 105-->
+<h5 class = "section"><a name = "chapXLI" id = "chapXLI">
+XLI</a></h5>
+
+<h5 class = "smallcaps">How to Find the Distance of a Given Figure or
+Point from the Base Line</h5>
+
+
+<p>Let <span class = "smallroman">P</span> be the given figure. Form
+scale <span class = "smallroman">ACS</span>, <span class =
+"smallroman">S</span> being the point of sight and <span class =
+"smallroman">D</span> the distance. Draw horizontal <i>do</i> through
+<span class = "smallroman">P</span>. From <span class =
+"smallroman">A</span> draw diagonal <span class = "smallroman">AD</span>
+to distance point, cutting <i>do</i> in <i>o</i>, through <i>o</i> draw
+<span class = "smallroman">SB</span> to base, and we now have a square
+<span class = "smallroman">A</span><i>do</i><span class =
+"smallroman">B</span> on the perspective plane; and as figure <span
+class = "smallroman">P</span> is standing on the far side of that square
+it must be the distance <span class = "smallroman">AB</span>, which is
+one side of it, from the base line&mdash;or picture plane. For figures
+very far away it might be necessary to make use of half-distance.</p>
+
+<p class = "illustration">
+<a name = "fig97" id = "fig97"> </a>
+<img src = "images/fig97.png" width = "328" height = "87"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 97.</p>
+
+
+
+
+<span class = "pagenum">90</span>
+<a name = "page90" id = "page90"> </a>
+<!--png 106-->
+<h5 class = "section"><a name = "chapXLII" id = "chapXLII">
+XLII</a></h5>
+
+<h5 class = "smallcaps">How to Measure the Height of Figures on Uneven
+Ground</h5>
+
+
+<p>In previous problems we have drawn figures on level planes, which is
+easy enough. We have now to represent some above and some below the
+perspective plane.</p>
+
+<p class = "illustration">
+<a name = "fig98" id = "fig98"> </a>
+<img src = "images/fig98.png" width = "333" height = "347"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 98.</p>
+
+<p><span class = "pagenum">91</span>
+<a name = "page91" id = "page91"> </a>
+<!--png 107-->
+Form scale <i>b</i><span class = "smallroman">S</span>, <i>c</i><span
+class = "smallroman">S</span>; mark off distances 20 feet, 40 feet,
+&amp;c. Suppose figure <span class = "smallroman">K</span> to be 60 feet
+off. From point at his feet draw horizontal to meet vertical <span class
+= "smallroman">O</span><i>n</i>, which is 60 feet distant. At the point
+<i>m</i> where this line meets the vertical, measure height <i>mn</i>
+equal to width of scale at that distance, transfer this to <span class =
+"smallroman">K</span>, and you have the required height of the figure in
+black.</p>
+
+<p>For the figures under the cliff 20 feet below the perspective plane,
+form scale <span class = "smallroman">FS</span>, <span class =
+"smallroman">GS</span>, making it the same width as the other, namely
+5&nbsp;feet, and proceed in the usual way to find the height of the
+figures on the sands, which are here supposed to be nearly on a level
+with the sea, of course making allowance for different heights and
+various other things.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXLIII" id = "chapXLIII">
+XLIII</a></h5>
+
+<h5 class = "smallcaps">Further Illustration of the Size of Figures at
+Different Distances and on Uneven Ground</h5>
+
+
+<p><span class = "pagenum">92</span>
+<a name = "page92" id = "page92"> </a>
+<!--png 108-->
+Let <i>ab</i> be the height of a figure, say 6&nbsp;feet. First form
+scale <i>a</i><span class = "smallroman">S</span>, <i>b</i><span class =
+"smallroman">S</span>, the lower line of which, <i>a</i><span class =
+"smallroman">S</span>, is on a level with the base or on the perspective
+plane. The figure marked <span class = "smallroman">C</span> is close to
+base, the group of three is farther off (24 feet), and 6&nbsp;feet
+higher up, so we measure the height on the vanishing scale and also
+above it. The two girls carrying fish are still farther off, and about
+12 feet below. To tell how far a figure is away, refer its measurements
+to the vanishing scale (see <a href = "#fig96">Fig.&nbsp;96</a>).</p>
+
+<p class = "illustration">
+<a name = "fig99" id = "fig99"> </a>
+<img src = "images/fig99.png" width = "338" height = "198"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 99.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXLIV" id = "chapXLIV">
+XLIV</a></h5>
+
+<h5 class = "smallcaps">Figures on a Descending Plane</h5>
+
+
+<p>In this case (Fig. 100) the same rule applies as in the previous
+problem, but as the road on the left is going down hill, the vanishing
+point of the inclined plane is below the horizon at point <span class =
+"smallroman">S�</span>; <span class = "smallroman">AS</span>, <span
+class = "smallroman">BS</span> is the vanishing scale on the level
+plane; and <span class = "smallroman">A�S�</span>, <span class =
+"smallroman">B�S�</span>, that on the incline.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[93a]</span>
+<a name = "page93" id = "page93"> </a>
+<!--png 109-->
+<a name = "fig100" id = "fig100"> </a>
+<img src = "images/fig100.png" width = "453" height = "196"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 100.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[93b]</span>
+<!--png 109-->
+<a name = "fig101" id = "fig101"> </a>
+<img src = "images/fig101.png" width = "461" height = "131"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption"><span class = "smallcaps">
+Fig. 101.</span>
+This is an outline of above figure to show the working more plainly.</p>
+
+<p>Note the wall to the left marked <span class = "smallroman">W</span>
+and the manner in which it appears to drop at certain intervals, its
+base corresponding with the inclined plane, but the upper lines of each
+division being made level are drawn to the point of sight, or to their
+vanishing point on the horizon; it is important to observe this, as it
+aids greatly in drawing a road going down hill.</p>
+
+
+
+
+<span class = "pagenum">95</span>
+<a name = "page95" id = "page95"> </a>
+<!--png 111-->
+<h5 class = "section"><a name = "chapXLV" id = "chapXLV">
+XLV</a></h5>
+
+<h5 class = "smallcaps">Further Illustration of the Descending
+Plane</h5>
+
+
+<p>In the centre of this picture (Fig. 102) we suppose the road to be
+descending till it reaches a tunnel which goes under a road or leads to
+a river (like one leading out of the Strand near Somerset House). It is
+drawn on the same principle as the foregoing figure. Of course to see
+the road the spectator must get pretty near to it, otherwise it will be
+out of sight. Also a level plane must be shown, as by its contrast to
+the other we perceive that the latter is going down hill.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[94]</span>
+<a name = "page94" id = "page94"> </a>
+<!--png 110-->
+<a name = "fig102" id = "fig102"> </a>
+<img src = "images/fig102.png" width = "456" height = "280"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 102.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXLVI" id = "chapXLVI">
+XLVI</a></h5>
+
+<h5 class = "smallcaps">Further Illustration of Uneven Ground</h5>
+
+<p>An extended view drawn from a height of about 30 feet from a road
+that descends about 45 feet.</p>
+
+<p class = "illustration">
+<a name = "fig103" id = "fig103"> </a>
+<img src = "images/fig103.png" width = "328" height = "240"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 103.</span> Farningham.</p>
+
+<p><span class = "pagenum">96</span>
+<a name = "page96" id = "page96"> </a>
+<!--png 112-->
+In drawing a landscape such as Fig. 103 we have to bear in mind the
+height of the horizon, which being exactly opposite the eye, shows us at
+once which objects are below and which are above us, and to draw them
+accordingly, especially roofs, buildings, walls, hedges, &amp;c.; also
+it is well to sketch in the different fields figures of men and cattle,
+as from the size of these we can judge of the rest.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXLVII" id = "chapXLVII">
+XLVII</a></h5>
+
+<h5 class = "smallcaps">The Picture Standing on the Ground</h5>
+
+
+<p>Let <span class = "smallroman">K</span> represent a frame placed
+vertically and at a given distance in front of us. If stood on the
+ground our foreground will touch
+<span class = "pagenum">97</span>
+<a name = "page97" id = "page97"> </a>
+<!--png 113-->
+the base line of the picture, and we can fix up a standard of
+measurement both on the base and on the side as in this sketch, taking
+6&nbsp;feet as about the height of the figures.</p>
+
+<p class = "illustration">
+<a name = "fig104" id = "fig104"> </a>
+<img src = "images/fig104.png" width = "272" height = "335"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 104.</span> Toledo.</p>
+
+
+
+<h5 class = "section"><a name = "chapXLVIII" id = "chapXLVIII">
+XLVIII</a></h5>
+
+<h5 class = "smallcaps">The Picture on a Height</h5>
+
+
+<p>If we are looking at a scene from a height, that is from a terrace,
+or a window, or a cliff, then the near foreground, unless it be the
+terrace, window-sill, &amp;c., would not come into the picture, and we
+could not see the near figures at <span class = "smallroman">A</span>,
+and the nearest to come into view would be those at <span class =
+"smallroman">B</span>, so that a view from a window, &amp;c., would be
+as it were without a foreground. Note that the figures at <span class =
+"smallroman">B</span> would be (according to this sketch) 30 feet from
+the picture plane and about 18 feet below the base line.</p>
+
+<p class = "illustration">
+<a name = "fig105" id = "fig105"> </a>
+<img src = "images/fig105.png" width = "333" height = "148"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 105.</p>
+
+
+
+
+<span class = "pagenum">98</span>
+<a name = "page98" id = "page98"> </a>
+<!--png 114-->
+<h3 class = "chapter">BOOK THIRD</h3>
+
+<h5 class = "section"><a name = "chapXLIX" id = "chapXLIX">
+XLIX</a></h5>
+
+<h5 class = "smallcaps">Angular Perspective</h5>
+
+
+<p>Hitherto we have spoken only of parallel perspective, which is
+comparatively easy, and in our first figure we placed the cube with one
+of its sides either touching or parallel to the transparent plane. We
+now place it so that one angle only (<i>ab</i>), touches the
+picture.</p>
+
+<p class = "illustration">
+<a name = "fig106" id = "fig106"> </a>
+<img src = "images/fig106.png" width = "342" height = "217"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 106.</p>
+
+<p>Its sides are no longer drawn to the point of sight as in <a href =
+"#fig7">Fig.&nbsp;7</a>, nor its diagonal to the point of distance, but
+to some other points on the horizon, although the same rule holds good
+as regards their parallelism; as for instance, in the case of <i>bc</i>
+and <i>ad</i>, which, if produced, would meet at <span class =
+"smallroman">V</span>, a&nbsp;point on the horizon called a
+<span class = "pagenum">99</span>
+<a name = "page99" id = "page99"> </a>
+<!--png 115-->
+vanishing point. In this figure only one vanishing point is seen, which
+is to the right of the point of sight <span class =
+"smallroman">S</span>, whilst the other is some distance to the left,
+and outside the picture. If the cube is correctly drawn, it will be
+found that the lines <i>ae</i>, <i>bg</i>, &amp;c., if produced, will
+meet on the horizon at this other vanishing point. This far-away
+vanishing point is one of the inconveniences of oblique or angular
+perspective, and therefore it will be a considerable gain to the
+draughtsman if we can dispense with it. This can be easily done, as in
+the above figure, and here our geometry will come to our assistance, as
+I shall show presently.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapL" id = "chapL">
+L</a></h5>
+
+<h5 class = "smallcaps">How to put a Given Point into Perspective</h5>
+
+
+<p>Let us place the given point <span class = "smallroman">P</span> on a
+geometrical plane, to show how far it is from the base line, and indeed
+in the exact position we wish it to be in the picture. The geometrical
+plane is supposed to face us, to hang down, as it were, from the base
+line <span class = "smallroman">AB</span>, like the side of a table, the
+top of which represents the perspective plane. It is to that perspective
+plane that we now have to transfer the point <span class =
+"smallroman">P</span>.</p>
+
+<p class = "illustration">
+<a name = "fig107" id = "fig107"> </a>
+<img src = "images/fig107.png" width = "300" height = "131"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 107.</p>
+
+<p>From <span class = "smallroman">P</span> raise perpendicular <span
+class = "smallroman">P</span><i>m</i> till it touches the base line at
+<i>m</i>. With centre <i>m</i> and radius <i>m</i><span class =
+"smallroman">P</span> describe arc <span class =
+"smallroman">P</span><i>n</i> so that <i>mn</i> is now the same length
+as <i>m</i><span class = "smallroman">P</span>. As point <span class =
+"smallroman">P</span> is opposite point <i>m</i>, so
+<span class = "pagenum">100</span>
+<a name = "page100" id = "page100"> </a>
+<!--png 116-->
+must it be in the perspective, therefore we draw a line at right angles
+to the base, that is to the point of sight, and somewhere on this line
+will be found the required point <span class = "smallroman">P�</span>.
+We now have to find how far from <i>m</i> must that point be. It must be
+the length of <i>mn</i>, which is the same as <i>m</i><span class =
+"smallroman">P</span>. We therefore from <i>n</i> draw <i>n</i><span
+class = "smallroman">D</span> to the point of distance, which being at
+an angle of 45�, or half a right angle, makes <i>m</i><span class =
+"smallroman">P�</span> the perspective length of <i>mn</i> by its
+intersection with <i>m</i><span class = "smallroman">S</span>, and thus
+gives us the point <span class = "smallroman">P�</span>, which is the
+perspective of the original point.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLI" id = "chapLI">
+LI</a></h5>
+
+<h5 class = "smallcaps">A Perspective Point being given, Find its
+Position on the Geometrical Plane</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig108" id = "fig108"> </a>
+<img src = "images/fig108.png" width = "221" height = "151"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 108.</td>
+</tr>
+</table>
+
+<p>To do this we simply reverse the foregoing problem. Thus let <span
+class = "smallroman">P</span> be the given perspective point. From point
+of sight <span class = "smallroman">S</span> draw a line through <span
+class = "smallroman">P</span> till it cuts <span class =
+"smallroman">AB</span> at <i>m</i>. From distance <span class =
+"smallroman">D</span> draw another line through <span class =
+"smallroman">P</span> till it cuts the base at <i>n</i>. From <i>m</i>
+drop perpendicular, and then with centre <i>m</i> and radius <i>mn</i>
+describe arc, and where it cuts that perpendicular is the required point
+<span class = "smallroman">P�</span>. We often have to make use of this
+problem.</p>
+
+
+
+
+<span class = "pagenum">101</span>
+<a name = "page101" id = "page101"> </a>
+<!--png 117-->
+<h5 class = "section"><a name = "chapLII" id = "chapLII">
+LII</a></h5>
+
+<h5 class = "smallcaps">How to put a Given Line into Perspective</h5>
+
+
+<p>This is simply a question of putting two points into perspective,
+instead of one, or like doing the previous problem twice over, for the
+two points represent the two extremities of the line. Thus we have to
+find the perspective of <span class = "smallroman">A</span> and <span
+class = "smallroman">B</span>, namely <i>a�b�</i>. Join those points,
+and we have the line required.</p>
+
+<p class = "illustration">
+<a name = "fig109" id = "fig109"> </a>
+<img src = "images/fig109.png" width = "298" height = "160"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 109.</p>
+
+<p>If one end touches the base, as at <span class =
+"smallroman">A</span> (Fig. 110), then we have
+<span class = "pagenum">102</span>
+<a name = "page102" id = "page102"> </a>
+<!--png 118-->
+but to find one point, namely <i>b</i>. We also find the perspective of
+the angle <i>m</i><span class = "smallroman">AB</span>, namely the
+shaded triangle <i>m</i><span class = "smallroman">A</span><i>b</i>.
+Note also that the perspective triangle equals the geometrical
+triangle.</p>
+
+<p class = "illustration">
+<a name = "fig110" id = "fig110"> </a>
+<img src = "images/fig110.png" width = "292" height = "125"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 110.</p>
+
+<p>When the line required is parallel to the base line of the picture,
+then the perspective of it is also parallel to that base (see
+Rule&nbsp;3).</p>
+
+<p class = "illustration">
+<a name = "fig111" id = "fig111"> </a>
+<img src = "images/fig111.png" width = "261" height = "161"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 111.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLIII" id = "chapLIII">
+LIII</a></h5>
+
+<h5 class = "smallcaps">To Find the Length of a Given Perspective
+Line</h5>
+
+
+<p>A perspective line <span class = "smallroman">AB</span> being given,
+find its actual length and the angle at which it is placed.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig112" id = "fig112"> </a>
+<img src = "images/fig112.png" width = "233" height = "153"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 112.</td>
+</tr>
+</table>
+
+<p>This is simply the reverse of the previous problem. Let <span class =
+"smallroman">AB</span> be the given line. From distance <span class =
+"smallroman">D</span> through <span class = "smallroman">A</span> draw
+<span class = "smallroman">DC</span>, and from <span class =
+"smallroman">S</span>, point of sight, through <span class =
+"smallroman">A</span> draw <span class = "smallroman">SO</span>. Drop
+<span class = "smallroman">OP</span> at right angles to base, making it
+equal to <span class = "smallroman">OC</span>. Join <span class =
+"smallroman">PB</span>, and line <span class = "smallroman">PB</span> is
+the actual length of <span class = "smallroman">AB</span>.</p>
+
+<p><span class = "pagenum">103</span>
+<a name = "page103" id = "page103"> </a>
+<!--png 119-->
+This problem is useful in finding the position of any given line or
+point on the perspective plane.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLIV" id = "chapLIV">
+LIV</a></h5>
+
+<h5 class = "smallcaps">To Find these Points when the Distance-Point is
+Inaccessible</h5>
+
+
+<p>If the distance-point is a long way out of the picture, then the same
+result can be obtained by using the half distance and half base, as
+already shown.</p>
+
+<p class = "illustration">
+<a name = "fig113" id = "fig113"> </a>
+<img src = "images/fig113.png" width = "303" height = "146"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 113.</p>
+
+<p><span class = "pagenum">104</span>
+<a name = "page104" id = "page104"> </a>
+<!--png 120-->
+From <i>a</i>, half of <i>m</i><span class = "smallroman">P�</span>,
+draw quadrant <i>ab</i>, from <i>b</i> (half base), draw line from
+<i>b</i> to half Dist., which intersects <span class =
+"smallroman">S</span><i>m</i> at <span class = "smallroman">P</span>,
+precisely the same point as would be obtained by using the whole
+distance.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLV" id = "chapLV">
+LV</a></h5>
+
+<h5 class = "smallcaps">How to put a Given Triangle or other Rectilineal
+Figure into Perspective</h5>
+
+
+<p>Here we simply put three points into perspective to obtain the given
+triangle <span class = "smallroman">A</span>, or five points to obtain
+the five-sided figure at <span class = "smallroman">B</span>. So can we
+deal with any number of figures placed at any angle.</p>
+
+<p class = "illustration">
+<a name = "fig114" id = "fig114"> </a>
+<img src = "images/fig114.png" width = "330" height = "152"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 114.</p>
+
+<p>Both the above figures are placed in the same diagram, showing how
+any number can be drawn by means of the same point of sight and the same
+point of distance, which makes them belong to the same picture.</p>
+
+<p>It is to be noted that the figures appear reversed in the
+perspective. That is, in the geometrical triangle the base at <i>ab</i>
+is uppermost, whereas in the perspective <i>ab</i> is lowermost, yet
+both are nearest to the ground line.</p>
+
+
+
+
+<span class = "pagenum">105</span>
+<a name = "page105" id = "page105"> </a>
+<!--png 121-->
+<h5 class = "section"><a name = "chapLVI" id = "chapLVI">
+LVI</a></h5>
+
+<h5 class = "smallcaps">How to put a Given Square into Angular
+Perspective</h5>
+
+
+<p>Let <span class = "smallroman">ABCD</span> (Fig. 115) be the given
+square on the geometrical plane, where we can place it as near or as far
+from the base and at any angle that we wish. We then proceed to find its
+perspective on the picture by finding the perspective of the four points
+<span class = "smallroman">ABCD</span> as already shown. Note that the
+two sides of the perspective square <i>dc</i> and <i>ab</i> being
+produced, meet at point <span class = "smallroman">V</span> on the
+horizon, which is their vanishing point, but to find the point on the
+horizon where sides <i>bc</i> and <i>ad</i> meet, we should have to go a
+long way to the left of the figure, which by this method is not
+necessary.</p>
+
+<p class = "illustration">
+<a name = "fig115" id = "fig115"> </a>
+<img src = "images/fig115.png" width = "348" height = "238"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 115.</p>
+
+
+
+
+<span class = "pagenum">106</span>
+<a name = "page106" id = "page106"> </a>
+<!--png 122-->
+<h5 class = "section"><a name = "chapLVII" id = "chapLVII">
+LVII</a></h5>
+
+<h5 class = "smallcaps">Of Measuring Points</h5>
+
+
+<p>We now have to find certain points by which to measure those
+vanishing or retreating lines which are no longer at right angles to the
+picture plane, as in parallel perspective, and have to be measured in a
+different way, and here geometry comes to our assistance.</p>
+
+<p class = "illustration">
+<a name = "fig116" id = "fig116"> </a>
+<img src = "images/fig116.png" width = "335" height = "146"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 116.</p>
+
+<p>Note that the perspective square <span class = "smallroman">P</span>
+equals the geometrical square <span class = "smallroman">K</span>, so
+that side <span class = "smallroman">AB</span> of the one equals side
+<i>ab</i> of the other. With centre <span class = "smallroman">A</span>
+and radius <span class = "smallroman">AB</span> describe arc <span class
+= "smallroman">B</span><i>m�</i> till it cuts the base line at
+<i>m�</i>. Now <span class = "smallroman">AB</span> = <span class =
+"smallroman">A</span><i>m�</i>, and if we join <i>bm�</i> then triangle
+<span class = "smallroman">BA</span><i>m�</i> is an isosceles triangle.
+So likewise if we join <i>m�b</i> in the perspective figure will
+<i>m�</i><span class = "smallroman">A</span><i>b</i> be the same
+isosceles triangle in perspective. Continue line <i>m�b</i> till it cuts
+the horizon in <i>m</i>, which point will be the measuring point for the
+vanishing line <span class = "smallroman">A</span><i>b</i><span class =
+"smallroman">V</span>. For if in an isosceles triangle we draw lines
+across it, parallel to its base from one side to the other, we divide
+both sides in exactly the same quantities and proportions, so that if we
+measure on the base line of the picture the spaces we require, such as
+1,&nbsp;2,&nbsp;3, on the length <span class =
+"smallroman">A</span><i>m�</i>, and then from these divisions draw lines
+to
+<span class = "pagenum">107</span>
+<a name = "page107" id = "page107"> </a>
+<!--png 123-->
+the measuring point, these lines will intersect the vanishing line <span
+class = "smallroman">A</span><i>b</i><span class = "smallroman">V</span>
+in the lengths and proportions required. To find a measuring point for
+the lines that go to the other vanishing point, we proceed in the same
+way. Of course great accuracy is necessary.</p>
+
+<p>Note that the dotted lines 1,1, 2,2, &amp;c., are parallel in the
+perspective, as in the geometrical figure. In the former the lines are
+drawn to the same point <i>m</i> on the horizon.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLVIII" id = "chapLVIII">
+LVIII</a></h5>
+
+<h5 class = "smallcaps">How to Divide any Given Straight Line into Equal
+or Proportionate Parts</h5>
+
+
+<p>Let <span class = "smallroman">AB</span> (Fig. 117) be the given
+straight line that we wish to divide into five equal parts. Draw <span
+class = "smallroman">AC</span> at any convenient angle, and measure off
+five equal parts with the compasses thereon, as 1, 2, 3, 4, 5. From
+5<span class = "smallroman">C</span> draw line to 5<span class =
+"smallroman">B</span>. Now from each division on <span class =
+"smallroman">AC</span> draw lines 4,&nbsp;4, 3,&nbsp;3, &amp;c.,
+parallel to 5,5. Then <span class = "smallroman">AB</span> will be
+divided into the required number of equal parts.</p>
+
+<p class = "illustration">
+<a name = "fig117" id = "fig117"> </a>
+<img src = "images/fig117.png" width = "226" height = "69"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 117.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLIX" id = "chapLIX">
+LIX</a></h5>
+
+<h5 class = "smallcaps">How to Divide a Diagonal Vanishing Line into any
+Number of Equal or Proportional Parts</h5>
+
+
+<p>In a previous figure (<a href = "#fig116">Fig. 116</a>) we have shown
+how to find a measuring point when the exact measure of a vanishing line
+is required, but if it suffices merely to divide a line into a given
+number of equal parts, then the following simple method can be
+adopted.</p>
+
+<p><span class = "pagenum">108</span>
+<a name = "page108" id = "page108"> </a>
+<!--png 124-->
+We wish to divide <i>ab</i> into five equal parts. From <i>a</i>,
+measure off on the ground line the five equal spaces required.
