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diff --git a/75399-h/75399-h.htm b/75399-h/75399-h.htm new file mode 100644 index 0000000..95dcc69 --- /dev/null +++ b/75399-h/75399-h.htm @@ -0,0 +1,6293 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + <meta charset="UTF-8"> + <title> + The Science of Beauty | Project Gutenberg + </title> + <link rel="icon" href="images/cover.jpg" type="image/x-cover"> + <style> + +a { + text-decoration: none; +} + +body { + margin-left: 10%; + margin-right: 10%; +} + +h1,h2,h3 { + text-align: center; + clear: both; +} + +h2.nobreak { + page-break-before: avoid; +} + +hr.chap { + margin-top: 2em; + margin-bottom: 2em; + clear: both; + width: 65%; + margin-left: 17.5%; + margin-right: 17.5%; +} + +img.w100 { + width: 100%; +} + +div.chapter { + page-break-before: always; +} + +ul { + list-style-type: none; +} + +li { + margin-top: .5em; + padding-left: 2em; + text-indent: -2em; +} + +p { + margin-top: 0.5em; + text-align: justify; + margin-bottom: 0.5em; + text-indent: 1em; +} + +table { + margin: 1em auto 1em auto; + max-width: 40em; + border-collapse: collapse; +} + +#contents { + max-width: 30em; +} + +td { + padding-left: 0.25em; + padding-right: 0.25em; + vertical-align: top; + text-align: center; +} + +th { + padding: 0.25em; + font-weight: normal; + font-size: 85%; +} + +.tdl { + text-align: left; +} + +.tdr { + text-align: right; +} + +#contents td { + padding-left: 2.25em; + padding-right: 0.25em; + vertical-align: top; + text-indent: -2em; + text-align: justify; +} + +#contents .tdpg { + vertical-align: bottom; + text-align: right; +} + +.borders td, .borders th { + border: thin solid black; +} + +.blockquote { + margin: 1.5em 10%; +} + +.caption p { + text-align: center; + margin-bottom: 1em; + font-size: 90%; + text-indent: 0em; +} + +.caption p.hanging { + text-align: justify; + padding-left: 2em; + text-indent: -2em; +} + +.center { + text-align: center; + text-indent: 0em; +} + +.dedication { + text-align: center; + text-indent: 0em; + line-height: 1.8em; +} + +.ditto { + margin-left: 1.75em; + margin-right: 1.75em; +} + +.figcenter { + margin: auto; + text-align: center; +} + +.footnotes { + margin-top: 1em; + border: dashed 1px; +} + +.footnote { + margin-left: 10%; + margin-right: 10%; + font-size: 0.9em; +} + +.footnote .label { + position: absolute; + right: 84%; + text-align: right; +} + +.fnanchor { + vertical-align: super; + font-size: .8em; + text-decoration: none; +} + +.larger { + font-size: 150%; +} + +.noindent { + text-indent: 0em; +} + +.nw { + white-space: nowrap; +} + +.pagenum { + position: absolute; + right: 4%; + font-size: smaller; + text-align: right; + font-style: normal; +} + +.poetry-container { + text-align: center; +} + +.poetry { + display: inline-block; + text-align: left; +} + +.poetry .stanza { + margin: 1em 0em 1em 0em; +} + +.poetry .verse { + padding-left: 3em; +} + +.poetry .indent0 { + text-indent: -3em; +} + +.poetry .indent8 { + text-indent: 1em; +} + +.right { + text-align: right; +} + +.sidenote { + width: 20%; + padding: 0.5em; + margin-left: 1em; + float: right; + clear: right; + font-size: smaller; + color: black; + background: #eeeeee; + border: dashed 1px; +} + +.smaller { + font-size: 80%; +} + +.smcap { + font-variant: small-caps; + font-style: normal; +} + +.allsmcap { + font-variant: small-caps; + font-style: normal; + text-transform: lowercase; +} + +.spacer { + margin-left: 8em; +} + +.titlepage { + text-align: center; + margin-top: 3em; + text-indent: 0em; +} + +.valign { + vertical-align: middle; +} + +.x-ebookmaker img { + max-width: 100%; + width: auto; + height: auto; +} + +.x-ebookmaker .poetry { + display: block; + margin-left: 1.5em; +} + +.x-ebookmaker .blockquote { + margin: 1.5em 5%; +} + +/* Illustration classes */ +.illowp100 {width: 100%;} +.illowp50 {width: 50%;} +.x-ebookmaker .illowp50 {width: 100%;} +.illowp80 {width: 80%;} +.x-ebookmaker .illowp80 {width: 100%;} + + </style> + </head> +<body> +<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 75399 ***</div> + +<p><span class="pagenum"><a id="Page_i"></a>[i]</span></p> + +<h1>THE SCIENCE OF BEAUTY.</h1> + +<hr class="chap x-ebookmaker-drop"> + +<p><span class="pagenum"><a id="Page_ii"></a>[ii]</span></p> + +<p class="center">EDINBURGH:<br> +PRINTED BY BALLANTYNE AND COMPANY,<br> +PAUL’S WORK.</p> + +<hr class="chap x-ebookmaker-drop"> + +<p><span class="pagenum"><a id="Page_iii"></a>[iii]</span></p> + +<p class="titlepage larger"><span class="smaller">THE</span><br> +<br> +<span class="larger">SCIENCE OF BEAUTY,</span><br> +<br> +AS DEVELOPED IN NATURE AND<br> +APPLIED IN ART.</p> + +<p class="titlepage"><span class="smaller">BY</span><br> +D. R. HAY, F.R.S.E.</p> + +<div class="blockquote"> + +<p>“The irregular combinations of fanciful invention may delight awhile, by that +novelty of which the common satiety of life sends us all in quest; the pleasures +of sudden wonder are soon exhausted, and the mind can only repose on the stability +of truth.”</p> + +<p class="right"><span class="smcap">Dr Johnson.</span></p> + +</div> + +<p class="titlepage">WILLIAM BLACKWOOD AND SONS,<br> +<span class="smaller">EDINBURGH AND LONDON.<br> +MDCCCLVI.</span></p> + +<p><span class="pagenum"><a id="Page_iv"></a>[iv]</span></p> + +<hr class="chap x-ebookmaker-drop"> + +<p><span class="pagenum"><a id="Page_v"></a>[v]</span></p> + +<p class="dedication"><span class="smaller">TO</span><br> +JOHN GOODSIR, ESQ., F.R.S S. L. & E.,<br> +<span class="smaller"><span class="allsmcap">PROFESSOR OF ANATOMY IN THE UNIVERSITY OF EDINBURGH,</span><br> +AS AN EXPRESSION OF GRATITUDE FOR VALUABLE ASSISTANCE,<br> +AS ALSO OF HIGH ESTEEM AND SINCERE REGARD,<br> +THIS VOLUME IS DEDICATED,<br> +BY</span><br> +<span class="spacer">THE AUTHOR.</span></p> + +<p><span class="pagenum"><a id="Page_vi"></a>[vi]</span></p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_vii"></a>[vii]</span></p> + +<h2 class="nobreak" id="PREFACE">PREFACE.</h2> + +</div> + +<p>My theory of beauty in form and colour being now +admitted by the best authorities to be based on truth, +I have of late been often asked, by those who wished +to become acquainted with its nature, and the manner +of its being applied in art, which of my publications I +would recommend for their perusal. This question +I have always found difficulty in answering; for +although the law upon which my theory is based is +characterised by unity, yet the subjects in which it is +applied, and the modes of its application, are equally +characterised by variety, and consequently occupy +several volumes.</p> + +<p>Under these circumstances, I consulted a highly +respected friend, whose mathematical talents and +good taste are well known, and to whom I have +been greatly indebted for much valuable assistance +during the course of my investigations. The advice +I received on this occasion, was to publish a <i>résumé</i> +of my former works, of such a character as not only<span class="pagenum"><a id="Page_viii"></a>[viii]</span> +to explain the nature of my theory, but to exhibit to +the general reader, by the most simple modes of illustration +and description, how it is developed in nature, +and how it may be extensively and with ease applied +in those arts in which beauty forms an essential +element.</p> + +<p>The following pages, with their illustrations, are +the results of an attempt to accomplish this object.</p> + +<p>To those who are already acquainted, through +my former works, with the nature, scope, and tendency +of my theory, I have the satisfaction to intimate +that I have been enabled to include in this <i>résumé</i> +much original matter, with reference both to form +and colour.</p> + +<p class="right">D. R. HAY.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_ix"></a>[ix]</span></p> + +<h2 class="nobreak" id="CONTENTS">CONTENTS.</h2> + +</div> + +<table id="contents"> + <tr> + <td></td> + <td class="tdpg smaller">PAGE</td> + </tr> + <tr> + <td><span class="smcap">Introduction</span></td> + <td class="tdpg"><a href="#INTRODUCTION">1</a></td> + </tr> + <tr> + <td><span class="smcap">The Science of Beauty, evolved from the + Harmonic Law of Nature, agreeably to the Pythagorean System of + Numerical Ratio</span></td> + <td class="tdpg"><a href="#EVOLVED_FROM_THE_HARMONIC_LAW_OF_NATURE">15</a></td> + </tr> + <tr> + <td><span class="smcap">The Science of Beauty, as applied to + Sounds</span></td> + <td class="tdpg"><a href="#AS_APPLIED_TO_SOUNDS">28</a></td> + </tr> + <tr> + <td><span class="smcap">The Science of Beauty, as applied to + Forms</span></td> + <td class="tdpg"><a href="#AS_APPLIED_TO_FORMS">34</a></td> + </tr> + <tr> + <td><span class="smcap">The Science of Beauty, as developed in + the Form of the Human Head and Countenance</span></td> + <td class="tdpg"><a href="#AS_DEVELOPED_IN_THE_HUMAN_HEAD_AND_COUNTENANCE">54</a></td> + </tr> + <tr> + <td><span class="smcap">The Science of Beauty, as developed in + the Form of the Human Figure</span></td> + <td class="tdpg"><a href="#AS_DEVELOPED_IN_THE_FORM_OF_THE_HUMAN_FIGURE">61</a></td> + </tr> + <tr> + <td><span class="smcap">The Science of Beauty, as developed in + Colours</span></td> + <td class="tdpg"><a href="#AS_DEVELOPED_IN_COLOURS">67</a></td> + </tr> + <tr> + <td><span class="smcap">The Science of Beauty applied to the + Forms and Proportions of Ancient Grecian Vases and Ornaments</span></td> + <td class="tdpg"><a href="#APPLIED_TO_THE_FORMS_AND_PROPORTIONS">82</a></td> + </tr> + <tr> + <td><span class="smcap">Appendix, No. I.</span></td> + <td class="tdpg"><a href="#APPENDIX_I">91</a></td> + </tr> + <tr> + <td><span class="smcap">Appendix, No. II.</span></td> + <td class="tdpg"><a href="#APPENDIX_II">99</a></td> + </tr> + <tr> + <td><span class="smcap">Appendix, No. III.</span></td> + <td class="tdpg"><a href="#APPENDIX_III">100</a></td> + </tr> + <tr> + <td><span class="smcap">Appendix, No. IV.</span></td> + <td class="tdpg"><a href="#APPENDIX_IV">100</a></td> + </tr> + <tr> + <td><span class="smcap">Appendix, No. V.</span></td> + <td class="tdpg"><a href="#APPENDIX_V">104</a></td> + </tr> + <tr> + <td><span class="smcap">Appendix, No. VI.</span></td> + <td class="tdpg"><a href="#APPENDIX_VI">105</a></td> + </tr> +</table> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_x"></a>[x]</span></p> + +<h2 class="nobreak" id="ILLUSTRATIONS">ILLUSTRATIONS.