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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> </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 [ ] 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> </p> +<hr class="full" noshade> + +<!--png 001--> + + +<p> <br> </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> <br> </p> + +<!--png 002--> +<h4>THE</h4> + +<h2>THEORY AND PRACTICE<br> +OF PERSPECTIVE</h2> + +<p> </p> + +<h6>BY</h6> + +<h4>G. A. STOREY, A.R.A.</h4> + +<h6>TEACHER OF PERSPECTIVE AT THE ROYAL ACADEMY</h6> + +<p> </p> + +<p class = "illustration"> +<img src = "images/titlepage.png" width = "128" height = "109" +alt = "drawing" title = "drawing"> +<br> +<span class = "smallroman">‘QUÎ FIT?’</span> +</p> + +<p> </p> + +<h5>OXFORD<br> +AT THE CLARENDON PRESS</h5> +<h6>1910</h6> + +<p> <br> </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> <br> </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, &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, &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, &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, &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, &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—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>‘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.’<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’s that it is well I +should here transcribe; he says: ‘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.’ 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 ‘everything in a hurry’, +<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 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 Dürer 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—over a hundred years ago—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 it.</p> + +<p>However, we are here chiefly concerned with the necessity of this +study, and of the necessity of starting our work with 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 +‘Wedding-feast at Cana’, by Paul Veronese, or ‘The +School of Athens’, by Raphael, the artist should set out his +floors, his walls, his colonnades, his balconies, his steps, &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 ‘The Descent +from the Cross’, 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 ‘The Family of Darius at the Feet of +Alexander’, by Paul Veronese, and ‘The Origin of the +“Via Lactea”’, 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. +‘In the first place,’ I said, ‘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.’ +<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’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’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—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. 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, &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 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 Dürer, 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’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>Traité Pratique de +Perspective</i> of M. A. Cassagne (Paris, 1873), which is +thoroughly artistic, and full of pictorial examples admirably done; and +to M. Henriet’s <i>Cours Rational de Dessin</i>. There are +many other foreign books of excellence, notably M. Thibault's +<i>Perspective</i>, and some German and Swiss books, and yet, +notwithstanding this imposing array of authors, I 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. 141, <a href = "#fig162">Fig. 162</a>, +&c.), a 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, &c., and is almost, if +not quite, as simple as parallel perspective (see p. 133, <a href = +"#fig150">Fig. 150</a>, &c.). How to measure distances by the square +and diagonal, and to draw interiors thereby (p. 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. 29, <a href = "#fig29">Fig. 29</a>). Then again, I 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. 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>), &c., &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 table, for instance, would be +represented like a ground-plan and the objects upon it in elevation or +standing up. A 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, +&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 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, &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 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 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—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 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 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 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—where the +horizontal-line is placed at one quarter the height of the canvas. +Indeed, a 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, +&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), +&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—</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 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 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 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, &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, &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, &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. 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 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 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. 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>—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, &c., forming so many squares. +Vertical line 2 viewed from <span class = "smallroman">S</span> +will appear on <span class = "smallroman">AB</span> but half its length, +vertical 3 will be only a third, vertical 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—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 7. It also applies to those on an +inclined plane, as in Rule 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 thorough +understanding of these few pages will make the labour now before us +simple and easy. I 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 think, a still greater pleasure than this, namely, in +invention and in overcoming difficulties—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 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 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>, &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>, &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, &c., which look quite different in perspective, +I 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 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 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 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 feet wide, +2 feet to the right of the vertical, and 1 foot from the +base.</p> + +<p>First measure from <span class = "smallroman">V</span>, 2 feet +to <i>e</i>, which gives the distance from the vertical. Second, from +<i>e</i> measure 3 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 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 2½ feet wide, 1 foot from the vertical and 2 feet +from the base, and is worked out in the same way.</p> + +<p><i>Note.</i>—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 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, &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 long and 3 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 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’s large altar-piece in the Sistine Chapel, ‘The +Last Judgement’; 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, +&c., we can make a fair calculation, such as the length of the nave, +if the picture is an altar-piece—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 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 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 ‘Infant’ 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>—Whilst on this subject, I 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:—</p> + +<h5><i>“I’ho già fatto un gozzo in questo +stento.”</i></h5> + +<p class = "verse"> +Now have I such a goitre ’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’ my back,</p> +<p class = "verse"> +My chest like any harpy’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"> +‘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>).’ Leonardo da Vinci’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 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 ft. +10⅞ 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 +‘occult’.</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 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, &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, &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>, &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, &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 common arrangement in brick floors, or tiles of an octagonal form +as at Fig. 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 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 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 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 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. 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. 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, &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, &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, &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>, &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 ½ base to ½ distance. But even this ½ 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 ¼ base to ¼ 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>, &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 feet. +But we found point ¼ <span class = "smallroman">D</span> by drawing 3D +from ¼ base to ¼ 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>, 1, 2, 3, we mark off on +<span class = "smallroman">SB</span> four spaces each equal to +4 feet, in all 16 feet, so that by taking the whole base and the ¼ +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, &c. The same rule applies +to this corridor (Fig. 87 and Fig. 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 measuring line for distances. Let us first of all divide +the base <span class = "smallroman">AB</span> into eight parts, each +part representing 5 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, &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>, &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>, &c., +to point of sight, and from each division on <span class = +"smallroman">SB</span>, such as 40, 80, &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, &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 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 on.</p> + +<p>Fig. 91 is a sketch made at Sandwich, Kent, and shows a somewhat +similar subject to <a href = "#fig84">Fig. 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, &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 feet instead of +3 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 ¼ 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 × 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>.—This figure (here much reduced) should be drawn +large by the student, so that the numbering, &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 ‘View from Richmond Hill’, 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—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, +&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 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 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 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. 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, &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 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, &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, &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. 