+From&nbsp;5, the point to which these measures extend (as they are taken
+at random), draw a line through <i>b</i> till it cuts the horizon at
+<span class = "smallroman">O</span>. Then proceed to draw lines from
+each division on the base to point <span class = "smallroman">O</span>,
+and they will intersect and divide <i>ab</i> into the required number of
+equal parts.</p>
+
+<p class = "illustration">
+<a name = "fig118" id = "fig118"> </a>
+<img src = "images/fig118.png" width = "338" height = "133"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 118.</p>
+
+<p>The same method applies to a given line to be divided into various
+proportions, as shown in this lower figure.</p>
+
+<p class = "illustration">
+<a name = "fig119" id = "fig119"> </a>
+<img src = "images/fig119.png" width = "344" height = "127"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 119.</p>
+
+
+<span class = "pagenum">110</span>
+<a name = "page110" id = "page110"> </a>
+<!--png 126-->
+
+<h5 class = "section"><a name = "chapLX" id = "chapLX">
+LX</a></h5>
+
+<h5 class = "smallcaps">Further Use of the Measuring Point O</h5>
+
+
+<p>One square in oblique or angular perspective being given, draw any
+number of other squares equal to it by means of this point <span class =
+"smallroman">O</span> and the diagonals.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[109a]</span>
+<a name = "page109" id = "page109"> </a>
+<!--png 125-->
+<a name = "fig120" id = "fig120"> </a>
+<img src = "images/fig120.png" width = "347" height = "97"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 120.</p>
+
+<p>Let <span class = "smallroman">ABCD</span> (Fig. 120) be the given
+square; produce its sides <span class = "smallroman">AB</span>, <span
+class = "smallroman">DC</span> till they meet at point <span class =
+"smallroman">V</span>. From <span class = "smallroman">D</span> measure
+off on base any number of equal spaces of any convenient length, as 1,
+2, 3, &amp;c.; from&nbsp;1, through corner of square <span class =
+"smallroman">C</span>, draw a line to meet the horizon at <span class =
+"smallroman">O</span>, and from <span class = "smallroman">O</span> draw
+lines to the several divisions on base line. These lines will divide the
+vanishing line <span class = "smallroman">DV</span> into the required
+number of parts equal to <span class = "smallroman">DC</span>, the side
+of the square. Produce the diagonal of the square <span class =
+"smallroman">DB</span> till it cuts the horizon at <span class =
+"smallroman">G</span>. From the divisions on line <span class =
+"smallroman">DV</span> draw diagonals to point <span class =
+"smallroman">G</span>: their intersections with the other vanishing line
+<span class = "smallroman">AV</span> will determine the direction of the
+cross-lines which form the bases of other squares without the necessity
+of drawing them to the other vanishing point, which in this case is some
+distance to the left of the picture. If we produce these cross-lines to
+the horizon we shall find that they all meet at the other vanishing
+point, to which of course it is easy to draw them when that point is
+accessible, as in Fig. 121; but if it is too far out of the picture,
+then this method enables us to do without&nbsp;it.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[109b]</span>
+<!--png 125-->
+<a name = "fig121" id = "fig121"> </a>
+<img src = "images/fig121.png" width = "409" height = "75"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 121.</p>
+
+<p>Figure 121 corroborates the above by showing the two vanishing points
+and additional squares. Note the working of the diagonals drawn to point
+<span class = "smallroman">G</span> <!--not small-capped in print-->, in
+both figures.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXI" id = "chapLXI">
+LXI</a></h5>
+
+<h5 class = "smallcaps">Further Use of the Measuring Point O</h5>
+
+
+<p>Suppose we wish to divide the side of a building, as in Fig. 123, or
+to draw a balcony, a&nbsp;series of windows, or columns, or what not,
+or, in other words, any line above the horizon, as <span class =
+"smallroman">AB</span>. Then from <span class = "smallroman">A</span> we
+draw <span class = "smallroman">AC</span> parallel to the horizon, and
+mark thereon
+<span class = "pagenum">111</span>
+<a name = "page111" id = "page111"> </a>
+<!--png 127-->
+the required divisions 5, 10, 15, &amp;c.: in this case twenty-five
+(Fig. 122). From <span class = "smallroman">C</span> draw a line through
+<span class = "smallroman">B</span> till it cuts the horizon at <span
+class = "smallroman">O</span>. Then proceed to draw the other lines from
+each division to <span class = "smallroman">O</span>, and thus divide
+the vanishing line <span class = "smallroman">AB</span> as required.</p>
+
+<p class = "illustration">
+<a name = "fig122" id = "fig122"> </a>
+<img src = "images/fig122.png" width = "331" height = "213"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption"><span class = "smallcaps">Fig. 122</span> is a
+front view of the portico, Fig. 123.</p>
+
+<p><span class = "pagenum">112</span>
+<a name = "page112" id = "page112"> </a>
+<!--png 128-->
+In this portico there are thirteen triglyphs with twelve spaces between
+them, making twenty-five divisions. The required number of parts to draw
+the columns can be obtained in the same way.</p>
+
+<p class = "illustration">
+<a name = "fig123" id = "fig123"> </a>
+<img src = "images/fig123.png" width = "316" height = "145"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 123.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXII" id = "chapLXII">
+LXII</a></h5>
+
+<h5 class = "smallcaps">Another Method of Angular Perspective, being
+that Adopted in our Art Schools</h5>
+
+
+<p>In the previous method we have drawn our squares by means of a
+geometrical plan, putting each point into perspective as required, and
+then by means of the perspective drawing thus obtained, finding our
+vanishing and measuring points. In this method we proceed in exactly the
+opposite way, setting out our points first, and drawing the square (or
+other figure) afterwards.</p>
+
+<p class = "illustration">
+<a name = "fig124" id = "fig124"> </a>
+<img src = "images/fig124.png" width = "335" height = "164"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 124.</p>
+
+<p>Having drawn the horizontal and base lines, and fixed upon the
+position of the point of sight, we next mark the position of the
+spectator by dropping a perpendicular, <span class =
+"smallcaps">S&nbsp;ST</span>, from that point of sight, making it the
+same length as the distance we suppose the spectator to be from the
+picture, and thus we make <span class = "smallroman">ST</span> the
+station-point.</p>
+
+<p><span class = "pagenum">113</span>
+<a name = "page113" id = "page113"> </a>
+<!--png 129-->
+To understand this figure we must first look upon it as a ground-plan or
+bird&rsquo;s-eye view, the line <span class =
+"smallroman">V</span><sup>2</sup><span class =
+"smallroman">V</span><sup>1</sup> or horizon line representing the
+picture seen edgeways, because of course the station-point cannot be in
+the picture itself, but a certain distance in front of it. The angle at
+<span class = "smallroman">ST</span>, that is the angle which decides
+the positions of the two vanishing points <span class =
+"smallroman">V</span><sup>1</sup>, <span class =
+"smallroman">V</span><sup>2</sup>, is always a right angle, and the two
+remaining angles on that side of the line, called the directing line,
+are together equal to a right angle or 90�. So that in fixing upon the
+angle at which the square or other figure is to be placed, we say
+&lsquo;let it be 60� and 30�, or 70� and 20�&rsquo;, &amp;c. Having
+decided upon the station-point and the angle at which the square is to
+be placed, draw <span class = "smallroman">TV</span><sup>1</sup> and
+<span class = "smallroman">TV</span><sup>2</sup>, till they cut the
+horizon at <span class = "smallroman">V</span><sup>1</sup> and <span
+class = "smallroman">V</span><sup>2</sup>. These are the two vanishing
+points to which the sides of the figure are respectively drawn. But we
+still want the measuring points for these two vanishing lines. We
+therefore take first, <span class = "smallroman">V</span><sup>1</sup> as
+centre and <span class = "smallroman">V</span><sup>1</sup><span class =
+"smallroman">T</span> as radius, and describe arc of circle till it cuts
+the horizon in <span class = "smallroman">M</span><sup>1</sup>, which is
+the measuring point for all lines drawn to <span class =
+"smallroman">V</span><sup>1</sup>. Then with radius <span class =
+"smallroman">V</span><sup>2</sup><span class = "smallroman">T</span>
+describe arc from centre <span class = "smallroman">V</span><sup>2</sup>
+till it cuts the horizon in <span class =
+"smallroman">M</span><sup>2</sup>, which is the measuring point for all
+vanishing lines drawn to <span class =
+"smallroman">V</span><sup>2</sup>. We have now set out our points. Let
+us proceed to draw the square <span class =
+"smallroman">A</span><i>bcd</i>. From <span class =
+"smallroman">A</span>, the nearest angle (in this instance touching the
+base line), measure on each side of it the equal lengths <span class =
+"smallroman">AB</span> and <span class = "smallroman">AE</span>, which
+represent the width or side of the square. Draw <span class =
+"smallroman">EM</span><sup>2</sup> and <span class =
+"smallroman">BM</span><sup>1</sup> from the two measuring points, which
+give us, by their intersections with the vanishing lines <span class =
+"smallroman">AV</span><sup>1</sup> and <span class =
+"smallroman">AV</span><sup>2</sup>, the perspective lengths of the sides
+of the square <span class = "smallroman">A</span><i>bcd</i>. Join
+<i>b</i> and <span class = "smallroman">V</span><sup>1</sup> and
+<i>d</i><span class = "smallroman">V</span><sup>2</sup>, which intersect
+each other at <span class = "smallroman">C</span>, then <span class =
+"smallroman">A</span><i>dcb</i> is the square required.</p>
+
+<p>This method, which is easy when you know it, has certain drawbacks,
+the chief one being that if we require a long-distance point, and a
+small angle, such as 10� on one side, and 80� on the other, then the
+size of the diagram becomes so large that it has to be carried out on
+the floor of the studio with long strings, &amp;c., which is a very
+clumsy and unscientific way of setting to work. The architects in such
+cases make use of the centrolinead, a&nbsp;clever mechanical contrivance
+for getting over the difficulty of the far-off vanishing point, but by
+the method I have shown you, and shall further illustrate, you will find
+that you can dispense with
+<span class = "pagenum">114</span>
+<a name = "page114" id = "page114"> </a>
+<!--png 130-->
+all this trouble, and do all your perspective either inside the picture
+or on a very small margin outside&nbsp;it.</p>
+
+<p>Perhaps another drawback to this method is that it is not
+self-evident, as in the former one, and being rather difficult to
+explain, the student is apt to take it on trust, and not to trouble
+about the reasons for its construction: but to show that it is equally
+correct, I&nbsp;will draw the two methods in one figure.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXIII" id = "chapLXIII">
+LXIII</a></h5>
+
+<h5 class = "smallcaps">Two Methods of Angular Perspective in one
+Figure</h5>
+
+
+<p><span class = "pagenum">115</span>
+<a name = "page115" id = "page115"> </a>
+<!--png 131-->
+It matters little whether the station-point is placed above or below the
+horizon, as the result is the same. In Fig. 125 it is placed above, as
+the lower part of the figure is occupied with the geometrical plan of
+the other method.</p>
+
+<p class = "illustration">
+<a name = "fig125" id = "fig125"> </a>
+<img src = "images/fig125.png" width = "329" height = "303"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 125.</p>
+
+<p>In each case we make the square <span class = "smallroman">K</span>
+the same size and at the same angle, its near corner being at <span
+class = "smallroman">A</span>. It must be seen that by whichever method
+we work out this perspective, the result is the same, so that both are
+correct: the great advantage of the first or geometrical system being,
+that we can place the square at any angle, as it is drawn without
+reference to vanishing points.</p>
+
+<p>We will, however, work out a few figures by the second method.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXIV" id = "chapLXIV">
+LXIV</a></h5>
+
+<h5 class = "smallcaps">To Draw a Cube, the Points being Given</h5>
+
+
+<p>As in a previous figure (<a href = "#fig124">124</a>) we found the
+various working points of angular perspective, we need now merely
+transfer them to the horizontal line in this figure, as in this case
+they will answer our purpose perfectly well.</p>
+
+<p class = "illustration">
+<a name = "fig126" id = "fig126"> </a>
+<img src = "images/fig126.png" width = "341" height = "86"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 126.</p>
+
+<p>Let <span class = "smallroman">A</span> be the nearest angle touching
+the base. Draw <span class = "smallroman">AV</span><sup>1</sup>, <span
+class = "smallroman">AV</span><sup>2</sup>. From <span class =
+"smallroman">A</span>, raise vertical <span class =
+"smallroman">A</span><i>e</i>, the height of the cube. From <i>e</i>
+draw <i>e</i><span class = "smallroman">V</span><sup>1</sup>,
+<i>e</i><span class = "smallroman">V</span><sup>2</sup>, from the other
+angles raise verticals <i>bf</i>, <i>dh</i>, <i>cg</i>, to meet
+<i>e</i><span class = "smallroman">V</span><sup>1</sup>, <i>e</i><span
+class = "smallroman">V</span><sup>2</sup>, <i>f</i><span class =
+"smallroman">V</span><sup>2</sup>, &amp;c., and the cube is
+complete.</p>
+
+
+
+
+<span class = "pagenum">116</span>
+<a name = "page116" id = "page116"> </a>
+<!--png 132-->
+<h5 class = "section"><a name = "chapLXV" id = "chapLXV">
+LXV</a></h5>
+
+<h5 class = "smallcaps">Amplification of the Cube Applied to Drawing a
+Cottage</h5>
+
+
+<p>Note that we have started this figure with the cube <span class =
+"smallroman">A</span><i>dhefb</i>. We have taken three times <span class
+= "smallroman">AB</span>, its width, for the front of our house, and
+twice <span class = "smallroman">AB</span> for the side, and have made
+it two cubes high, not counting the roof. Note also the use of the
+measuring-points in connexion with the measurements on the base line,
+and the upper measuring line <span class = "smallroman">TPK</span>.</p>
+
+<p class = "illustration">
+<a name = "fig127" id = "fig127"> </a>
+<img src = "images/fig127.png" width = "336" height = "143"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 127.</p>
+
+
+
+
+<span class = "pagenum">117</span>
+<a name = "page117" id = "page117"> </a>
+<!--png 133-->
+<h5 class = "section"><a name = "chapLXVI" id = "chapLXVI">
+LXVI</a></h5>
+
+<h5 class = "smallcaps">How to Draw an Interior at an Angle</h5>
+
+
+<p>Here we make use of the same points as in a previous figure, with the
+addition of the point <span class = "smallroman">G</span>, which is the
+vanishing point of the diagonals of the squares on the floor.</p>
+
+<p class = "illustration">
+<a name = "fig128" id = "fig128"> </a>
+<img src = "images/fig128.png" width = "321" height = "181"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 128.</p>
+
+<p>From <span class = "smallroman">A</span> draw square <span class =
+"smallroman">A</span><i>bcd</i>, and produce its sides in all
+directions; again from <span class = "smallroman">A</span>, through the
+opposite angle of the square <span class = "smallroman">C</span>, draw a
+diagonal till it cuts the horizon at <span class =
+"smallroman">G</span>. From <span class = "smallroman">G</span> draw
+diagonals through <i>b</i> and <i>d</i>, cutting the base at <i>o</i>,
+<i>o</i>, make spaces <i>o</i>, <i>o</i>, equal to <span class =
+"smallroman">A</span><i>o</i> all along the base, and from them draw
+diagonals to <span class = "smallroman">G</span>; through the points
+where these diagonals intersect the vanishing lines drawn in the
+direction of <span class = "smallroman">A</span><i>b</i>, <i>dc</i> and
+<span class = "smallroman">A</span><i>d</i>, <i>bc</i>, draw lines to
+the other vanishing point <span class =
+"smallroman">V</span><sup>1</sup>, thus completing the squares, and so
+cover the floor with them; they will then serve to measure width of
+door, windows, &amp;c. Of course horizontal lines on wall&nbsp;1 are
+drawn to <span class = "smallroman">V</span><sup>1</sup>, and those on
+wall&nbsp;2 to <span class = "smallroman">V</span><sup>2</sup>.</p>
+
+<p>In order to see this drawing properly, the eye should be placed about
+3&nbsp;inches from it, and opposite the point of sight; it will then
+stand out like a stereoscopic picture, and appear as actual space, but
+otherwise the perspective seems deformed, and the
+<span class = "pagenum">118</span>
+<a name = "page118" id = "page118"> </a>
+<!--png 134-->
+angles exaggerated. To make this drawing look right from a reasonable
+distance, the point of distance should be at least twice as far off as
+it is here, and this would mean altering all the other points and
+sending them a long way out of the picture; this is why artists use
+those long strings referred to above. I&nbsp;would however, advise them
+to make their perspective drawing on a small scale, and then square it
+up to the size of the canvas.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXVII" id = "chapLXVII">
+LXVII</a></h5>
+
+<h5 class = "smallcaps">How to Correct Distorted Perspective by Doubling
+the Line of Distance</h5>
+
+
+<p>Here we have the same interior as the foregoing, but drawn with
+double the distance, so that the perspective is not so violent and the
+objects are truer in proportion to each other.</p>
+
+<p class = "illustration">
+<a name = "fig129" id = "fig129"> </a>
+<img src = "images/fig129.png" width = "342" height = "202"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 129.</p>
+
+<p>To redraw the whole figure double the size, including the
+station-point, would require a very large diagram, that we could not get
+into this book without a folding plate, but it comes to the same thing
+if we double the distances between the various
+<span class = "pagenum">119</span>
+<a name = "page119" id = "page119"> </a>
+<!--png 135-->
+points. Thus, if from <span class = "smallroman">S</span> to <span class
+= "smallroman">G</span> in the small diagram is 1&nbsp;inch, in the
+larger one make it 2&nbsp;inches. If from <span class =
+"smallroman">S</span> to <span class = "smallroman">M</span><sup>2</sup>
+is 2&nbsp;inches, in the larger make it&nbsp;4, and so&nbsp;on.</p>
+
+<p>Or this form may be used: make <span class = "smallroman">AB</span>
+twice the length of <span class = "smallroman">AC</span> (Fig. 130), or
+in any other proportion required. On <span class =
+"smallroman">AC</span> mark the points as in the drawing you wish to
+enlarge. Make <span class = "smallroman">AB</span> the length that you
+wish to enlarge to, draw <span class = "smallroman">CB</span>, and then
+from each division on <span class = "smallroman">AC</span> draw lines
+parallel to <span class = "smallroman">CB</span>, and <span class =
+"smallroman">AB</span> will be divided in the same proportions, as I
+have already shown (<a href = "#fig117">Fig. 117</a>).</p>
+
+<p class = "illustration">
+<a name = "fig130" id = "fig130"> </a>
+<img src = "images/fig130.png" width = "226" height = "76"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 130.</p>
+
+<p>There is no doubt that it is easier to work direct from the vanishing
+points themselves, especially in complicated architectural work, but at
+the same time I will now show you how we can dispense with, at all
+events, one of them, and that the farthest away.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXVIII" id = "chapLXVIII">
+LXVIII</a></h5>
+
+<h5 class = "smallcaps">How to Draw a Cube on a Given Square, using only
+One Vanishing Point</h5>
+
+
+<p><span class = "smallroman">ABCD</span> is the given square (Fig.
+131). At <span class = "smallroman">A</span> raise vertical <span class
+= "smallroman">A</span><i>a</i> equal to side of square <span class =
+"smallroman">AB</span>�, from <i>a</i> draw <i>ab</i> to the vanishing
+point. Raise <span class = "smallroman">B</span><i>b</i>. Produce <span
+class = "smallroman">VD</span> to <span class = "smallroman">E</span> to
+touch the base line. From <span class = "smallroman">E</span> raise
+vertical <span class = "smallroman">EF</span>, making it equal to <span
+class = "smallroman">A</span><i>a</i>. From <span class =
+"smallroman">F</span> draw <span class = "smallroman">FV</span>. Raise
+<span class = "smallroman">D</span><i>d</i> and <span class =
+"smallroman">C</span><i>c</i>, their heights being determined by the
+line <span class = "smallroman">FV</span>. Join <i>da</i> and the cube
+is complete. It will be seen that the verticals raised at each corner of
+the square are equal perspectively, as they are drawn between parallels
+which start from equal heights, namely, from <span class =
+"smallroman">EF</span> and <span class = "smallroman">A</span><i>a</i>
+to the same point <span class = "smallroman">V</span>, the vanishing
+point. Any other
+<span class = "pagenum">120</span>
+<a name = "page120" id = "page120"> </a>
+<!--png 136-->
+line, such as <span class = "smallroman">OO�</span>, can be directed to
+the inaccessible vanishing point in the same way as <i>ad</i>,
+&amp;c.</p>
+
+<p class = "illustration">
+<a name = "fig131" id = "fig131"> </a>
+<img src = "images/fig131.png" width = "332" height = "100"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 131.</p>
+
+<p><i>Note.</i> This is only one of many original figures and problems
+in this book which have been called up by the wish to facilitate the
+work of the artist, and as it were by necessity.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXIX" id = "chapLXIX">
+LXIX</a></h5>
+
+<h5 class = "smallcaps">A Courtyard or Cloister Drawn with one Vanishing
+Point</h5>
+
+
+<p>In this figure I have first drawn the pavement by means of the
+diagonals <span class = "smallroman">GA</span>, <span class =
+"smallroman">G</span><i>o</i>, <span class =
+"smallroman">G</span><i>o</i>, &amp;c., and the vanishing point <span
+class = "smallroman">V</span>, the square at <span class =
+"smallroman">A</span> being given. From <span class =
+"smallroman">A</span> draw diagonal through opposite corner till it cuts
+the horizon at <span class = "smallroman">G</span>. From this same point
+<span class = "smallroman">G</span> draw
+<span class = "pagenum">121</span>
+<a name = "page121" id = "page121"> </a>
+<!--png 137-->
+lines through the other corners of the square till they cut the ground
+line at <i>o</i>, <i>o</i>. Take this measurement <span class =
+"smallroman">A</span><i>o</i> and mark it along the base right and left
+of <span class = "smallroman">A</span>, and the lines drawn from these
+points <i>o</i> to point <span class = "smallroman">G</span> will give
+the diagonals of all the squares on the pavement. Produce sides of
+square <span class = "smallroman">A</span>, and where these lines are
+intersected by the diagonals <span class = "smallroman">G</span><i>o</i>
+draw lines from the vanishing point <span class = "smallroman">V</span>
+to base. These will give us the outlines of the squares lying between
+them and also guiding points that will enable us to draw as many more as
+we please. These again will give us our measurements for the widths of
+the arches, &amp;c., or between the columns. Having fixed the height of
+wall or dado, we make use of <span class = "smallroman">V</span> point
+to draw the sides of the building, and by means of proportionate
+measurement complete the rest, as in <a href = "#fig128">Fig.
+128</a>.</p>
+
+<p class = "illustration">
+<a name = "fig132" id = "fig132"> </a>
+<img src = "images/fig132.png" width = "332" height = "154"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 132.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXX" id = "chapLXX">
+LXX</a></h5>
+
+<h5 class = "smallcaps">How to Draw Lines which shall Meet at a Distant
+Point, by Means of Diagonals</h5>
+
+
+<p>This is in a great measure a repetition of the foregoing figure, and
+therefore needs no further explanation.</p>
+
+<p class = "illustration">
+<a name = "fig133" id = "fig133"> </a>
+<img src = "images/fig133.png" width = "347" height = "98"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 133.</p>
+
+<p>I must, however, point out the importance of the point <span class =
+"smallroman">G</span>. In angular perspective it in a measure takes the
+place of the point of distance in parallel perspective, since it is the
+vanishing point of diagonals at 45� drawn between parallels such as
+<span class = "smallroman">AV</span>, <span class =
+"smallroman">DV</span>, drawn to a vanishing point <span class =
+"smallroman">V</span>. The method of dividing line <span class =
+"smallroman">AV</span> into a number of parts equal to <span class =
+"smallroman">AB</span>, the side of the square, is also shown in a
+previous figure (<a href = "#fig120">Fig. 120</a>).</p>
+
+
+
+
+<span class = "pagenum">122</span>
+<a name = "page122" id = "page122"> </a>
+<!--png 138-->
+<h5 class = "section"><a name = "chapLXXI" id = "chapLXXI">
+LXXI</a></h5>
+
+<h5 class = "smallcaps">How to Divide a Square Placed at an Angle into a
+Given Number of Small Squares</h5>
+
+
+<p><span class = "smallroman">ABCD</span> is the given square, and only
+one vanishing point is accessible. Let us divide it into sixteen small
+squares. Produce side <span class = "smallroman">CD</span> to base at
+<span class = "smallroman">E</span>. Divide <span class =
+"smallroman">EA</span> into four equal parts. From each division draw
+lines to vanishing point <span class = "smallroman">V</span>. Draw
+diagonals <span class = "smallroman">BD</span> and <span class =
+"smallroman">AC</span>, and produce the latter till it cuts the horizon
+in <span class = "smallroman">G</span>. Draw the three cross-lines
+through the intersections made by the diagonals and the lines drawn to
+<span class = "smallroman">V</span>, and thus divide the square into
+sixteen.</p>
+
+<p class = "illustration">
+<a name = "fig134" id = "fig134"> </a>
+<img src = "images/fig134.png" width = "333" height = "147"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 134.</p>
+
+<p>This is to some extent the reverse of the previous problem. It also
+shows how the long vanishing point can be dispensed with, and the
+perspective drawing brought within the picture.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXII" id = "chapLXXII">
+LXXII</a></h5>
+
+<h5 class = "smallcaps">Further Example of how to Divide a Given Oblique
+Square into a Given Number of Equal Squares, say Twenty-five</h5>
+
+
+<p>Having drawn the square <span class = "smallroman">ABCD</span>, which
+is enclosed, as will be seen, in a dotted square in parallel
+perspective, I&nbsp;divide the line
+<span class = "pagenum">123</span>
+<a name = "page123" id = "page123"> </a>
+<!--png 139-->
+<span class = "smallroman">EA</span> into five equal parts instead of
+four (Fig. 135), and have made use of the device for that purpose by
+measuring off the required number on line <span class =
+"smallroman">EF</span>, &amp;c. Fig. 136 is introduced here simply to
+show that the square can be divided into any number of smaller squares.
+Nor need the figure be necessarily a square; it is just as easy to make
+it an oblong, as <span class = "smallroman">ABEF</span> (Fig. 136); for
+although we begin with a square we can extend it in any direction we
+please, as here shown.</p>
+
+<p class = "illustration">
+<a name = "fig135" id = "fig135"> </a>
+<img src = "images/fig135.png" width = "264" height = "170"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 135.</p>
+
+<p class = "illustration">
+<a name = "fig136" id = "fig136"> </a>
+<img src = "images/fig136.png" width = "340" height = "129"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 136.</p>
+
+
+
+
+<span class = "pagenum">124</span>
+<a name = "page124" id = "page124"> </a>
+<!--png 140-->
+<h5 class = "section"><a name = "chapLXXIII" id = "chapLXXIII">
+LXXIII</a></h5>
+
+<h5 class = "smallcaps">Of Parallels and Diagonals</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig137" id = "fig137"> </a>
+<img src = "images/fig137a.png" width = "125" height = "132"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 137 A.</td>
+</tr>
+<tr>
+<td class = "picture">
+<img src = "images/fig137b.png" width = "124" height = "136"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 137 B.</td>
+</tr>
+</table>
+
+<p>To find the centre of a square or other rectangular figure we have
+but to draw its two diagonals, and their intersection will give us the
+centre of the figure (see 137 <span class = "smallroman">A</span>). We
+do the same with perspective figures, as at <span class =
+"smallroman">B</span>. In Fig. <span class = "smallroman">C</span> is
+shown how a diagonal, drawn from one angle of a square <span class =
+"smallroman">B</span> through the centre <span class =
+"smallroman">O</span> of the opposite side of the square, will enable us
+to find a second square lying between the same parallels, then a third,
+a&nbsp;fourth, and so on. At figure <span class = "smallroman">K</span>
+lying on the ground, I&nbsp;have divided the farther side of the square
+<i>mn</i> into �, &#x2153;, �. If I draw
+<span class = "pagenum">125</span>
+<a name = "page125" id = "page125"> </a>
+<!--png 141-->
+a diagonal from <span class = "smallroman">G</span> (at the base)
+through the half of this line I cut off on <span class =
+"smallroman">FS</span> the lengths or sides of two squares; if through
+the quarter I cut off the length of four squares on the vanishing line
+<span class = "smallroman">FS</span>, and so on. In Fig. 137 <span class
+= "smallroman">D</span> is shown how easily any number of objects at any
+equal distances apart, such as posts, trees, columns, &amp;c., can be
+drawn by means of diagonals between parallels, guided by a central line
+<span class = "smallroman">GS</span>.</p>
+
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<img src = "images/fig137c.png" width = "222" height = "131"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig137d.png" width = "187" height = "109"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 137 C.</td>
+<td class = "caption smallcaps">
+Fig. 137 D.</td>
+</tr>
+</table>
+
+
+
+
+
+<h5 class = "section"><a name = "chapLXXIV" id = "chapLXXIV">
+LXXIV</a></h5>
+
+<h5 class = "smallcaps">The Square, the Oblong, and their Diagonals</h5>
+
+
+<p>Having found the centre of a square or oblong, such as Figs. 138 and
+139, if we draw a third line through that centre at a given angle and
+then at each of its extremities draw perpendiculars <span class =
+"smallroman">AB</span>, <span class = "smallroman">DC</span>, we divide
+that square or oblong into three parts, the two outer portions being
+equal to each other, and the centre one either
+<span class = "pagenum">126</span>
+<a name = "page126" id = "page126"> </a>
+<!--png 142-->
+larger or smaller as desired; as, for instance, in the triumphal arch we
+make the centre portion larger than the two outer sides. When certain
+architectural details and spaces are to be put into perspective,
+a&nbsp;scale such as that in Fig. 123 will be found of great
+convenience; but if only a ready division of the principal proportions
+is required, then these diagonals will be found of the greatest use.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture" colspan = "2">
+<a name = "fig138" id = "fig138"> </a>
+<a name = "fig139" id = "fig139"> </a>
+<img src = "images/fig138_139.png" width = "344" height = "100"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 138.</td>
+<td class = "caption smallcaps">
+Fig. 139.</td>
+</tr>
+</table>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXV" id = "chapLXXV">
+LXXV</a></h5>
+
+<h5 class = "smallcaps">Showing the Use of the Square and Diagonals in
+Drawing Doorways, Windows, and other Architectural Features</h5>
+
+
+<p>This example is from Serlio's <i>Architecture</i> (1663), showing
+what excellent proportion can be obtained by the square and diagonals.
+The width of the door is one-third of the base of square, the height
+two-thirds. As a further illustration we have drawn the same figure in
+perspective.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig140" id = "fig140"> </a>
+<img src = "images/fig140.png" width = "157" height = "157"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig141" id = "fig141"> </a>
+<img src = "images/fig141.png" width = "81" height = "155"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 140.</td>
+<td class = "caption smallcaps">
+Fig. 141.</td>
+</tr>
+</table>
+
+
+
+
+<span class = "pagenum">127</span>
+<a name = "page127" id = "page127"> </a>
+<!--png 143-->
+<h5 class = "section"><a name = "chapLXXVI" id = "chapLXXVI">
+LXXVI</a></h5>
+
+<h5 class = "smallcaps">How to Measure Depths by Diagonals</h5>
+
+
+<p>If we take any length on the base of a square, say from <span class =
+"smallroman">A</span> to <i>g</i>, and from <i>g</i> raise a
+perpendicular till it cuts the diagonal <span class =
+"smallroman">AB</span> in <span class = "smallroman">O</span>, then from
+<span class = "smallroman">O</span> draw horizontal <span class =
+"smallroman">O</span><i>g�</i>, we form a square <span class =
+"smallroman">A</span><i>g</i><span class =
+"smallroman">O</span><i>g�</i>, and thus measure on one side of the
+square the distance or depth <span class =
+"smallroman">A</span><i>g�</i>. So can we measure any other length, such
+as <i>fg</i>, in like manner.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig142" id = "fig142"> </a>
+<img src = "images/fig142.png" width = "149" height = "144"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig143" id = "fig143"> </a>
+<img src = "images/fig143.png" width = "231" height = "163"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 142.</td>
+<td class = "caption smallcaps">
+Fig. 143.</td>
+</tr>
+</table>
+
+<p>To do this in perspective we pursue precisely the same method, as
+shown in this figure (143).</p>
+
+<p><span class = "pagenum">128</span>
+<a name = "page128" id = "page128"> </a>
+<!--png 144-->
+To measure a length <span class = "smallroman">A</span><i>g</i> on the
+side of square <span class = "smallroman">AC</span>, we draw a line from
+<i>g</i> to the point of sight <span class = "smallroman">S</span>, and
+where it crosses diagonal <span class = "smallroman">AB</span> at <span
+class = "smallroman">O</span> we draw horizontal <span class =
+"smallroman">O</span><i>g</i>, and thus find the required depth <span
+class = "smallroman">A</span><i>g</i> in the picture.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXVII" id = "chapLXXVII">
+LXXVII</a></h5>
+
+<h5 class = "smallcaps">How to Measure Distances by the Square and
+Diagonal</h5>
+
+
+<p>It may sometimes be convenient to have a ready method by which to
+measure the width and length of objects standing against the wall of a
+gallery, without referring to distance-points, &amp;c.</p>
+
+<p class = "illustration">
+<a name = "fig144" id = "fig144"> </a>
+<img src = "images/fig144.png" width = "333" height = "328"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 144.</p>
+
+<p><span class = "pagenum">129</span>
+<a name = "page129" id = "page129"> </a>
+<!--png 145-->
+In Fig. 144 the floor is divided into two large squares with their
+diagonals. Suppose we wish to draw a fireplace or a piece of furniture
+<span class = "smallroman">K</span>, we measure its base <i>ef</i> on
+<span class = "smallroman">AB</span>, as far from <span class =
+"smallroman">B</span> as we wish it to be in the picture; draw <i>eo</i>
+and <i>fo</i> to point of sight, and proceed as in the previous figure
+by drawing parallels from <span class = "smallroman">O</span><i>o</i>,
+&amp;c.</p>
+
+<p>Let it be observed that the great advantage of this method is, that
+we can use it to measure such distant objects as <span class =
+"smallroman">XY</span> just as easily as those near to&nbsp;us.</p>
+
+<p>There is, however, a&nbsp;still further advantage arising from it,
+and that is that it introduces us to a new and simpler method of
+perspective, to which I have already referred, and it will, I&nbsp;hope,
+be found of infinite use to the artist.</p>
+
+<p><i>Note.</i>&mdash;As we have founded many of these figures on a
+given square in angular perspective, it is as well to have a ready and
+certain means of drawing that square without the elaborate setting out
+of a geometrical plan, as in the first method, or the more cumbersome
+and extended system of the second method. I&nbsp;shall therefore show
+you another method equally correct, but much simpler than either, which
+I have invented for our use, and which indeed forms one of the chief
+features of this book.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXVIII" id = "chapLXXVIII">
+LXXVIII</a></h5>
+
+<h5 class = "smallcaps">How by Means of the Square and Diagonal we can
+Determine the Position of Points in Space</h5>
+
+
+<p>Apart from the aid that perspective affords the draughtsman, there is
+a further value in it, in that it teaches us almost a new science, which
+we might call the mystery of aspect, and how it is that the objects
+around us take so many different forms, or rather appearances, although
+they themselves remain the same. And also that it enables us, with,
+I&nbsp;think, great pleasure to ourselves, to fathom space, to work out
+difficult problems by simple reasoning, and to exercise those inventive
+and critical faculties which give strength and enjoyment to mental
+life.</p>
+
+<p><span class = "pagenum">130</span>
+<a name = "page130" id = "page130"> </a>
+<!--png 146-->
+And now, after this brief excursion into philosophy, let us come down to
+the simple question of the perspective of a point.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig145" id = "fig145"> </a>
+<img src = "images/fig145a.png" width = "154" height = "130"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig145b.png" width = "148" height = "135"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption" colspan = "2">
+Fig. 145.</td>
+</tr>
+</table>
+
+<p>Here, for instance, are two aspects of the same thing: the
+geometrical square <span class = "smallroman">A</span>, which is facing
+us, and the perspective square <span class = "smallroman">B</span>,
+which we suppose to lie flat on the table, or rather on the perspective
+plane. Line <span class = "smallroman">A�C�</span> is the perspective of
+line <span class = "smallroman">AC</span>. On the geometrical square we
+can make what measurements we please with the compasses, but on the
+perspective square <span class = "smallroman">B�</span> the only line we
+can actually measure is the base line. In both figures this base line is
+the same length. Suppose we want to find the
+<span class = "pagenum">131</span>
+<a name = "page131" id = "page131"> </a>
+<!--png 147-->
+perspective of point <span class = "smallroman">P</span> (Fig. 146), we
+make use of the diagonal <span class = "smallroman">CA</span>. From
+<span class = "smallroman">P</span> in the geometrical square draw <span
+class = "smallroman">PO</span> to meet the diagonal in <span class =
+"smallroman">O</span>; through <span class = "smallroman">O</span> draw
+perpendicular <i>fe</i>; transfer length <i>f</i><span class =
+"smallroman">B</span>, so found, to the base of the perspective square;
+from <i>f</i> draw <i>f</i><span class = "smallroman">S</span> to point
+of sight; where it cuts the diagonal in <span class =
+"smallroman">O</span>, draw horizontal <span class =
+"smallroman">OP�</span>, which gives us the point required. In the same
+way we can find the perspective of any number of points on any side of
+the square.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig146" id = "fig146"> </a>
+<img src = "images/fig146a.png" width = "134" height = "133"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig146b.png" width = "156" height = "135"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption" colspan = "2">
+Fig. 146.</td>
+</tr>
+</table>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXIX" id = "chapLXXIX">
+LXXIX</a></h5>
+
+<h5 class = "smallcaps">Perspective of a Point Placed in any Position
+within the Square</h5>
+
+
+<p>Let the point <span class = "smallroman">P</span> be the one we wish
+to put into perspective. We have but to repeat the process of the
+previous problem, making use of our measurements on the base, the
+diagonals, &amp;c.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig147" id = "fig147"> </a>
+<img src = "images/fig147a.png" width = "112" height = "121"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig147b.png" width = "149" height = "132"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption" colspan = "2">
+Fig. 147.</td>
+</tr>
+</table>
+
+<p>Indeed these figures are so plain and evident that further
+description of them is hardly necessary, so I will here give two
+drawings of triangles which explain themselves. To put a triangle into
+perspective we have but to find three points, such as <i>f</i><span
+class = "smallroman">EP</span>, Fig. 148 <span class =
+"smallroman">A</span>, and then transfer these points to the perspective
+square 148 <span class = "smallroman">B</span>, as there shown, and form
+the perspective triangle; but these figures explain themselves. Any
+other triangle or rectilineal
+<span class = "pagenum">132</span>
+<a name = "page132" id = "page132"> </a>
+<!--png 148-->
+figure can be worked out in the same way, which is not only the simplest
+method, but it carries its mathematical proof with&nbsp;it.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig148" id = "fig148"> </a>
+<img src = "images/fig148a.png" width = "145" height = "157"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig148b.png" width = "153" height = "154"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 148 A.</td>
+<td class = "caption smallcaps">
+Fig. 148 B.</td>
+</tr>
+</table>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig149" id = "fig149"> </a>
+<img src = "images/fig149a.png" width = "153" height = "153"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig149b.png" width = "174" height = "145"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 149 A.</td>
+<td class = "caption smallcaps">
+Fig. 149 B.</td>
+</tr>
+</table>
+
+
+
+
+<span class = "pagenum">133</span>
+<a name = "page133" id = "page133"> </a>
+<!--png 149-->
+<h5 class = "section"><a name = "chapLXXX" id = "chapLXXX">
+LXXX</a></h5>
+
+<h5 class = "smallcaps">Perspective of a Square Placed at an Angle New
+Method</h5>
+
+
+<p>As we have drawn a triangle in a square so can we draw an oblique
+square in a parallel square. In Figure 150 <span class =
+"smallroman">A</span> we have drawn the oblique square <span class =
+"smallroman">GEP</span><i>n</i>. We find the points on the base <span
+class = "smallroman">A</span><i>m</i>, as in the previous figures, which
+enable us to construct the oblique perspective square <i>n�</i><span
+class = "smallroman">G�E�P�</span> in the parallel perspective square
+Fig. 150 <span class = "smallroman">B</span>. But it is not necessary to
+construct the geometrical figure, as I will show presently. It is here
+introduced to explain the method.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig150" id = "fig150"> </a>
+<img src = "images/fig150a.png" width = "158" height = "156"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig150b.png" width = "187" height = "155"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 150 A.</td>
+<td class = "caption smallcaps">
+Fig. 150 B.</td>
+</tr>
+</table>
+
+<p>Fig. 150 <span class = "smallroman">B</span>. To test the accuracy of
+the above, produce sides <span class = "smallroman">G�E�</span> and
+<i>n�</i><span class = "smallroman">P�</span> of perspective square till
+they touch the horizon, where they will meet at <span class =
+"smallroman">V</span>, their vanishing point, and again produce the
+other sides <i>n�</i><span class = "smallroman">G�</span> and <span
+class = "smallroman">P�E�</span> till they meet on the horizon at the
+other vanishing point, which they must do if the figure is correctly
+drawn.</p>
+
+<p>In any parallel square construct an oblique square from
+<span class = "pagenum">134</span>
+<a name = "page134" id = "page134"> </a>
+<!--png 150-->
+a given point&mdash;given the parallel square at Fig. 150 <span class =
+"smallroman">B</span>, and given point <i>n�</i> on base. Make <span
+class = "smallroman">A�</span><i>f�</i> equal to <i>n�m�</i>, draw
+<i>f�</i><span class = "smallroman">S</span> and <i>n�</i><span class =
+"smallroman">S</span> to point of sight. Where these lines cut the
+diagonal <span class = "smallroman">AC</span> draw horizontals to <span
+class = "smallroman">P�</span> and <span class = "smallroman">G�</span>,
+and so find the four points <span class =
+"smallroman">G�E�P�</span><i>n�</i> through which to draw the
+square.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXXI" id = "chapLXXXI">
+LXXXI</a></h5>
+
+<h5 class = "smallcaps">On a Given Line Placed at an Angle to the Base
+Draw a Square in Angular Perspective, the Point of Sight, and Distance,
+being given.</h5>
+
+
+<p>Let <span class = "smallroman">AB</span> be the given line, <span
+class = "smallroman">S</span> the point of sight, and <span class =
+"smallroman">D</span> the distance (Fig. 151,&nbsp;1). Through <span
+class = "smallroman">A</span> draw <span class = "smallroman">SC</span>
+from point of sight to base (Fig. 151, 2&nbsp;and&nbsp;3). From <span
+class = "smallroman">C</span> draw <span class = "smallroman">CD</span>
+to point of distance. Draw <span class = "smallroman">A</span><i>o</i>
+parallel to base till it cuts <span class = "smallroman">CD</span> at
+<i>o</i>, through <span class = "smallcaps">o</span> draw <span class =
+"smallroman">SP</span>, from <span class = "smallroman">B</span> mark
+off <span class = "smallroman">BE</span> equal to <span class =
+"smallroman">CP</span>. From <span class = "smallroman">E</span> draw
+<span class = "smallroman">ES</span> intersecting <span class =
+"smallroman">CD</span> at <span class = "smallroman">K</span>, from
+<span class = "smallroman">K</span> draw <span class =
+"smallroman">KM</span>, thus completing the outer parallel square.