</h2> + +</div> + +<h3>PLATES</h3> + +<figure class="figcenter illowp100" id="plate01" style="max-width: 62.5em;"> + <img class="w100" src="images/plate01.jpg" alt=""> + <figcaption class="caption"><p class="hanging">I. Three Scales of the + Elementary Rectilinear Figures, viz., the Scalene Triangle, the Isosceles + Triangle, and the Rectangle, comprising twenty-seven varieties of each, + according to the harmonic parts of the Right Angle from ¹⁄₂ to ¹⁄₁₆.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp100" id="plate02" style="max-width: 62.5em;"> + <img class="w100" src="images/plate02.jpg" alt=""> + <figcaption class="caption"><p class="hanging">II. Diagram of the + Rectilinear Orthography of the Principal Front of the Parthenon of + Athens, in which its Proportions are determined by harmonic parts of + the Right Angle.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp100" id="plate03" style="max-width: 62.5em;"> + <img class="w100" src="images/plate03.jpg" alt=""> + <figcaption class="caption"><p class="hanging">III. Diagram of the + Rectilinear Orthography of the Portico of the Temple of Theseus at Athens, + in which its Proportions are determined by harmonic parts of the Right Angle.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate04" style="max-width: 40.625em;"> + <img class="w100" src="images/plate04.jpg" alt=""> + <figcaption class="caption"><p class="hanging">IV. Diagram of the + Rectilinear Orthography of the East End of Lincoln Cathedral, in which + its Proportions are determined by harmonic parts of the Right Angle.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp100" id="plate05" style="max-width: 62.5em;"> + <img class="w100" src="images/plate05.jpg" alt=""> + <figcaption class="caption"><p class="hanging">V. Four Ellipses described + from Foci, determined by harmonic parts of the Right Angle, shewing in each + the Scalene Triangle, the Isosceles Triangle, and the Rectangle to which + it belongs.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate06" style="max-width: 39.0625em;"> + <img class="w100" src="images/plate06.jpg" alt=""> + <figcaption class="caption"><p class="hanging">VI. The Composite Ellipse + of ¹⁄₆ and ¹⁄₈ of the Right Angle, shewing its greater and lesser Axis, + its various Foci, and the Isosceles Triangle in which they are placed.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate07" style="max-width: 35.9375em;"> + <img class="w100" src="images/plate07.jpg" alt=""> + <figcaption class="caption"><p class="hanging">VII. The Composite Ellipse + of ¹⁄₄₈ and ¹⁄₆₄ of the Right Angle, shewing how it forms the Entasis of + the Columns of the Parthenon of Athens.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate08" style="max-width: 39.0625em;"> + <img class="w100" src="images/plate08.jpg" alt=""> + <figcaption class="caption"><p class="hanging">VIII. Sectional Outlines of + two Mouldings of the Parthenon of Athens, full size, shewing the harmonic + nature of their Curves, and the simple manner of their Construction.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp100" id="plate09" style="max-width: 62.5em;"> + <img class="w100" src="images/plate09.jpg" alt=""> + <figcaption class="caption"><p class="hanging">IX. Three Diagrams, giving + a Vertical, a Front, and a Side Aspect of the Geometrical Construction of + the Human Head and Countenance, in which the Proportions are determined + by harmonic parts of the Right Angle.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<p><span class="pagenum"><a id="Page_xi"></a>[xi]</span></p> + +<figure class="figcenter illowp50" id="plate10" style="max-width: 35.9375em;"> + <img class="w100" src="images/plate10.jpg" alt=""> + <figcaption class="caption"><p class="hanging">X. Diagram in which the + Symmetrical Proportions of the Human Figure are determined by harmonic + parts of the Right Angle.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate11" style="max-width: 37.5em;"> + <img class="w100" src="images/plate11.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XI. The Contour of the + Human Figure as viewed in Front and in Profile, its Curves being determined + by Ellipses, whose Foci are determined by harmonic parts of the Right Angle.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate12" style="max-width: 35.9375em;"> + <img class="w100" src="images/plate12.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XII. Rectilinear Diagram, + shewing the Proportions of the Portland Vase, as determined by harmonic + parts of the Right Angle, and the outline of its form by an Elliptic Curve + harmonically described.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate13" style="max-width: 43.75em;"> + <img class="w100" src="images/plate13.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XIII. Rectilinear Diagram + of the Proportions and Curvilinear Outline of the form of an ancient + Grecian Vase, the proportions determined by harmonic parts of the Right + Angle, and the melody of the form by Curves of two Ellipses.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp100" id="plate14" style="max-width: 62.5em;"> + <img class="w100" src="images/plate14.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XIV. Rectilinear Diagram + of the Proportions and Curvilinear Outline of the form an ancient Grecian + Vase, the proportions determined by harmonic parts of the Right Angle, + and the melody of the form by an Elliptic Curve.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp100" id="plate15" style="max-width: 62.5em;"> + <img class="w100" src="images/plate15.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XV. Two Diagrams of Etruscan + Vases, the harmony of Proportions and melody of the Contour determined, + respectively, by parts of the Right Angle and an Elliptic Curve.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp100" id="plate16" style="max-width: 62.5em;"> + <img class="w100" src="images/plate16.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XVI. Two Diagrams of Etruscan + Vases, whose harmony of Proportion and melody of Contour are determined as + above.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate17" style="max-width: 40.625em;"> + <img class="w100" src="images/plate17.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XVII. Diagram shewing the + Geometric Construction of an Ornament belonging to the Parthenon at Athens.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate18" style="max-width: 37.5em;"> + <img class="w100" src="images/plate18.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XVIII. Diagram of the + Geometrical Construction of the ancient Grecian Ornament called the + Honeysuckle.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate19" style="max-width: 39.0625em;"> + <img class="w100" src="images/plate19.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XIX. An additional Illustration + of the Contour of the Human Figure, as viewed in Front and in Profile.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate20" style="max-width: 35.9375em;"> + <img class="w100" src="images/plate20.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XX. Diagram shewing the manner + in which the Elliptic Curves are arranged in order to produce an Outline of + the Form of the Human Figure as viewed in Front.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate21" style="max-width: 39.0625em;"> + <img class="w100" src="images/plate21.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XXI. Diagram of a variation + on the Form of the Portland Vase.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate22" style="max-width: 42.1875em;"> + <img class="w100" src="images/plate22.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XXII. Diagram of a second + variation on the Form of the Portland Vase.</p> <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<figure class="figcenter illowp50" id="plate23" style="max-width: 42.1875em;"> + <img class="w100" src="images/plate23.jpg" alt=""> + <figcaption class="caption"><p class="hanging">XXIII. Diagram of a third + variation on the Form of the Portland Vase.</p> + <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption> +</figure> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_1"></a>[1]</span></p> + +<h2 class="nobreak" id="INTRODUCTION">INTRODUCTION.</h2> + +</div> + +<p>Twelve years ago, one of our most eminent philosophers,<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a> +through the medium of the <i>Edinburgh Review</i>,<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> gave the +following account of what was then the state of the fine arts +as connected with science:—“The disposition to introduce +into the intellectual community the principles of free intercourse, +is by no means general; but we are confident that +Art will not sufficiently develop her powers, nor Science attain +her most commanding position, till the practical knowledge of +the one is taken in return for the sound deductions of the +other.... It is in the fine arts, principally, and in the +speculations with which they are associated, that the controlling +power of scientific truth has not exercised its legitimate +influence. In discussing the principles of painting, sculpture, +architecture, and landscape gardening, philosophers have renounced +science as a guide, and even as an auxiliary; and a +school has arisen whose speculations will brook no restraint, +and whose decisions stand in opposition to the strongest convictions +of our senses. That the external world, in its gay +colours and lovely forms, is exhibited to the mind only as a +tinted mass, neither within nor without the eye, neither touching +it nor distant from it—an ubiquitous chaos, which experience<span class="pagenum"><a id="Page_2"></a>[2]</span> +only can analyse and transform into the realities which +compose it; that the beautiful and sublime in nature and in +art derive their power over the mind from association alone, +are among the philosophical doctrines of the present day, +which, if it be safe, it is scarcely prudent to question. Nor +are these opinions the emanations of poetical or ill-trained +minds, which ingenuity has elaborated, and which fashion sustains. +They are conclusions at which most of our distinguished +philosophers have arrived. They have been given to the +world with all the authority of demonstrated truth; and in +proportion to the hold which they have taken of the public +mind, have they operated as a check upon the progress of +knowledge.”