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 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>, &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 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, 2, 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, &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, 4, 3, 3, &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 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, &c.; from 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 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 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, &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 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’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 +‘let it be 60° and 30°, or 70° and 20°’, &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, &c., which is a very +clumsy and unscientific way of setting to work. The architects in such +cases make use of the centrolinead, a 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 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 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>, &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, &c. Of course horizontal lines on wall 1 are +drawn to <span class = "smallroman">V</span><sup>1</sup>, and those on +wall 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 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 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 inch, in the +larger one make it 2 inches. If from <span class = +"smallroman">S</span> to <span class = "smallroman">M</span><sup>2</sup> +is 2 inches, in the larger make it 4, and so 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>, +&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>, &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, &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 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>, &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 fourth, and so on. At figure <span class = "smallroman">K</span> +lying on the ground, I have divided the farther side of the square +<i>mn</i> into ¼, ⅓, ½. 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, &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 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, &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>, +&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 us.</p> + +<p>There is, however, a 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 hope, +be found of infinite use to the artist.</p> + +<p><i>Note.</i>—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 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 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, &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 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—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, 1). Through <span +class = "smallroman">A</span> draw <span class = "smallroman">SC</span> +from point of sight to base (Fig. 151, 2 and 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)—a box, +for instance—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, 3, +<span class = "pagenum">138</span> +<a name = "page138" id = "page138"> </a> +<!--png 154--> +2, 1, 3; these we transfer to base of the horizontal perspective +square, and then construct shaded triangle 1, 2, 3, and raise +to the height required as shown at 1·, 2·, 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 2 3 4, then by means of the diagonal 1 3 and 2 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 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 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, +&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, &c. Let us start with base <span class = +"smallroman">AB</span>, a 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 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 <span class = "smallroman">O</span>, 3 <span class = +"smallroman">O</span>, 5 <span class = "smallroman">O</span>, +&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, +&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, 2, 3, 4, &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>, &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 3 +is much wider than column 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 ½-<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, &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, 2, 3, 4, &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 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, &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>, +&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 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>, +&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>, &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, &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, &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. 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 2 3, &c., +we can obtain smaller octagons inside the larger ones. Note how these +are carried out in the second figure (202 <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 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 ½ 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 <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 1, 2 2, &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, 2, 3, 4, &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, &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, 2, 3, 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, &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, &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 1, +draw a line to apex, marked 2. Again, with 1 as centre and 1 2 +as radius, describe arc 2 3. Now with 2 as centre and 2 3 as +radius describe arc 3 4, which will cut the circumference at +point 4. Then line 2 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, 2, 4, &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> 3 2, &c. Take base of +triangle 1 2, measure it round the circle, and so find the five +points through which to draw the pentagon. The angles at 1 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, &c. The cone is +drawn in the same way, so also is any other figure, whether octagonal, +hexangular, triangular, &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> ½ <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>, +&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, 3, +&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>, &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, &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>, &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, &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, &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, 10, 12, &c., each part representing 1 foot. (In +this case the base is 10 feet and the horizon 5 feet.) Set out +floor by means of ¼ distance. Divide it into squares of +1 foot, so that there will be 8 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 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>, &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 ¼ distance and ¼ 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>, &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, 2, 3, &c., on base line equal to <span class = +"smallcaps">B 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 2, its intersection with <span class += "smallroman">AR</span>, draw line 2 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, 2. +On the other side, using the other scales, we have dropped +perpendiculars in the same way from the diagonal to 3, 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> 1 2 O 3 4 <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, 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, 2, 3, &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>, &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>, &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>, &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>, &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, &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—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, 2, 3, +&c., cutting the line <span class = "smallroman">AC</span> at +1, 2, 3, &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, 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 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 1, +2 2, &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>, &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, &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 1, and thus complete the first step. On <span +class = "smallroman">DE</span>, measure heights equal to <span class = +"smallroman">D</span> 1. Draw 2 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 2, when we come to the sixth step, the other end +of which is against the wall opposite to us. Steps 6, 7, 8, 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>, &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, 2, 3, 4, &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 ‘Paris during the Siege’, exhibited some years +ago; ‘The Battle of Trafalgar’, 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 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—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 may tell it here as the origin of +sciagraphy. A 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, +&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. 1 is cast upon step No. 2, and that of No. 2 on +No. 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, 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, 2, 3, &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>—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 2, 2 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, &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, 2, 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>, +&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, &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 2 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> 1 2 3 <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> 1 2 <span class = +"smallroman">B</span>, &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, &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, &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, &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—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—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 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—electric +and otherwise—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 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>—<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 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, &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, &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, +&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, 0, 2, &c., to the surface of the reflecting plane, and +then measuring the same lengths downwards to corresponding points, +1, 0, 2, &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 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 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’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 stood at +arm’s length from it and with a brush and Chinese white, +I 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 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 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 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 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 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, &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">Dürer</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, &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> </p> +<p> </p> +<hr class="full" noshade> +<p>***END OF THE PROJECT GUTENBERG EBOOK THE THEORY AND PRACTICE OF PERSPECTIVE***</p> +<p>******* This file should be named 20165-h.txt or 20165-h.zip *******</p> +<p>This and all associated files of various formats will be found in:<br /> +<a href="http://www.gutenberg.org/dirs/2/0/1/6/20165">http://www.gutenberg.org/2/0/1/6/20165</a></p> +<p>Updated editions will replace the previous one--the old editions +will be renamed.</p> + +<p>Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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