+Through <span class = "smallroman">F</span>, where <span class =
+"smallroman">PS</span> intersects <span class = "smallroman">MK</span>,
+draw <span class = "smallroman">AV</span> till it cuts the horizon in
+<span class = "smallroman">V</span>, its vanishing point. From <span
+class = "smallroman">V</span> draw <span class = "smallroman">VB</span>
+cutting side <span class = "smallroman">KE</span> of outer square in
+<span class = "smallroman">G</span>, and we have the four points
+<span class = "pagenum">135</span>
+<a name = "page135" id = "page135"> </a>
+<!--png 151-->
+<span class = "smallroman">AFGB</span>, which are the four angles of the
+square required. Join <span class = "smallroman">FG</span>, and the
+figure is complete.</p>
+
+<p class = "illustration">
+<a name = "fig151" id = "fig151"> </a>
+<img src = "images/fig151.png" width = "307" height = "136"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 151.</p>
+
+<p>Any other side of the square might be given, such as <span class =
+"smallroman">AF</span>. First through <span class =
+"smallroman">A</span> and <span class = "smallroman">F</span> draw <span
+class = "smallroman">SC</span>, <span class = "smallroman">SP</span>,
+then draw <span class = "smallroman">A</span><i>o</i>, then through
+<i>o</i> draw <span class = "smallroman">CD</span>. From <span class =
+"smallroman">C</span> draw base of parallel square <span class =
+"smallroman">CE</span>, and at <span class = "smallroman">M</span>
+through <span class = "smallroman">F</span> draw <span class =
+"smallroman">MK</span> cutting diagonal at <span class =
+"smallroman">K</span>, which gives top of square. Now through <span
+class = "smallroman">K</span> draw <span class = "smallroman">SE</span>,
+giving <span class = "smallroman">KE</span> the remaining side thereof,
+produce <span class = "smallroman">AF</span> to <span class =
+"smallroman">V</span>, from <span class = "smallroman">V</span> draw
+<span class = "smallroman">VB</span>. Join <span class =
+"smallroman">FG</span>, <span class = "smallroman">GB</span>, and <span
+class = "smallroman">BA</span>, and the square required is complete.</p>
+
+<p>The student can try the remaining two sides, and he will find they
+work out in a similar way.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXXII" id = "chapLXXXII">
+LXXXII</a></h5>
+
+<h5 class = "smallcaps">How to Draw Solid Figures at any Angle by the
+New Method</h5>
+
+
+<p>As we can draw planes by this method so can we draw solids, as shown
+in these figures. The heights of the corners of the triangles are
+obtained by means of the vanishing scales <span class =
+"smallroman">AS</span>, <span class = "smallroman">OS</span>, which have
+already been explained.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig152" id = "fig152"> </a>
+<img src = "images/fig152.png" width = "151" height = "134"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig153" id = "fig153"> </a>
+<img src = "images/fig153.png" width = "143" height = "142"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 152.</td>
+<td class = "caption smallcaps">
+Fig. 153.</td>
+</tr>
+</table>
+
+<p>In the same manner we can draw a cubic figure (Fig. 154)&mdash;a box,
+for instance&mdash;at any required angle. In this case, besides the
+scale <span class = "smallroman">AS</span>, <span class =
+"smallroman">OS</span>, we have made use of the vanishing lines <span
+class = "smallroman">DV</span>, <span class = "smallroman">BV</span>,
+<span class = "pagenum">136</span>
+<a name = "page136" id = "page136"> </a>
+<!--png 152-->
+to corroborate the scale, but they can be dispensed with in these simple
+objects, or we can use a scale on each side of the figure as
+<i>a�o�</i><span class = "smallroman">S</span>, should both vanishing
+points be inaccessible. Let it be noted that in the scale <span class =
+"smallroman">AOS</span>, <span class = "smallroman">AO</span> is made
+equal to <span class = "smallroman">BC</span>, the height of the
+box.</p>
+
+<p class = "illustration">
+<a name = "fig154" id = "fig154"> </a>
+<img src = "images/fig154.png" width = "306" height = "157"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 154.</p>
+
+<p>By a similar process we draw these two figures, one on the square,
+the other on the circle.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig155" id = "fig155"> </a>
+<img src = "images/fig155.png" width = "160" height = "96"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig156" id = "fig156"> </a>
+<img src = "images/fig156.png" width = "142" height = "119"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 155.</td>
+<td class = "caption smallcaps">
+Fig. 156.</td>
+</tr>
+</table>
+
+
+
+
+<span class = "pagenum">137</span>
+<a name = "page137" id = "page137"> </a>
+<!--png 153-->
+<h5 class = "section"><a name = "chapLXXXIII" id = "chapLXXXIII">
+LXXXIII</a></h5>
+
+<h5 class = "smallcaps">Points in Space</h5>
+
+
+<p>The chief use of these figures is to show how by means of diagonals,
+horizontals, and perpendiculars almost any figure in space can be set
+down. Lines at any slope and at any angle can be drawn by this
+descriptive geometry.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig157" id = "fig157"> </a>
+<img src = "images/fig157.png" width = "222" height = "102"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 157.</td>
+</tr>
+</table>
+
+<p>The student can examine these figures for himself, and will
+understand their working from what has gone before. Here (Fig. 157) in
+the geometrical square we have a vertical plane <span class =
+"smallroman">A</span><i>ab</i><span class = "smallroman">B</span>
+standing on its base <span class = "smallroman">AB</span>. We wish to
+place a projection of this figure at a certain distance and at a given
+angle in space. First of all we transfer it to the side of the cube,
+where it is seen in perspective, whilst at its side is another
+perspective square lying flat, on which we have to stand our figure. By
+means of the diagonal of this flat square, horizontals from figure on
+side of cube, and lines drawn from point of sight (as already
+explained), we obtain the direction of base line <span class =
+"smallroman">AB</span>, and also by means of lines <i>aa�</i> and
+<i>bb�</i> we obtain the two points in space <i>a�b�</i>. Join <span
+class = "smallroman">A</span><i>a�</i>, <i>a�b�</i> and <span class =
+"smallroman">B</span><i>b�</i>, and we have the projection required, and
+which may be said to possess the third dimension.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig158" id = "fig158"> </a>
+<img src = "images/fig158.png" width = "211" height = "102"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 158.</td>
+</tr>
+</table>
+
+
+<p>In this other case (Fig. 158) we have a wedge-shaped figure standing
+on a triangle placed on the ground, as in the previous figure, its three
+corners being the same height. In the vertical geometrical square we
+have a ground-plan of the figure, from which we draw lines to diagonal
+and to base, and notify by numerals 1,&nbsp;3,
+<span class = "pagenum">138</span>
+<a name = "page138" id = "page138"> </a>
+<!--png 154-->
+2, 1,&nbsp;3; these we transfer to base of the horizontal perspective
+square, and then construct shaded triangle 1,&nbsp;2,&nbsp;3, and raise
+to the height required as shown at 1�,&nbsp;2�,&nbsp;3�. Although we may
+not want to make use of these special figures, they show us how we could
+work out almost any form or object suspended in space.</p>
+
+
+
+<h5 class = "section"><a name = "chapLXXXIV" id = "chapLXXXIV">
+LXXXIV</a></h5>
+
+<h5 class = "smallcaps">The Square and Diagonal Applied to Cubes And
+Solids Drawn Therein</h5>
+
+
+<p>As we have made use of the square and diagonal to draw figures at
+various angles so can we make use of cubes either in parallel or angular
+perspective to draw other solid figures within
+<span class = "pagenum">139</span>
+<a name = "page139" id = "page139"> </a>
+<!--png 155-->
+them, as shown in these drawings, for this is simply an amplification of
+that method. Indeed we might invent many more such things. But subjects
+for perspective treatment will constantly present themselves to the
+artist or draughtsman in the course of his experience, and while I
+endeavour to show him how to grapple with any new difficulty or subject
+that may arise, it is impossible to set down all of them in this
+book.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig159" id = "fig159"> </a>
+<img src = "images/fig159.png" width = "184" height = "157"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig160" id = "fig160"> </a>
+<img src = "images/fig160.png" width = "293" height = "174"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 159.</td>
+<td class = "caption smallcaps">
+Fig. 160.</td>
+</tr>
+</table>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXXV" id = "chapLXXXV">
+LXXXV</a></h5>
+
+<h5 class = "smallcaps">To Draw an Oblique Square in Another Oblique
+Square without Using Vanishing Points</h5>
+
+
+<p>It is not often that both vanishing points are inaccessible, still it
+is well to know how to proceed when this is the case. We first draw the
+square <span class = "smallroman">ABCD</span> inside the parallel
+square, as in previous figures. To draw the smaller square <span class =
+"smallroman">K</span> we simply draw a smaller parallel square <i>h h h
+h</i>, and within that, guided by the intersections of the diagonals
+therewith, we obtain the four points through which to draw square <span
+class = "smallroman">K</span>. To raise a solid figure on these squares
+we can make use of the vanishing scales as
+<span class = "pagenum">140</span>
+<a name = "page140" id = "page140"> </a>
+<!--png 156-->
+shown on each side of the figure, thus obtaining the upper square
+1&nbsp;2 3&nbsp;4, then by means of the diagonal 1&nbsp;3 and 2&nbsp;4
+and verticals raised from each corner of square <span class =
+"smallroman">K</span> to meet them we obtain the smaller upper square
+corresponding to <span class = "smallroman">K</span>.</p>
+
+<p>It might be said that all this can be done by using the two vanishing
+points in the usual way. In the first place, if they were as far off as
+required for this figure we could not get them into a page unless it
+were three or four times the width of this one, and to use shorter
+distances results in distortion, so that the real use of this system is
+that we can make our figures look quite natural and with much less
+trouble than by the other method.</p>
+
+<p class = "illustration">
+<a name = "fig161" id = "fig161"> </a>
+<img src = "images/fig161.png" width = "297" height = "296"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 161.</p>
+
+
+
+
+<span class = "pagenum">141</span>
+<a name = "page141" id = "page141"> </a>
+<!--png 157-->
+<h5 class = "section"><a name = "chapLXXXVI" id = "chapLXXXVI">
+LXXXVI</a></h5>
+
+<h5 class = "smallcaps">Showing How a Pedestal can be Drawn by the New
+Method</h5>
+
+
+<p>This is a repetition of the previous problem, or rather the
+application of it to architecture, although when there are many details
+it may be more convenient to use vanishing points or the
+centrolinead.</p>
+
+<p class = "illustration">
+<a name = "fig162" id = "fig162"> </a>
+<img src = "images/fig162.png" width = "343" height = "329"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 162.</p>
+
+
+
+
+<span class = "pagenum">143</span>
+<a name = "page143" id = "page143"> </a>
+<!--png 159-->
+<h5 class = "section"><a name = "chapLXXXVII" id = "chapLXXXVII">
+LXXXVII</a></h5>
+
+<h5 class = "smallcaps">Scale on Each Side of the Picture</h5>
+
+
+<p>As one of my objects in writing this book is to facilitate the
+working of our perspective, partly for the comfort of the artist, and
+partly that he may have no excuse for neglecting it, I&nbsp;will here
+show you how you may, by a very simple means, secure the general
+correctness of your perspective when sketching or painting out of
+doors.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[142]</span>
+<a name = "page142" id = "page142"> </a>
+<!--png 158-->
+<a name = "fig163" id = "fig163"> </a>
+<img src = "images/fig163.png" width = "296" height = "454"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 163.</span> Honfleur.</p>
+
+<p>Let us take this example from a sketch made at Honfleur (Fig. 163),
+and in which my eye was my only guide, but it stands the test of the
+rule. First of all note that line <span class = "smallroman">HH</span>,
+drawn from one side of the picture to the other, is the horizontal line;
+below that is a wall and a pavement marked <i>a</i><span class =
+"smallroman">V</span>, also going from one side of the picture to the
+other, and being lower down at <i>a</i> than at <span class =
+"smallroman">V</span> it runs up as it were to meet the horizon at some
+distant point. In order to form our scale I take first the length of
+<span class = "smallroman">H</span><i>a</i>, and measure it above and
+below the horizon, along the side to our left as many times as required,
+in this case four or five. I&nbsp;now take the length <span class =
+"smallroman">HV</span> on the right side of the picture and measure it
+above and below the horizon, as in the other case; and then from these
+divisions obtain dotted lines crossing the picture from one side to the
+other which must all meet at some distant point on the horizon. These
+act as guiding lines, and are sufficient to give us the direction of any
+vanishing lines going to the same point. For those that go in the
+opposite direction we proceed in the same way, as from <i>b</i> on the
+right to <span class = "smallroman">V�</span> on the left. They are here
+put in faintly, so as not to interfere with the drawing. In the sketch
+of Toledo (Fig. 164) the same thing is shown by double lines on each
+side to separate the two sets of lines, and to make the principle more
+evident.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[144]</span>
+<a name = "page144" id = "page144"> </a>
+<!--png 160-->
+<a name = "fig164" id = "fig164"> </a>
+<img src = "images/fig164.png" width = "267" height = "413"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 164.</span> Toledo.</p>
+
+
+
+
+<span class = "pagenum">145</span>
+<a name = "page145" id = "page145"> </a>
+<!--png 161-->
+<h5 class = "section"><a name = "chapLXXXVIII" id = "chapLXXXVIII">
+LXXXVIII</a></h5>
+
+<h5 class = "smallcaps">The Circle</h5>
+
+
+<p>If we inscribe a circle in a square we find that it touches that
+square at four points which are in the middle of each side, as at <i>a b
+c d</i>. It will also intersect the two diagonals at the four points
+<i>o</i> (Fig. 165). If, then, we put this square and its diagonals,
+&amp;c., into perspective we shall have eight guiding points through
+which to trace the required circle, as shown in Fig. 166, which has the
+same base as Fig. 165.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig165" id = "fig165"> </a>
+<img src = "images/fig165.png" width = "120" height = "120"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig166" id = "fig166"> </a>
+<img src = "images/fig166.png" width = "185" height = "130"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 165.</td>
+<td class = "caption smallcaps">
+Fig. 166.</td>
+</tr>
+</table>
+
+
+
+
+<h5 class = "section"><a name = "chapLXXXIX" id = "chapLXXXIX">
+LXXXIX</a></h5>
+
+<h5 class = "smallcaps">The Circle in Perspective a True Ellipse</h5>
+
+
+<p>Although the circle drawn through certain points must be a freehand
+drawing, which requires a little practice to make it true, it is
+sufficient for ordinary purposes and on a small scale, but to be
+mathematically true it must be an ellipse. We will first draw an ellipse
+(Fig. 167). Let <i>ee</i> be its long, or transverse, diameter, and
+<i>db</i> its short or conjugate diameter. Now take half of the long
+diameter <i>e</i><span class = "smallroman">E</span>, and from point
+<i>d</i> with <i>c</i><span class = "smallroman">E</span> for radius
+mark on <i>ee</i> the two points <i>ff</i>, which are the foci of the
+ellipse. At each focus fix a pin, then make a loop of fine string that
+does not stretch and of such a length that when drawn out the double
+<span class = "pagenum">146</span>
+<a name = "page146" id = "page146"> </a>
+<!--png 162-->
+thread will reach from <i>f</i> to <i>e</i>. Now place this double
+thread round the two pins at the foci <i>ff�</i> and distend it with the
+pencil point until it forms triangle <i>fdf�</i>, then push the pencil
+along and right round the two foci, which being guided by the thread
+will draw the curve, which is a true ellipse, and will pass through the
+eight points indicated in our first figure. This will be a sufficient
+proof that the circle in perspective and the ellipse are identical
+curves. We must also remember that the ellipse is an oblique projection
+of a circle, or an oblique section of a cone. The difference between the
+two figures consists in their centres not being in the same place, that
+of the perspective circle being at <i>c</i>, higher up than <i>e</i> the
+centre of the ellipse. The latter being a geometrical figure, its long
+diameter is exactly in the centre of the figure, whereas the centre
+<i>c</i> and the diameter of the perspective are at the intersection of
+the diagonals of the perspective square in which it is inscribed.</p>
+
+<p class = "illustration">
+<a name = "fig167" id = "fig167"> </a>
+<img src = "images/fig167.png" width = "334" height = "143"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 167.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXC" id = "chapXC">
+XC</a></h5>
+
+<h5 class = "smallcaps">Further Illustration of the Ellipse</h5>
+
+
+<p>In order to show that the ellipse drawn by a loop as in the previous
+figure is also a circle in perspective we must reconstruct around it the
+square and its eight points by means of which it was drawn in the first
+instance. We start with nothing but
+<span class = "pagenum">147</span>
+<a name = "page147" id = "page147"> </a>
+<!--png 163-->
+the ellipse itself. We have to find the points of sight and distance,
+the base, &amp;c. Let us start with base <span class =
+"smallroman">AB</span>, a&nbsp;horizontal tangent to the curve extending
+beyond it on either side. From <span class = "smallroman">A</span> and
+<span class = "smallroman">B</span> draw two other tangents so that they
+shall touch the curve at points such as <span class =
+"smallroman">TT�</span> a little above the transverse diameter and on a
+level with each other. Produce these tangents till they meet at point
+<span class = "smallroman">S</span>, which will be the point of sight.
+Through this point draw horizontal line <span class =
+"smallroman">H</span>. Now draw tangent <span class =
+"smallroman">CD</span> parallel to <span class = "smallroman">AB</span>.
+Draw diagonal <span class = "smallroman">AD</span> till it cuts the
+horizon at the point of distance, this will cut through diameter of
+circle at its centre, and so proceed to find the eight points through
+which the perspective circle passes, when it will be found that they all
+lie on the ellipse we have drawn with the loop, showing that the two
+curves are identical although their centres are distinct.</p>
+
+<p class = "illustration">
+<a name = "fig168" id = "fig168"> </a>
+<img src = "images/fig168.png" width = "302" height = "172"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 168.</p>
+
+
+
+
+<span class = "pagenum">148</span>
+<a name = "page148" id = "page148"> </a>
+<!--png 164-->
+<h5 class = "section"><a name = "chapXCI" id = "chapXCI">
+XCI</a></h5>
+
+<h5 class = "smallcaps">How To Draw a Circle in Perspective Without a
+Geometrical Plan</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig169" id = "fig169"> </a>
+<img src = "images/fig169.png" width = "174" height = "193"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 169.</td>
+</tr>
+<tr>
+<td class = "picture">
+<a name = "fig170" id = "fig170"> </a>
+<img src = "images/fig170.png" width = "186" height = "184"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 170.</td>
+</tr>
+<tr>
+<td class = "picture">
+<a name = "fig171" id = "fig171"> </a>
+<img src = "images/fig171.png" width = "185" height = "218"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 171.</td>
+</tr>
+</table>
+
+<p>Divide base <span class = "smallroman">AB</span> into four equal
+parts. At <span class = "smallroman">B</span> drop perpendicular <span
+class = "smallroman">B</span><i>n</i>, making <span class =
+"smallroman">B</span><i>n</i> equal to <span class =
+"smallroman">B</span><i>m</i>, or one-fourth of base. Join <i>mn</i> and
+transfer this measurement to each side of <i>d</i> on base line; that
+is, make <i>df</i> and <i>df�</i> equal to <i>mn</i>. Draw <i>f</i><span
+class = "smallroman">S</span> and <i>f�</i><span class =
+"smallroman">S</span>, and the intersections of these lines with the
+diagonals of square will give us the four points <i>o o
+o&nbsp;o</i>.</p>
+
+<p>The reason of this is that <i>ff�</i> is the measurement on the base
+<span class = "smallroman">AB</span> of another square <i>o o o o</i>
+which is exactly half of the outer square. For if we inscribe a circle
+in a square and then inscribe a second square in that circle, this
+second square will be exactly half the area of the larger one; for its
+side will be equal to half the diagonal of the larger square, as can be
+seen by studying
+<span class = "pagenum">149</span>
+<a name = "page149" id = "page149"> </a>
+<!--png 165-->
+the following figures. In Fig. 170, for instance, the side of small
+square <span class = "smallroman">K</span> is half the diagonal of large
+square <i>o</i>.</p>
+
+<p>In Fig. 171, <span class = "smallroman">CB</span> represents half of
+diagonal <span class = "smallroman">EB</span> of the outer square in
+which the circle is inscribed. By taking a fourth
+<span class = "pagenum">150</span>
+<a name = "page150" id = "page150"> </a>
+<!--png 166-->
+of the base <i>m</i><span class = "smallroman">B</span> and drawing
+perpendicular <i>mh</i> we cut <span class = "smallroman">CB</span> at
+<i>h</i> in two equal parts, <span class =
+"smallroman">C</span><i>h</i>, <i>h</i><span class =
+"smallroman">B</span>. It will be seen that <i>h</i><span class =
+"smallroman">B</span> is equal to <i>mn</i>, one-quarter of the
+diagonal, so if we measure <i>mn</i> on each side of <span class =
+"smallroman">D</span> we get <i>ff�</i> equal to <span class =
+"smallroman">CB</span>, or half the diagonal. By drawing <i>ff</i>,
+<i>f�f</i> passing through the diagonals we get the four points <i>o o o
+o</i> through which to draw the smaller square. Without referring to
+geometry we can see at a glance by Fig. 172, where we have simply turned
+the square <i>o o o o</i> on its centre so that its angles touch the
+sides of the outer square, that it is exactly half of square <span class
+= "smallroman">ABEF</span>, since each quarter of it, such as <span
+class = "smallroman">E</span><i>o</i><span class =
+"smallroman">C</span><i>o</i>, is bisected by its diagonal
+<i>oo</i>.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig172" id = "fig172"> </a>
+<img src = "images/fig172.png" width = "146" height = "139"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig173" id = "fig173"> </a>
+<img src = "images/fig173.png" width = "142" height = "140"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 172.</td>
+<td class = "caption smallcaps">
+Fig. 173.</td>
+</tr>
+</table>
+
+
+
+
+<span class = "pagenum">151</span>
+<a name = "page151" id = "page151"> </a>
+<!--png 167-->
+<h5 class = "section"><a name = "chapXCII" id = "chapXCII">
+XCII</a></h5>
+
+<h5 class = "smallcaps">How to Draw a Circle in Angular Perspective</h5>
+
+
+<p>Let <span class = "smallroman">ABCD</span> be the oblique square.
+Produce <span class = "smallroman">VA</span> till it cuts the base line
+at <span class = "smallroman">G</span>.</p>
+
+<p class = "illustration">
+<a name = "fig174" id = "fig174"> </a>
+<img src = "images/fig174.png" width = "328" height = "176"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 174.</p>
+
+<p>Take <i>m</i><span class = "smallroman">D</span>, the fourth of the
+base. Find <i>mn</i> as in Fig. 171, measure it on each side of <span
+class = "smallroman">E</span>, and so obtain <span class =
+"smallroman">E</span><i>f</i> and <span class =
+"smallroman">E</span><i>f�</i>, and proceed to draw <i>f</i><span class
+= "smallroman">V</span>, <span class = "smallroman">EV</span>,
+<i>f�</i><span class = "smallroman">V</span> and the diagonals, whose
+intersections with these lines will give us the eight points through
+which to draw the circle. In fact the process is the same as in parallel
+perspective, only instead of making our divisions on the actual base
+<span class = "smallroman">AD</span> of the square, we make them on
+<span class = "smallroman">GD</span>, the base line.</p>
+
+<p>To obtain the central line <i>hh</i> passing through <span class =
+"smallroman">O</span>, we can make use of diagonals of the half squares;
+that is, if the other vanishing point is inaccessible, as in this
+case.</p>
+
+
+
+
+<span class = "pagenum">152</span>
+<a name = "page152" id = "page152"> </a>
+<!--png 168-->
+<h5 class = "section"><a name = "chapXCIII" id = "chapXCIII">
+XCIII</a></h5>
+
+<h5 class = "smallcaps">How to Draw a Circle in Perspective more
+Correctly, by Using Sixteen Guiding Points</h5>
+
+
+<p>First draw square <span class = "smallroman">ABCD</span>. From <span
+class = "smallroman">O</span>, the middle of the base, draw semicircle
+<span class = "smallroman">AKB</span>, and divide it into eight equal
+parts. From each division raise perpendiculars to the base, such as
+2&nbsp;<span class = "smallroman">O</span>, 3&nbsp;<span class =
+"smallroman">O</span>, 5&nbsp;<span class = "smallroman">O</span>,
+&amp;c., and from divisions <span class = "smallroman">O</span>, <span
+class = "smallroman">O</span>, <span class = "smallroman">O</span> draw
+lines to point of sight, and where these lines cut the diagonals <span
+class = "smallroman">AC</span>, <span class = "smallroman">DB</span>,
+draw horizontals parallel to base <span class = "smallroman">AB</span>.