</p> + +<p>Such, then, was the state of art as connected with science +twelve years ago. But although the causes which then placed +science and the fine arts at variance have since been gradually +diminishing, yet they are still far from being removed. In +proof of this I may refer to what took place at the annual +distribution of the prizes to the students attending our Scottish +Metropolitan School of Design, in 1854, the pupils in which +amount to upwards of two hundred. The meeting on +that occasion included, besides the pupils, a numerous and +highly respectable assemblage of artists and men of science. +The chairman, a Professor in our University, and editor of +one of the most voluminous works on art, science, and +literature ever produced in this country, after extolling the +general progress of the pupils, so far as evinced by the drawings +exhibited on the occasion, drew the attention of the +meeting to a discovery made by the head master of the architectural +and ornamental department of the school, viz.—That +the ground-plan of the Parthenon at Athens had been constructed +by the application of the <i>mysterious</i> ovoid or <i>Vesica<span class="pagenum"><a id="Page_3"></a>[3]</span> +Piscis</i> of the middle ages, subdivided by the <i>mythic</i> numbers 3 +and 7, and their intermediate odd number 5. Now, it may be +remarked, that the figure thus referred to is not an ovoid, neither +is it in any way of a mysterious nature, being produced simply +by two equal circles cutting each other in their centres. Neither +can it be shewn that the numbers 3 and 7 are in any way more +mythic than other numbers. In fact, the terms <i>mysterious</i> +and <i>mythic</i> so applied, can only be regarded as a remnant of +an ancient terminology, calculated to obscure the simplicity of +scientific truth, and when used by those employed to teach—for +doubtless the chairman only gave the description he received—must +tend to retard the connexion of that truth with +the arts of design. I shall now give a specimen of the manner +in which a knowledge of the philosophy of the fine arts is at +present inculcated upon the public mind generally. In the same +metropolis there has likewise existed for upwards of ten years +a Philosophical Institution of great importance and utility, +whose members amount to nearly three thousand, embracing a +large proportion of the higher classes of society, both in respect +to talent and wealth. At the close of the session of this +Institution, in 1854, a learned and eloquent philologus, who +occasionally lectures upon beauty, was appointed to deliver +the closing address, and touching upon the subject of the +beautiful, he thus concluded—</p> + +<p>“In the worship of the beautiful, and in that alone, we are +inferior to the Greeks. Let us therefore be glad to borrow +from them; not slavishly, but with a wise adaptation—not +exclusively, but with a cunning selection; in art, as in religion, +let us learn to prove all things, and hold fast that which is +good—not merely one thing which is good, but all good +things—Classicalism, Mediævalism, Modernism—let us have +and hold them all in one wide and lusty embrace. Why<span class="pagenum"><a id="Page_4"></a>[4]</span> +should the world of art be more narrow, more monotonous, +than the world of nature? Did God make all the flowers of +one pattern, to please the devotees of the rose or the lily; and +did He make all the hills, with the green folds of their queenly +mantles, all at one slope, to suit the angleometer of the most +mathematical of decorators? I trow not. Let us go and do +likewise.”</p> + +<p>I here take for granted, that what the lecturer meant by +“the worship of the beautiful,” is the production and appreciation +of works of art in which beauty should be a primary +element; and judging from the remains which we possess of +such works as were produced by the ancient Grecians, our inferiority +to them in these respects cannot certainly be denied. +But I must reiterate what I have often before asserted, that it +is not by borrowing from them, however cunning our selection, +or however wise our adaptations, that this inferiority is to be +removed, but by a re-discovery of the science which these +ancient artists must have employed in the production of that +symmetrical beauty and chaste elegance which pervaded all +their works for a period of nearly three hundred years. And +I hold, that as in religion, so in art, there is only one truth, a +grain of which is worth any amount of philological eloquence.</p> + +<p>I also take for granted, that what is meant by Classicalism +in the above quotation, is the ancient Grecian style of art; by +Mediævalism, the semi-barbaric style of the middle ages; and +by Modernism, that chaotic jumble of all previous styles and +fashions of art, which is the peculiar characteristic of our present +school, and which is, doubtless, the result of a system of +education based upon plagiarism and mere imitation. Therefore +a recommendation to embrace with equal fervour “as +good things,” these very opposite artic<i>isms</i> must be a doctrine +as mischievous in art as it would be in religion to recommend<span class="pagenum"><a id="Page_5"></a>[5]</span> +as equally good things the various <i>isms</i> into which it has also +been split in modern times.</p> + +<p>Now, “the world of nature” and “the world of art” have +not that equality of scope which this lecturer on beauty ascribes +to them, but differ very decidedly in that particular. Neither +will it be difficult to shew why “the world of art <i>should</i> be +more narrow than the world of nature”—that it should be +thereby rendered more monotonous does not follow.</p> + +<p>It is well known, that the “world of nature” consists of +productions, including objects of every degree of beauty from +the very lowest to the highest, and calculated to suit not only +the tastes arising from various degrees of intellect, but those +arising from the natural instincts of the lower animals. On +the other hand, “the world of art,” being devoted to the +gratification and improvement of intelligent minds only, is +therefore narrowed in its scope by the exclusion from its productions +of the lower degrees of beauty—even mediocrity is +inadmissible; and we know that the science of the ancient +Greek artists enabled them to excel the highest individual +productions of nature in the perfection of symmetrical beauty. +Consequently, all objects in nature are not equally well adapted +for artistic study, and it therefore requires, on the part of the +artist, besides true genius, much experience and care to enable +him to choose proper subjects from nature; and it is in the +choice of such subjects, and not in plagiarism from the ancients, +that he should select with knowledge and adapt with wisdom. +Hence, all such latitudinarian doctrines as those I have +quoted must act as a check upon the progress of knowledge +in the scientific truth of art. I have observed in some of my +works, that in this country a course had been followed in our +search for the true science of beauty not differing from that +by which the alchymists of the middle ages conducted their<span class="pagenum"><a id="Page_6"></a>[6]</span> +investigations; for our ideas of visible beauty are still undefined, +and our attempts to produce it in the various branches +of art are left dependant, in a great measure, upon chance. +Our schools are conducted without reference to any first principles +or definite laws of beauty, and from the drawing of a +simple architectural moulding to the intricate combinations of +form in the human figure, the pupils trust to their hands +and eyes alone, servilely and mechanically copying the works +of the ancients, instead of being instructed in the unerring +principles of science, upon which the beauty of those works +normally depends. The instruction they receive is imparted +without reference to the judgment or understanding, and they +are thereby led to imitate effects without investigating causes. +Doubtless, men of great genius sometimes arrive at excellence +in the arts of design without a knowledge of the principles +upon which beauty of form is based; but it should be kept +in mind, that true genius includes an intuitive perception of +those principles along with its creative power. It is, therefore, +to the generality of mankind that instruction in the definite +laws of beauty will be of most service, not only in improving +the practice of those who follow the arts professionally, but +in enabling all of us to distinguish the true from the false, +and to exercise a sound and discriminating taste in forming +our judgment upon artistic productions. Æsthetic culture +should consequently supersede servile copying, as the basis +of instruction in our schools of art. Many teachers of drawing, +however, still assert, that, by copying the great works +of the ancients, the mind of the pupil will become imbued +with ideas similar to theirs—that he will imbibe their feeling +for the beautiful, and thereby become inspired with their +genius, and think as they thought. To study carefully and +to investigate the principles which constitute the excellence<span class="pagenum"><a id="Page_7"></a>[7]</span> +of the works of the ancients, is no doubt of much benefit +to the student; but it would be as unreasonable to suppose +that he should become inspired with artistic genius by merely +copying them, as it would be to imagine, that, in literature, +poetic inspiration could be created by making boys +transcribe or repeat the works of the ancient poets. Sir +Joshua Reynolds considered copying as a delusive kind of +industry, and has observed, that “Nature herself is not to be +too closely copied,” asserting that “there are excellences in +the art of painting beyond what is commonly called the +imitation of nature,” and that “a mere copier of nature can +never produce any thing great.” Proclus, an eminent philosopher +and mathematician of the later Platonist school (<span class="allsmcap">A.D.</span> +485), says, that “he who takes for his model such forms as +nature produces, and confines himself to an exact imitation +of these, will never attain to what is perfectly beautiful. For +the works of nature are full of disproportion, and fall very +short of the true standard of beauty.”</p> + +<p>It is remarked by Mr. J. C. Daniel, in the introduction +to his translation of M. Victor Cousin’s “Philosophy of the +Beautiful,” that “the English writers have advocated no +theory which allows the beautiful to be universal and absolute; +nor have they professedly founded their views on original +and ultimate principles. Thus the doctrine of the English +school has for the most part been, that beauty is mutable and +special, and the inference that has been drawn from this +teaching is, that all tastes are equally just, provided that each +man speaks of what he feels.” He then observes, that the +German, and some of the French writers, have thought far +differently; for with them the beautiful is “simple, immutable, +absolute, though its <i>forms</i> are manifold.”</p> + +<p>So far back as the year 1725, the same truths advanced by<span class="pagenum"><a id="Page_8"></a>[8]</span> +the modern German and French writers, and so eloquently +illustrated by M. Cousin, were given to the world by Hutchison +in his “Inquiry into the Original of our Ideas of Beauty +and Virtue.” This author says—“We, by absolute beauty, +understand only that beauty which we perceive in objects, +without comparison to any thing external, of which the object +is supposed an imitation or picture, such as the beauty perceived +from the works of nature, artificial forms, figures, +theorems. Comparative or relative beauty is that which we +perceive in objects commonly considered as imitations or +resemblances of something else.”</p> + +<p>Dr. Reid also, in his “Intellectual Powers of Man,” says—“That +taste, which we may call rational, is that part of our +constitution by which we are made to receive pleasure from +the contemplation of what we conceive to be excellent in its +kind, the pleasure being annexed to this judgment, and regulated +by it. This taste may be true or false, according as it +is founded on a true or false judgment. And if it may be +true or false, it must have first principles.”</p> + +<p>M. Victor Cousin’s opinion upon this subject is, however, +still more conclusive. He observes—“If the idea of the beautiful +is not absolute, like the idea of the true—if it is nothing +more than the expression of individual sentiment, the rebound +of a changing sensation, or the result of each person’s fancy—then +the discussions on the fine arts waver without support, +and will never end. For a theory of the fine arts to be possible, +there must be something absolute in beauty, just as there must +be something absolute in the idea of goodness, to render morals +a possible science.”</p> + +<p>The basis of the science of beauty must thus be founded +upon fixed principles, and when these principles are evolved +with the same care which has characterised the labours of investigators<span class="pagenum"><a id="Page_9"></a>[9]</span> +in natural science, and are applied in the fine arts +as the natural sciences have been in the useful arts, a solid +foundation will be laid, not only for correct practice, but also +for a just appreciation of productions in every branch of the +arts of design.</p> + +<p>We know that the mind receives pleasure through the sense +of hearing, not only from the music of nature, but from the +euphony of prosaic composition, the rhythm of poetic measure, +the artistic composition of successive harmony in simple melody, +and the combined harmony of counterpoint in the more complex +works of that art. We know, also, that the mind is +similarly gratified through the sense of seeing, not only by the +visible beauties of nature, but by those of art, whether in symmetrical +or picturesque compositions of forms, or in harmonious +arrangements of gay or sombre colouring.</p> + +<p>Now, in respect to the first of these modes of sensation, we +know, that from the time of Pythagoras, the fact has been +established, that in whatever manner nature or art may address +the ear, the degree of obedience paid to the fundamental law +of harmony will determine the presence and degree of that +beauty with which a perfect organ can impress a well-constituted +mind; and it is my object in this, as it has been in +former attempts, to prove it consistent with scientific truth, +that that beauty which is addressed to the mind by objects of +nature and art, through the eye, is similarly governed. In +short, to shew that, as in compositions of sounds, there can be +no true beauty in the absence of a strict obedience to this great +law of nature, neither can there exist, in compositions of forms +or colours, that principle of unity in variety which constitutes +beauty, unless such compositions are governed by the same law.</p> + +<p>Although in the songs of birds, the gurgling of brooks, the +sighing of the gentle summer winds, and all the other beautiful<span class="pagenum"><a id="Page_10"></a>[10]</span> +music of nature, no analysis might be able to detect the operation +of any precise system of harmony, yet the pleasure thus +afforded to the human mind we know to arise from its responding +to every development of an obedience to this law. When, +in like manner, we find even in those compositions of forms +and colours which constitute the wildest and most rugged of +Nature’s scenery, a species of picturesque grandeur and beauty +to which the mind as readily responds as to her more mild +and pleasing aspects, or to her sweetest music, we may rest +assured that this beauty is simply another development of, +and response to, the same harmonic law, although the precise +nature of its operation may be too subtle to be easily detected.</p> + +<p>The <i>résumé</i> of the various works I have already published +upon the subject, along with the additional illustrations I am +about to lay before my readers, will, I trust, point out a system +of harmony, which, in formative art, as well as in that of +colouring, will rise superior to the idiosyncracies of different +artists, and bring back to one common type the sensations of +the eye and the ear, thereby improving that knowledge of +the laws of the universe which it is as much the business of +science to combine with the ornamental as with the useful arts.</p> + +<p>In attempting this, however, I beg it may be understood, +that I do not believe any system, based even upon the laws of +nature, capable of forming a royal road to the perfection of +art, or of “mapping the mighty maze of a creative mind.” +At the same time, however, I must continue to reiterate the +fact, that the diffusion of a general knowledge of the science +of visible beauty will afford latent artistic genius just such +a vantage ground as that which the general knowledge of +philology diffused throughout this country affords its latent +literary genius. Although <i>mere learning</i> and <i>true genius</i> differ +as much in the practice of art as they do in the practice of<span class="pagenum"><a id="Page_11"></a>[11]</span> +literature, yet a precise and systematic education in the true +science of beauty must certainly be as useful in promoting +the practice and appreciation of the one, as a precise and systematic +education in the science of philology is in promoting +the practice and appreciation of the other.</p> + +<p>As all beauty is the result of harmony, it will be requisite +here to remark, that harmony is not a simple quality, but, as +Aristotle defines it, “the union of contrary principles having +a ratio to each other.” Harmony thus operates in the production +of all that is beautiful in nature, whether in the combinations, +in the motions, or in the affinities of the elements +of matter.</p> + +<p>The contrary principles to which Aristotle alludes, are those +of uniformity and variety; for, according to the predominance +of the one or the other of these principles, every kind of beauty +is characterised. Hence the difference between symmetrical +and picturesque beauty:—the first allied to the principle of +uniformity, in being based upon precise laws that may be +taught so as to enable men of ordinary capacity to produce it +in their works—the second allied to the principle of variety +often to so great a degree that they yield an obedience to the +precise principles of harmony so subtilely, that they cannot be +detected in its constitution, but are only felt in the response +by which true genius acknowledges their presence. The +generality of mankind may be capable of perceiving this latter +kind of beauty, and of feeling its effects upon the mind, but +men of genius, only, can impart it to works of art, whether +addressed to the eye or the ear. Throughout the sounds, +forms, and colours of nature, these two kinds of beauty are +found not only in distinct developments, but in every degree +of amalgamation. We find in the songs of some birds, such +as those of the chaffinch, thrush, &c., a rhythmical division,<span class="pagenum"><a id="Page_12"></a>[12]</span> +resembling in some measure the symmetrically precise arrangements +of parts which characterises all artistic musical composition; +while in the songs of other birds, and in the other +numerous melodies with which nature charms and soothes the +mind, there is no distinct regularity in the division of their +parts. In the forms of nature, too, we find amongst the +innumerable flowers with which the surface of the earth is so +profusely decorated, an almost endless variety of systematic +arrangements of beautiful figures, often so perfectly symmetrical +in their combination, that the most careful application +of the angleometer could scarcely detect the slightest deviation +from geometrical precision; while, amongst the masses of +foliage by which the forms of many trees are divided and subdivided +into parts, as also amongst the hills and valleys, the +mountains and ravines, which divide the earth’s surface, we +find in every possible variety of aspect the beauty produced +by that irregular species of symmetry which characterises the +picturesque.