+Then through the points thus obtained draw the circle as shown in this
+figure, which also shows us how the circumference of a circle in
+perspective may be divided into any number of equal parts.</p>
+
+<p class = "illustration">
+<a name = "fig175" id = "fig175"> </a>
+<img src = "images/fig175.png" width = "257" height = "291"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 175.</p>
+
+
+
+
+<span class = "pagenum">153</span>
+<a name = "page153" id = "page153"> </a>
+<!--png 169-->
+<h5 class = "section"><a name = "chapXCIV" id = "chapXCIV">
+XCIV</a></h5>
+
+<h5 class = "smallcaps">How to Divide a Perspective Circle into any
+Number of Equal Parts</h5>
+
+
+<p>This is simply a repetition of the previous figure as far as its
+construction is concerned, only in this case we have divided the
+semicircle into twelve parts and the perspective into twenty-four.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig176" id = "fig176"> </a>
+<img src = "images/fig176.png" width = "216" height = "165"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig177" id = "fig177"> </a>
+<img src = "images/fig177.png" width = "185" height = "169"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 176.</td>
+<td class = "caption smallcaps">
+Fig. 177.</td>
+</tr>
+</table>
+
+<p><span class = "pagenum">154</span>
+<a name = "page154" id = "page154"> </a>
+<!--png 170-->
+We have raised perpendiculars from the divisions on the semicircle, and
+proceeded as before to draw lines to the point of sight, and have thus
+by their intersections with the circumference already drawn in
+perspective divided it into the required number of equal parts, to which
+from the centre we have drawn the radii. This will show us how to draw
+traceries in Gothic windows, columns in a circle, cart-wheels,
+&amp;c.</p>
+
+<p>The geometrical figure (177) will explain the construction of the
+perspective one by showing how the divisions are obtained on the line
+<span class = "smallroman">AB</span>, which represents base of square,
+from the divisions on the semicircle <span class =
+"smallroman">AKB</span>.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXCV" id = "chapXCV">
+XCV</a></h5>
+
+<h5 class = "smallcaps">How to Draw Concentric Circles</h5>
+
+
+<p>First draw a square with its diagonals (Fig. 178), and from its
+centre <span class = "smallroman">O</span> inscribe a circle; in this
+circle inscribe a square, and in this again inscribe a second circle,
+and so on. Through their intersections with the diagonals draw lines to
+base, and
+<span class = "pagenum">155</span>
+<a name = "page155" id = "page155"> </a>
+<!--png 171-->
+number them 1,&nbsp;2, 3,&nbsp;4, &amp;c.; transfer these measurements
+to the base of the perspective square (Fig. 179), and proceed to
+construct the circles as before, drawing lines from each point on the
+base to the point of sight, and drawing the curves through the
+inter-sections of these lines with the diagonals.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig178" id = "fig178"> </a>
+<img src = "images/fig178.png" width = "180" height = "189"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig179" id = "fig179"> </a>
+<img src = "images/fig179.png" width = "177" height = "171"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 178.</td>
+<td class = "caption smallcaps">
+Fig. 179.</td>
+</tr>
+</table>
+
+<p>Should it be required to make the circles at equal distances, as for
+steps for instance, then the geometrical plan should be made
+accordingly.</p>
+
+<p>Or we may adopt the method shown at Fig. 180, by taking quarter base
+of both outer and inner square, and finding the measurement <i>mn</i> on
+each side of <span class = "smallroman">C</span>, &amp;c.</p>
+
+<p class = "illustration">
+<a name = "fig180" id = "fig180"> </a>
+<img src = "images/fig180.png" width = "271" height = "138"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 180.</p>
+
+
+
+
+<span class = "pagenum">156</span>
+<a name = "page156" id = "page156"> </a>
+<!--png 172-->
+<h5 class = "section"><a name = "chapXCVI" id = "chapXCVI">
+XCVI</a></h5>
+
+<h5 class = "smallcaps">The Angle of the Diameter of the Circle in
+Angular and Parallel Perspective</h5>
+
+
+<p>The circle, whether in angular or parallel perspective, is always an
+ellipse. In angular perspective the angle of the circle's diameter
+varies in accordance with the angle of the square in which it is placed,
+as in Fig. 181, <i>cc</i> is the diameter of the circle and <i>ee</i>
+the diameter of the ellipse. In parallel perspective the diameter of the
+circle always remains horizontal, although the long diameter of the
+ellipse varies in inclination according to the distance it is from the
+point of sight, as shown in Fig. 182, in which the third circle is much
+elongated and distorted, owing to its being outside the angle of
+vision.</p>
+
+<p class = "illustration">
+<a name = "fig181" id = "fig181"> </a>
+<img src = "images/fig181.png" width = "319" height = "118"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 181.</p>
+
+<p class = "illustration">
+<a name = "fig182" id = "fig182"> </a>
+<img src = "images/fig182.png" width = "338" height = "118"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 182.</p>
+
+
+
+
+<span class = "pagenum">157</span>
+<a name = "page157" id = "page157"> </a>
+<!--png 173-->
+<h5 class = "section"><a name = "chapXCVII" id = "chapXCVII">
+XCVII</a></h5>
+
+<h5 class = "smallcaps">How to Correct Disproportion in the Width of
+Columns</h5>
+
+
+<p>The disproportion in the width of columns in Fig. 183 arises from the
+point of distance being too near the point of sight, or, in other words,
+taking too wide an angle of vision. It will be seen that column&nbsp;3
+is much wider than column&nbsp;1. <!--column S, column 2, column 3. No
+column 1.--></p>
+
+<p class = "illustration">
+<a name = "fig183" id = "fig183"> </a>
+<img src = "images/fig183.png" width = "323" height = "164"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 183.</p>
+
+<p><span class = "pagenum">158</span>
+<a name = "page158" id = "page158"> </a>
+<!--png 174-->
+In our second figure (184) is shown how this defect is remedied, by
+doubling the distance, or by counting the same distance as half, which
+is easily effected by drawing the diagonal from <span class =
+"smallroman">O</span> to �-<span class = "smallroman">D</span>, instead
+of from <span class = "smallroman">A</span>, as in the other figure,
+<span class = "smallroman">O</span> being at half base. Here the squares
+lie much more level, and the columns are nearly the same width, showing
+the advantage of a long distance.</p>
+
+<p class = "illustration">
+<a name = "fig184" id = "fig184"> </a>
+<img src = "images/fig184.png" width = "333" height = "183"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 184.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapXCVIII" id = "chapXCVIII">
+XCVIII</a></h5>
+
+<h5 class = "smallcaps">How to Draw a Circle over a Circle or a
+Cylinder</h5>
+
+
+<table class = "float left" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig185" id = "fig185"> </a>
+<img src = "images/fig185.png" width = "192" height = "163"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 185.</td>
+</tr>
+</table>
+
+<p>First construct square and circle <span class =
+"smallroman">ABE</span>, then draw square <span class =
+"smallroman">CDF</span> with its diagonals. Then find the various points
+<span class = "smallroman">O</span>, and from these raise perpendiculars
+to meet the diagonals of the upper square at points <span class =
+"smallroman">P</span>, which, with the other points will be sufficient
+guides to draw the circle required. This can be applied to towers,
+columns, &amp;c. The size of the circles can be varied so that the upper
+portion of a cylinder or column shall be smaller than the lower.</p>
+
+
+
+
+<span class = "pagenum">159</span>
+<a name = "page159" id = "page159"> </a>
+<!--png 175-->
+<h5 class = "section"><a name = "chapXCIX" id = "chapXCIX">
+XCIX</a></h5>
+
+<h5 class = "smallcaps">To Draw a Circle Below a Given Circle</h5>
+
+
+<table class = "float left" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig186" id = "fig186"> </a>
+<img src = "images/fig186.png" width = "164" height = "162"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 186.</td>
+</tr>
+</table>
+
+<p>Construct the upper square and circle as before, then by means of the
+vanishing scale <span class = "smallroman">POV</span>, which should be
+made the depth required, drop perpendiculars from the various points
+marked <span class = "smallroman">O</span>, obtained by the diagonals,
+making them the right depth by referring them to the vanishing scale, as
+shown in this figure. This can be used for drawing garden fountains,
+basins, and various architectural objects.</p>
+
+
+
+
+<span class = "pagenum">160</span>
+<a name = "page160" id = "page160"> </a>
+<!--png 176-->
+<h5 class = "section"><a name = "chapC" id = "chapC">
+C</a></h5>
+
+<h5 class = "smallcaps">Application of Previous Problem</h5>
+
+
+<p>That is, to draw a circle above a circle. In Fig. 187 can be seen how
+by means of the vanishing scale at the side we obtain the height of the
+verticals 1,&nbsp;2, 3,&nbsp;4, &amp;c., which determine the direction
+of the upper circle; and in this second figure, how we resort to the
+same means to draw circular steps.</p>
+
+<p class = "illustration">
+<a name = "fig187" id = "fig187"> </a>
+<img src = "images/fig187.png" width = "284" height = "145"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 187.</p>
+
+<p class = "illustration">
+<a name = "fig188" id = "fig188"> </a>
+<img src = "images/fig188.png" width = "292" height = "147"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 188.</p>
+
+
+
+
+<span class = "pagenum">161</span>
+<a name = "page161" id = "page161"> </a>
+<!--png 177-->
+<h5 class = "section"><a name = "chapCI" id = "chapCI">
+CI</a></h5>
+
+<h5 class = "smallcaps">Doric Columns</h5>
+
+
+<p>It is as well for the art student to study the different orders of
+architecture, whether architect or not, as he frequently has to
+introduce them into his pictures, and at least must know their
+proportions, and how columns diminish from base to capital, as shown in
+this illustration.</p>
+
+<p class = "illustration">
+<a name = "fig189" id = "fig189"> </a>
+<img src = "images/fig189.png" width = "289" height = "375"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 189.</p>
+
+
+
+
+<span class = "pagenum">162</span>
+<a name = "page162" id = "page162"> </a>
+<!--png 178-->
+<h5 class = "section"><a name = "chapCII" id = "chapCII">
+CII</a></h5>
+
+<h5 class = "smallcaps">To Draw Semicircles Standing upon a Circle at
+any Angle</h5>
+
+
+<p>Given the circle <span class = "smallroman">ACBH</span>, on diagonal
+<span class = "smallroman">AB</span> draw semicircle <span class =
+"smallroman">AKB</span>, and on the same line <span class =
+"smallroman">AB</span> draw rectangle <span class =
+"smallroman">AEFB</span>, its height being determined by radius <span
+class = "smallroman">OK</span> of semicircle. From centre <span class =
+"smallroman">O</span> draw <span class = "smallroman">OF</span> to
+corner of rectangle. Through <i>f�</i>, where that line intersects the
+semicircle, draw <i>mn</i> parallel to <span class =
+"smallroman">AB</span>. This will give intersection <span class =
+"smallroman">O</span>� on the vertical <span class =
+"smallroman">OK</span>, through which all such horizontals as
+<i>m�n�</i>, level with <i>mn</i>, must pass. Now take any other
+diameter, such as <span class = "smallroman">GH</span>, and thereon
+raise rectangle <span class = "smallroman">G</span><i>gh</i><span class
+= "smallroman">H</span>, the same height as the other. The manner of
+doing this is to produce diameter <span class = "smallroman">GH</span>
+to the horizon till it finds its vanishing point at <span class =
+"smallroman">V</span>. From <span class = "smallroman">V</span> through
+<span class = "pagenum">163</span>
+<a name = "page163" id = "page163"> </a>
+<!--png 179-->
+<span class = "smallroman">K</span> draw <i>hg</i>, and through <span
+class = "smallroman">O</span>� draw <i>n�m�</i>. From <span class =
+"smallroman">O</span> draw the two diagonals <i>og</i> and <i>oh</i>,
+intersecting <i>m�n�</i> at <span class = "smallroman">O</span>, <span
+class = "smallroman">O</span>, and thus we have the five points <span
+class = "smallroman">GOKOH</span> through which to draw the required
+semicircle.</p>
+
+<p class = "illustration">
+<a name = "fig190" id = "fig190"> </a>
+<img src = "images/fig190.png" width = "339" height = "243"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 190.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCIII" id = "chapCIII">
+CIII</a></h5>
+
+<h5 class = "smallcaps">A Dome Standing on a Cylinder</h5>
+
+
+<p>This figure is a combination of the two preceding it. A&nbsp;cylinder
+is first raised on the circle, and on the top of that we draw
+semicircles from the different divisions on the circumference of the
+<span class = "pagenum">164</span>
+<a name = "page164" id = "page164"> </a>
+<!--png 180-->
+upper circle. This, however, only represents a small half-globular
+object. To draw the dome of a cathedral, or other building high above
+us, is another matter. From outside, where we can get to a distance, it
+is not difficult, but from within it will tax all our knowledge of
+perspective to give it effect.</p>
+
+<p>We shall go more into this subject when we come to archways and
+vaulted roofs, &amp;c.</p>
+
+<p class = "illustration">
+<a name = "fig191" id = "fig191"> </a>
+<img src = "images/fig191.png" width = "253" height = "322"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 191.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCIV" id = "chapCIV">
+CIV</a></h5>
+
+<h5 class = "smallcaps">Section of a Dome or Niche</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig192" id = "fig192"> </a>
+<img src = "images/fig192.png" width = "259" height = "187"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 192.</td>
+</tr>
+</table>
+
+<p>First draw outline of the niche <span class =
+"smallroman">GFDBA</span> (Fig. 193), then at its base draw square and
+circle <span class = "smallroman">GOA</span>, <span class =
+"smallroman">S</span> being the point of sight, and divide the
+circumference of the circle into the required number of parts. Then draw
+semicircle <span class = "smallroman">FOB</span>, and over that another
+semicircle <span class = "smallroman">EOC</span>. The manner of drawing
+them is shown in Fig. 192. From the divisions on the circle <span class
+= "smallroman">GOA</span> raise verticals to semicircle <span class =
+"smallroman">FOB</span>, which will divide it in the same way. Divide
+the smaller semicircle <span class = "smallroman">EOC</span> into the
+same number of parts as the others,
+<span class = "pagenum">165</span>
+<a name = "page165" id = "page165"> </a>
+<!--png 181-->
+which divisions will serve as guiding points in drawing the curves of
+the dome that are drawn towards <span class = "smallroman">D</span>, but
+the shading must assist greatly in giving the effect of the recess.</p>
+
+<p class = "illustration">
+<a name = "fig193" id = "fig193"> </a>
+<img src = "images/fig193.png" width = "210" height = "397"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 193.</p>
+
+<p><span class = "pagenum">166</span>
+<a name = "page166" id = "page166"> </a>
+<!--png 182-->
+In Fig. 192 will be seen how to draw semicircles in perspective. We
+first draw the half squares by drawing from centres <span class =
+"smallroman">O</span> of their diameters diagonals to distance-point, as
+<span class = "smallroman">OD</span>, which cuts the vanishing line
+<span class = "smallroman">BS</span> at <i>m</i>, and gives us the depth
+of the square, and in this we draw the semicircle in the usual way.</p>
+
+
+
+
+<span class = "pagenum">167</span>
+<a name = "page167" id = "page167"> </a>
+<!--png 183-->
+
+<h5 class = "section"><a name = "chapCV" id = "chapCV">
+<ins class = "correction" title = "C missing">C</ins>V</a></h5>
+
+<h5 class = "smallcaps">A Dome</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig194" id = "fig194"> </a>
+<img src = "images/fig194.png" width = "189" height = "352"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption">
+<span class = "smallcaps">Fig. 194.</span> A Dome.</td>
+</tr>
+</table>
+
+<p>First draw a section of the dome <span class =
+"smallroman">ACEDB</span> (Fig. 194) the shape required. Draw <span
+class = "smallroman">AB</span> at its base and <span class =
+"smallroman">CD</span> at some distance above it. Keeping these as
+central lines, form squares thereon by drawing <span class =
+"smallroman">SA</span>, <span class = "smallroman">SB</span>, <span
+class = "smallroman">SC</span>, <span class = "smallroman">SD</span>,
+&amp;c., from point of sight, and determining their lengths by diagonals
+<i>fh</i>, <i>f�h�</i> from point of distance, passing through <span
+class = "smallroman">O</span>. Having formed the two squares, draw
+perspective circles in each, and divide their circumferences into twelve
+or whatever number of parts are needed. To complete the figure draw from
+each division in the lower circle curves passing through the
+corresponding divisions in the upper one, to the apex. But as these are
+freehand lines, it requires some taste and knowledge to draw them
+properly, and of course in a large drawing several more squares and
+circles might be added to aid the draughtsman. The interior of the dome
+can be drawn in the same way.</p>
+
+<p class = "illustration section">
+<a name = "fig194_x" id = "fig194_x"> </a>
+<img src = "images/fig194_x.png" width = "298" height = "130"
+alt = "figure" title = "figure">
+</p>
+
+
+
+
+<span class = "pagenum">169</span>
+<a name = "page169" id = "page169"> </a>
+<!--png 185-->
+<h5 class = "section"><a name = "chapCVI" id = "chapCVI">
+CVI</a></h5>
+
+<h5 class = "smallcaps">How to Draw Columns Standing in a Circle</h5>
+
+
+<p>In Fig. 195 are sixteen cylinders or columns standing in a circle.
+First draw the circle on the ground, then divide it into sixteen equal
+parts, and let each division be the centre of the circle on which to
+raise the column. The question is how to make each one the right width
+in accordance with its position, for it is evident that a near column
+must appear wider than the opposite one. On the right of the figure is
+the vertical scale <span class = "smallroman">A</span>, which gives the
+heights of the columns, and at its foot is a horizontal scale, or a
+scale of widths <span class = "smallroman">B</span>. Now, according to
+the line on which the column stands, we find its apparent width marked
+on the scale. Thus take the small square and circle at 15, without its
+column, or the broken column at 16; and note that on each side of its
+centre <span class = "smallroman">O</span> I have measured <i>oa</i>,
+<i>ob</i>, equal to spaces marked 3 on the same horizontal in the scale
+<span class = "smallroman">B</span>. Through these points <i>a</i> and
+<i>b</i> I have drawn lines towards point of sight <span class =
+"smallroman">S</span>. Through their intersections with diagonal
+<i>e</i>, which is directed to point of distance, draw the farther and
+nearer sides of the square in which to describe the circle and the
+cylinder or column thereon. I&nbsp;have made all the squares thus
+obtained in parallel perspective, but they do not represent the bases of
+columns arranged in circles, which should converge towards the centre,
+and I believe in some cases are modified in form to suit that
+design.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[168]</span>
+<a name = "page168" id = "page168"> </a>
+<!--png 184-->
+<a name = "fig195" id = "fig195"> </a>
+<img src = "images/fig195.png" width = "408" height = "204"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 195.</p>
+
+
+
+
+<span class = "pagenum">170</span>
+<a name = "page170" id = "page170"> </a>
+<!--png 186-->
+<h5 class = "section"><a name = "chapCVII" id = "chapCVII">
+CVII</a></h5>
+
+<h5 class = "smallcaps">Columns and Capitals</h5>
+
+
+<p>This figure shows the application of the square and diagonal in
+drawing and placing columns in angular perspective.</p>
+
+<p class = "illustration">
+<a name = "fig196" id = "fig196"> </a>
+<img src = "images/fig196.png" width = "327" height = "232"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 196.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCVIII" id = "chapCVIII">
+CVIII</a></h5>
+
+<h5 class = "smallcaps">Method of Perspective Employed by
+Architects</h5>
+
+
+<p>The architects first draw a plan and elevation of the building to be
+put into perspective. Having placed the plan at the required angle to
+the picture plane, they fix upon the point of sight, and the distance
+from which the drawing is to be viewed. They then draw a line <span
+class = "smallroman">SP</span> at right angles to the picture plane
+<span class = "smallroman">VV�</span>, which represents that distance so
+that <span class = "smallroman">P</span> is the station-point. The eye
+is generally considered to be the station-point, but when lines are
+drawn to that point from the ground-plan, the station-point
+<span class = "pagenum">171</span>
+<a name = "page171" id = "page171"> </a>
+<!--png 191-->
+is placed on the ground, and is in fact the trace or projection exactly
+under the point at which the eye is placed. From this station-point
+<span class = "smallroman">P</span>, draw lines <span class =
+"smallroman">PV</span> and <span class = "smallroman">PV�</span>
+parallel to the two sides of the plan <i>ba</i> and <i>ad</i> (which
+will be at right angles to each other), and produce them to the horizon,
+which they will touch at points <span class = "smallroman">V</span> and
+<span class = "smallroman">V�</span>. These points thus obtained will be
+the two vanishing points.</p>
+
+<p>The next operation is to draw lines from the principal points of the
+plan to the station-point <span class = "smallroman">P</span>, such as
+<i>b</i><span class = "smallroman">P</span>, <i>c</i><span class =
+"smallroman">P</span>, <i>d</i><span class = "smallroman">P</span>,
+&amp;c., and where these lines intersect the picture plane (<span class
+= "smallroman">VV�</span> here represents it as well as the horizon),
+drop perpendiculars <i>b�</i><span class = "smallroman">B</span>,
+<i>a</i><span class = "smallroman">A</span>, <i>d�</i><span class =
+"smallroman">D</span>, &amp;c., to meet the vanishing lines <span class
+= "smallroman">AV</span>, <span class = "smallroman">AV�</span>, which
+will determine the points <span class = "smallroman">A</span>, <span
+class = "smallroman">B</span>, <span class = "smallroman">C</span>,
+<span class = "smallroman">D</span>, 1, 2, 3, &amp;c., and also the
+perspective lengths of the sides of the figure <span class =
+"smallroman">AB</span>, <span class = "smallroman">AD</span>, and the
+divisions <span class = "smallroman">B</span>, 1, 2, &amp;c. Taking the
+height of the figure <span class = "smallroman">AE</span> from the
+elevation, we measure it on <span class = "smallroman">A</span><i>a</i>;
+as in this instance <span class = "smallroman">A</span> touches the
+ground line, it may be used as a line of heights.</p>
+
+<p class = "illustration">
+<a name = "fig197" id = "fig197"> </a>
+<img src = "images/fig197thumb.png" width = "408" height = "275"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 197.</span> A method of angular
+Perspective employed by architects.<br>
+[<i>To face p.&nbsp;171</i>]</p>
+
+<p class = "caption">
+<a href = "images/fig197large.png"><i>Larger View</i></a></p>
+
+<p>I have here placed the perspective drawing under the ground plan to
+show the relation between the two, and how the perspective is worked
+out, but the general practice is to find the required measurements as
+here shown, to mark them on a straight edge of card or paper, and
+transfer them to the paper on which the drawing is to be made.</p>
+
+<p>This of course is the simplest form of a plan and elevation. It is
+easy to see, however, that we could set out an elaborate building in the
+same way as this figure, but in that case we should not place the
+drawing underneath the ground-plan, but transfer the measurements to
+another sheet of paper as mentioned above.</p>
+
+
+
+
+<span class = "pagenum">172</span>
+<a name = "page172" id = "page172"> </a>
+<!--png 192-->
+<h5 class = "section"><a name = "chapCIX" id = "chapCIX">
+CIX</a></h5>
+
+<h5 class = "smallcaps">The Octagon</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig198" id = "fig198"> </a>
+<img src = "images/fig198.png" width = "112" height = "108"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 198.</td>
+</tr>
+<tr>
+<td class = "picture">
+<a name = "fig199" id = "fig199"> </a>
+<img src = "images/fig199.png" width = "138" height = "187"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 199.</td>
+</tr>
+</table>
+
+<p>To draw the geometrical figure of an octagon contained in a square,
+take half of the diagonal of that square as radius, and from each corner
+describe a quarter circle. At the eight points where they touch the
+sides of the square, draw the eight sides of the octagon.</p>
+
+<p>To put this into perspective take the base of the square <span class
+= "smallroman">AB</span> and thereon form the perspective square <span
+class = "smallroman">ABCD</span>. From either extremity of that base
+(say <span class = "smallroman">B</span>) drop perpendicular <span class
+= "smallroman">BF</span>, draw diagonal <span class =
+"smallroman">AF</span>, and then from <span class =
+"smallroman">B</span> with radius <span class = "smallroman">BO</span>,
+half that diagonal, describe arc <span class = "smallroman">EOE</span>.
+This will give us the measurement <span class = "smallroman">AE</span>.
+Make <span class = "smallroman">GB</span> equal to <span class =
+"smallroman">AE</span>. Then draw lines from <span class =
+"smallroman">G</span> and <span class = "smallroman">E</span> towards
+<span class = "smallroman">S</span>, and by means of the diagonals find
+the transverse lines <span class = "smallroman">KK</span>, <i>hh</i>,
+which will give us the eight points through which to draw the
+octagon.</p>
+
+
+
+
+<span class = "pagenum">173</span>
+<a name = "page173" id = "page173"> </a>
+<!--png 193-->
+<h5 class = "section"><a name = "chapCX" id = "chapCX">
+CX</a></h5>
+
+<h5 class = "smallcaps">How to Draw the Octagon in Angular
+Perspective</h5>
+
+
+<p>Form square <span class = "smallroman">ABCD</span> (new method),
+produce sides <span class = "smallroman">BC</span> and <span class =
+"smallroman">AD</span> to the horizon at <span class =
+"smallroman">V</span>, and produce <span class = "smallroman">VA</span>
+to <i>a�</i> on base. Drop perpendicular from <span class =
+"smallroman">B</span> to <span class = "smallroman">F</span> the same
+length as <i>a�</i><span class = "smallroman">B</span>, and proceed as
+in the previous figure to find the eight points on the oblique square
+through which to draw the octagon.</p>
+
+<p class = "illustration">
+<a name = "fig200" id = "fig200"> </a>
+<img src = "images/fig200.png" width = "290" height = "259"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 200.</p>
+
+<p>It will be seen that this operation is very much the same as in
+parallel perspective, only we make our measurements on the base line
+<i>a�</i><span class = "smallroman">B</span> as we cannot measure the
+vanishing line <span class = "smallroman">BA</span> otherwise.</p>
+
+
+
+
+<span class = "pagenum">174</span>
+<a name = "page174" id = "page174"> </a>
+<!--png 194-->
+<h5 class = "section"><a name = "chapCXI" id = "chapCXI">
+CXI</a></h5>
+
+<h5 class = "smallcaps">How to Draw an Octagonal Figure in Angular
+Perspective</h5>
+
+
+<p>In this figure in angular perspective we do precisely the same thing
+as in the previous problem, taking our measurements on the base line
+<span class = "smallroman">EB</span> instead of on the vanishing line
+<span class = "smallroman">BA</span>. If we wish to raise a figure on
+this octagon the height of <span class = "smallroman">EG</span> we form
+the vanishing scale <span class = "smallroman">EGO</span>, and from the
+eight points on the ground draw horizontals to <span class =
+"smallroman">EO</span> and thus find all the points that give us the
+perspective height of each angle of the octagonal figure.</p>
+
+<p class = "illustration">
+<a name = "fig201" id = "fig201"> </a>
+<img src = "images/fig201.png" width = "349" height = "148"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 201.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXII" id = "chapCXII">
+CXII</a></h5>
+
+<h5 class = "smallcaps">How to Draw Concentric Octagons, with
+Illustration of a Well</h5>
+
+<p>The geometrical figure 202 <span class = "smallroman">A</span> shows
+how by means of diagonals <span class = "smallroman">AC</span> and <span
+class = "smallroman">BD</span> and the radii 1&nbsp;2&nbsp;3, &amp;c.,
+we can obtain smaller octagons inside the larger ones. Note how these
+are carried out in the second figure (202&nbsp;<span class =
+"smallroman">B</span>), and their application to this drawing of an
+octagonal well on an octagonal base.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<span class = "pagenum">[175]</span>
+<a name = "page175" id = "page175"> </a>
+<!--png 195-->
+<a name = "fig202" id = "fig202"> </a>
+<img src = "images/fig202a.png" width = "147" height = "143"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig202b.png" width = "157" height = "140"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 202 A.</td>
+<td class = "caption smallcaps">
+Fig. 202 B.</td>
+</tr>
+</table>
+
+<p class = "illustration">
+<a name = "fig203" id = "fig203"> </a>
+<img src = "images/fig203.png" width = "300" height = "269"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 203.</p>
+
+
+
+
+<span class = "pagenum">176</span>
+<a name = "page176" id = "page176"> </a>
+<!--png 196-->
+<h5 class = "section"><a name = "chapCXIII" id = "chapCXIII">
+CXIII</a></h5>
+
+<h5 class = "smallcaps">A Pavement Composed of Octagons and Small
+Squares</h5>
+
+
+<p>To draw a pavement with octagonal tiles we will begin with an octagon
+contained in a square <i>abcd</i>. Produce diagonal <i>ac</i> to <span
+class = "smallroman">V</span>. This will be the vanishing point for the
+sides of the small squares directed towards it. The other sides are
+directed to an inaccessible point out of the picture, but their
+directions are determined by the lines drawn from divisions on base to
+<span class = "smallroman">V</span><sup>2</sup> (see back, <a href =
+"#fig133">Fig. 133</a>).</p>
+
+<p class = "illustration">
+<a name = "fig204" id = "fig204"> </a>
+<img src = "images/fig204.png" width = "349" height = "130"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 204.</p>
+
+<p>I have drawn the lower figure to show how the squares which contain
+the octagons are obtained by means of the diagonals,
+<span class = "pagenum">177</span>
+<a name = "page177" id = "page177"> </a>
+<!--png 197-->
+<span class = "smallroman">BD</span>, <span class =
+"smallroman">AC</span>, and the central line <span class =
+"smallroman">OV</span><sup>2</sup>. Given the square <span class =
+"smallroman">ABCD</span>. From <span class = "smallroman">D</span> draw
+diagonal to <span class = "smallroman">G</span>, then from <span class =
+"smallroman">C</span> through centre <i>o</i> draw <span class =
+"smallroman">CE</span>, and so on all the way up the floor until
+sufficient are obtained. It is easy to see how other squares on each
+side of these can be produced.</p>
+
+<p class = "illustration">
+<a name = "fig205" id = "fig205"> </a>
+<img src = "images/fig205.png" width = "333" height = "123"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 205.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXIV" id = "chapCXIV">
+CXIV</a></h5>
+
+<h5 class = "smallcaps">The Hexagon</h5>
+
+
+<p>The hexagon is a six-sided figure which, if inscribed in a circle,
+will have each of its sides equal to the radius of that circle (Fig.
+206). If inscribed in a rectangle <span class =
+"smallroman">ABCD</span>, that rectangle will be equal in length to two
+sides of the hexagon or two radii of the circle, as <span class =
+"smallroman">EF</span>, and its width will be twice the height of an
+equilateral triangle <i>mon</i>.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture middle">
+<a name = "fig206" id = "fig206"> </a>
+<img src = "images/fig206.png" width = "188" height = "159"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig207" id = "fig207"> </a>
+<img src = "images/fig207.png" width = "275" height = "182"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 206.</td>
+<td class = "caption smallcaps">
+Fig. 207.</td>
+</tr>
+</table>
+
+<p>To put the hexagon into perspective, draw base of quadrilateral <span
+class = "smallroman">AD</span>, divide it into four equal parts, and
+from each division draw lines to point of sight. From <i>h</i> drop
+perpendicular <i>ho</i>, and form equilateral triangle <i>mno</i>. Take
+the height <i>ho</i> and measure it twice along the base from <span
+class = "smallroman">A</span> to&nbsp;2. From 2 draw line
+<span class = "pagenum">178</span>
+<a name = "page178" id = "page178"> </a>
+<!--png 198-->
+to point of distance, or from 1 to �&nbsp;distance, and so find length
+of side <span class = "smallroman">AB</span> equal to <span class =
+"smallroman">A</span>2. Draw <span class = "smallroman">BC</span>, and
+<span class = "smallroman">EF</span> through centre&nbsp;<i>o�</i>, and
+thus we have the six points through which to draw the hexagon.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXV" id = "chapCXV">
+CXV</a></h5>
+
+<h5 class = "smallcaps">A Pavement Composed of Hexagonal Tiles</h5>
+
+
+<p>In drawing pavements, except in the cases of square tiles, it is
+necessary to make a plan of the required design, as in this figure
+composed of hexagons. First set out the hexagon as at <span class =
+"smallroman">A</span>, then draw parallels 1&nbsp;1, 2&nbsp;2, &amp;c.,
+to mark the horizontal ends of the tiles and the intermediate lines
+<i>oo</i>. Divide the base into the required number of parts, each equal
+to one side of the hexagon, as 1,&nbsp;2, 3,&nbsp;4, &amp;c.; from these
+draw perpendiculars as shown in the figure, and also the diagonals
+passing through their intersections. Then mark with a strong line the
+outlines of the hexagonals, shading some of them; but the figure
+explains itself.</p>
+
+<p class = "illustration">
+<a name = "fig208" id = "fig208"> </a>
+<img src = "images/fig208.png" width = "296" height = "240"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 208.</p>
+
+<p>It is easy to put all these parallels, perpendiculars, and diagonals
+into perspective, and then to draw the hexagons.</p>
+
+<p>First draw the hexagon on <span class = "smallroman">AD</span> as in
+the previous figure, dividing
+<span class = "pagenum">179</span>
+<a name = "page179" id = "page179"> </a>
+<!--png 199-->
+<span class = "smallroman">AD</span> into four, &amp;c., set off right
+and left spaces equal to these fourths, and from each division draw
+lines to point of sight. Produce sides <i>me</i>, <i>nf</i> till they
+touch the horizon in points <span class = "smallroman">V</span>, <span
+class = "smallroman">V�</span>; these will be the two vanishing points
+for all the sides of the tiles that are receding from us. From each
+division on base draw lines to each of these vanishing points, then draw
+parallels through their intersections as shown on the figure. Having all
+these guiding lines it will not be difficult to draw as many hexagons as
+you please.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[180]</span>
+<a name = "page180" id = "page180"> </a>
+<!--png 200-->
+<a name = "fig209" id = "fig209"> </a>
+<img src = "images/fig209.png" width = "444" height = "203"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 209.</p>
+
+<p>Note that the vanishing points should be at equal distances from
+<span class = "smallroman">S</span>, also that the parallelogram in
+which each tile is contained is oblong, and not square, as already
+pointed out.</p>
+
+<p>We have also made use of the triangle <i>omn</i> to ascertain the
+length and width of that oblong. Another thing to note is that we have
+made use of the half distance, which enables us to make our pavement
+look flat without spreading our lines outside the picture.</p>
+
+
+
+<span class = "pagenum">181</span>
+<a name = "page181" id = "page181"> </a>
+<!--png 201-->
+<h5 class = "section"><a name = "chapCXVI" id = "chapCXVI">
+CXVI</a></h5>
+
+<h5 class = "smallcaps">A Pavement of Hexagonal Tiles in Angular
+Perspective</h5>
+
+
+<p>This is more difficult than the previous figure, as we only make use
+of one vanishing point; but it shows how much can be done by diagonals,
+as nearly all this pavement is drawn by their aid. First make a
+geometrical plan <span class = "smallroman">A</span> at the angle
+required. Then draw its perspective <span class = "smallroman">K</span>.
+Divide line 4<i>b</i> into four equal parts, and continue these
+measurements all along the base: from each division draw lines to <span
+class = "smallroman">V</span>, and draw the hexagon <span class =
+"smallroman">K</span>. Having this one to start with we produce its
+sides right and left, but first to the left to find point <span class =
+"smallroman">G</span>, the vanishing point of the
+<span class = "pagenum">182</span>
+<a name = "page182" id = "page182"> </a>
+<!--png 202-->
+diagonals. Those to the right, if produced far enough, would meet at a
+distant vanishing point not in the picture. But the student should study
+this figure for himself, and refer back to <a href = "#fig204">Figs.
+204</a> and <a href = "#fig205">205</a>.</p>
+
+<p class = "illustration">
+<a name = "fig210" id = "fig210"> </a>
+<img src = "images/fig210.png" width = "329" height = "251"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 210.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXVII" id = "chapCXVII">
+CXVII</a></h5>
+
+<h5 class = "smallcaps">Further Illustration of the Hexagon</h5>
+
+
+<p>To draw the hexagon in perspective we must first find the rectangle
+in which it is inscribed, according to the view we take of it. That at
+<span class = "smallroman">A</span> we have already drawn. We will now
+work out that at <span class = "smallroman">B</span>. Divide the base
+<span class = "smallroman">AD</span> into four equal parts and transfer
+those measurements to the perspective figure <span class =
+"smallroman">C</span>, as at <span class = "smallroman">AD</span>,
+measuring other equal spaces along the base. To find the depth <span
+class = "smallroman">A</span><i>n</i> of the rectangle, make <span class
+= "smallroman">DK</span> equal to base of square. Draw <span class =
+"smallroman">KO</span> to distance-point, cutting <span class =
+"smallroman">DO</span> at <span class = "smallroman">O</span>, and thus
+find line <span class = "smallroman">LO</span>. Draw diagonal <span
+class = "smallroman">D</span><i>n</i>, and through its intersections
+with the
+<span class = "pagenum">183</span>
+<a name = "page183" id = "page183"> </a>
+<!--png 203-->
+lines 1,&nbsp;2, 3,&nbsp;4 draw lines parallel to the base, and we shall
+thus have the framework, as it were, by which to draw the pavement.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig211" id = "fig211"> </a>
+<img src = "images/fig211a.png" width = "148" height = "163"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<img src = "images/fig211b.png" width = "149" height = "151"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 211 A.</td>
+<td class = "caption smallcaps">
+Fig. 211 B.</td>
+</tr>
+</table>
+
+<p class = "illustration">
+<a name = "fig212" id = "fig212"> </a>
+<img src = "images/fig212.png" width = "352" height = "143"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 212.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXVIII" id = "chapCXVIII">
+CXVIII</a></h5>
+
+<h5 class = "smallcaps">Another View of the Hexagon in Angular
+Perspective</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig213" id = "fig213"> </a>
+<img src = "images/fig213.png" width = "229" height = "122"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 213.</td>
+</tr>
+</table>
+
+<p>Given the rectangle <span class = "smallroman">ABCD</span> in angular
+perspective, produce side <span class = "smallroman">DA</span> to <span
+class = "smallroman">E</span> on base line. Divide <span class =
+"smallroman">EB</span> into four equal parts, and from each division
+draw lines to vanishing point, then by means of diagonals, &amp;c., draw
+the hexagon.</p>
+
+<p><span class = "pagenum">184</span>
+<a name = "page184" id = "page184"> </a>
+<!--png 204-->
+In Fig. 214 we have first drawn a geometrical plan, <span class =
+"smallroman">G</span>, for the sake of clearness, but the one above
+shows that this is not necessary.</p>
+
+<p class = "illustration">
+<a name = "fig214" id = "fig214"> </a>
+<img src = "images/fig214.png" width = "269" height = "238"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 214.</p>
+
+<p>To raise the hexagonal figure <span class = "smallroman">K</span> we
+have made use of the vanishing scale <span class = "smallroman">O</span>
+and the vanishing point <span class = "smallroman">V</span>. Another
+method could be used by drawing two hexagons one over the other at the
+required height.</p>
+
+
+
+
+<span class = "pagenum">185</span>
+<a name = "page185" id = "page185"> </a>
+<!--png 205-->
+<h5 class = "section"><a name = "chapCXIX" id = "chapCXIX">
+CXIX</a></h5>
+
+<h5 class = "smallcaps">Application of the Hexagon to Drawing a
+Kiosk</h5>
+
+
+<p>This figure is built up from the hexagon standing on a rectangular
+base, from which we have raised verticals, &amp;c. Note how the jutting
+portions of the roof are drawn from <i>o�</i>. But the figure explains
+itself, so there is no necessity to repeat descriptions already given in
+the foregoing problems.</p>
+
+<p class = "illustration">
+<a name = "fig215" id = "fig215"> </a>
+<img src = "images/fig215.png" width = "333" height = "340"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 215.</p>
+
+
+
+
+<span class = "pagenum">186</span>
+<a name = "page186" id = "page186"> </a>
+<!--png 206-->
+<h5 class = "section"><a name = "chapCXX" id = "chapCXX">
+CXX</a></h5>
+
+<h5 class = "smallcaps">The Pentagon</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig216" id = "fig216"> </a>
+<img src = "images/fig216.png" width = "188" height = "186"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 216.</td>
+</tr>
+</table>
+
+<p>The pentagon is a figure with five equal sides, and if inscribed in a
+circle will touch its circumference at five equidistant points. With any
+convenient radius describe circle. From half this radius, marked&nbsp;1,
+draw a line to apex, marked&nbsp;2. Again, with 1 as centre and 1&nbsp;2
+as radius, describe arc 2&nbsp;3. Now with 2 as centre and 2&nbsp;3 as
+radius describe arc 3&nbsp;4, which will cut the circumference at
+point&nbsp;4. Then line 2&nbsp;4 will be one of the sides of the
+pentagon, which we can measure round the circle and so produce the
+required figure.</p>
+
+<p>To put this pentagon into parallel perspective inscribe the circle in
+which it is drawn in a square, and from its five angles
+4,&nbsp;2,&nbsp;4, &amp;c., drop perpendiculars to base and number them
+as in the figure. Then draw the perspective square (Fig. 217) and
+transfer these measurements to its base. From these draw lines to point
+of sight, then by their aid and the two diagonals proceed to construct
+the pentagon in the same way that we did the triangles and other
+figures. Should it be required to place this
+<span class = "pagenum">187</span>
+<a name = "page187" id = "page187"> </a>
+<!--png 207-->
+pentagon in the opposite position, then we can transfer our measurements
+to the far side of the square, as in Fig. 218.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig217" id = "fig217"> </a>
+<img src = "images/fig217.png" width = "192" height = "212"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig218" id = "fig218"> </a>
+<img src = "images/fig218.png" width = "187" height = "192"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 217.</td>
+<td class = "caption smallcaps">
+Fig. 218.</td>
+</tr>
+</table>
+
+<p><span class = "pagenum">188</span>
+<a name = "page188" id = "page188"> </a>
+<!--png 208-->
+Or if we wish to put it into angular perspective we adopt the same
+method as with the hexagon, as shown at Fig. 219.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig219" id = "fig219"> </a>
+<img src = "images/fig219.png" width = "255" height = "172"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig220" id = "fig220"> </a>
+<img src = "images/fig220.png" width = "151" height = "151"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 219.</td>
+<td class = "caption smallcaps">
+Fig. 220.</td>
+</tr>
+</table>
+
+<p>Another way of drawing a pentagon (Fig. 220) is to draw an isosceles
+triangle with an angle of 36� at its apex, and from centre of each side
+of the triangle draw perpendiculars to meet at <i>o</i>, which will be
+the centre of the circle in which it is inscribed. From this centre and
+with radius <span class = "smallroman">OA</span> describe circle <span
+class = "smallroman">A</span>&nbsp;3&nbsp;2, &amp;c. Take base of
+triangle 1&nbsp;2, measure it round the circle, and so find the five
+points through which to draw the pentagon. The angles at 1&nbsp;2 will
+each be 72�, double that at <span class = "smallroman">A</span>, which
+is 36�.</p>
+
+
+
+
+<span class = "pagenum">189</span>
+<a name = "page189" id = "page189"> </a>
+<!--png 209-->
+<h5 class = "section"><a name = "chapCXXI" id = "chapCXXI">
+CXXI</a></h5>
+
+<h5 class = "smallcaps">The Pyramid</h5>
+
+
+<p>Nothing can be more simple than to put a pyramid into perspective.
+Given the base (<i>abc</i>), raise from its centre a perpendicular
+(<span class = "smallroman">OP</span>) of the required height, then draw
+lines from the corners of that base to a point <span class =
+"smallroman">P</span> on the vertical line, and the thing is done. These
+pyramids can be used in drawing roofs, steeples, &amp;c. The cone is
+drawn in the same way, so also is any other figure, whether octagonal,
+hexangular, triangular, &amp;c.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture" colspan = "3">
+<a name = "fig221" id = "fig221"> </a>
+<a name = "fig222" id = "fig222"> </a>
+<a name = "fig223" id = "fig223"> </a>
+<img src = "images/fig221_222_223.png" width = "321" height = "151"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 221.</td>
+<td class = "caption smallcaps">
+Fig. 222.</td>
+<td class = "caption smallcaps">
+Fig. 223.</td>
+</tr>
+</table>
+
+
+
+
+<span class = "pagenum">191</span>
+<a name = "page191" id = "page191"> </a>
+<!--png 211-->
+<h5 class = "section"><a name = "chapCXXII" id = "chapCXXII">
+CXXII</a></h5>
+
+<h5 class = "smallcaps">The Great Pyramid</h5>
+
+
+<p>This enormous structure stands on a square base of over thirteen
+acres, each side of which measures, or did measure, 764 feet. Its
+original height was 480 feet, each side being an equilateral triangle.
+Let us see how we can draw this gigantic mass on our little sheet of
+paper.</p>
+
+<p>In the first place, to take it all in at one view we must put it very
+far back, and in the second the horizon must be so low down that we
+cannot draw the square base of thirteen acres on the perspective plane,
+that is on the ground, so we must draw it in the air, and also to a very
+small scale.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[190]</span>
+<a name = "page190" id = "page190"> </a>
+<!--png 210-->
+<a name = "fig224" id = "fig224"> </a>
+<img src = "images/fig224.png" width = "440" height = "242"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 224.</p>
+
+<p>Divide the base <span class = "smallroman">AB</span> into ten equal
+parts, and suppose each of these parts to measure 10 feet, <span class =
+"smallroman">S</span>, the point of sight, is placed on the left of the
+picture near the side, in order that we may get a long line of distance,
+<span class = "smallroman">S</span> � <span class =
+"smallroman">D</span>; but even this line is only half the distance we
+require. Let us therefore take the 16th distance, as shown in our
+previous illustration of the lighthouse (Fig. 92), which enables us to
+measure sixteen times the length of base <span class =
+"smallroman">AB</span>, or 1,600 feet. The base <i>ef</i> of the pyramid
+is 1,600 feet from the base line of the picture, and is, according to
+our 10-foot scale, 764 feet long.</p>
+
+<p>The next thing to consider is the height of the pyramid. We make a
+scale to the right of the picture measuring 50 feet from <span class =
+"smallroman">B</span> to 50 at point where <span class =
+"smallroman">BP</span> intersects base of pyramid, raise perpendicular
+<span class = "smallroman">CG</span> and thereon measure 480 feet. As we
+cannot obtain a palpable square on the ground, let us draw one 480 feet
+above the ground. From <i>e</i> and <i>f</i> raise verticals
+<i>e</i><span class = "smallroman">M</span> and <i>f</i><span class =
+"smallroman">N</span>, making them equal to perpendicular <span class =
+"smallroman">G</span>, and draw line <span class =
+"smallroman">MN</span>, which will be the same length as base, or 764
+feet. On this line form square <span class = "smallroman">MNK</span>
+parallel to the perspective plane, find its centre <span class =
+"smallroman">O�</span> by means of diagonals, and <span class =
+"smallroman">O�</span> will be the central height of the pyramid and
+exactly over the centre of the base. From this point <span class =
+"smallroman">O�</span> draw sloping lines <span class =
+"smallroman">O�</span><i>f</i>, <span class =
+"smallroman">O�</span><i>e</i>, <span class = "smallcaps">O�y</span>,
+&amp;c., and the figure is complete.</p>
+
+<p><span class = "pagenum">192</span>
+<a name = "page192" id = "page192"> </a>
+<!--png 212-->
+Note the way in which we find the measurements on base of pyramid and on
+line <span class = "smallroman">MN</span>. By drawing <span class =
+"smallroman">AS</span> and <span class = "smallroman">BS</span> to point
+of sight we find <span class = "smallroman">T</span><i>e</i>, which
+measures 100 feet at a distance of 1,600 feet. We mark off seven of
+these lengths, and an additional 64 feet by the scale, and so obtain the
+required length. The position of the third corner of the base is found
+by dropping a perpendicular from <span class = "smallroman">K</span>,
+till it meets the line <i>e</i><span class = "smallroman">S</span>.</p>
+
+<p>Another thing to note is that the side of the pyramid that faces us,
+although an equilateral triangle, does not appear so, as its top angle
+is 382 feet farther off than its base owing to its leaning position.</p>
+
+
+
+
+<span class = "pagenum">193</span>
+<a name = "page193" id = "page193"> </a>
+<!--png 213-->
+<h5 class = "section"><a name = "chapCXXIII" id = "chapCXXIII">
+CXXIII</a></h5>
+
+<h5 class = "smallcaps">The Pyramid in Angular Perspective</h5>
+
+
+<p>In order to show the working of this proposition I have taken a much
+higher horizon, which immediately detracts from the impression of the
+bigness of the pyramid.</p>
+
+<p class = "illustration">
+<a name = "fig225" id = "fig225"> </a>
+<img src = "images/fig225.png" width = "298" height = "210"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 225.</p>
+
+<p>We proceed to make our ground-plan <i>abcd</i> high above the horizon
+instead of below it, drawing first the parallel square and then the
+oblique one. From all the principal points drop perpendiculars to the
+ground and thus find the points through which to draw the base of the
+pyramid. Find centres <span class = "smallroman">OO�</span> and decide
+upon the height <span class = "smallroman">OP</span>. Draw the sloping
+lines from <span class = "smallroman">P</span> to the corners of the
+base, and the figure is complete.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXIV" id = "chapCXXIV">
+CXXIV</a></h5>
+
+<h5 class = "smallcaps">To Divide the Sides of the Pyramid
+Horizontally</h5>
+
+
+<p>Having raised the pyramid on a given oblique square, divide the
+vertical line <span class = "smallroman">OP</span> into the required
+number of parts. From
+<span class = "pagenum">194</span>
+<a name = "page194" id = "page194"> </a>
+<!--png 214-->
+<span class = "smallroman">A</span> through <span class =
+"smallroman">C</span> draw <span class = "smallroman">AG</span> to
+horizon, which gives us <span class = "smallroman">G</span>, the
+vanishing point of all the diagonals of squares parallel to and at the
+same angle as <span class = "smallroman">ABCD</span>. From <span class =
+"smallroman">G</span> draw lines through the divisions 2,&nbsp;3,
+&amp;c., on <span class = "smallroman">OP</span> cutting the lines <span
+class = "smallroman">PA</span> and <span class = "smallroman">PC</span>,
+thus dividing them into the required parts. Through the points thus
+found draw from <span class = "smallroman">V</span> all those sides of
+the squares that have <span class = "smallroman">V</span> for their
+vanishing point, as <i>ab</i>, <i>cd</i>, &amp;c. Then join <i>bd</i>,
+<i>ac</i>, and the rest, and thus make the horizontal divisions
+required.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig226" id = "fig226"> </a>
+<img src = "images/fig226.png" width = "304" height = "181"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig227" id = "fig227"> </a>
+<img src = "images/fig227.png" width = "158" height = "109"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 226.</td>
+<td class = "caption smallcaps">
+Fig. 227.</td>
+</tr>
+</table>
+
+<p>The same method will apply to drawing steps, square blocks, &amp;c.,
+as shown in Fig. 227, which is at the same angle as the above.</p>
+
+
+
+
+<span class = "pagenum">195</span>
+<a name = "page195" id = "page195"> </a>
+<!--png 215-->
+<h5 class = "section"><a name = "chapCXXV" id = "chapCXXV">
+CXXV</a></h5>
+
+<h5 class = "smallcaps">Of Roofs</h5>
+
+
+<p>The pyramidal roof (Fig. 228) is so simple that it explains itself.
+The chief thing to be noted is the way in which the diagonals are
+produced beyond the square of the walls, to give the width of the eaves,
+according to their position.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig228" id = "fig228"> </a>
+<img src = "images/fig228.png" width = "195" height = "256"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig229" id = "fig229"> </a>
+<img src = "images/fig229.png" width = "184" height = "286"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 228.</td>
+<td class = "caption smallcaps">
+Fig. 229.</td>
+</tr>
+</table>
+
+<p>Another form of the pyramidal roof is here given (Fig. 229). First
+draw the cube <i>edcba</i> at the required height, and on the side
+facing us, <i>adcb</i>, draw triangle <span class =
+"smallroman">K</span>, which represents the end of a gable roof. Then
+draw similar triangles on the other sides of the cube (see <a href =
+"#fig159">Fig. 159</a>, LXXXIV). Join the opposite triangles
+<span class = "pagenum">196</span>
+<a name = "page196" id = "page196"> </a>
+<!--png 216-->
+at the apex, and thus form two gable roofs crossing each other at right
+angles. From <i>o</i>, centre of base of cube, raise vertical <span
+class = "smallroman">OP</span>, and then from <span class =
+"smallroman">P</span> draw sloping lines to each corner of base
+<i>a</i>, <i>b</i>, &amp;c., and by means of central lines drawn from
+<span class = "smallroman">P</span> to half base, find the points where
+the gable roofs intersect the central spire or pyramid. Any other
+proportions can be obtained by adding to or altering the cube.</p>
+
+<p class = "illustration">
+<a name = "fig230" id = "fig230"> </a>
+<img src = "images/fig230.png" width = "314" height = "144"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 230.</p>
+
+<p>To draw a sloping or hip-roof which falls back at each end we must
+first draw its base, <span class = "smallroman">CBDA</span> (Fig. 230).
+Having found the centre <span class = "smallroman">O</span> and central
+line <span class = "smallroman">SP</span>, and how far the roof is to
+fall back at each end, namely the distance <span class =
+"smallroman">P</span><i>m</i>, draw horizontal line <span class =
+"smallroman">RB</span> through <i>m</i>. Then from <span class =
+"smallroman">B</span> through <span class = "smallroman">O</span> draw
+diagonal <span class = "smallroman">BA</span>, and from
+<span class = "pagenum">197</span>
+<a name = "page197" id = "page197"> </a>
+<!--png 217-->
+<span class = "smallroman">A</span> draw horizontal <span class =
+"smallroman">AD</span>, which gives us point <i>n</i>. From these two
+points <i>m</i> and <i>n</i> raise perpendiculars the height required
+for the roof, and from these draw sloping lines to the corners of the
+base. Join <i>ef</i>, that is, draw the top line of the roof, which
+completes it. Fig. 231 shows a plan or bird's-eye view of the roof and
+the diagonal <span class = "smallroman">AB</span> passing through centre
+<span class = "smallroman">O</span>. But there are so many varieties of
+roofs they would take almost a book to themselves to illustrate them,
+especially the cottages and farm-buildings, barns, &amp;c., besides
+churches, old mansions, and others. There is also such irregularity
+about some of them that perspective rules, beyond those few here given,
+are of very little use. So that the best thing for an artist to do is to
+sketch them from the real whenever he has an opportunity.</p>
+
+<p class = "illustration">
+<a name = "fig231" id = "fig231"> </a>
+<img src = "images/fig231.png" width = "222" height = "94"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 231.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXVI" id = "chapCXXVI">
+CXXVI</a></h5>
+
+<h5 class = "smallcaps">Of Arches, Arcades, Bridges, &amp;c.</h5>
+
+
+<p><span class = "pagenum">199</span>
+<a name = "page199" id = "page199"> </a>
+<!--png 219-->
+For an arcade or cloister (Fig. 232) first set up the outer frame <span
+class = "smallroman">ABCD</span> according to the proportions required.
+For round arches the height may be twice that of the base, varying to
+one and a half. In Gothic arches the height may be about three times the
+width, all of which proportions are chosen to suit the different
+purposes and effects required. Divide the base <span class =
+"smallroman">AB</span> into the desired number of parts,
+8,&nbsp;10,&nbsp;12, &amp;c., each part representing 1&nbsp;foot. (In
+this case the base is 10 feet and the horizon 5&nbsp;feet.) Set out
+floor by means of �&nbsp;distance. Divide it into squares of
+1&nbsp;foot, so that there will be 8&nbsp;feet between each column or
+pilaster, supposing we make them to stand on a square foot. Draw the
+first archway <span class = "smallroman">EKF</span> facing us, and its
+inner semicircle <i>gh</i>, with also its thickness or depth of
+1&nbsp;foot. Draw the span of the archway <span class =
+"smallroman">EF</span>, then central line <span class =
+"smallroman">PO</span> to point of sight. Proceed to raise as many other
+arches as required at the given distances. The intersections of the
+central line with the chords <i>mn</i>, &amp;c., will give the centres
+from which to describe the semicircles.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[198]</span>
+<a name = "page198" id = "page198"> </a>
+<!--png 218-->
+<a name = "fig232" id = "fig232"> </a>
+<img src = "images/fig232.png" width = "328" height = "417"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 232.</p>
+
+
+
+
+<span class = "pagenum">200</span>
+<a name = "page200" id = "page200"> </a>
+<!--png 220-->
+<h5 class = "section"><a name = "chapCXXVII" id = "chapCXXVII">
+CXXVII</a></h5>
+
+<h5 class = "smallcaps">Outline of an Arcade with Semicircular
+Arches</h5>
+
+
+<p>This is to show the method of drawing a long passage, corridor, or
+cloister with arches and columns at equal distances, and is worked in
+the same way as the previous figure, using � distance and � base. The
+floor consists of five squares; the semicircles of the arches are
+described from the numbered points on the central line <span class =
+"smallroman">OS</span>, where it intersects the chords of the
+arches.</p>
+
+<p class = "illustration">
+<a name = "fig233" id = "fig233"> </a>
+<img src = "images/fig233.png" width = "264" height = "300"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 233.</p>
+
+
+
+
+<span class = "pagenum">201</span>
+<a name = "page201" id = "page201"> </a>
+<!--png 221-->
+<h5 class = "section"><a name = "chapCXXVIII" id = "chapCXXVIII">
+CXXVIII</a></h5>
+
+<h5 class = "smallcaps">Semicircular Arches on a Retreating Plane</h5>
+
+
+<p>First draw perspective square <i>abcd</i>. Let <i>ae�</i> be the
+height of the figure. Draw <i>ae�f�b</i> and proceed with the rest of
+the outline. To draw the arches begin with the one facing us, <span
+class = "smallroman">E</span><i>o�</i><span class =
+"smallroman">F</span> enclosed in the quadrangle <span class =
+"smallroman">E</span><i>e�f�</i><span class = "smallroman">F</span>.
+With centre <span class = "smallroman">O</span> describe the semicircle
+and across it draw the diagonals <i>e�</i><span class =
+"smallroman">F</span>, <span class = "smallroman">E</span><i>f�</i>, and
+through <i>nn</i>, where these lines intersect the semicircle, draw
+horizontal <span class = "smallroman">KK</span> and also <span class =
+"smallroman">KS</span> to point of sight. It will be seen that the
+half-squares at the side are the same size in perspective as the one
+facing us, and we carry out in them much the same operation; that is, we
+draw the diagonals, find the point <span class = "smallroman">O</span>,
+and the points <i>nn</i>, &amp;c., through which to draw our arches. See
+perspective of the circle (Fig. 165).</p>
+
+<p class = "illustration">
+<a name = "fig234" id = "fig234"> </a>
+<img src = "images/fig234.png" width = "297" height = "183"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 234.</p>
+
+<p>If more points are required an additional diagonal from <span class =
+"smallroman">O</span> to
+<span class = "pagenum">202</span>
+<a name = "page202" id = "page202"> </a>
+<!--png 222-->
+<span class = "smallroman">K</span> may be used, as shown in the figure,
+which perhaps explains itself. The method is very old and very simple,
+and of course can be applied to any kind of arch, pointed or stunted, as
+in this drawing of a pointed arch (Fig. 235).</p>
+
+<p class = "illustration">
+<a name = "fig235" id = "fig235"> </a>
+<img src = "images/fig235.png" width = "341" height = "191"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 235.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXIX" id = "chapCXXIX">
+CXXIX</a></h5>
+
+<h5 class = "smallcaps">An Arcade in Angular Perspective</h5>
+
+
+<p>First draw the perspective square <span class =
+"smallroman">ABCD</span> at the angle required, by new method. Produce
+sides <span class = "smallroman">AD</span> and <span class =
+"smallroman">BC</span> to <span class = "smallroman">V</span>. Draw
+diagonal BD and produce to point <span class = "smallroman">G</span>,
+from whence we draw the other diagonals to <i>cfh</i>. Make spaces
+1,&nbsp;2,&nbsp;3, &amp;c., on base line equal to <span class =
+"smallcaps">B&nbsp;1</span> to obtain sides of squares. Raise vertical
+<span class = "smallroman">BM</span> the height required. Produce <span
+class = "smallroman">DA</span> to <span class = "smallroman">O</span> on
+base line, and from <span class = "smallroman">O</span> raise vertical
+OP equal to <span class = "smallroman">BM</span>. This line enables us
+to dispense with the long vanishing point to the left; its working has
+been explained at Fig. 131. From <span class = "smallroman">P</span>
+draw <span class = "smallroman">PRV</span> to vanishing point <span
+class = "smallroman">V</span>, which will intersect vertical <span class
+= "smallroman">AR</span> at <span class = "smallroman">R</span>. Join
+<span class = "smallroman">MR</span>, and this line, if produced, would
+meet the horizon at the other vanishing point.
+<span class = "pagenum">203</span>
+<a name = "page203" id = "page203"> </a>
+<!--png 223-->
+In like manner make <span class = "smallroman">O</span>2 equal to <span
+class = "smallroman">B</span>2�. From 2 draw line to <span class =
+"smallroman">V</span>, and at&nbsp;2, its intersection with <span class
+= "smallroman">AR</span>, draw line 2&nbsp;2, which will also meet the
+horizon at the other vanishing point. By means of the quarter-circle
+<span class = "smallroman">A</span> we can obtain the points through
+which to draw the semicircular arches in the same way as in the previous
+figure.</p>
+
+<p class = "illustration">
+<a name = "fig236" id = "fig236"> </a>
+<img src = "images/fig236.png" width = "328" height = "235"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 236.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXX" id = "chapCXXX">
+CXXX</a></h5>
+
+<h5 class = "smallcaps">A Vaulted Ceiling</h5>
+
+
+<p>From the square ceiling <span class = "smallroman">ABCD</span> we
+have, as it were, suspended two arches from the two diagonals <span
+class = "smallroman">DB</span>, <span class = "smallroman">AC</span>,
+which spring from the four corners of the square <span class =
+"smallroman">EFGH</span>, just underneath it. The curves of these
+arches, which are not semicircular but elongated, are obtained by means
+of the vanishing scales <i>m</i><span class = "smallroman">S</span>,
+<i>n</i><span class = "smallroman">S</span>. Take any two convenient
+points <span class = "smallroman">P</span>, <span class =
+"smallroman">R</span>, on each side of the semicircle, and
+<span class = "pagenum">204</span>
+<a name = "page204" id = "page204"> </a>
+<!--png 224-->
+raise verticals <span class = "smallroman">P</span><i>m</i>, <span class
+= "smallroman">R</span><i>n</i> to <span class = "smallroman">AB</span>,
+and on these verticals form the scales. Where <i>m</i><span class =
+"smallroman">S</span> and <i>n</i><span class = "smallroman">S</span>
+cut the diagonal <span class = "smallroman">AC</span> drop
+perpendiculars to meet the lower line of the scale at points 1,&nbsp;2.
+On the other side, using the other scales, we have dropped
+perpendiculars in the same way from the diagonal to 3,&nbsp;4. These
+points, together
+<span class = "pagenum">205</span>
+<a name = "page205" id = "page205"> </a>
+<!--png 225-->
+with <span class = "smallroman">EOG</span>, enable us to trace the curve
+<span class = "smallroman">E</span>&nbsp;1 2 O 3 4&nbsp;<span class =
+"smallroman">G</span>. We draw the arch under the other diagonal in
+precisely the same way.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig237" id = "fig237"> </a>
+<img src = "images/fig237.png" width = "222" height = "366"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig238" id = "fig238"> </a>
+<img src = "images/fig238.png" width = "226" height = "376"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 237.</td>
+<td class = "caption smallcaps">
+Fig. 238.</td>
+</tr>
+</table>
+
+<p>The reason for thus proceeding is that the cross arches, although
+elongated, hang from their diagonals just as the semicircular arch <span
+class = "smallroman">EKF</span> hangs from <span class =
+"smallroman">AB</span>, and the lines <i>mn</i>, touching the circle at
+<span class = "smallroman">PR</span>, are represented by 1,&nbsp;2,
+hanging from the diagonal <span class = "smallroman">AC</span>.</p>
+
+<p><span class = "pagenum">206</span>
+<a name = "page206" id = "page206"> </a>
+<!--png 226-->
+Figure 238, which is practically the same as the preceding only
+differently shaded, is drawn in the following manner. Draw arch <span
+class = "smallroman">EGF</span> facing us, and proceed with the rest of
+the corridor, but first finding the flat ceiling above the square on the
+ground <span class = "smallroman">AB</span><i>cd</i>. Draw diagonals
+<i>ac</i>, <i>bd</i>, and the curves pending from them. But we no longer
+see the clear arch as in the other drawing, for the spaces between the
+curves are filled in and arched across.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXXI" id = "chapCXXXI">
+CXXXI</a></h5>
+
+<h5 class = "smallcaps">A Cloister, from a Photograph</h5>
+
+
+<p>This drawing of a cloister from a photograph shows the correctness of
+our perspective, and the manner of applying it to practical work.</p>
+
+<p class = "illustration">
+<a name = "fig239" id = "fig239"> </a>
+<img src = "images/fig239.png" width = "224" height = "270"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 239.</p>
+
+
+
+
+<span class = "pagenum">207</span>
+<a name = "page207" id = "page207"> </a>
+<!--png 227-->
+<h5 class = "section"><a name = "chapCXXXII" id = "chapCXXXII">
+CXXXII</a></h5>
+
+<h5 class = "smallcaps">The Low or Elliptical Arch</h5>
+
+
+<p>Let <span class = "smallroman">AB</span> be the span of the arch and
+<span class = "smallroman">O</span><i>h</i> its height. From centre
+<span class = "smallroman">O</span>, with <span class =
+"smallroman">OA</span>, or half the span, for radius, describe outer
+semicircle. From same centre and <i>oh</i> for radius describe the inner
+semicircle. Divide outer circle into a convenient number of parts,
+1,&nbsp;2,&nbsp;3, &amp;c., to which draw radii from centre <span class
+= "smallroman">O</span>. From each division drop perpendiculars. Where
+the radii intersect the inner circle, as at <i>gkmo</i>, draw
+horizontals <i>op</i>, <i>mn</i>, <i>kj</i>, &amp;c., and
+<span class = "pagenum">208</span>
+<a name = "page208" id = "page208"> </a>
+<!--png 228-->
+through their intersections with the perpendiculars <i>f</i>, <i>j</i>,
+<i>n</i>, <i>p</i>, draw the curve of the flattened arch. Transfer this
+to the lower figure, and proceed to draw the tunnel. Note how the
+vanishing scale is formed on either side by horizontals <i>ba</i>,
+<i>fe</i>, &amp;c., which enable us to make the distant arches similar
+to the near ones.</p>
+
+<p class = "illustration">
+<a name = "fig240" id = "fig240"> </a>
+<img src = "images/fig240.png" width = "255" height = "147"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 240.</p>
+
+<p class = "illustration">
+<a name = "fig241" id = "fig241"> </a>
+<img src = "images/fig241.png" width = "261" height = "144"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 241.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXXIII" id = "chapCXXXIII">
+CXXXIII</a></h5>
+
+<h5 class = "smallcaps">Opening or Arched Window in a Vault</h5>
+
+
+<p>First draw the vault <span class = "smallroman">AEB</span>. To
+introduce the window <span class = "smallroman">K</span>, the upper part
+of which follows the form of the vault, we first decide on its width,
+which is <i>mn</i>, and its height from floor <span class =
+"smallroman">B</span><i>a</i>. On line <span class =
+"smallroman">B</span><i>a</i> at the side of the arch form scales
+<i>aa�</i><span class = "smallroman">S</span>, <i>bb�</i><span class =
+"smallroman">S</span>, &amp;c. Raise the semicircular arch <span class =
+"smallroman">K</span>, shown by a dotted line. The scale at the side
+will give the lengths <i>aa�</i>, <i>bb�</i>, &amp;c., from different
+parts of this dotted arch to corresponding points in the curved archway
+or window required.</p>
+
+<p class = "illustration">
+<a name = "fig242" id = "fig242"> </a>
+<img src = "images/fig242.png" width = "245" height = "205"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 242.</p>
+
+<p>Note that to obtain the width of the window <span class =
+"smallroman">K</span> we have used
+<span class = "pagenum">209</span>
+<a name = "page209" id = "page209"> </a>
+<!--png 229-->
+the diagonals on the floor and width <i>m n</i> on base. This method of
+measurement is explained at Fig. 144, and is of ready application in a
+case of this kind.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXXIV" id = "chapCXXXIV">
+CXXXIV</a></h5>
+
+<h5 class = "smallcaps">Stairs, Steps, &amp;c.</h5>
+
+
+<p>Having decided upon the incline or angle, such as <span class =
+"smallroman">CBA</span>, at which the steps are to be placed, and the
+height <span class = "smallroman">B</span><i>m</i> of each step, draw
+<i>mn</i> to <span class = "smallroman">CB</span>, which will give the
+width. Then measure along base <span class = "smallroman">AB</span> this
+width equal to <span class = "smallroman">DB</span>, which will give
+that for all the other steps. Obtain length <span class =
+"smallroman">BF</span> of steps, and draw <span class =
+"smallroman">EF</span> parallel to <span class = "smallroman">CB</span>.