</p> + +<p>In like manner, we find in wild as well as cultivated flowers +the most symmetrical distributions of colours accompanying +an equally precise species of harmony in their various kinds of +contrasts, often as mathematically regular as the geometric +diagrams by which writers upon colour sometimes illustrate +their works; while in the general colouring of the picturesque +beauties of nature, there is an endless variety in its distributions, +its blendings, and its modifications. In the forms and +colouring of animals, too, the same endless variety of regular +and irregular symmetry is to be found. But the highest +degree of beauty in nature is the result of an equal balance +of uniformity with variety. Of this the human figure is an +example; because, when it is of those proportions universally +acknowledged to be the most perfect, its uniformity bears to<span class="pagenum"><a id="Page_13"></a>[13]</span> +its variety an apparently equal ratio. The harmony of combination +in the normal proportions of its parts, and the beautifully +simple harmony of succession in the normal melody of +its softly undulating outline, are the perfection of symmetrical +beauty, while the innumerable changes upon the contour which +arise from the actions and attitudes occasioned by the various +emotions of the mind, are calculated to produce every species +of picturesque beauty, from the softest and most pleasing to +the grandest and most sublime.</p> + +<p>Amongst the purely picturesque objects of inanimate nature, +I may, as in a former work, instance an ancient oak tree, for +its beauty is enhanced by want of apparent symmetry. Thus, +the more fantastically crooked its branches, and the greater +the dissimilarity and variety it exhibits in its masses of foliage, +the more beautiful it appears to the artist and the amateur; +and, as in the human figure, any attempt to produce variety +in the proportions of its lateral halves would be destructive of +its symmetrical beauty, so in the oak tree any attempt to +produce palpable similarity between any of its opposite sides +would equally deteriorate its picturesque beauty. But picturesque +beauty is not the result of the total absence of +symmetry; for, as none of the irregularly constructed music +of nature could be pleasing to the ear unless there existed in +the arrangement of its notes an obedience, however subtle, to +the great harmonic law of Nature, so neither could any object +be picturesquely beautiful, unless the arrangement of its parts +yields, although it may be obscurely, an obedience to the +same law.</p> + +<p>However symmetrically beautiful any architectural structure +may be, when in a complete and perfect state, it must, as it +proceeds towards ruin, blend the picturesque with the symmetrical; +but the type of its beauty will continue to be the<span class="pagenum"><a id="Page_14"></a>[14]</span> +latter, so long as a sufficient portion of it remains to convey +an idea of its original perfection. It is the same with the +human form and countenance; for age does not destroy their +original beauty, but in both only lessens that which is symmetrical, +while it increases that which is picturesque.</p> + +<p>In short, as a variety of simultaneously produced sounds, +which do not relate to each other agreeably to this law, can +only convey to the mind a feeling of mere noise; so a variety +of forms or colours simultaneously exposed to the eye under +similar circumstances, can only convey to the mind a feeling +of chaotic confusion, or what may be termed <i>visible</i> discord. +As, therefore, the two principles of uniformity and variety, or +similarity and dissimilarity, are in operation in every harmonious +combination of the elements of sound, of form, and of colour, +we must first have recourse to numbers in the abstract before +we can form a proper basis for a universal science of beauty.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_15"></a>[15]</span></p> + +<h2 class="nobreak" id="EVOLVED_FROM_THE_HARMONIC_LAW_OF_NATURE">THE SCIENCE OF BEAUTY EVOLVED FROM THE HARMONIC LAW OF NATURE, AGREEABLY +TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO.</h2> + +</div> + +<p>The scientific principles of beauty appear to have been well +known to the ancient Greeks; and it must have been by the +practical application of that knowledge to the arts of Design, +that that people continued for a period of upwards of three +hundred years to execute, in every department of these arts, +works surpassing in chaste beauty any that had ever before +appeared, and which have not been equalled during the two +thousand years which have since elapsed.</p> + +<p>Æsthetic science, as the science of beauty is now termed, is +based upon that great harmonic law of nature which pervades +and governs the universe. It is in its nature neither absolutely +physical nor absolutely metaphysical, but of an intermediate +nature, assimilating in various degrees, more or less, +to one or other of those opposite kinds of science. It +specially embodies the inherent principles which govern impressions +made upon the mind through the senses of hearing +and seeing. Thus, the æsthetic pleasure derived from +listening to the beautiful in musical composition, and from +contemplating the beautiful in works of formative art, is in +both cases simply a response in the human mind to artistic<span class="pagenum"><a id="Page_16"></a>[16]</span> +developments of the great harmonic law upon which the +science is based.</p> + +<p>Although the eye and the ear are two different senses, +and, consequently, various in their modes of receiving impressions; +yet the sensorium is but one, and the mind by +which these impressions are perceived and appreciated is also +characterised by unity. There appears, likewise, a striking +analogy between the natural constitution of the two kinds of +beauty, which is this, that the more physically æsthetic +elements of the highest works of musical composition are +melody, harmony, and tone, whilst those of the highest works +of formative art are contour, proportion, and colour. The +melody or theme of a musical composition and its harmony +are respectively analogous,—1st, To the outline of an artistic +work of formative art; and 2d, To the proportion which +exists amongst its parts. To the careful investigator these +analogies become identities in their effect upon the mind, like +those of the more metaphysically æsthetic emotions produced +by expression in either of these arts.</p> + +<p>Agreeably to the first analogy, the outline and contour of +an object, suppose that of a building in shade when viewed +against a light background, has a similar effect upon the mind +with that of the simple melody of a musical composition when +addressed to the ear unaccompanied by the combined harmony +of counterpoint. Agreeably to the second analogy, the +various parts into which the surface of the supposed elevation +is divided being simultaneously presented to the eye, will, if +arranged agreeably to the same great law, affect the mind +like that of an equally harmonious arrangement of musical +notes accompanying the supposed melody.</p> + +<p>There is, however, a difference between the construction of +these two organs of sense, viz., that the ear must in a great<span class="pagenum"><a id="Page_17"></a>[17]</span> +degree receive its impressions involuntarily; while the eye, on +the other hand, is provided by nature with the power of +either dwelling upon, or instantly shutting out or withdrawing +itself from an object. The impression of a sound, whether +simple or complex, when made upon the ear, is instantaneously +conveyed to the mind; but when the sound ceases, +the power of observation also ceases. But the eye can dwell +upon objects presented to it so long as they are allowed to +remain pictured on the retina; and the mind has thereby the +power of leisurely examining and comparing them. Hence +the ear guides more as a mere sense, at once and without +reflection; whilst the eye, receiving its impressions gradually, +and part by part, is more directly under the influence of +mental analysis, consequently producing a more metaphysically +æsthetic emotion. Hence, also, the acquired power of +the mind in appreciating impressions made upon it through +the organ of sight under circumstances, such as perspective, +&c., which to those who take a hasty view of the subject +appear impossible.</p> + +<p>Dealing as this science therefore does, alike with the sources +and the resulting principles of beauty, it is scarcely less dependent +on the accuracy of the senses than on the power of +the understanding, inasmuch as the effect which it produces +is as essential a property of objects, as are its laws inherent in +the human mind. It necessarily comprehends a knowledge +of those first principles in art, by which certain combinations +of sounds, forms, and colours produce an effect upon the +mind, connected, in the first instance, with sensation, and in +the second with the reasoning faculty. It is, therefore, not +only the basis of all true practice in art, but of all sound +judgment on questions of artistic criticism, and necessarily +includes those laws whereon a correct taste must be based.<span class="pagenum"><a id="Page_18"></a>[18]</span> +Doubtless many eloquent and ingenious treatises have been +written upon beauty and taste; but in nearly every case, with +no other effect than that of involving the subject in still +greater uncertainty. Even when restricted to the arts of +design, they have failed to exhibit any definite principles +whereby the true may be distinguished from the false, and +some natural and recognised laws of beauty reduced to +demonstration. This may be attributed, in a great degree, +to the neglect of a just discrimination between what is merely +agreeable, or capable of exciting pleasurable sensations, and +what is essentially beautiful; but still more to the confounding +of the operations of the understanding with those of the +imagination. Very slight reflection, however, will suffice to +shew how essentially distinct these two faculties of the mind +are; the former being regulated, in matters of taste, by irrefragable +principles existing in nature, and responded to by +an inherent principle existing in the human mind; while the +latter operates in the production of ideal combinations of its +own creation, altogether independent of any immediate impression +made upon the senses. The beauty of a flower, for +example, or of a dew-drop, depends on certain combinations +of form and colour, manifestly referable to definite and systematic, +though it may be unrecognised, laws; but when +Oberon, in “Midsummer Night’s Dream,” is made to exclaim—</p> + +<div class="poetry-container"> +<div class="poetry"> + <div class="stanza"> + <div class="verse indent0">“And that same dew, which sometimes on the buds</div> + <div class="verse indent0">Was wont to swell, like round and orient pearls,</div> + <div class="verse indent0">Stood now within the pretty floweret’s eyes,</div> + <div class="verse indent0">Like tears that did their own disgrace bewail,”—</div> + </div> +</div> +</div> + +<p class="noindent">the poet introduces a new element of beauty equally legitimate, +yet altogether distinct from, although accompanying +that which constitutes the more precise science of æsthetics<span class="pagenum"><a id="Page_19"></a>[19]</span> +as here defined. The composition of the rhythm is an operation +of the understanding, but the beauty of the poetic +fancy is an operation of the imagination.