+These lines will aid in securing the exactness of the figure.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig243" id = "fig243"> </a>
+<img src = "images/fig243.png" width = "261" height = "115"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture middle">
+<a name = "fig244" id = "fig244"> </a>
+<img src = "images/fig244.png" width = "187" height = "87"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 243.</td>
+<td class = "caption smallcaps">
+Fig. 244.</td>
+</tr>
+</table>
+
+
+
+
+<span class = "pagenum">210</span>
+<a name = "page210" id = "page210"> </a>
+<!--png 230-->
+<h5 class = "section"><a name = "chapCXXXV" id = "chapCXXXV">
+CXXXV</a></h5>
+
+<h5 class = "smallcaps">Steps, Front View</h5>
+
+
+<p>In this figure the height of each step is measured on the vertical
+line <span class = "smallroman">AB</span> (this line is sometimes called
+the line of heights), and their depth is found by diagonals drawn to the
+point of distance <span class = "smallroman">D</span>. The rest of the
+figure explains itself.</p>
+
+<p class = "illustration">
+<a name = "fig245" id = "fig245"> </a>
+<img src = "images/fig245.png" width = "340" height = "157"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 245.</p>
+
+
+
+
+<span class = "pagenum">211</span>
+<a name = "page211" id = "page211"> </a>
+<!--png 231-->
+<h5 class = "section"><a name = "chapCXXXVI" id = "chapCXXXVI">
+CXXXVI</a></h5>
+
+<h5 class = "smallcaps">Square Steps</h5>
+
+
+<p>Draw first step <span class = "smallroman">ABEF</span> and its two
+diagonals. Raise vertical <span class = "smallroman">AH</span>, and
+measure thereon the required height of each step, and thus form scale.
+Let the second step <span class = "smallroman">CD</span> be less all
+round than the first by <span class = "smallroman">A</span><i>o</i> or
+<span class = "smallroman">B</span><i>o</i>. Draw <i>o</i><span class =
+"smallroman">C</span> till it cuts the diagonal, and proceed to draw the
+second step, guided by the diagonals and taking its height from the
+scale as shown. Draw the third step in the same way.</p>
+
+<p class = "illustration">
+<a name = "fig246" id = "fig246"> </a>
+<img src = "images/fig246.png" width = "251" height = "129"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 246.</p>
+
+
+
+
+<span class = "pagenum">212</span>
+<a name = "page212" id = "page212"> </a>
+<!--png 232-->
+<h5 class = "section"><a name = "chapCXXXVII" id = "chapCXXXVII">
+CXXXVII</a></h5>
+
+<h5 class = "smallcaps">To Divide an Inclined Plane into Equal
+Parts&mdash;such as a Ladder Placed against a Wall</h5>
+
+
+<p>Divide the vertical <span class = "smallroman">EC</span> into the
+required number of parts, and draw lines from point of sight <span class
+= "smallroman">S</span> through these divisions 1,&nbsp;2,&nbsp;3,
+&amp;c., cutting the line <span class = "smallroman">AC</span> at
+1,&nbsp;2,&nbsp;3, &amp;c. Draw parallels to <span class =
+"smallroman">AB</span>, such as <i>mn</i>, from <span class =
+"smallroman">AC</span> to <span class = "smallroman">BD</span>, which
+will represent the steps of the ladder.</p>
+
+<p class = "illustration">
+<a name = "fig247" id = "fig247"> </a>
+<img src = "images/fig247.png" width = "216" height = "210"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 247.</p>
+
+
+
+
+<span class = "pagenum">213</span>
+<a name = "page213" id = "page213"> </a>
+<!--png 233-->
+<h5 class = "section"><a name = "chapCXXXVIII" id = "chapCXXXVIII">
+CXXXVIII</a></h5>
+
+<h5 class = "smallcaps">Steps and the Inclined Plane</h5>
+
+
+<p>In Fig. 248 we treat a flight of steps as if it were an inclined
+plane. Draw the first and second steps as in Fig. 245. Then through
+1,&nbsp;2, draw 1<span class = "smallroman">V</span>, <span class =
+"smallroman">AV</span> to <span class = "smallroman">V</span>, the
+vanishing point on the vertical line <span class =
+"smallroman">SV</span>. These two lines and the corresponding ones at
+<span class = "smallroman">BV</span> will form a kind of vanishing
+scale, giving the height of each step as we ascend. It is especially
+useful when we pass the horizontal line and we no longer see the upper
+surface of the step, the scale on the right showing us how to proceed in
+that case.</p>
+
+<p class = "illustration">
+<a name = "fig248" id = "fig248"> </a>
+<img src = "images/fig248.png" width = "332" height = "205"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 248.</p>
+
+<p><span class = "pagenum">214</span>
+<a name = "page214" id = "page214"> </a>
+<!--png 234-->
+In Fig. 249 we have an example of steps ascending and descending. First
+set out the ground-plan, and find its vanishing point <span class =
+"smallroman">S</span> (point of sight). Through <span class =
+"smallroman">S</span> draw vertical <span class =
+"smallroman">BA</span>, and make <span class = "smallroman">SA</span>
+equal to <span class = "smallroman">SB</span>. Set out the first step
+<span class = "smallroman">CD</span>. Draw <span class =
+"smallroman">EA</span>, <span class = "smallroman">CA</span>, <span
+class = "smallroman">DA</span>, and <span class =
+"smallroman">GA</span>, for the ascending guiding lines. Complete the
+steps facing us, at central line <span class = "smallroman">OO</span>.
+Then draw guiding line <span class = "smallroman">FB</span> for the
+descending steps (see Rule&nbsp;8).</p>
+
+<p class = "illustration">
+<a name = "fig249" id = "fig249"> </a>
+<img src = "images/fig249.png" width = "282" height = "235"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 249.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXXXIX" id = "chapCXXXIX">
+CXXXIX</a></h5>
+
+<h5 class = "smallcaps">Steps in Angular Perspective</h5>
+
+
+<p>First draw the base <span class = "smallroman">ABCD</span> (Fig. 251)
+at the required angle by the new method (Fig. 250). Produce <span class
+= "smallroman">BC</span> to the horizon, and thus find vanishing point
+<span class = "smallroman">V</span>. At this point raise vertical <span
+class = "smallroman">VV�</span>. Construct
+<span class = "pagenum">215</span>
+<a name = "page215" id = "page215"> </a>
+<!--png 235-->
+first step <span class = "smallroman">AB</span>, refer its height at
+<span class = "smallroman">B</span> to line of heights <i>h</i><span
+class = "smallroman">I</span> on left, and thus obtain height of step at
+<span class = "smallroman">A</span>. Draw lines from <span class =
+"smallroman">A</span> and <span class = "smallroman">F</span> to <span
+class = "smallroman">V�</span>. From <i>n</i> draw diagonal through
+<span class = "smallroman">O</span> to <span class =
+"smallroman">G</span>. Raise vertical at <span class =
+"smallroman">O</span> to represent the height of the next step, its
+height being determined by the scale of heights at the side. From <span
+class = "smallroman">A</span> and <span class = "smallroman">F</span>
+draw lines to <span class = "smallroman">V�</span>, and also similar
+lines from <span class = "smallroman">B</span>, which will serve as
+guiding lines to determine the height of the steps at either end as we
+raise them to the required number.</p>
+
+<p class = "illustration">
+<a name = "fig250" id = "fig250"> </a>
+<img src = "images/fig250.png" width = "301" height = "92"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 250.</p>
+
+<p class = "illustration">
+<a name = "fig251" id = "fig251"> </a>
+<img src = "images/fig251.png" width = "368" height = "191"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 251.</p>
+
+
+
+
+<span class = "pagenum">216</span>
+<a name = "page216" id = "page216"> </a>
+<!--png 236-->
+<h5 class = "section"><a name = "chapCXL" id = "chapCXL">
+CXL</a></h5>
+
+<h5 class = "smallcaps">A Step Ladder at an Angle</h5>
+
+
+<p>First draw the ground-plan <span class = "smallroman">G</span> at the
+required angle, using vanishing and measuring points. Find the height
+<i>h</i><span class = "smallroman">H</span>, and width at top <span
+class = "smallroman">HH�</span>, and draw the sides <span class =
+"smallroman">HA</span> and <span class = "smallroman">H�E</span>. Note
+that <span class = "smallroman">AE</span> is wider than <span class =
+"smallroman">HH�</span>, and also that the back legs are not at the same
+angle as the front ones, and that they overlap them. From <span class =
+"smallroman">E</span> raise vertical <span class =
+"smallroman">EF</span>, and divide into as many parts as you require
+rounds to the ladder. From these divisions draw lines 1&nbsp;1,
+2&nbsp;2, &amp;c., towards the other vanishing point (not in the
+picture), but
+<span class = "pagenum">217</span>
+<a name = "page217" id = "page217"> </a>
+<!--png 237-->
+having obtained their direction from the ground-plan in perspective at
+line <span class = "smallroman">E</span><i>e</i>, you may set up a
+second vertical <i>ef</i> at any point on <span class =
+"smallroman">E</span><i>e</i> and divide it into the same number of
+parts, which will be in proportion to those on <span class =
+"smallroman">EF</span>, and you will obtain the same result by drawing
+lines from the divisions on <span class = "smallroman">EF</span> to
+those on <i>ef</i> as in drawing them to the vanishing point.</p>
+
+<p class = "illustration">
+<a name = "fig252" id = "fig252"> </a>
+<img src = "images/fig252.png" width = "313" height = "276"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 252.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXLI" id = "chapCXLI">
+CXLI</a></h5>
+
+<h5 class = "smallcaps">Square Steps Placed over each Other</h5>
+
+
+<p>This figure shows the other method of drawing steps, which is simple
+enough if we have sufficient room for our vanishing points.</p>
+
+<p class = "illustration">
+<a name = "fig253" id = "fig253"> </a>
+<img src = "images/fig253.png" width = "453" height = "86"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 253.</p>
+
+<p>The manner of working it is shown at <a href = "#fig124">Fig.
+124</a>.</p>
+
+
+
+
+<span class = "pagenum">218</span>
+<a name = "page218" id = "page218"> </a>
+<!--png 238-->
+<h5 class = "section"><a name = "chapCXLII" id = "chapCXLII">
+CXLII</a></h5>
+
+<h5 class = "smallcaps">Steps and a Double Cross Drawn by Means of
+Diagonals and one Vanishing Point</h5>
+
+
+<p>Although in this figure we have taken a longer distance-point than in
+the previous one, we are able to draw it all within the page.</p>
+
+<p class = "illustration">
+<a name = "fig254" id = "fig254"> </a>
+<img src = "images/fig254.png" width = "348" height = "184"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 254.</p>
+
+<p>Begin by setting out the square base at the angle required. Find
+point <span class = "smallroman">G</span> by means of diagonals, and
+produce <span class = "smallroman">AB</span> to <span class =
+"smallroman">V</span>, &amp;c. Mark height of step <span class =
+"smallroman">A</span><i>o</i>, and proceed to draw the steps as already
+shown. Then by the diagonals and measurements on base draw the second
+step and the square inside it on which to stand the foot of the cross.
+To draw the cross, raise verticals from the four corners of its base,
+and a line <span class = "smallroman">K</span> from its centre. Through
+any
+<span class = "pagenum">219</span>
+<a name = "page219" id = "page219"> </a>
+<!--png 239-->
+point on this central line, if we draw a diagonal from point <span class
+= "smallroman">G</span> we cut the two opposite verticals of the shaft
+at <i>mn</i> (see Fig. 255), and by means of the vanishing point <span
+class = "smallroman">V</span> we cut the other two verticals at the
+opposite corners and thus obtain the four points through which to draw
+the other sides of the square, which go to the distant or inaccessible
+vanishing point. It will be seen by carefully examining the figure that
+by this means we are enabled to draw the double cross standing on its
+steps.</p>
+
+<p class = "illustration">
+<a name = "fig255" id = "fig255"> </a>
+<img src = "images/fig255.png" width = "263" height = "170"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 255.</p>
+
+
+
+
+<span class = "pagenum">221</span>
+<a name = "page221" id = "page221"> </a>
+<!--png 241-->
+<h5 class = "section"><a name = "chapCXLIII" id = "chapCXLIII">
+CXLIII</a></h5>
+
+<h5 class = "smallcaps">A Staircase Leading to a Gallery</h5>
+
+
+<p>In this figure we have made use of the devices already set forth in
+the foregoing figures of steps, &amp;c., such as the side scale on the
+left of the figure to ascertain the height of the steps, the double
+lines drawn to the high vanishing point of the inclined plane, and so
+on; but the principal use of this diagram is to show on the perspective
+plane, which as it were runs under the stairs, the trace or projection
+of the flights of steps, the landings and positions of other objects,
+which will be found very useful in placing figures in a composition of
+this kind. It will be seen that these underneath measurements, so to
+speak, are obtained by the half-distance.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[220]</span>
+<a name = "page220" id = "page220"> </a>
+<!--png 240-->
+<a name = "fig256" id = "fig256"> </a>
+<img src = "images/fig256.png" width = "266" height = "369"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 256.</p>
+
+
+
+
+<span class = "pagenum">222</span>
+<a name = "page222" id = "page222"> </a>
+<!--png 242-->
+<h5 class = "section"><a name = "chapCXLIV" id = "chapCXLIV">
+CXLIV</a></h5>
+
+<h5 class = "smallcaps">Winding Stairs in a Square Shaft</h5>
+
+
+<p>Draw square <span class = "smallroman">ABCD</span> in parallel
+perspective. Divide each side into four, and raise verticals from each
+division. These verticals will mark the positions of the steps on each
+wall, four in number. From centre <span class = "smallroman">O</span>
+raise vertical <span class = "smallroman">OP</span>, around which the
+steps are to wind. Let <span class = "smallroman">AF</span> be the
+height of each step. Form scale <span class = "smallroman">AB</span>,
+which will give the height of each step according to its position. Thus
+at <i>mn</i> we find the height at the centre of the square, so if we
+transfer this measurement to the central line <span class =
+"smallroman">OP</span> and repeat it upwards, say to fourteen, then we
+have the height of each step on the line where they all meet. Starting
+then with the first on the right, draw the rectangle <i>g</i><span class
+= "smallroman">D</span>1<i>f</i>, the height of <span class =
+"smallroman">AF</span>, then draw to the central line <i>go</i>,
+<i>f</i>1, and 1&nbsp;1, and thus complete the first step. On <span
+class = "smallroman">DE</span>, measure heights equal to <span class =
+"smallroman">D</span>&nbsp;1. Draw 2&nbsp;2 towards central line, and
+2<i>n</i> towards point of sight till it meets the second vertical
+<i>n</i><span class = "smallroman">K</span>. Then draw <i>n</i>2 to
+centre, and so complete the second step. From 3 draw 3<i>a</i> to third
+vertical, from 4 to fourth, and so on, thus obtaining the height of each
+ascending step on the wall to the right, completing them in the same way
+as numbers 1 and&nbsp;2, when we come to the sixth step, the other end
+of which is against the wall opposite to us. Steps 6,&nbsp;7, 8,&nbsp;9
+are all on this wall, and are therefore equal in height all along, as
+they are equally distant. Step 10 is turned towards us, and abuts on the
+wall to our left; its measurement is taken on the scale <span class =
+"smallroman">AB</span> just underneath it, and on the same line to which
+it is drawn. Step 11 is just over the centre of base <i>mo</i>, and is
+therefore parallel to it, and its height is <i>mn</i>. The widths of
+steps 12 and 13 seem gradually to increase as they come towards us, and
+as they rise above the horizon we begin to see underneath them. Steps
+13, 14, 15, 16 are against the wall on this side of the picture, which
+we may suppose has been removed to show the working of the drawing, or
+they might be an open flight as we sometimes see in shops and galleries,
+although in that case they are generally enclosed in a cylindrical
+shaft.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[223]</span>
+<a name = "page223" id = "page223"> </a>
+<!--png 243-->
+<a name = "fig257" id = "fig257"> </a>
+<img src = "images/fig257.png" width = "341" height = "431"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 257.</p>
+
+
+
+
+<span class = "pagenum">225</span>
+<a name = "page225" id = "page225"> </a>
+<!--png 245-->
+<h5 class = "section"><a name = "chapCXLV" id = "chapCXLV">
+CXLV</a></h5>
+
+<h5 class = "smallcaps">Winding Stairs in a Cylindrical Shaft</h5>
+
+
+<p>First draw the circular base <span class = "smallroman">CD</span>.
+Divide the circumference into equal parts, according to the number of
+steps in a complete round, say twelve. Form scale <span class =
+"smallroman">ASF</span> and the larger scale <span class =
+"smallroman">ASB</span>, on which is shown the perspective measurements
+of the steps according to their positions; raise verticals such as
+<i>ef</i>, <span class = "smallroman">G</span><i>h</i>, &amp;c. From
+divisions on circumference measure out the central line <span class =
+"smallroman">OP</span>, as in the other figure, and find the heights of
+the steps 1,&nbsp;2, 3,&nbsp;4, &amp;c., by the corresponding numbers in
+the large scale to the left; then proceed in much the same way as in the
+previous figure. Note the central column <span class =
+"smallroman">OP</span> cuts off a small portion of the steps at that
+end.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[224]</span>
+<a name = "page224" id = "page224"> </a>
+<!--png 244-->
+<a name = "fig258" id = "fig258"> </a>
+<img src = "images/fig258.png" width = "335" height = "406"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 258.</p>
+
+<p><span class = "pagenum">226</span>
+<a name = "page226" id = "page226"> </a>
+<!--png 246-->
+In ordinary cases only a small portion of a winding staircase is
+actually seen, as in this sketch.</p>
+
+<p class = "illustration">
+<a name = "fig259" id = "fig259"> </a>
+<img src = "images/fig259.png" width = "298" height = "347"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption">
+<span class = "smallcaps">Fig. 259.</span> Sketch of Courtyard in
+Toledo.</p>
+
+
+
+
+<span class = "pagenum">227</span>
+<a name = "page227" id = "page227"> </a>
+<!--png 247-->
+<h5 class = "section"><a name = "chapCXLVI" id = "chapCXLVI">
+CXLVI</a></h5>
+
+<h5 class = "smallcaps">Of the Cylindrical Picture or Diorama</h5>
+
+
+<p>Although illusion is by no means the highest form of art, there is no
+picture painted on a flat surface that gives such a wonderful appearance
+of truth as that painted on a cylindrical canvas, such as those
+panoramas of &lsquo;Paris during the Siege&rsquo;, exhibited some years
+ago; &lsquo;The Battle of Trafalgar&rsquo;, only lately shown at Earl's
+Court; and many others. In these pictures the spectator is in the centre
+of a cylinder, and although he turns round to look at the scene the
+point of sight is always in front of him, or nearly so. I&nbsp;believe
+on the canvas these points are from 12 to 16 feet apart.</p>
+
+<p class = "illustration">
+<a name = "fig260" id = "fig260"> </a>
+<img src = "images/fig260.png" width = "311" height = "260"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 260.</p>
+
+<p><span class = "pagenum">228</span>
+<a name = "page228" id = "page228"> </a>
+<!--png 248-->
+The reason of this look of truth may be explained thus. If we place
+three globes of equal size in a straight line, and trace their apparent
+widths on to a straight transparent plane, those at the sides, as
+<i>a</i> and <i>b</i>, will appear much wider than the centre one at
+<i>c</i>. Whereas, if we trace them on a semicircular glass they will
+appear very nearly equal and, of the three, the central one <i>c</i>
+will be rather the largest, as may be seen by this figure.</p>
+
+<p>We must remember that, in the first case, when we are looking at a
+globe or a circle, the visual rays form a cone, with a globe at its
+base. If these three cones are intersected by a straight glass <span
+class = "smallroman">GG</span>, and looked at from point <span class =
+"smallroman">S</span>, the intersection of <span class =
+"smallroman">C</span> will be a circle, as the cone is cut straight
+across. The other two being intersected at an angle, will each be an
+ellipse. At the same time, if we look at them from the station point,
+with one eye only, then the three globes (or tracings of them) will
+appear equal and perfectly round.</p>
+
+<p>Of course the cylindrical canvas is necessary for panoramas; but we
+have, as a rule, to paint our pictures and wall-decorations on flat
+surfaces, and therefore must adapt our work to these conditions.</p>
+
+<p>In all cases the artist must exercise his own judgement both in the
+arrangement of his design and the execution of the work, for there is
+perspective even in the touch&mdash;a painting to be looked at from a
+distance requires a bold and broad handling; in small cabinet pictures
+that we live with in our own rooms we look for the exquisite workmanship
+of the best masters.</p>
+
+
+
+
+<span class = "pagenum">229</span>
+<a name = "page229" id = "page229"> </a>
+<!--png 249-->
+<h3 class = "chapter">BOOK FOURTH</h3>
+
+<h5 class = "section"><a name = "chapCXLVII" id = "chapCXLVII">
+CXLVII</a></h5>
+
+<h5 class = "smallcaps">The Perspective of Cast Shadows</h5>
+
+
+<p>There is a pretty story of two lovers which is sometimes told as the
+origin of art; at all events, I&nbsp;may tell it here as the origin of
+sciagraphy. A&nbsp;young shepherd was in love with the daughter of a
+potter, but it so happened that they had to part, and were passing their
+last evening together, when the girl, seeing the shadow of her lover's
+profile cast from a lamp on to some wet plaster or on the wall, took a
+metal point, perhaps some sort of iron needle, and traced the outline of
+the face she loved on to the plaster, following carefully the outline of
+the features, being naturally anxious to make it as like as possible.
+The old potter, the father of the girl, was so struck with it that he
+began to ornament his wares by similar devices, which gave them
+increased value by the novelty and beauty thus imparted to them.</p>
+
+<p>Here then we have a very good illustration of our present subject and
+its three elements. First, the light shining on the wall; second, the
+wall or the plane of projection, or plane of shade; and third, the
+intervening object, which receives as much light on itself as it
+deprives the wall of. So that the dark portion thus caused on the plane
+of shade is the cast shadow of the intervening object.</p>
+
+<p>We have to consider two sorts of shadows: those cast by a luminary a
+long way off, such as the sun; and those cast by artificial light, such
+as a lamp or candle, which is more or less close to the object. In the
+first case there is no perceptible divergence of rays, and the outlines
+of the sides of the shadows of regular objects, as cubes, posts,
+&amp;c., will be parallel. In the second case, the rays diverge
+according to the nearness of the light, and consequently the lines of
+the shadows, instead of being parallel, are spread out.</p>
+
+
+
+
+<span class = "pagenum">230</span>
+<a name = "page230" id = "page230"> </a>
+<!--png 250-->
+<h5 class = "section"><a name = "chapCXLVIII" id = "chapCXLVIII">
+CXLVIII</a></h5>
+
+<h5 class = "smallcaps">The Two Kinds of Shadows</h5>
+
+
+<p>In Figs. 261 and 262 is seen the shadow cast by the sun by parallel
+rays.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig261" id = "fig261"> </a>
+<img src = "images/fig261.png" width = "194" height = "131"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig262" id = "fig262"> </a>
+<img src = "images/fig262.png" width = "171" height = "126"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 261.</td>
+<td class = "caption smallcaps">
+Fig. 262.</td>
+</tr>
+</table>
+
+<p>Fig. 263 shows the shadows cast by a candle or lamp, where the rays
+diverge from the point of light to meet corresponding diverging lines
+which start from the foot of the luminary on the ground.</p>
+
+<p class = "illustration">
+<a name = "fig263" id = "fig263"> </a>
+<img src = "images/fig263.png" width = "293" height = "220"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 263.</p>
+
+<p>The simple principle of cast shadows is that the rays coming from the
+point of light or luminary pass over the top of the intervening object
+which casts the shadow on to the plane of shade to meet the horizontal
+trace of those rays on that plane, or the
+<span class = "pagenum">231</span>
+<a name = "page231" id = "page231"> </a>
+<!--png 251-->
+lines of light proceed from the point of light, and the lines of the
+shadow are drawn from the foot or trace of the point of light.</p>
+
+<p>Fig. 264 shows this in profile. Here the sun is on the same plane as
+the picture, and the shadow is cast sideways.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig264" id = "fig264"> </a>
+<img src = "images/fig264.png" width = "163" height = "146"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig265" id = "fig265"> </a>
+<img src = "images/fig265.png" width = "189" height = "239"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 264.</td>
+<td class = "caption smallcaps">
+Fig. 265.</td>
+</tr>
+</table>
+
+<p>Fig. 265 shows the same thing, but the sun being behind the
+<span class = "pagenum">232</span>
+<a name = "page232" id = "page232"> </a>
+<!--png 252-->
+object, casts its shadow forwards. Although the lines of light are
+parallel, they are subject to the laws of perspective, and are therefore
+drawn from their respective vanishing points.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCXLIX" id = "chapCXLIX">
+CXLIX</a></h5>
+
+<h5 class = "smallcaps">Shadows Cast by the Sun</h5>
+
+
+<p>Owing to the great distance of the sun, we have to consider the rays
+of light proceeding from it as parallel, and therefore subject to the
+same laws as other parallel lines in perspective, as already noted. And
+for the same reason we have to place the foot of the luminary on the
+horizon. It is important to remember this, as these two things make the
+difference between shadows cast by the sun and those cast by artificial
+light.</p>
+
+<p>The sun has three principal positions in relation to the picture. In
+the first case it is supposed to be in the same plane either to the
+right or to the left, and in that case the shadows will be
+<span class = "pagenum">233</span>
+<a name = "page233" id = "page233"> </a>
+<!--png 253-->
+parallel with the base of the picture. In the second position it is on
+the other side of it, or facing the spectator, when the shadows of
+objects will be thrown forwards or towards him. In the third, the sun is
+in front of the picture, and behind the spectator, so that the shadows
+are thrown in the opposite direction, or towards the horizon, the
+objects themselves being in full light.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCL" id = "chapCL">
+CL</a></h5>
+
+<h5 class = "smallcaps">The Sun in the Same Plane as the Picture</h5>
+
+
+<p>Besides being in the same plane, the sun in this figure is at an
+angle of 45� to the horizon, consequently the shadows will be the same
+length as the figures that cast them are high. Note that the shadow of
+step No.&nbsp;1 is cast upon step No.&nbsp;2, and that of No.&nbsp;2 on
+No.&nbsp;3, the top of each of these becoming a plane of shade.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig266" id = "fig266"> </a>
+<img src = "images/fig266.png" width = "229" height = "81"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig267" id = "fig267"> </a>
+<img src = "images/fig267.png" width = "102" height = "101"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 266.</td>
+<td class = "caption smallcaps">
+Fig. 267.</td>
+</tr>
+</table>
+
+<p>When the shadow of an object such as <span class =
+"smallroman">A</span>, Fig. 268, which would fall upon the plane, is
+interrupted by another object <span class = "smallroman">B</span>, then
+the
+<span class = "pagenum">234</span>
+<a name = "page234" id = "page234"> </a>
+<!--png 254-->
+outline of the shadow is still drawn on the plane, but being interrupted
+by the surface <span class = "smallroman">B</span> at <span class =
+"smallroman">C</span>, the shadow runs up that plane till it meets the
+rays 1,&nbsp;2, which define the shadow on plane <span class =
+"smallroman">B</span>. This is an important point, but is quite
+explained by the figure.</p>
+
+<p class = "illustration">
+<a name = "fig268" id = "fig268"> </a>
+<img src = "images/fig268.png" width = "302" height = "108"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 268.</p>
+
+<p>Although we have said that the rays pass over the top of the object
+casting the shadow, in the case of an archway or similar figure they
+pass underneath it; but the same principle holds good, that is, we draw
+lines from the guiding points in the arch, 1,&nbsp;2,&nbsp;3, &amp;c.,
+at the same angle of 45� to meet the traces of those rays on the plane
+of shade, and so get the shadow of the archway, as here shown.</p>
+
+<p class = "illustration">
+<a name = "fig269" id = "fig269"> </a>
+<img src = "images/fig269.png" width = "308" height = "188"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 269.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCLI" id = "chapCLI">
+CLI</a></h5>
+
+<h5 class = "smallcaps">The Sun Behind the Picture</h5>
+
+
+<p>We have seen that when the sun's altitude is at an angle of 45� the
+shadows on the horizontal plane are the same length as the height of the
+objects that cast them. Here (Fig. 270), the sun still being at 45�
+altitude, although behind the picture, and consequently throwing the
+shadow of <span class = "smallroman">B</span> forwards, that shadow must
+be the same length as the height of cube <span class =
+"smallroman">B</span>, which will be seen is the case, for the shadow
+<span class = "smallroman">C</span> is a square in perspective.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[235]</span>
+<a name = "page235" id = "page235"> </a>
+<!--png 255-->
+<a name = "fig270" id = "fig270"> </a>
+<img src = "images/fig270.png" width = "281" height = "399"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 270.</p>
+
+<p><span class = "pagenum">236</span>
+<a name = "page236" id = "page236"> </a>
+<!--png 256-->
+To find the angle of altitude and the angle of the sun to the picture,
+we must first find the distance of the spectator from the foot of the
+luminary.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig271" id = "fig271"> </a>
+<img src = "images/fig271.png" width = "153" height = "82"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 271.</td>
+</tr>
+</table>
+
+<p>From point of sight <span class = "smallroman">S</span> (Fig. 270)
+drop perpendicular to <span class = "smallroman">T</span>, the
+station-point. From <span class = "smallroman">T</span> draw <span class
+= "smallroman">TF</span> at 45� to meet horizon at <span class =
+"smallroman">F</span>. With radius <span class = "smallroman">FT</span>
+make <span class = "smallroman">FO</span> equal to it. Then <span class
+= "smallroman">O</span> is the position of the spectator. From <span
+class = "smallroman">F</span> raise vertical <span class =
+"smallroman">FL</span>, and from <span class = "smallroman">O</span>
+draw a line at 45� to meet <span class = "smallroman">FL</span> at <span
+class = "smallroman">L</span>, which is the luminary at an altitude of
+45�, and at an angle of 45� to the picture.</p>
+
+<p>Fig. 272 is similar to the foregoing, only the angles of altitude and
+of the sun to the picture are altered.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[237]</span>
+<a name = "page237" id = "page237"> </a>
+<!--png 257-->
+<a name = "fig272" id = "fig272"> </a>
+<img src = "images/fig272.png" width = "347" height = "440"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 272.</p>
+
+<p><i>Note.</i>&mdash;The sun being at 50� to the picture instead of
+45�, is nearer the point of sight; at 90� it would be exactly opposite
+the spectator, and so on. Again, the elevation being less (40� instead
+of 45�) the shadow is longer. Owing to the changed position of the sun
+two sides of the cube throw a shadow. Note also that the outlines of the
+shadow, 1&nbsp;2, 2&nbsp;3, are drawn to the same vanishing points as
+the cube itself.</p>
+
+<p>It will not be necessary to mark the angles each time we make a
+drawing, as it must be seen we can place the luminary in any position
+that suits our convenience.</p>
+
+
+
+
+<span class = "pagenum">238</span>
+<a name = "page238" id = "page238"> </a>
+<!--png 258-->
+<h5 class = "section"><a name = "chapCLII" id = "chapCLII">
+CLII</a></h5>
+
+<h5 class = "smallcaps">Sun Behind the Picture, Shadows Thrown on a
+Wall</h5>
+
+
+<p>As here we change the conditions we must also change our procedure.
+An upright wall now becomes the plane of shade, therefore as the
+principle of shadows must always remain the same we have to change the
+relative positions of the luminary and the foot thereof.</p>
+
+<p>At <span class = "smallroman">S</span> (point of sight) raise
+vertical <span class = "smallroman">SF�</span>, making it equal to
+<i>f</i><span class = "smallroman">L</span>. <span class =
+"smallroman">F�</span> becomes the foot of the luminary, whilst the
+luminary itself still remains at <span class =
+"smallroman">L</span>.</p>
+
+<p class = "illustration">
+<a name = "fig273" id = "fig273"> </a>
+<img src = "images/fig273.png" width = "326" height = "231"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 273.</p>
+
+<p>We have but to turn this page half round and look at it from the
+right, and we shall see that <span class = "smallroman">SF�</span>
+becomes as it were the horizontal line. The luminary <span class =
+"smallroman">L</span> is at the right side of point <span class =
+"smallroman">S</span> instead of the left, and the foot thereof is, as
+before, the trace of the luminary, as it is just underneath it. We shall
+also see that by
+<span class = "pagenum">239</span>
+<a name = "page239" id = "page239"> </a>
+<!--png 259-->
+proceeding as in previous figures we obtain the same results on the wall
+as we did on the horizontal plane. Fig. <span class =
+"smallroman">B</span> being on the horizontal plane is treated as
+already shown. The steps have their shadows partly on the wall and
+partly on the horizontal plane, so that the shadows on the wall are
+outlined from <span class = "smallroman">F�</span> and those on the
+ground from <i>f</i>. Note shadow of roof <span class =
+"smallroman">A</span>, and how the line drawn from <span class =
+"smallroman">F�</span> through <span class = "smallroman">A</span> is
+met by the line drawn from the luminary <span class =
+"smallroman">L</span>, at the point <span class = "smallroman">P</span>,
+and how the lower line of the shadow is directed to point of sight <span
+class = "smallroman">S</span>.</p>
+
+<p>Fig. 274 is a larger drawing of the steps, &amp;c., in further
+illustration of the above.</p>
+
+<p class = "illustration">
+<a name = "fig274" id = "fig274"> </a>
+<img src = "images/fig274.png" width = "326" height = "323"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 274.</p>
+
+
+
+
+<span class = "pagenum">240</span>
+<a name = "page240" id = "page240"> </a>
+<!--png 260-->
+<h5 class = "section"><a name = "chapCLIII" id = "chapCLIII">
+CLIII</a></h5>
+
+<h5 class = "smallcaps">Sun Behind the Picture Throwing Shadow on an
+Inclined Plane</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig275" id = "fig275"> </a>
+<img src = "images/fig275.png" width = "227" height = "267"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 275.</td>
+</tr>
+</table>
+
+<p>The vanishing point of the shadows on an inclined plane is on a
+vertical dropped from the luminary to a point (<span class =
+"smallroman">F</span>) on a level with the vanishing point (<span class
+= "smallroman">P</span>) of that inclined plane. Thus <span class =
+"smallroman">P</span> is the vanishing point of the inclined plane <span
+class = "smallroman">K</span>. Draw horizontal <span class =
+"smallroman">PF</span> to meet <i>f</i><span class =
+"smallroman">L</span> (the line drawn from the luminary to the horizon).
+Then <span class = "smallroman">F</span> will be the vanishing point of
+the shadows on the inclined plane. To find the shadow of <span class =
+"smallroman">M</span> draw lines from <span class =
+"smallroman">F</span> through the
+<span class = "pagenum">241</span>
+<a name = "page241" id = "page241"> </a>
+<!--png 261-->
+base <i>eg</i> to <i>cd</i>. From luminary <span class =
+"smallroman">L</span> draw lines through <i>ab</i>, also to <i>cd</i>,
+where they will meet those drawn from <span class =
+"smallroman">F</span>. Draw <span class = "smallroman">CD</span>, which
+determines the length of the shadow <i>egcd</i>.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCLIV" id = "chapCLIV">
+CLIV</a></h5>
+
+<h5 class = "smallcaps">The Sun in Front of the Picture</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig277" id = "fig277"> </a>
+<img src = "images/fig277.png" width = "148" height = "134"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 277.</td>
+</tr>
+</table>
+
+<p>When the sun is in front of the picture we have exactly the opposite
+effect to that we have just been studying. The shadows, instead of
+coming towards us, are retreating from us, and the objects throwing them
+are in full light, consequently we have to reverse our treatment. Let us
+suppose the sun to be placed
+<span class = "pagenum">242</span>
+<a name = "page242" id = "page242"> </a>
+<!--png 262-->
+above the horizon at <span class = "smallroman">L�</span>, on the right
+of the picture and behind the spectator (Fig. 276). If we transport the
+length <span class = "smallroman">L�</span><i>f�</i> to the opposite
+side and draw the vertical downwards from the horizon, as at <span class
+= "smallroman">FL</span>, we can then suppose point <span class =
+"smallroman">L</span> to be exactly opposite the sun, and if we make
+that the vanishing point for the sun's rays we shall find that we obtain
+precisely the same result. As in Fig. 277, if we wish to find the length
+of <span class = "smallroman">C</span>, which we may suppose to be the
+shadow of <span class = "smallroman">P</span>, we can either draw a line
+from <span class = "smallroman">A</span> through <span class =
+"smallroman">O</span> to <span class = "smallroman">B</span>, or from
+<span class = "smallroman">B</span> through <span class =
+"smallroman">O</span> to <span class = "smallroman">A</span>, for the
+result is the same. And as we cannot make use of a point that is behind
+us and out of the picture, we have to resort to this very ingenious
+device.</p>
+
+<p class = "illustration">
+<a name = "fig276" id = "fig276"> </a>
+<img src = "images/fig276.png" width = "310" height = "274"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 276.</p>
+
+<p>In Fig. 276 we draw lines <span class = "smallroman">L</span>1, <span
+class = "smallroman">L</span>2, <span class = "smallroman">L</span>3
+from the luminary to the top of the object to meet those drawn from the
+foot <span class = "smallroman">F</span>, namely <span class =
+"smallroman">F</span>1, <span class = "smallroman">F</span>2, <span
+class = "smallroman">F</span>3, in the same way as in the figures we
+have already drawn.</p>
+
+<span class = "pagenum">243</span>
+<a name = "page243" id = "page243"> </a>
+<!--png 263-->
+<p>Fig. 278 gives further illustration of this problem.</p>
+
+<p class = "illustration">
+<a name = "fig278" id = "fig278"> </a>
+<img src = "images/fig278.png" width = "348" height = "252"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 278.</p>
+
+
+
+
+<span class = "pagenum">244</span>
+<a name = "page244" id = "page244"> </a>
+<!--png 264-->
+<h5 class = "section"><a name = "chapCLV" id = "chapCLV">
+CLV</a></h5>
+
+<h5 class = "smallcaps">The Shadow of an Inclined Plane</h5>
+
+
+<p>The two portions of this inclined plane which cast the shadow are
+first the side <i>fbd</i>, and second the farther end <i>abcd</i>. The
+points we have to find are the shadows of <i>a</i> and <i>b</i>. From
+luminary <span class = "smallroman">L</span> draw <span class =
+"smallroman">L</span><i>a</i>, <span class =
+"smallroman">L</span><i>b</i>, and from <span class =
+"smallroman">F</span>, the foot, draw <span class =
+"smallroman">F</span><i>c</i>, <span class =
+"smallroman">F</span><i>d</i>. The intersection of these lines will be
+at <i>a�b�</i>. If we join <i>fb�</i> and <i>db�</i> we have the shadow
+of the side <i>fbd</i>, and if we join <i>ca�</i> and <i>a�b�</i> we
+have the shadow of <i>abcd</i>, which together form that of the
+figure.</p>
+
+<p class = "illustration">
+<a name = "fig279" id = "fig279"> </a>
+<img src = "images/fig279.png" width = "320" height = "239"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 279.</p>
+
+
+
+
+<span class = "pagenum">245</span>
+<a name = "page245" id = "page245"> </a>
+<!--png 265-->
+<h5 class = "section"><a name = "chapCLVI" id = "chapCLVI">
+CLVI</a></h5>
+
+<h5 class = "smallcaps">Shadow on a Roof or Inclined Plane</h5>
+
+
+<p>To draw the shadow of the figure <span class = "smallroman">M</span>
+on the inclined plane <span class = "smallroman">K</span> (or a chimney
+on a roof). First find the vanishing point <span class =
+"smallroman">P</span> of the inclined plane and draw horizontal <span
+class = "smallroman">PF</span> to meet vertical raised from <span class
+= "smallroman">L</span>, the luminary. Then <span class =
+"smallroman">F</span> will be the vanishing point of the shadow. From
+<span class = "smallroman">L</span> draw <span class =
+"smallroman">L</span>1, <span class = "smallroman">L</span>2, <span
+class = "smallroman">L</span>3 to top of figure <span class =
+"smallroman">M</span>, and from the base of <span class =
+"smallroman">M</span> draw 1<span class = "smallroman">F</span>, 2<span
+class = "smallroman">F</span>, 3<span class = "smallroman">F</span> to
+<span class = "smallroman">F</span>, the vanishing point of the shadow.
+The intersections of these lines at 1,&nbsp;2,&nbsp;3 on <span class =
+"smallroman">K</span> will determine the length and form of the
+shadow.</p>
+
+<p class = "illustration">
+<a name = "fig280" id = "fig280"> </a>
+<img src = "images/fig280.png" width = "288" height = "193"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 280.</p>
+
+
+
+
+<span class = "pagenum">246</span>
+<a name = "page246" id = "page246"> </a>
+<!--png 266-->
+<h5 class = "section"><a name = "chapCLVII" id = "chapCLVII">
+CLVII</a></h5>
+
+<h5 class = "smallcaps">To Find the Shadow of a Projection or Balcony on
+a Wall</h5>
+
+
+<p>To find the shadow of the object <span class = "smallroman">K</span>
+on the wall <span class = "smallroman">W</span>, drop verticals <span
+class = "smallroman">OO</span> till they meet the base line <span class
+= "smallroman">B�B�</span> of the wall. Then from the point of sight
+<span class = "smallroman">S</span> draw lines through <span class =
+"smallroman">OO</span>, also drop verticals <span class =
+"smallroman">D</span><i>d�</i>, <span class =
+"smallroman">C</span><i>c�</i>, to meet these lines in <i>d�c�</i>; draw
+<i>c�</i><span class = "smallroman">F</span> and <i>d�</i><span class =
+"smallroman">F</span> to foot of luminary. From the points <i>xx</i>
+where these lines cut the base <span class = "smallroman">B</span> raise
+perpendiculars <i>xa�</i>, <i>xb�</i>. From <span class =
+"smallroman">D</span>, <span class = "smallroman">A</span>, and <span
+class = "smallroman">B</span> draw lines to the luminary <span class =
+"smallroman">L</span>. These lines or rays intersecting the verticals
+raised from <i>xx</i> at <i>a�b�</i> will give the respective points of
+the shadow.</p>
+
+<p class = "illustration">
+<a name = "fig281" id = "fig281"> </a>
+<img src = "images/fig281.png" width = "299" height = "277"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 281.</p>
+
+<p>The shadow of the eave of a roof can be obtained in the same way.
+Take any point thereon, mark its trace on the ground, and then proceed
+as above.</p>
+
+
+
+
+<span class = "pagenum">247</span>
+<a name = "page247" id = "page247"> </a>
+<!--png 267-->
+<h5 class = "section"><a name = "chapCLVIII" id = "chapCLVIII">
+CLVIII</a></h5>
+
+<h5 class = "smallcaps">Shadow on a Retreating Wall, Sun in Front</h5>
+
+
+<p>Let <span class = "smallroman">L</span> be the luminary. Raise
+vertical <span class = "smallroman">LF</span>. <span class =
+"smallroman">F</span> will be the vanishing point of the shadows on the
+ground. Draw <span class = "smallroman">L</span><i>f�</i> parallel to
+<span class = "smallroman">FS</span>. Drop <span class =
+"smallroman">S</span><i>f�</i> from point of sight; <i>f�</i> (so found)
+is the vanishing point of the shadows on the wall. For shadow of roof
+draw <span class = "smallroman">LE</span> and <i>f�</i><span class =
+"smallroman">B</span>, giving us <i>e</i>, the shadow of <span class =
+"smallroman">E</span>. Join <span class = "smallroman">B</span><i>e</i>,
+&amp;c., and so draw shadow of eave of roof.</p>
+
+<p><span class = "pagenum">248</span>
+<a name = "page248" id = "page248"> </a>
+<!--png 268-->
+For shadow of <span class = "smallroman">K</span> draw lines from
+luminary <span class = "smallroman">L</span> to meet those from
+<i>f�</i> the foot, &amp;c.</p>
+
+<p>The shadow of <span class = "smallroman">D</span> over the door is
+found in a similar way to that of the roof.</p>
+
+<p class = "illustration">
+<a name = "fig282" id = "fig282"> </a>
+<img src = "images/fig282.png" width = "263" height = "318"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 282.</p>
+
+<p>Figure 283 shows how the shadow of the old man in the preceding
+drawing is found.</p>
+
+<p class = "illustration">
+<a name = "fig283" id = "fig283"> </a>
+<img src = "images/fig283.png" width = "188" height = "168"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 283.</p>
+
+
+
+
+<span class = "pagenum">249</span>
+<a name = "page249" id = "page249"> </a>
+<!--png 269-->
+<h5 class = "section"><a name = "chapCLIX" id = "chapCLIX">
+CLIX</a></h5>
+
+<h5 class = "smallcaps">Shadow of an Arch, Sun in Front</h5>
+
+
+<p>Having drawn the arch, divide it into a certain number of parts, say
+five. From these divisions drop perpendiculars to base line. From
+divisions on <span class = "smallroman">AB</span> draw lines to <span
+class = "smallroman">F</span> the foot, and from those on the semicircle
+draw lines to <span class = "smallroman">L</span> the luminary. Their
+intersections will give the points through which to draw the shadow of
+the arch.</p>
+
+<p class = "illustration">
+<a name = "fig284" id = "fig284"> </a>
+<img src = "images/fig284.png" width = "266" height = "184"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 284.</p>
+
+
+
+
+<span class = "pagenum">250</span>
+<a name = "page250" id = "page250"> </a>
+<!--png 270-->
+<h5 class = "section"><a name = "chapCLX" id = "chapCLX">
+CLX</a></h5>
+
+<h5 class = "smallcaps">Shadow in a Niche or Recess</h5>
+
+
+<p>In this figure a similar method to that just explained is adopted.
+Drop perpendiculars from the divisions of the arch 1&nbsp;2&nbsp;3 to
+the base. From the foot of each draw 1<span class =
+"smallroman">S</span>, 2<span class = "smallroman">S</span>, 3<span
+class = "smallroman">S</span> to foot of luminary <span class =
+"smallroman">S</span>, and from the top of each, <span class =
+"smallroman">A</span>&nbsp;1&nbsp;2&nbsp;3&nbsp;<span class =
+"smallroman">B</span>, draw lines to <span class = "smallroman">L</span>
+as before. Where the former intersect the curve on the floor of the
+niche raise verticals to meet the latter at <span class =
+"smallroman">P</span>&nbsp;1&nbsp;2&nbsp;<span class =
+"smallroman">B</span>, &amp;c. These points will indicate about the
+position of the shadow; but the niche being semicircular and domed at
+the top the shadow gradually loses itself in a gradated and somewhat
+serpentine half-tone.</p>
+
+<p class = "illustration">
+<a name = "fig285" id = "fig285"> </a>
+<img src = "images/fig285.png" width = "336" height = "258"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 285.</p>
+
+
+
+
+<span class = "pagenum">251</span>
+<a name = "page251" id = "page251"> </a>
+<!--png 271-->
+<h5 class = "section"><a name = "chapCLXI" id = "chapCLXI">
+CLXI</a></h5>
+
+<h5 class = "smallcaps">Shadow in an Arched Doorway</h5>
+
+
+<p><span class = "pagenum">252</span>
+<a name = "page252" id = "page252"> </a>
+<!--png 272-->
+This is so similar to the last figure in many respects that I need not
+repeat a description of the manner in which it is done. And surely an
+artist after making a few sketches from the actual thing will hardly
+require all this machinery to draw a simple shadow.</p>
+
+<p class = "illustration">
+<a name = "fig286" id = "fig286"> </a>
+<img src = "images/fig286.png" width = "312" height = "407"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 286.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCLXII" id = "chapCLXII">
+CLXII</a></h5>
+
+<h5 class = "smallcaps">Shadows Produced by Artificial Light</h5>
+
+
+<p>Shadows thrown by artificial light, such as a candle or lamp, are
+found by drawing lines from the seat of the luminary through the feet of
+the objects to meet lines representing rays of light drawn from the
+luminary itself over the tops or the corners of the objects; very much
+as in the cases of sun-shadows, but with
+<span class = "pagenum">253</span>
+<a name = "page253" id = "page253"> </a>
+<!--png 273-->
+this difference, that whereas the foot of the luminary in this latter
+case is supposed to be on the horizon an infinite distance away, the
+foot in the case of a lamp or candle may be on the floor or on a table
+close to us. First draw the table and chair, &amp;c. (Fig. 287), and let
+<span class = "smallroman">L</span> be the luminary. For objects on the
+table such as <span class = "smallroman">K</span> the foot will be at
+<i>f</i> on the table. For the shadows on the floor, of the chair and
+table itself, we must find the foot of the luminary on the floor. Draw
+<span class = "smallroman">S</span><i>o</i>, find trace of the edge of
+the table, drop vertical <i>o</i><span class = "smallroman">P</span>,
+draw <span class = "smallroman">PS</span> to point of sight, drop
+vertical from foot of candlestick to meet <span class =
+"smallroman">PS</span> in <span class = "smallroman">F</span>. Then
+<span class = "smallroman">F</span> is the foot of the luminary on the
+floor. From this point draw lines through the feet or traces of objects
+such as the corners of the table, &amp;c., to meet other lines drawn
+from the point of light, and so obtain the shadow.</p>
+
+<p class = "illustration">
+<a name = "fig287" id = "fig287"> </a>
+<img src = "images/fig287.png" width = "344" height = "212"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 287.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCLXIII" id = "chapCLXIII">
+CLXIII</a></h5>
+
+<h5 class = "smallcaps">Some Observations on Real Light and Shade</h5>
+
+
+<p>Although the figures we have been drawing show the principles on
+which sun-shadows are shaped, still there are so many more laws to be
+considered in the great art of light and shade that it is better to
+observe them in Nature herself or under the teaching of the real sun. In
+the study of a kitchen and scullery in an old house in Toledo (Fig. 288)
+we have an example of the many things to be considered besides the mere
+shapes of shadows of regular forms. It will be seen that the light is
+dispersed in all directions, and although there is a good deal of
+half-shade there are scarcely any cast shadows except on the floor; but
+the light on the white walls in the outside gallery is so reflected into
+the cast shadows that they are extremely faint. The luminosity of this
+part of the sketch is greatly enhanced by the contrast of the dark legs
+of the bench and the shadows in the roof. The warm glow of all this
+portion is contrasted by the grey door and its frame.</p>
+
+<span class = "pagenum">254</span>
+<a name = "page254" id = "page254"> </a>
+<!--png 274-->
+<p class = "illustration">
+<a name = "fig288" id = "fig288"> </a>
+<img src = "images/fig288.png" width = "296" height = "376"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 288.</p>
+
+<p>Note that the door itself is quite luminous, and lighted up by the
+reflection of the sun from the tiled floor, so that the bars in the
+upper part throw distinct shadows, besides the mystery of colour thus
+introduced. The little window to the left, though not admitting much
+direct sunlight, is evidence of the brilliant glare outside; for the
+reflected light is very conspicuous on the
+<span class = "pagenum">255</span>
+<a name = "page255" id = "page255"> </a>
+<!--png 275-->
+top and on the shutters on each side; indeed they cast distinct shadows
+up and down, while some clear daylight from the blue sky is reflected on
+the window-sill. As to the sink, the table, the wash-tubs, &amp;c.,
+although they seem in strong light and shade they really receive little
+or no direct light from a single point; but from the strong reflected
+light re-reflected into them from the wall of the doorway. There are
+many other things in such effects as this which the artist will observe,
+and which can only be studied from real light and shade. Such is the
+character of reflected light, varying according to the angle and
+intensity of the luminary and a hundred other things. When we come to
+study light in the open air we get into another region, and have to deal
+with it accordingly, and yet we shall find that our sciagraphy will be a
+help to us even in this bewilderment; for it will explain in a manner
+the innumerable shapes of sun-shadows that we observe out of doors among
+hills and dales, showing up their forms and structure; its play in the
+woods and gardens, and its value among buildings, showing all their
+juttings and abuttings, recesses, doorways, and all the other
+architectural details. Nor must we forget light's most glorious display
+of all on the sea and in the clouds and in the sunrises and the sunsets
+down to the still and lovely moonlight.</p>
+
+<p>These sun-shadows are useful in showing us the principle of light and
+shade, and so also are the shadows cast by artificial light; but they
+are only the beginning of that beautiful study, that exquisite art of
+tone or <i>chiaro-oscuro</i>, which is infinite in its variety, is full
+of the deepest mystery, and is the true poetry of art. For this the
+student must go to Nature herself, must study her in all her moods from
+early dawn to sunset, in the twilight and when night sets in. No
+mathematical rules can help him, but only the thoughtful contemplation,
+the silent watching, and the mental notes that he can make and commit to
+memory, combining them with the sentiments to which they in turn give
+rise. The <i>plein air</i>, or broad daylight effects, are but one item
+of the great range of this ever-changing and deepening
+mystery&mdash;from the hard reality to the soft blending of evening when
+form almost disappears, even to the merging of the whole landscape, nay,
+the whole world, into a dream&mdash;which is felt
+<span class = "pagenum">256</span>
+<a name = "page256" id = "page256"> </a>
+<!--png 276-->
+rather than seen, but possesses a charm that almost defies the pencil of
+the painter, and can only be expressed by the deep and sweet notes of
+the poet and the musician. For love and reverence are necessary to
+appreciate and to present&nbsp;it.</p>
+
+<p>There is also much to learn about artificial light. For here, again,
+the study is endless: from the glare of a hundred lights&mdash;electric
+and otherwise&mdash;to the single lamp or candle. Indeed a whole volume
+could be filled with illustrations of its effects. To those who aim at
+producing intense brilliancy, refusing to acknowledge any limitations to
+their capacity, a&nbsp;hundred or a thousand lights commend themselves;
+and even though wild splashes of paint may sometimes be the result,
+still the effort is praiseworthy. But those who prefer the mysterious
+lighting of a Rembrandt will find, if they sit contemplating in a room
+lit with one lamp only, that an endless depth of mystery surrounds them,
+full of dark recesses peopled by fancy and sweet thought, whilst the
+most beautiful gradations soften the forms without distorting them; and
+at the same time he can detect the laws of this science of light and
+shade a thousand times repeated and endless in its variety.</p>
+
+<p><i>Note.</i>&mdash;<a href = "#fig288">Fig. 288</a> must be looked upon as a rough sketch
+which only gives the general effect of the original drawing; to render
+all the delicate tints, tones and reflections described in the text
+would require a highly-finished reproduction in half-tone or in
+colour.</p>
+
+<p>As many of the figures in this book had to be re-drawn, not a light
+task, I&nbsp;must here thank Miss Margaret L. Williams, one of our
+Academy students, for kindly coming to my assistance and volunteering
+her careful co-operation.</p>
+
+
+
+
+<span class = "pagenum">257</span>
+<a name = "page257" id = "page257"> </a>
+<!--png 277-->
+<h5 class = "section"><a name = "chapCLXIV" id = "chapCLXIV">
+CLXIV</a></h5>
+
+<h5 class = "smallcaps">Reflection</h5>
+
+
+<p>Reflections in still water can best be illustrated by placing some
+simple object, such as a cube, on a looking-glass laid horizontally on a
+table, or by studying plants, stones, banks, trees, &amp;c., reflected
+in some quiet pond. It will then be seen that the reflection is the
+counterpart of the object reversed, and having the same vanishing points
+as the object itself.</p>
+
+<p class = "illustration">
+<a name = "fig289" id = "fig289"> </a>
+<img src = "images/fig289.png" width = "339" height = "171"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 289.</p>
+
+<p>Let us suppose <span class = "smallroman">R</span> (Fig. 289) to be
+standing on the water or reflecting plane. To find its reflection make
+square <ins class = "correction" title = "upside-down R">[<span class =
+"smallcaps">R</span>]</ins> equal to the original square <span class =
+"smallroman">R</span>. Complete the reversed cube by drawing its other
+sides, &amp;c. It is evident that this lower cube is the reflection of
+the one above it, although it differs in one respect, for whereas in
+figure <span class = "smallroman">R</span> the top of the cube is seen,
+in its reflection [<span class = "smallcaps">R</span>] it is hidden,
+&amp;c. In figure <span class = "smallroman">A</span> of a semicircular
+arch we see the
+<span class = "pagenum">258</span>
+<a name = "page258" id = "page258"> </a>
+<!--png 278-->
+underneath portion of the arch reflected in the water, but we do not see
+it in the actual object. However, these things are obvious. Note that
+the reflected line must be equal in length to the actual one, or the
+reflection of a square would not be a square, nor that of a semicircle a
+semicircle. The apparent lengthening of reflections in water is owing to
+the surface being broken by wavelets, which, leaping up near to us,
+catch some of the image of the tree, or whatever it is, that it is
+reflected.</p>
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig290" id = "fig290"> </a>
+<img src = "images/fig290.png" width = "238" height = "206"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 290.</td>
+</tr>
+</table>
+
+<p>In this view of an arch (Fig. 290) note that the reflection is
+obtained by dropping perpendiculars from certain points on the arch,
+1,&nbsp;0,&nbsp;2, &amp;c., to the surface of the reflecting plane, and
+then measuring the same lengths downwards to corresponding points,
+1,&nbsp;0,&nbsp;2, &amp;c., in the reflection.</p>
+
+
+
+
+<span class = "pagenum">259</span>
+<a name = "page259" id = "page259"> </a>
+<!--png 279-->
+<h5 class = "section"><a name = "chapCLXV" id = "chapCLXV">
+CLXV</a></h5>
+
+<h5 class = "smallcaps">Angles of Reflection</h5>
+
+
+<p>In Fig. 291 we take a side view of the reflected object in order to
+show that at whatever angle the visual ray strikes the reflecting
+surface it is reflected from it at the same angle.</p>
+
+<p class = "illustration">
+<a name = "fig291" id = "fig291"> </a>
+<img src = "images/fig291.png" width = "335" height = "131"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 291.</p>
+
+<p>We have seen that the reflected line must be equal to the original
+line, therefore <i>m</i><span class = "smallroman">B</span> must equal
+<span class = "smallroman">M</span><i>a</i>. They are also at right
+angles to <span class = "smallroman">MN</span>, the plane of reflection.
+We will now draw the visual ray passing from <span class =
+"smallroman">E</span>, the eye, to <span class = "smallroman">B</span>,
+which is the reflection of <span class = "smallroman">A</span>; and just
+underneath it passes through <span class = "smallroman">MN</span> at
+<span class = "smallroman">O</span>, which is the point where the visual
+ray strikes the reflecting surface. Draw <span class =
+"smallroman">OA</span>. This line represents the ray reflected from it.
+We have now two triangles, <span class = "smallroman">OA</span><i>m</i>
+and <span class = "smallroman">O</span><i>m</i><span class =
+"smallroman">B</span>, which are right-angled triangles and equal,
+therefore angle <i>a</i> equals angle <i>b</i>. But angle <i>b</i>
+equals angle <i>c</i>. Therefore angle <span class =
+"smallroman">E</span><i>c</i><span class = "smallroman">M</span> equals
+angle <span class = "smallroman">A</span><i>am</i>, and the angle at
+which the ray strikes the reflecting plane is equal to the angle at
+which it is reflected from&nbsp;it.</p>
+
+
+
+
+<span class = "pagenum">260</span>
+<a name = "page260" id = "page260"> </a>
+<!--png 280-->
+<h5 class = "section"><a name = "chapCLXVI" id = "chapCLXVI">
+CLXVI</a></h5>
+
+<h5 class = "smallcaps">Reflections of Objects at Different
+Distances</h5>
+
+
+<p>In this sketch the four posts and other objects are represented
+standing on a plane level or almost level with the water, in order to
+show the working of our problem more clearly. It will be seen that the
+post <span class = "smallroman">A</span> is on the brink of the
+reflecting plane, and therefore is entirely reflected; <span class =
+"smallroman">B</span> and <span class = "smallroman">C</span> being
+farther back are only partially seen, whereas the reflection of <span
+class = "smallroman">D</span> is not seen at all. I&nbsp;have made all
+the posts the same height, but with regard to the houses, where the
+length of the vertical lines varies, we obtain their reflections by
+measuring from the points <i>oo</i> upwards and downwards as in the
+previous figure.</p>
+
+<p class = "illustration">
+<a name = "fig292" id = "fig292"> </a>
+<img src = "images/fig292.png" width = "332" height = "205"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 292.</p>
+
+<p>Of course these reflections vary according to the position they are
+viewed from; the lower we are down, the more do we see of the
+reflections of distant objects, and vice versa. When the figures are on
+a higher plane than the water, that is, above the plane of reflection,
+we have to find their perspective position,
+<span class = "pagenum">261</span>
+<a name = "page261" id = "page261"> </a>
+<!--png 281-->
+and drop a perpendicular <span class = "smallroman">AO</span> (Fig. 293)
+till it comes in contact with the plane of reflection, which we suppose
+to run under the ground, then measure the same length downwards, as in
+this figure of a girl on the top of the steps. Point <i>o</i> marks the
+point of contact with the plane, and by measuring downwards to <i>a�</i>
+we get the length of her reflection, or as much as is seen of it. Note
+the reflection of the steps and the sloping bank, and the application of
+the inclined plane ascending and descending.</p>
+
+<p class = "illustration">
+<a name = "fig293" id = "fig293"> </a>
+<img src = "images/fig293.png" width = "355" height = "354"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 293.</p>
+
+
+
+
+<span class = "pagenum">262</span>
+<a name = "page262" id = "page262"> </a>
+<!--png 282-->
+<h5 class = "section"><a name = "chapCLXVII" id = "chapCLXVII">
+CLXVII</a></h5>
+
+<h5 class = "smallcaps">Reflection in a Looking-glass</h5>
+
+
+<table class = "float right" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig294" id = "fig294"> </a>
+<img src = "images/fig294.png" width = "131" height = "179"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 294.</td>
+</tr>
+</table>
+
+<p>I had noticed that some of the figures in Titian&rsquo;s pictures
+were only half life-size, and yet they looked natural; and one day,
+thinking I would trace myself in an upright mirror, I&nbsp;stood at
+arm&rsquo;s length from it and with a brush and Chinese white,
+I&nbsp;made a rough outline of my face and figure, and when I measured
+it I found that my drawing was exactly half as long and half as wide as
+nature. I&nbsp;went closer to the glass, but the same outline fitted me.