</p> + +<p>Our physical and mental powers, æsthetically considered, +may therefore be classed under three heads, in their relation +to the fine arts, viz., the receptive, the perceptive, and the +conceptive.</p> + +<p>The senses of hearing and seeing are respectively, in the +degree of their physical power, receptive of impressions made +upon them, and of these impressions the sensorium, in the +degree of its mental power, is perceptive. This perception +enables the mind to form a judgment whereby it appreciates +the nature and quality of the impression originally made on +the receptive organ. The mode of this operation is intuitive, +and the quickness and accuracy with which the nature and +quality of the impression is apprehended, will be in the degree +of the intellectual vigour of the mind by which it is perceived. +Thus we are, by the cultivation of these intuitive faculties, +enabled to decide with accuracy as to harmony or discord, +proportion or deformity, and assign sound reasons for our +judgment in matters of taste. But mental conception is the +intuitive power of constructing original ideas from these +materials; for after the receptive power has acted, the perception +operates in establishing facts, and then the judgment +is formed upon these operations by the reasoning +powers, which lead, in their turn, to the creations of the +imagination.</p> + +<p>The power of forming these creations is the true characteristic +of genius, and determines the point at which art is +placed beyond all determinable canons,—at which, indeed, +æsthetics give place to metaphysics.</p> + +<p>In the science of beauty, therefore, the human mind is the<span class="pagenum"><a id="Page_20"></a>[20]</span> +subject, and the effect of external nature, as well as of works +of art, the object. The external world, and the individual +mind, with all that lies within the scope of its powers, may be +considered as two separate existences, having a distinct relation +to each other. The subject is affected by the object, through +that inherent faculty by which it is enabled to respond to every +development of the all-governing harmonic law of nature; and +the media of communication are the sensorium and its inlets—the +organs of sense.</p> + +<p>This harmonic law of nature was either originally discovered +by that illustrious philosopher Pythagoras, upwards of five +hundred years before Christ, or a knowledge of it obtained by +him about that period, from the Egyptian or Chaldean priests. +For after having been initiated into all the Grecian and barbarian +sacred mysteries, he went to Egypt, where he remained +upwards of twenty years, studying in the colleges of its priests; +and from Egypt he went into the East, and visited the Persian +and Chaldean magi.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></p> + +<p>By the generality of the biographers of Pythagoras, it is +said to be difficult to give a clear idea of his philosophy, as it +is almost certain he never committed it to writing, and that it +has been disfigured by the fantastic dreams and chimeras of +later Pythagoreans. Diogenes Laërtius, however, whose “Lives +of the Philosophers” was supposed to be written about the +end of the second century of our era, says “there are three +volumes extant written by Pythagoras. One on education, +one on politics, and one on natural philosophy.” And adds, +that there were several other books extant, attributed to +Pythagoras, but which were not written by him. Also, in his +“Life of Philolaus,” that Plato wrote to Dion to take care and +purchase the books of Pythagoras.<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a> But whether this great<span class="pagenum"><a id="Page_21"></a>[21]</span> +philosopher committed his discoveries to writing or not, his +doctrines regarding the philosophy of beauty are well-known +to be, that he considered numbers as the essence and the +principle of all things, and attributed to them a real and distinct +existence; so that, in his view, they were the elements +out of which the universe was constructed, and to which it +owed its beauty. Diogenes Laërtius gives the following account +of this law:—“That the monad was the beginning of +everything. From the monad proceeds an indefinite duad, +which is subordinate to the monad as to its cause. That from +the monad and indefinite duad proceeds numbers. That the +part of science to which Pythagoras applied himself above all +others, was arithmetic; and that he taught ‘that from numbers +proceed signs, and from these latter, lines, of which plane +figures consist; that from plane figures are derived solid +bodies; that of all plane figures the most beautiful was the +circle, and of all solid bodies the most beautiful was the +sphere.’ He discovered the numerical relations of sounds on +a single string; and taught that everything owes its existence +and consistency to harmony. In so far as I know, the most +condensed account of all that is known of the Pythagorian +system of numbers is the following:—‘The monad or unity is +that quantity, which, being deprived of all number, remains +fixed. It is the fountain of all number. The duad is imperfect +and passive, and the cause of increase and division. +The triad, composed of the monad and duad, partakes of +the nature of both. The tetrad, tetractys, or quaternion +number is most perfect. The decad, which is the sum of +the four former, comprehends all arithmetical and musical +proportions.’”<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a></p> + +<p>These short quotations, I believe, comprise all that is known,<span class="pagenum"><a id="Page_22"></a>[22]</span> +for certain, of the manner in which Pythagoras systematised +the law of numbers. Yet, from the teachings of this great +philosopher and his disciples, the harmonic law of nature, in +which the fundamental principles of beauty are embodied, +became so generally understood and universally applied in +practice throughout all Greece, that the fragments of their +works, which have reached us through a period of two thousand +years, are still held to be examples of the highest artistic +excellence ever attained by mankind. In the present state of +art, therefore, a knowledge of this law, and of the manner in +which it may again be applied in the production of beauty in +all works of form and colour, must be of singular advantage; +and the object of this work is to assist in the attainment of +such a knowledge.</p> + +<p>It has been remarked, with equal comprehensiveness and +truth, by a writer<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a> in the <i>British and Foreign Medical Review</i>, +that “there is harmony of numbers in all nature—in +the force of gravity—in the planetary movements—in the +laws of heat, light, electricity, and chemical affinity—in the +forms of animals and plants—in the perceptions of the mind. +The direction, indeed, of modern natural and physical science +is towards a generalization which shall express the fundamental +laws of all by one simple numerical ratio. And we +think modern science will soon shew that the mysticism of +Pythagoras was mystical only to the unlettered, and that it +was a system of philosophy founded on the then existing +mathematics, which latter seem to have comprised more of the +philosophy of numbers than our present.” Many years of +careful investigation have convinced me of the truth of this +remark, and of the great advantage derivable from an application +of the Pythagorean system in the arts of design. For so<span class="pagenum"><a id="Page_23"></a>[23]</span> +simple is its nature, that any one of an ordinary capacity of +mind, and having a knowledge of the most simple rules of +arithmetic, may, in a very short period, easily comprehend its +nature, and be able to apply it in practice.</p> + +<p>The elements of the Pythagorean system of harmonic +number, so far as can be gathered from the quotations I +have given above, seem to be simply the indivisible monad +(1); the duad (2), arising from the union of one monad with +another; the triad (3), arising from the union of the monad +with the duad; and the tetrad (4), arising from the union of +one duad with another, which tetrad is considered a perfect +number. From the union of these four elements arises the +decad (10), the number, which, agreeably to the Pythagorean +system, comprehends all arithmetical and harmonic proportions. +If, therefore, we take these elements and unite them +progressively in the following order, we shall find the series +of harmonic numbers (2), (3), (5), and (7), which, with their +multiples, are the complete numerical elements of all harmony, +thus:—</p> + +<table> + <tr> + <td>1</td> + <td>+</td> + <td>1</td> + <td>=</td> + <td>2</td> + </tr> + <tr> + <td>1</td> + <td>+</td> + <td>2</td> + <td>=</td> + <td>3</td> + </tr> + <tr> + <td>2</td> + <td>+</td> + <td>3</td> + <td>=</td> + <td>5</td> + </tr> + <tr> + <td>3</td> + <td>+</td> + <td>4</td> + <td>=</td> + <td>7</td> + </tr> +</table> + +<p>In order to render an extended series of harmonic numbers +useful, it must be divided into scales; and it is a rule in the +formation of these scales, that the first must begin with the +monad (1) and end with the duad (2), the second begin with +the duad (2) and end with the tetrad (4), and that the beginning +and end of all other scales must be continued in the same +arithmetical progression. These primary elements will then +form the foundation of a series of such scales.</p> + +<p><span class="pagenum"><a id="Page_24"></a>[24]</span></p> + +<table> + <tr> + <td class="tdr">I.</td> + <td>(1)</td> + <td></td> + <td></td> + <td></td> + <td></td> + <td></td> + <td></td> + <td></td> + <td>(2)</td> + </tr> + <tr> + <td class="tdr">II.</td> + <td>(2)</td> + <td></td> + <td></td> + <td></td> + <td>(3)</td> + <td></td> + <td></td> + <td></td> + <td>(4)</td> + </tr> + <tr> + <td class="tdr">III.</td> + <td>(4)</td> + <td></td> + <td>(5)</td> + <td></td> + <td>(6)</td> + <td></td> + <td>(7)</td> + <td></td> + <td>(8)</td> + </tr> + <tr> + <td class="tdr">IV.</td> + <td>(8)</td> + <td>(9)</td> + <td>(10)</td> + <td>( )</td> + <td>(12)</td> + <td>( )</td> + <td>(14)</td> + <td>(15)</td> + <td>(16)</td> + </tr> +</table> + +<p>The first of these scales has in (1) and (2) a beginning and +an end; but the second has in (2), (3), and (4) the essential +requisites demanded by Aristotle in every composition, viz., +“a beginning, a middle, and an end;” while the third has not +only these essential requisites, but two intermediate parts (5) +and (7), by which the beginning, the middle, and the end are +united. In the fourth scale, however, the arithmetical progression +is interrupted by the omission of numbers 11 and 13, +which, not being multiples of either (2), (3), (5), or (7), are +inadmissible.</p> + +<p>Such is the nature of the harmonic law which governs the +progressive scales of numbers by the simple multiplication of +the monad.</p> + +<p>I shall now use these numbers as divisors in the formation +of a series of four such scales of parts, which has for its +primary element, instead of the indivisible monad, a quantity +which may be indefinitely divided, but which cannot be added +to or multiplied. Like the monad, however, this quantity is +represented by (1). The following is this series of four scales +of harmonic parts:—</p> + +<table> + <tr> + <td class="tdr">I.</td> + <td>(1)</td> + <td></td> + <td></td> + <td></td> + <td></td> + <td></td> + <td></td> + <td></td> + <td>(¹⁄₂)</td> + </tr> + <tr> + <td class="tdr">II.</td> + <td>(¹⁄₂)</td> + <td></td> + <td></td> + <td></td> + <td>(¹⁄₃)</td> + <td></td> + <td></td> + <td></td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdr">III.</td> + <td>(¹⁄₄)</td> + <td></td> + <td>(¹⁄₅)</td> + <td></td> + <td>(¹⁄₆)</td> + <td></td> + <td>(¹⁄₇)</td> + <td></td> + <td>(¹⁄₈)</td> + </tr> + <tr> + <td class="tdr">IV.</td> + <td>(¹⁄₈)</td> + <td>(¹⁄₉)</td> + <td>(¹⁄₁₀)</td> + <td>( )</td> + <td>(¹⁄₁₂)</td> + <td>( )</td> + <td>(¹⁄₁₄)</td> + <td>(¹⁄₁₅)</td> + <td>(¹⁄₁₆)</td> + </tr> +</table> + +<p>The scales I., II., and III. may now be rendered as complete<span class="pagenum"><a id="Page_25"></a>[25]</span> +as scale IV., simply by multiplying upwards by 2 from +(¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇), and (¹⁄₁₅), thus:—</p> + +<table> + <tr> + <td class="tdr">I.</td> + <td>(1)</td> + <td>(⁸⁄₉)</td> + <td>(⁴⁄₅)</td> + <td></td> + <td>(²⁄₃)</td> + <td></td> + <td>(⁴⁄₇)</td> + <td>(⁸⁄₁₅)</td> + <td>(¹⁄₂)</td> + </tr> + <tr> + <td class="tdr">II.</td> + <td>(¹⁄₂)</td> + <td>(⁴⁄₉)</td> + <td>(²⁄₅)</td> + <td></td> + <td>(¹⁄₃)</td> + <td></td> + <td>(²⁄₇)</td> + <td>(⁴⁄₁₅)</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdr">III.</td> + <td>(¹⁄₄)</td> + <td>(²⁄₉)</td> + <td>(¹⁄₅)</td> + <td></td> + <td>(¹⁄₆)</td> + <td></td> + <td>(¹⁄₇)</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₈)</td> + </tr> + <tr> + <td class="tdr">IV.</td> + <td>(¹⁄₈)</td> + <td>(¹⁄₉)</td> + <td>(¹⁄₁₀)</td> + <td>( )</td> + <td>(¹⁄₁₂)</td> + <td>( )</td> + <td>(¹⁄₁₄)</td> + <td>(¹⁄₁₅)</td> + <td>(¹⁄₁₆)</td> + </tr> +</table> + +<p>We now find between the beginning and the end of scale I. +the quantities (⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅).</p> + +<p>The three first of these quantities we find to be the remainders +of the whole indefinite quantity contained in (1), after +subtracting from it the primary harmonic quantities (¹⁄₉), (¹⁄₅), +and (¹⁄₃); we, however, find also amongst these harmonic +quantities that of (¹⁄₄), which being subtracted from (1) leaves +(³⁄₄), a quantity the most suitable whereby to fill up the hiatus +between (⁴⁄₅) and (²⁄₃) in scale I., which arises from the omission +of (¹⁄₁₁) in scale IV. In like manner we find the two last of +these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest of the +two parts into which 7 and 15 are susceptible of being divided. +Finding the number 5 to be divisible into parts more unequal +than (2) to (3) and less unequal than (4) to (7), (³⁄₅) naturally +fills up the hiatus between these quantities in scale I., which +hiatus arises from the omission of (¹⁄₁₃) in scale IV. Thus:—</p> + +<table> + <tr> + <td class="tdr">I.</td> + <td>(1)</td> + <td>(⁸⁄₉)</td> + <td>(⁴⁄₅)</td> + <td>(³⁄₄)</td> + <td>(²⁄₃)</td> + <td>(³⁄₅)</td> + <td>(⁴⁄₇)</td> + <td>(⁸⁄₁₅)</td> + <td>(¹⁄₂)</td> + </tr> + <tr> + <td class="tdr">II.</td> + <td>(¹⁄₂)</td> + <td>(⁴⁄₉)</td> + <td>(²⁄₅)</td> + <td>( )</td> + <td>(¹⁄₃)</td> + <td>( )</td> + <td>(²⁄₇)</td> + <td>(⁴⁄₁₅)</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdr">III.</td> + <td>(¹⁄₄)</td> + <td>(²⁄₉)</td> + <td>(¹⁄₅)</td> + <td>( )</td> + <td>(¹⁄₆)</td> + <td>( )</td> + <td>(¹⁄₇)</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₈)</td> + </tr> + <tr> + <td class="tdr">IV.</td> + <td>(¹⁄₈)</td> + <td>(¹⁄₉)</td> + <td>(¹⁄₁₀)</td> + <td>( )</td> + <td>(¹⁄₁₂)</td> + <td>( )</td> + <td>(¹⁄₁₄)</td> + <td>(¹⁄₁₅)</td> + <td>(¹⁄₁₆)</td> + </tr> +</table> + +<p>Scale I. being now complete, we have only to divide these<span class="pagenum"><a id="Page_26"></a>[26]</span> +latter quantities by (2) downwards in order to complete the +other three. Thus:—</p> + +<table> + <tr> + <td class="tdr">I.</td> + <td>(1)</td> + <td>(⁸⁄₉)</td> + <td>(⁴⁄₅)</td> + <td>(³⁄₄)</td> + <td>(²⁄₃)</td> + <td>(³⁄₅)</td> + <td>(⁴⁄₇)</td> + <td>(⁸⁄₁₅)</td> + <td>(¹⁄₂)</td> + </tr> + <tr> + <td class="tdr">II.</td> + <td>(¹⁄₂)</td> + <td>(⁴⁄₉)</td> + <td>(²⁄₅)</td> + <td>(³⁄₈)</td> + <td>(¹⁄₃)</td> + <td>(³⁄₁₀)</td> + <td>(²⁄₇)</td> + <td>(⁴⁄₁₅)</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdr">III.</td> + <td>(¹⁄₄)</td> + <td>(²⁄₉)</td> + <td>(¹⁄₅)</td> + <td>(³⁄₁₆)</td> + <td>(¹⁄₆)</td> + <td>(³⁄₂₀)</td> + <td>(¹⁄₇)</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₈)</td> + </tr> + <tr> + <td class="tdr">IV.</td> + <td>(¹⁄₈)</td> + <td>(¹⁄₉)</td> + <td>(¹⁄₁₀)</td> + <td>(³⁄₃₂)</td> + <td>(¹⁄₁₂)</td> + <td>(³⁄₄₀)</td> + <td>(¹⁄₁₄)</td> + <td>(¹⁄₁₅)</td> + <td>(¹⁄₁₆)</td> + </tr> +</table> + +<p>The harmony existing amongst these numbers or quantities +consists of the numerical relations which the parts bear to the +whole and to each other; and the more simple these relations +are, the more perfect is the harmony. The following are +the numerical harmonic ratios which the parts bear to the +whole:—</p> + +<table> + <tr> + <td class="tdr">I.</td> + <td>(1:1)</td> + <td>(8:9)</td> + <td>(4:5)</td> + <td>(3:4)</td> + <td>(2:3)</td> + <td>(3:5)</td> + <td>(4:7)</td> + <td>(8:15)</td> + <td>(1:2)</td> + </tr> + <tr> + <td class="tdr">II.</td> + <td>(1:2)</td> + <td>(4:9)</td> + <td>(2:5)</td> + <td>(3:8)</td> + <td>(1:3)</td> + <td>(3:10)</td> + <td>(2:7)</td> + <td>(4:15)</td> + <td>(1:4)</td> + </tr> + <tr> + <td class="tdr">III.</td> + <td>(1:4)</td> + <td>(2:9)</td> + <td>(1:5)</td> + <td>(3:16)</td> + <td>(1:6)</td> + <td>(3:20)</td> + <td>(1:7)</td> + <td>(2:15)</td> + <td>(1:8)</td> + </tr> + <tr> + <td class="tdr">IV.</td> + <td>(1:8)</td> + <td>(1:9)</td> + <td>(1:10)</td> + <td>(3:32)</td> + <td>(1:12)</td> + <td>(3:40)</td> + <td>(1:14)</td> + <td>(1:15)</td> + <td>(1:16)</td> + </tr> +</table> + +<p>The following are the principal numerical relations which +the parts in each scale bear to one another:—</p> + +<table> + <tr> + <td>(¹⁄₂):(⁴⁄₇)</td> + <td>=</td> + <td>(7:8)</td> + </tr> + <tr> + <td>(⁴⁄₅):(⁸⁄₉)</td> + <td>=</td> + <td>(9:10)</td> + </tr> + <tr> + <td>(²⁄₃):(⁴⁄₅)</td> + <td>=</td> + <td>(5:6)</td> + </tr> + <tr> + <td>(⁴⁄₇):(²⁄₃)</td> + <td>=</td> + <td>(6:7)</td> + </tr> + <tr> + <td>(⁸⁄₁₅):(⁴⁄₇)</td> + <td>=</td> + <td>(14:15)</td> + </tr> + <tr> + <td>(¹⁄₂):(⁸⁄₁₅)</td> + <td>=</td> + <td>(15:16)</td> + </tr> +</table> + +<p>Although these relations are exemplified by parts of scale I., +the same ratios exist between the relative parts of scales II.,<span class="pagenum"><a id="Page_27"></a>[27]</span> +III., and IV., and would exist between the parts of any other +scales that might be added to that series.</p> + +<p>These are the simple elements of the science of that +harmony which pervades the universe, and by which the +various kinds of beauty æsthetically impressed upon the senses +of hearing and seeing are governed.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_28"></a>[28]</span></p> + +<h2 class="nobreak" id="AS_APPLIED_TO_SOUNDS">THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.</h2> + +</div> + +<p>It is well-known that all sounds arise from a peculiar action +of the air, and that this action may be excited by the concussion +resulting from the sudden displacement of a portion of +the atmosphere itself, or by the rapid motions of bodies, or of +confined columns of air; in all which cases, when the motions +are irregular, and the force great, the sound conveyed to the +sensorium is called a noise. But that musical sounds are the +result of equal and regular vibratory motions, either of an +elastic body, or of a column of air in a tube, exciting in the +surrounding atmosphere a regular and equal pulsation. The +ear is the medium of communication between those varieties +of atmospheric action and the seat of consciousness. To describe +fully the beautiful arrangement of the various parts of +this organ, and their adaptation to the purpose of collecting +and conveying these undulatory motions of the atmosphere, is +as much beyond the scope of my present attempt as it is beyond +my anatomical knowledge; but I may simply remark, +that within the ear, and most carefully protected in the construction +of that organ, there is a small cavity containing a +pellucid fluid, in which the minute extremities of the auditory +nerve float; and that this fluid is the last of the media +through which the action producing the sensation of sound is<span class="pagenum"><a id="Page_29"></a>[29]</span> +conveyed to the nerve, and thence to the sensorium, where its +nature becomes perceptible to the mind.</p> + +<p>The impulses which produce musical notes must arrive at a +certain frequency before the ear loses the intervals of silence +between them, and is impressed by only one continued sound; +and as they increase in frequency the sound becomes more +acute upon the ear. The pitch of a musical note is, therefore, +determined by the frequency of these impulses; but, on the +other hand, its intensity or loudness will depend upon the +violence and the quality of its tone on the material employed +in producing them. All such sounds, therefore, whatever be +their loudness or the quality of their tone in which the impulses +occur with the same frequency are in perfect unison, +having the same pitch. Upon this the whole doctrine of +harmonies is founded, and by this the laws of numerical ratio +are found to operate in the production of harmony, and the +theory of music rendered susceptible of exact reasoning.