+Then I retreated several paces, and still the same outline surrounded
+me. Although a little surprising at first, the reason is obvious. The
+image in the glass retreats or advances exactly in the same measure as
+the spectator.</p>
+
+<p>Suppose him to represent one end of a parallelogram <i>e�s�</i>, and
+his image <i>a�b�</i> to represent the other. The mirror <span class =
+"smallroman">AB</span> is a perpendicular half-way between them, the
+diagonal <i>e�b�</i> is the visual ray
+<span class = "pagenum">263</span>
+<a name = "page263" id = "page263"> </a>
+<!--png 283-->
+passing from the eye of the spectator to the foot of his image, and is
+the diagonal of a rectangle, therefore it cuts <span class =
+"smallroman">AB</span> in the centre <i>o</i>, and <span class =
+"smallroman">AO</span> represents <i>a�b�</i> to the spectator. This is
+an experiment that any one may try for himself. Perhaps the above fact
+may have something to do with the remarks I made about Titian at the
+beginning of this chapter.</p>
+
+<p class = "illustration">
+<a name = "fig295" id = "fig295"> </a>
+<img src = "images/fig295.png" width = "333" height = "133"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 295.</p>
+
+<p class = "illustration">
+<a name = "fig296" id = "fig296"> </a>
+<img src = "images/fig296.png" width = "233" height = "109"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 296.</p>
+
+
+
+
+<span class = "pagenum">264</span>
+<a name = "page264" id = "page264"> </a>
+<!--png 284-->
+<h5 class = "section"><a name = "chapCLXVIII" id = "chapCLXVIII">
+CLXVIII</a></h5>
+
+<h5 class = "smallcaps">The Mirror at an Angle</h5>
+
+
+<p>If an object or line <span class = "smallroman">AB</span> is inclined
+at an angle of 45� to the mirror <span class = "smallroman">RR</span>,
+then the angle <span class = "smallroman">BAC</span> will be a right
+angle, and this angle is exactly divided in two by the reflecting plane
+<span class = "smallroman">RR</span>. And whatever the angle of the
+object or line makes with its reflection that angle will also be exactly
+divided.</p>
+
+<table class = "illustration" summary = "illustration">
+<tr>
+<td class = "picture">
+<a name = "fig297" id = "fig297"> </a>
+<img src = "images/fig297.png" width = "145" height = "120"
+alt = "figure" title = "figure">
+</td>
+<td class = "picture">
+<a name = "fig298" id = "fig298"> </a>
+<img src = "images/fig298.png" width = "266" height = "118"
+alt = "figure" title = "figure">
+</td>
+</tr>
+<tr>
+<td class = "caption smallcaps">
+Fig. 297.</td>
+<td class = "caption smallcaps">
+Fig. 298.</td>
+</tr>
+</table>
+
+<p>Now suppose our mirror to be standing on a horizontal plane and on a
+pivot, so that it can be inclined either way. Whatever angle the mirror
+is to the plane the reflection of that plane in the mirror will be at
+the same angle on the other side of it, so that if the mirror <span
+class = "smallroman">OA</span> (Fig. 298) is at 45� to the plane <span
+class = "smallroman">RR</span> then the
+<span class = "pagenum">265</span>
+<a name = "page265" id = "page265"> </a>
+<!--png 285-->
+reflection of that plane in the mirror will be 45� on the other side of
+it, or at right angles, and the reflected plane will appear
+perpendicular, as shown in Fig. 299, where we have a front view of a
+mirror leaning forward at an angle of 45� and reflecting the square
+<i>aob</i> with a cube standing upon it, only in the reflection the cube
+appears to be projecting from an upright plane or wall.</p>
+
+<p class = "illustration">
+<a name = "fig299" id = "fig299"> </a>
+<img src = "images/fig299.png" width = "223" height = "193"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 299.</p>
+
+<p>If we increase the angle from 45� to 60�, then the reflection of the
+plane and cube will lean backwards as shown in Fig. 300. If we place it
+on a level with the original plane, the cube will be standing upright
+twice the distance away. If the mirror is still farther tilted till it
+makes an angle of 135� as at <span class = "smallroman">E</span> (Fig.
+298), or 45� on the other side of the vertical <span class =
+"smallroman">O</span><i>c</i>, then the plane and cube would disappear,
+and objects exactly over that plane, such as the ceiling, would come
+into view.</p>
+
+<p>In Fig. 300 the mirror is at 60� to the plane <i>mn</i>, and the
+plane itself at about 15� to the plane <i>an</i> (so that here we are
+using angular perspective, <span class = "smallroman">V</span> being the
+accessible vanishing point). The reflection of the plane and cube is
+seen leaning back at an
+<span class = "pagenum">266</span>
+<a name = "page266" id = "page266"> </a>
+<!--png 286-->
+angle of 60�. Note the way the reflection of this cube is found by the
+dotted lines on the plane, on the surface of the mirror, and also on the
+reflection.</p>
+
+<p class = "illustration">
+<a name = "fig300" id = "fig300"> </a>
+<img src = "images/fig300.png" width = "345" height = "258"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 300.</p>
+
+
+
+
+<h5 class = "section"><a name = "chapCLXIX" id = "chapCLXIX">
+CLXIX</a></h5>
+
+<h5 class = "smallcaps">The Upright Mirror at an Angle of 45� to the
+Wall</h5>
+
+
+<p>In Fig. 301 the mirror is vertical and at an angle of 45� to the wall
+opposite the spectator, so that it reflects a portion of that wall as
+though it were receding from us at right angles; and the wall with the
+pictures upon it, which appears to be facing us, in reality is on our
+left.</p>
+
+<p class = "illustration">
+<span class = "pagenum">[267]</span>
+<a name = "page267" id = "page267"> </a>
+<!--png 287-->
+<a name = "fig301" id = "fig301"> </a>
+<img src = "images/fig301.png" width = "188" height = "450"
+alt = "figure" title = "figure">
+</p>
+
+<p class = "caption smallcaps">
+Fig. 301.</p>
+
+<p><span class = "pagenum">268</span>
+<a name = "page268" id = "page268"> </a>
+<!--png 288-->
+An endless number of complicated problems could be invented of the
+inclined mirror, but they would be mere puzzles calculated rather to
+deter the student than to instruct him. What we chiefly have to bear in
+mind is the simple principle of reflections. When a mirror is vertical
+and placed at the end or side of a room it reflects that room and gives
+the impression that we are in one double the size. If two mirrors are
+placed opposite to each other at each end of a room they reflect and
+reflect, so that we see an endless number of rooms.</p>
+
+<p>Again, if we are sitting in a gallery of pictures with a hand mirror,
+we can so turn and twist that mirror about that we can bring any picture
+in front of us, whether it is behind us, at the side, or even on the
+ceiling. Indeed, when one goes to those old palaces and churches where
+pictures are painted on the ceiling, as in the Sistine Chapel or the
+Louvre, or the palaces at Venice, it is not a bad plan to take a hand
+mirror with us, so that we can see those elevated works of art in
+comfort.</p>
+
+<p>There are also many uses for the mirror in the studio, well known to
+the artist. One is to look at one's own picture reversed, when faults
+become more evident; and another, when the model is required to be at a
+longer distance than the dimensions of the studio will admit, by drawing
+his reflection in the glass we double the distance he is
+from&nbsp;us.</p>
+
+<p>The reason the mirror shows the fault of a work to which the eye has
+become accustomed is that it doubles it. Thus if a line that should be
+vertical is leaning to one side, in the mirror it will lean to the
+other; so that if it is out of the perpendicular to the left, its
+reflection will be out of the perpendicular to the right, making a
+double divergence from one to the other.</p>
+
+
+
+
+<span class = "pagenum">269</span>
+<a name = "page269" id = "page269"> </a>
+<!--png 289-->
+<h5 class = "section"><a name = "chapCLXX" id = "chapCLXX">
+CLXX</a></h5>
+
+<h5 class = "smallcaps">Mental Perspective</h5>
+
+
+<p>Before we part, I&nbsp;should like to say a word about mental
+perspective, for we must remember that some see farther than others, and
+some will endeavour to see even into the infinite. To see Nature in all
+her vastness and magnificence, the thought must supplement and must
+surpass the eye. It is this far-seeing that makes the great poet, the
+great philosopher, and the great artist. Let the student bear this in
+mind, for if he possesses this quality or even a share of it, it will
+give immortality to his work.</p>
+
+<p>To explain in detail the full meaning of this suggestion is beyond
+the province of this book, but it may lead the student to think this
+question out for himself in his solitary and imaginative moments, and
+should, I&nbsp;think, give a charm and virtue to his work which he
+should endeavour to make of value, not only to his own time but to the
+generations that are to follow. Cultivate, therefore, this mental
+perspective, without forgetting the solid foundation of the science I
+have endeavoured to impart to you.</p>
+
+<hr class = "mid">
+
+<h4 class = "section">Footnotes</h4>
+
+<p class = "footnote">
+<a name = "note1" id = "note1" href = "#tag1">1.</a>
+Leonardo da Vinci's <i>Treatise on Painting</i>.</p>
+
+<p class = "footnote">
+<a name = "note2" id = "note2" href = "#tag2">2.</a>
+There is another book called <i>The Jesuit's Perspective</i> which I
+have not yet seen, but which I hear is a fine work.</p>
+
+<p class = "footnote">
+<a name = "note3" id = "note3" href = "#tag3">3.</a>
+In a sea-view, owing to the rotundity of the earth, the real horizontal
+line is slightly below the sea line, which is noted in
+Chapter&nbsp;I.</p>
+
+<p class = "footnote">
+<a name = "note4" id = "note4" href = "#tag4">4.</a>
+Some will tell us that Nature abhors a straight line, that all long
+straight lines in space appear curved, &amp;c., owing to certain optical
+conditions; but this is not apparent in short straight lines, so if our
+drawing is small it would be wrong to curve them; if it is large, like a
+scene or diorama, the same optical condition which applies to the line
+in space would also apply to the line in the picture.</p>
+
+<hr class = "mid">
+
+<span class = "pagenum">270</span>
+<a name = "page270" id = "page270"> </a>
+<!--png 290-->
+
+<h4 class = "section"><a name = "index" id = "index">INDEX</a></h4>
+
+<p class = "mynote">
+Index citations in the original book referred to page numbers. Where
+possible, links will lead directly to a chapter header or illustration.
+Note that the last two entries for Toledo are figure numbers rather than
+pages; these have not been corrected.</p>
+
+<div class = "index">
+
+<p class = "letterhead">A</p>
+<p>Albert <ins class = "correction" title = "umlaut missing">D�rer</ins>,
+<a href = "#page2">2</a>,
+<a href = "#page9">9</a>.</p>
+<p>Angles of Reflection,
+<a href = "#chapCLXV">259</a>.</p>
+<p>Angular Perspective,
+<a href = "#chapXLIX">98</a>-<a href = "#chapLXXII">123</a>,
+<a href = "#chapLXXX">133</a>,
+<a href = "#page170">170</a>.</p>
+<p><span class = "invisible">Ang</span>"<span class = "invisible">lar
+Persp</span>"<span class = "invisible">ctive,</span> New Method,
+<a href = "#chapLXXX">133</a>,
+<a href = "#chapLXXXI">134</a>,
+<a href = "#chapLXXXII">135</a>,
+<a href = "#chapLXXXIII">136</a>.</p>
+<p>Arches, Arcades, &amp;c.,
+<a href = "#chapCXXVI">198</a>,
+<a href = "#chapCXXVII">200</a>-<a href = "#chapCXXIII">208</a>.</p>
+<p>Architect's Perspective,
+<a href = "#chapCVIII">170</a>,
+<a href = "#fig197">171</a>.</p>
+<p>Art Schools Perspective,
+<a href = "#chapLXII">112</a>-<a href = "#chapLXVI">118</a>,
+<a href = "#chapCXLI">217</a>.</p>
+<p>Atmosphere,
+<a href = "#page1">1</a>,
+<a href = "#chapXXX">74</a>.</p>
+
+<p class = "letterhead">B</p>
+<p>Balcony, Shadow of,
+<a href = "#chapCLVII">246</a>.</p>
+<p>Base or groundline,
+<a href = "#chapXLI">89</a>.</p>
+
+<p class = "letterhead">C</p>
+<p>Campanile Florence,
+<a href = "#page5">5</a>,
+<a href = "#page59">59</a>.</p>
+<p>Cast Shadows,
+<a href = "#chapCXLVII">229</a>-<a href = "#chapCLXII">253</a>.</p>
+<p>Centre of Vision,
+<a href = "#chapII">15</a>.</p>
+<p>Chessboard,
+<a href = "#chapXXXI">74</a>.</p>
+<p>Chinese Art,
+<a href = "#page11">11</a>.</p>
+<p>Circle,
+<a href = "#chapLXXXVIII">145</a>,
+<a href = "#chapXCII">151</a>-<a href = "#chapXCVI">156</a>,
+<a href = "#chapXCIX">159</a>.</p>
+<p>Columns,
+<a href = "#chapXCVII">157</a>,
+<a href = "#chapXCIX">159</a>,
+<a href = "#chapCI">161</a>,
+<a href = "#chapCVI">169</a>,
+<a href = "#chapCVII">170</a>.</p>
+<p>Conditions of Perspective,
+<a href = "#chapVII">24</a>,
+<a href = "#rule1">25</a>.</p>
+<p>Cottage in Angular Perspective,
+<a href = "#chapLXV">116</a>.</p>
+<p>Cube,
+<a href = "#chapXVII">53</a>,
+<a href = "#chapXXIII">65</a>,
+<a href = "#chapLXIV">115</a>,
+<a href = "#chapLXVIII">119</a>.</p>
+<p>Cylinder,
+<a href = "#chapXCVIII">158</a>,
+<a href = "#chapCXIX">159</a>.</p>
+<p>Cylindrical picture<ins class = "correction"
+title = "comma missing">,</ins> <a href = "#chapCXLVI">227</a>.</p>
+
+<p class = "letterhead">D</p>
+<p>De Hoogh,
+<a href = "#page2">2</a>,
+<a href = "#fig68">62</a>,
+<a href = "#fig82">73</a>.</p>
+<p>Depths, How to measure by diagonals,
+<a href = "#chapLXXVI">127</a>,
+<a href = "#chapLXXVII">128</a>.</p>
+<p>Descending plane,
+<a href = "#chapXLIV">92</a>-<a href = "#chapXLV">95</a>.</p>
+<p>Diagonals,
+<a href = "#page45">45</a>,
+<a href = "#chapLXXIII">124</a>,
+<a href = "#chapLXXIV">125</a>,
+<a href = "#chapLXXV">126</a>.</p>
+<p>Disproportion, How to correct,
+<a href = "#page35">35</a>,
+<a href = "#chapLXVII">118</a>,
+<a href = "#chapXCVII">157</a>.</p>
+<p>Distance,
+<a href = "#chapIII">16</a>,
+<a href = "#chapXXXIII">77</a>,
+<a href = "#chapXXXIV">78</a>,
+<a href = "#chapXXXVII">85</a>,
+<a href = "#chapXXXIX">87</a>,
+<a href = "#chapLIV">103</a>,
+<a href = "#chapLXXVII">128</a>.</p>
+<p>Distorted perspective, How to correct,
+<a href = "#chapLXVII">118</a>.</p>
+<p>Dome,
+<a href = "#chapCIII">163</a>-<a href = "#chapCV">167</a>.</p>
+<p>Double Cross,
+<a href = "#chapCXLII">218</a>.</p>
+
+<p class = "letterhead">E</p>
+<p>Ellipse,
+<a href = "#chapLXXXIX">145</a>,
+<a href = "#chapXC">146</a>,
+<a href = "#fig168">147</a>.</p>
+<p>Elliptical Arch,
+<a href = "#chapCXXXII">207</a>.</p>
+
+<p class = "letterhead">F</p>
+<p>Farningham,
+<a href = "#fig103">95</a>.</p>
+<p>Figures on descending plane,
+<a href = "#chapXLIV">92</a>,
+<a href = "#fig100">93</a>,
+<a href = "#fig102">94</a>,
+<a href = "#chapXLV">95</a>.</p>
+<p><span class = "invisible">Fig</span>"<span class = "invisible">res
+</span>"<span class = "invisible">n</span> an inclined plane,
+<a href = "#chapXL">88</a>.</p>
+<p><span class = "invisible">Fig</span>"<span class = "invisible">res
+</span>"<span class = "invisible">n</span> a level plane,
+<a href = "#fig79">70</a>,
+<a href = "#chapXXVIII">71</a>,
+<a href = "#fig81">72</a>,
+<a href = "#fig82">73</a>,
+<a href = "#chapXXX">74</a>,
+<a href = "#chapXXXI">75</a>.</p>
+<p><span class = "invisible">Fig</span>"<span class = "invisible">res
+</span>"<span class = "invisible">n</span> uneven ground,
+<a href = "#chapXLII">90</a>,
+<a href = "#chapXLIII">91</a>.</p>
+
+<p class = "letterhead">G</p>
+<p>Geometrical and Perspective figures contrasted,
+<a href = "#chapXII">46</a>-<a href = "#page48">48</a>.</p>
+<p><span class = "invisible">Geom</span>"<span class =
+"invisible">trical</span> plane,
+<a href = "#chapL">99</a>.</p>
+<p>Giovanni da Pistoya, Sonnet to, by Michelangelo,
+<a href = "#page60">60</a>.</p>
+<p>Great Pyramid,
+<a href = "#chapCXXII">190</a>.</p>
+
+<p class = "letterhead">H</p>
+<p>Hexagon,
+<a href = "#chapCXIV">177</a>,
+<a href = "#chapCXVII">183</a>,
+<a href = "#chapCXIX">185</a>.</p>
+<p>Hogarth,
+<a href = "#page9">9</a>.</p>
+<p>Honfleur,
+<a href = "#fig92">83</a>,
+<a href = "#fig163">142</a>.</p>
+<p>Horizon,
+<a href = "#page3">3</a>,
+<a href = "#page4">4</a>,
+<a href = "#chapII">15</a>,
+<a href = "#page20">20</a>,
+<a href = "#chapXX">59</a>,
+<a href = "#fig66">60</a>.</p>
+<p>Horizontal line,
+<a href = "#chapI">13</a>,
+<a href = "#chapII">15</a>.</p>
+<p>Horizontals,
+<a href = "#rule6">30</a>,
+<a href = "#rule7">31</a>,
+<a href = "#rule10">36</a>.</p>
+
+<span class = "pagenum">271</span>
+<a name = "page271" id = "page271"> </a>
+<!--png 291-->
+
+<p class = "letterhead">I</p>
+<p>Inaccessible vanishing points,
+<a href = "#chapXXXII">77</a>,
+<a href = "#chapXXXIII">78</a>,
+<a href = "#page136">136</a>,
+<a href = "#page140">140</a>-<a href = "#page144">144</a>.</p>
+<p>Inclined plane,
+<a href = "#rule8">33</a>,
+<a href = "#page118">118</a>,
+<a href = "#chapCXXXVIII">213</a>,
+<a href = "#chapXLV">244</a>,
+<a href = "#chapXLVI">245</a>.</p>
+<p>Interiors,
+<a href = "#chapXXI">62</a>,
+<a href = "#chapLXVI">117</a>,
+<a href = "#chapLXVII">118</a>,
+<a href = "#page128">128</a>.</p>
+
+<p class = "letterhead">J</p>
+<p>Japanese Art,
+<a href = "#page11">11</a>.</p>
+<p>Jesuit of Paris, Practice of Perspective by,
+<a href = "#page9">9</a>.</p>
+
+<p class = "letterhead">K</p>
+<p>Kiosk, Application of Hexagon,
+<a href = "#chapXCIX">185</a>.</p>
+<p>Kirby, Joshua, Perspective made Easy (?),
+<a href = "#page9">9</a>.</p>
+
+<p class = "letterhead">L</p>
+<p>Ladder, Step,
+<a href = "#chapCXXXVII">212</a>,
+<a href = "#chapCXL">216</a>.</p>
+<p>Landscape Perspective,
+<a href = "#chapXXX">74</a>.</p>
+<p>Landseer, Sir Edwin,
+<a href = "#page1">1</a>.</p>
+<p>Leonardo da Vinci,
+<a href = "#page1">1</a>,
+<a href = "#page61">61</a>.</p>
+<p>Light, Observations on,
+<a href = "#chapCLXIII">253</a>.</p>
+<p>Light-house,
+<a href = "#chapXXXVII">84</a>.</p>
+<p>Long distances,
+<a href = "#chapXXXVIII">85</a>,
+<a href = "#chapXXXIX">87</a>.</p>
+
+<p class = "letterhead">M</p>
+<p>Measure distances by square and diagonal,
+<a href = "#chapXLI">89</a>,
+<a href = "#chapLXXVII">128</a>,
+<a href = "#page129">129</a>.</p>
+<p><span class = "invisible">Mea</span>"<span class =
+"invisible">ure</span> vanishing lines, How to,
+<a href = "#chapXIV">49</a>,
+<a href = "#chapXV">50</a>.</p>
+<p>Measuring points,
+<a href = "#chapLVII">106</a>,
+<a href = "#page113">113</a>.</p>
+<p><span class = "invisible">Meas</span>"<span class =
+"invisible">ring</span> point O,
+<a href = "#page108">108</a>,
+<a href = "#page109">109</a>,
+<a href = "#chapLX">110</a>.</p>
+<p>Mental Perspective,
+<a href = "#chapCLXX">269</a>.</p>
+<p>Michelangelo,
+<a href = "#page5">5</a>,
+<a href = "#page57">57</a>,
+<a href = "#page58">58</a>,
+<a href = "#page60">60</a>.</p>
+
+<p class = "letterhead">N</p>
+<p>Natural Perspective,
+<a href = "#page12">12</a>,
+<a href = "#fig91">82</a>,
+<a href = "#fig103">95</a>,
+<a href = "#fig163">142</a>,
+<a href = "#fig164">144</a>.</p>
+<p>New Method of Angular Perspective,
+<a href = "#chapLXXX">133</a>,
+<a href = "#chapLXXXI">134</a>,
+<a href = "#chapLXXXII">135</a>,
+<a href = "#chapLXXXVI">141</a>,
+<a href = "#chapCXXXIX">215</a>,
+<a href = "#page219">219</a>.</p>
+<p>Niche,
+<a href = "#chapCIV">164</a>,
+<a href = "#fig193">165</a>,
+<a href = "#chapCLX">250</a>.</p>
+
+<p class = "letterhead">O</p>
+<p>Oblique Square,
+<a href = "#chapLXXXV">139</a>.</p>
+<p>Octagon,
+<a href = "#chapCIX">172</a>-<a href = "#fig202">175</a>.</p>
+<p>O, measuring point,
+<a href = "#chapLX">110</a>.</p>
+<p>Optic Cone,
+<a href = "#chapIV">20</a>.</p>
+
+<p class = "letterhead">P</p>
+<p>Parallels and Diagonals,
+<a href = "#chapLXXIII">124</a>-<a href = "#chapLXXVI">128</a>.</p>
+<p>Paul Potter, cattle,
+<a href = "#fig16">19</a>.</p>
+<p>Paul Veronese,
+<a href = "#page4">4</a>.</p>
+<p>Pavements,
+<a href = "#chapXXII">64</a>,
+<a href = "#chapXXIV">66</a>,
+<a href = "#chapCXIII">176</a>,
+<a href = "#chapCXV">178</a>,
+<a href = "#fig209">180</a>,
+<a href = "#chapCXVI">181</a>,
+<a href = "#chapCXVII">183</a>.</p>
+<p>Pedestal,
+<a href = "#chapLXXXVI">141</a>,
+<a href = "#chapCI">161</a>.</p>
+<p>Pentagon,
+<a href = "#chapCXX">186</a>,
+<a href = "#fig217">187</a>,
+<a href = "#fig219">188</a>.</p>
+<p>Perspective, Angular,
+<a href = "#chapXLIX">98</a>-<a href = "#chapLXXII">123</a>.</p>
+<p><span class = "invisible">Persp</span>"<span class =
+"invisible">ctive,</span> Definitions,
+<a href = "#chapI">13</a>-<a href = "#chapVI">23</a>.</p>
+<p><span class = "invisible">Persp</span>"<span class =
+"invisible">ctive,</span> Necessity of,
+<a href = "#page1">1</a>.</p>
+<p><span class = "invisible">Persp</span>"<span class =
+"invisible">ctive,</span> Parallel,
+<a href = "#practice">42</a>-<a href = "#chapXLVII">97</a>.</p>
+<p><span class = "invisible">Persp</span>"<span class =
+"invisible">ctive,</span>
+Rules and Conditions of,
+<a href = "#chapVII">24</a>-<a href = "#rule10">41</a>.</p>
+<p><span class = "invisible">Persp</span>"<span class =
+"invisible">ctive,</span>
+Scientific definition of,
+<a href = "#chapVI">22</a>.</p>
+<p><span class = "invisible">Persp</span>"<span class =
+"invisible">ctive,</span> Theory of,
+<a href = "#theory">13</a>-<a href = "#chapVI">24</a>.</p>
+<p><span class = "invisible">Persp</span>"<span class =
+"invisible">ctive,</span> What is it?
+<a href = "#what_is">6</a>-<a href = "#page12">12</a>.</p>
+<p>Pictures painted according to positions they are to occupy,
+<a href = "#chapXX">59</a>.</p>
+<p>Point of Distance,
+<a href = "#chapIII">16</a>-<a href = "#chapIV">21</a>.</p>
+<p><span class = "invisible">Po</span>"<span class = "invisible">nt
+</span>"<span class = "invisible">f</span> Sight,
+<a href = "#page12">12</a>,
+<a href = "#chapII">15</a>.</p>
+<p>Points in Space,
+<a href = "#chapLXXVIII">129</a>,
+<a href = "#chapLXXXIII">137</a>.</p>
+<p>Portico,
+<a href = "#fig122">111</a>.</p>
+<p>Projection,
+<a href = "#chapV">21</a>,
+<a href = "#page137">137</a>.</p>
+<p>Pyramid,
+<a href = "#chapCXXI">189</a>,
+<a href = "#fig224">190</a>,
+<a href = "#chapCXXII">191</a>,
+<a href = "#chapCXXIII">193</a>-<a href = "#chapCXXV">196</a>.</p>
+
+<p class = "letterhead">R</p>
+<p>Raphael,
+<a href = "#page3">3</a>.</p>
+<p>Reduced distance,
+<a href = "#chapXXXIII">77</a>,
+<a href = "#chapXXXIV">78</a>,
+<a href = "#chapXXXV">79</a>,
+<a href = "#fig90">84</a>.</p>
+<p>Reflection,
+<a href = "#chapCLXIV">257</a>-<a href = "#chapCLXIX">268</a>.</p>
+<p>Rembrandt,
+<a href = "#chapXX">59</a>,
+<a href = "#page256">256</a>.</p>
+<p>Reynolds, Sir Joshua,
+<a href = "#page9">9</a>,
+<a href = "#page60">60</a>.</p>
+<p>Rubens,
+<a href = "#page4">4</a>.</p>
+<p>Rules of Perspective,
+<a href = "#rule1">24</a>-<a href = "#rule10">41</a>.</p>
+
+<span class = "pagenum">272</span>
+<a name = "page272" id = "page272"> </a>
+<!--png 292-->
+
+<p class = "letterhead">S</p>
+<p>Scale on each side of Picture,
+<a href = "#chapLXXXVII">141</a>,
+<a href = "#fig163">142</a>-<a href = "#fig164">144</a>.</p>
+<p><span class = "invisible">Sc</span>"<span class =
+"invisible">le</span> Vanishing,
+<a href = "#chapXXVI">69</a>,
+<a href = "#chapXXVII">71</a>,
+<a href = "#chapXXXVI">81</a>,
+<a href = "#fig90">84</a>.</p>
+<p>Serlio,
+<a href = "#page5">5</a>,
+<a href = "#chapLXXV">126</a>.</p>
+<p>Shadows cast by sun,
+<a href = "#chapCXLVII">229</a>-<a href = "#chapCLXI">252</a>.</p>
+<p><span class = "invisible">Sha</span>"<span class = "invisible">ows
+ca</span>"<span class = "invisible">st
+</span>"<span class = "invisible">y</span> artificial light,
+<a href = "#chapCLXII">252</a>.</p>
+<p>Sight, Point of,
+<a href = "#page12">12</a>,
+<a href = "#chapII">15</a>.</p>
+<p>Sistine Chapel,
+<a href = "#page60">60</a>.</p>
+<p>Solid figures,
+<a href = "#chapLXXXII">135</a>-<a href = "#chapLXXXV">140</a>.</p>
+<p>Square in Angular Perspective,
+<a href = "#chapLVI">105</a>,
+<a href = "#chapLVII">106</a>,
+<a href = "#fig120">109</a>,
+<a href = "#chapLXII">112</a>,
+<a href = "#chapLXIII">114</a>,
+<a href = "#chapLXX">121</a>,
+<a href = "#chapLXXI">122</a>,
+<a href = "#chapLXXII">123</a>,
+<a href = "#chapLXXX">133</a>,
+<a href = "#chapLXXXI">134</a>,
+<a href = "#chapLXXXV">139</a>.</p>
+<p><span class = "invisible">Sq</span>"<span class =
+"invisible">are</span> and diagonals,
+<a href = "#chapLXXIV">125</a>,
+<a href = "#chapLXXXIV">138</a>,
+<a href = "#chapLXXXV">139</a>,
+<a href = "#chapLXXXVI">141</a>.</p>
+<p><span class = "invisible">Sq</span>"<span class =
+"invisible">are</span> of the hypotenuse (fig. 170),
+<a href = "#fig170">149</a>.</p>
+<p><span class = "invisible">Sq</span>"<span class =
+"invisible">are</span> in Parallel Perspective,
+<a href = "#chapIX">42</a>,
+<a href = "#chapX">43</a>,
+<a href = "#chapXV">50</a>,
+<a href = "#chapXVII">53</a>,
+<a href = "#chapXIX">54</a>.</p>
+<p><span class = "invisible">Sq</span>"<span class =
+"invisible">are</span> at 45�,
+<a href = "#chapXXII">64</a>-<a href = "#chapXXIV">66</a>.</p>
+<p>Staircase leading to a Gallery,
+<a href = "#chapCXLIII">221</a>.</p>
+<p>Stairs, Winding,
+<a href = "#chapCXLIV">222</a>,
+<a href = "#chapCXLV">225</a>.</p>
+<p>Station Point,
+<a href = "#chapI">13</a>.</p>
+<p>Steps,
+<a href = "#chapCXXXIV">209</a>-<a href = "#chapCXLII">218</a>.</p>
+
+<p class = "letterhead">T</p>
+<p><ins class = "correction"
+title = "text reads 'Tadeo'">Taddeo</ins> Gaddi,
+<a href = "#page5">5</a>.</p>
+<p>Terms made use of,
+<a href = "#chapXIII">48</a>.</p>
+<p>Tiles,
+<a href = "#chapCXIII">176</a>,
+<a href = "#chapCXV">178</a>,
+<a href = "#chapCXVI">181</a>.</p>
+<p>Tintoretto,
+<a href = "#page4">4</a>.</p>
+<p><ins class = "correction"
+title = "text reads 'Titien'">Titian</ins>,
+<a href = "#chapXX">59</a>,
+<a href = "#chapCLXVII">262</a>.</p>
+<p>Toledo,
+<a href = "#fig104">96</a>,
+<a href = "#fig164">144</a>,
+<a href = "#fig259">259</a>,
+<a href = "#fig288">288</a>.</p>
+<p>Trace and projection,
+<a href = "#chapV">21</a>.</p>
+<p>Transposed distance,
+<a href = "#chapXVIII">53</a>.</p>
+<p>Triangles,
+<a href = "#chapLV">104</a>,
+<a href = "#chapLVII">106</a>,
+<a href = "#fig148">132</a>,
+<a href = "#fig151">135</a>,
+<a href = "#fig158">138</a>.</p>
+<p>Turner,
+<a href = "#page2">2</a>,
+<a href = "#fig95">87</a>.</p>
+
+<p class = "letterhead">U</p>
+<p>Ubaldus, Guidus,
+<a href = "#page9">9</a>.</p>
+
+<p class = "letterhead">V</p>
+<p>Vanishing lines,
+<a href = "#chapXIV">49</a>.</p>
+<p><span class = "invisible">Vani</span>"<span class =
+"invisible">hing</span> point,
+<a href = "#chapLXVIII">119</a>.</p>
+<p><span class = "invisible">Vani</span>"<span class =
+"invisible">hing</span>
+scale<ins class = "correction" title = "comma missing">,</ins>
+<a href = "#chapXXV">68</a>-<a href = "#chapXXVIII">72</a>,
+<a href = "#chapXXX">74</a>,
+<a href = "#chapXXXII">77</a>,
+<a href = "#chapXXXV">79</a>,
+<a href = "#fig90">84</a>.</p>
+<p>Vaulted Ceiling,
+<a href = "#chapCXXX">203</a>.</p>
+<p>Velasquez,
+<a href = "#chapXX">59</a>.</p>
+<p>Vertical plane,
+<a href = "#chapI">13</a>.</p>
+<p>Visual rays,
+<a href = "#chapIV">20</a>.</p>
+
+<p class = "letterhead">W</p>
+<p>Winding Stairs,
+<a href = "#chapCXLIV">222</a>-<a href = "#chapCXLV">225</a>.</p>
+<p>Water, Reflections in,
+<a href = "#chapCLXIV">257</a>,
+<a href = "#chapCLXV">258</a>,
+<a href = "#chapCLXVI">260</a>,
+<a href = "#fig293">261</a>.</p>
+
+</div>
+
+
+<p>&nbsp;</p>
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author	Roger Frank <rfrank@pglaf.org>	2025-10-15 01:19:41 -0700
committer	Roger Frank <rfrank@pglaf.org>	2025-10-15 01:19:41 -0700
commit	ad22660321889062bdd0c15887fe07bc8bd4af96 (patch)
tree	d147d49e8340483b76fedb596398a5939f94199c /20165-h