</p> + +<p>The mechanical means by which such sounds can be produced +are extremely various; but, as it is my purpose simply +to shew the nature of harmony of sound as related to, or as +evolving numerical harmonic ratio, I shall confine myself to +the most simple mode of illustration—namely, that of the +monochord. This is an instrument consisting of a string of a +given length stretched between two bridges standing upon a +graduated scale. Suppose this string to be stretched until its +tension is such that, when drawn a little to a side and suddenly +let go, it would vibrate at the rate of 64 vibrations in a +second of time, producing to a certain distance in the surrounding +atmosphere a series of pulsations of the same frequency.</p> + +<p>These pulsations will communicate through the ear a musical +note which would, therefore, be the fundamental note of such<span class="pagenum"><a id="Page_30"></a>[30]</span> +a string. Now, the phenomenon said to be discovered by +Pythagoras is well known to those acquainted with the science +of acoustics, namely, that immediately after the string is thus +put into vibratory motion, it spontaneously divides itself, by +a node, into two equal parts, the vibrations of each of which +occur with a double frequency—namely, 128 in a second +of time, and, consequently, produce a note doubly acute in +pitch, although much weaker as to intensity or loudness; +that it then, while performing these two series of vibrations, +divides itself, by two nodes, into three parts, each of which +vibrates with a frequency triple that of the whole string; +that is, performs 192 vibrations in a second of time, and +produces a note corresponding in increase of acuteness, but +still less intense than the former, and that this continues to +take place in the arithmetical progression of 2, 3, 4, &c. +Simultaneous vibrations, agreeably to the same law of progression, +which, however, seem to admit of no other primes +than the numbers 2, 3, 5, and 7, are easily excited upon any +stringed instrument, even by the lightest possible touch of +any of its strings while in a state of vibratory motion, and the +notes thus produced are distinguished by the name of harmonics. +It follows, then, that one-half of a musical string, +when divided from the whole by the pressure of the finger, or +any other means, and put into vibratory motion, produces a +note doubly acute to that produced by the vibratory motion +of the whole string; the third part, similarly separated, a note +trebly acute; and the same with every part into which any +musical string may be divided. This is the fundamental principle +by which all stringed instruments are made to produce +harmony. It is the same with wind instruments, the sounds +of which are produced by the frequency of the pulsations +occasioned in the surrounding atmosphere by agitating a<span class="pagenum"><a id="Page_31"></a>[31]</span> +column of air confined within a tube as in an organ, in which +the frequency of pulsation becomes greater in an inverse ratio +to the length of the pipes. But the following series of four +successive scales of musical notes will give the reader a more +comprehensive view of the manner in which they follow the +law of numerical ratio just explained than any more lengthened +exposition.</p> + +<p>It is here requisite to mention, that in the construction of +these scales, I have not only adopted the old German or literal +mode of indicating the notes, but have included, as the Germans +do, the note termed by us B flat as B natural, and the +note we term B natural as H. Now, although this arrangement +differs from that followed in the construction of our +modern Diatonic scale, yet as the ratio of 4:7 is more closely +related to that of 1:2 than that of 8:15, and as it is offered +by nature in the spontaneous division of the monochord, I +considered it quite admissible. The figures give the parts of +the monochord which would produce the notes.</p> + +<table> + <tr> + <td rowspan="2" class="tdr valign">I.</td> + <td>{</td> + <td>(1)</td> + <td>(⁸⁄₉)</td> + <td>(⁴⁄₅)</td> + <td>(³⁄₄)</td> + <td>(²⁄₃)</td> + <td>(³⁄₅)</td> + <td>(⁴⁄₇)</td> + <td>(⁸⁄₁₅)</td> + <td>(¹⁄₂)*</td> + </tr> + <tr> + <td>{</td> + <td>C</td> + <td>D</td> + <td>E</td> + <td>F</td> + <td>G</td> + <td>A</td> + <td>B</td> + <td>H</td> + <td><i>c</i></td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">II.</td> + <td>{</td> + <td>(¹⁄₂)*</td> + <td>(⁴⁄₉)</td> + <td>(²⁄₅)</td> + <td>(³⁄₈)</td> + <td>(¹⁄₃)*</td> + <td>(³⁄₁₀)</td> + <td>(²⁄₇)</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₄)*</td> + </tr> + <tr> + <td>{</td> + <td><i>c</i></td> + <td><i>d</i></td> + <td><i>e</i></td> + <td><i>f</i></td> + <td><i>g</i></td> + <td><i>a</i></td> + <td><i>b</i></td> + <td><i>h</i></td> + <td><i>c′</i></td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">III.</td> + <td>{</td> + <td>(¹⁄₄)*</td> + <td>(²⁄₉)</td> + <td>(¹⁄₅)*</td> + <td>(³⁄₁₆)</td> + <td>(¹⁄₆)*</td> + <td>(³⁄₂₀)</td> + <td>(¹⁄₇)*</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₈)*</td> + </tr> + <tr> + <td>{</td> + <td><i>c′</i></td> + <td><i>d′</i></td> + <td><i>e′</i></td> + <td><i>f′</i></td> + <td><i>g′</i></td> + <td><i>a′</i></td> + <td><i>b′</i></td> + <td><i>h′</i></td> + <td><i>c′′</i></td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">IV.</td> + <td>{</td> + <td>(¹⁄₈)*</td> + <td>(¹⁄₉)*</td> + <td>(¹⁄₁₀)*</td> + <td>(³⁄₃₂)</td> + <td>(¹⁄₁₂)*</td> + <td>(³⁄₄₀)</td> + <td>(¹⁄₁₄)*</td> + <td>(¹⁄₁₅)*</td> + <td>(¹⁄₁₆)*</td> + </tr> + <tr> + <td>{</td> + <td><i>c′′</i></td> + <td><i>d′′</i></td> + <td><i>e′′</i></td> + <td><i>f′′</i></td> + <td><i>g′′</i></td> + <td><i>a′′</i></td> + <td><i>b′′</i></td> + <td><i>h′′</i></td> + <td><i>c′′′</i></td> + </tr> +</table> + +<p><span class="pagenum"><a id="Page_32"></a>[32]</span></p> + +<p>The notes marked (*) are the harmonics which naturally +arise from the division of the string by 2, 3, 5, and 7, and the +multiples of these primes.</p> + +<p>Thus every musical sound is composed of a certain number +of parts called pulsations, and these parts must in every scale +relate harmonically to some fundamental number. When +these parts are multiples of the fundamental number by 2, 4, +8, &c., like the pulsations of the sounds indicated by <i>c</i>, <i>c′</i>, <i>c′′</i>, <i>c′′′</i>, +they are called tonic notes, being the most consonant; when +the pulsations are similar multiples by 3, 6, 12, &c., like those +of the sounds indicated by <i>g</i>, <i>g′</i>, <i>g′′</i>, they are called dominant +notes, being the next most consonant; and multiples by 5, +10, &c., like those of the sounds indicated by <i>e</i>, <i>e′</i>, <i>e′′</i>, they are +called mediant notes, from a similar cause. In harmonic combinations +of musical sounds, the æsthetic feeling produced by +their agreement depends upon the relations they bear to each +other with reference to the number of pulsations produced in +a given time by the fundamental note of the scale to which +they belong; and it will be observed, that the more simple +the numerical ratios are amongst the pulsations of any number +of notes simultaneously produced, the more perfect their agreement. +Hence the origin of the common chord or fundamental +concord in the united sounds of the tonic, the dominant, and +the mediant notes, the ratios and coincidences of whose pulsations +2:1, 3:2, 5:4, may thus be exemplified:—</p> + +<figure class="figcenter illowp100" id="illus1" style="max-width: 37.5em;"> + <img class="w100" src="images/illus1.jpg" alt=""> +</figure> + +<p><span class="pagenum"><a id="Page_33"></a>[33]</span></p> + +<p>In musical composition, the law of number also governs its +division into parts, in order to produce upon the ear, along +with the beauty of harmony, that of rhythm. Thus a piece of +music is divided into parts each of which contains a certain +number of other parts called bars, which may be divided +and subdivided into any number of notes, and the performance +of each bar is understood to occupy the same portion +of time, however numerous the notes it contains may be; so +that the music of art is regularly symmetrical in its structure; +while that of nature is in general as irregular and indefinite in +its rhythm as it is in its harmony.</p> + +<p>Thus I have endeavoured briefly to explain the manner in +which the law of numerical ratio operates in that species of +beauty perceived through the ear.</p> + +<p>The definite principles of the art of music founded upon +this law have been for ages so systematised that those who are +instructed in them advance steadily in proportion to their +natural endowments, while those who refuse this instruction +rarely attain to any excellence. In the sister arts of form +and colour, however, a system of tuition, founded upon this +law, is still a desideratum, and a knowledge of the scientific +principles by which these arts are governed is confined to a +very few, and scarcely acknowledged amongst those whose +professions most require their practical application.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_34"></a>[34]</span></p> + +<h2 class="nobreak" id="AS_APPLIED_TO_FORMS">THE SCIENCE OF BEAUTY AS APPLIED TO FORMS.</h2> + +</div> + +<p>It is justly remarked, in the “Illustrated Record of the +New York Exhibition of 1853,” that “it is a question worthy +of consideration how far the mediocrity of the present day is +attributable to an overweening reliance on natural powers +and a neglect of the lights of science;” and there is expressed +a thorough conviction of the fact that, besides the evils of the +copying system, “much genius is now wasted in the acquirement +of rudimentary knowledge in the slow school of practical +experiment, and that the excellence of the ancient Greek school +of design arose from a thoroughly digested canon of form, and +the use of geometrical formulas, which make the works even +of the second and third-rate genius of that period the wonder +and admiration of the present day.”</p> + +<p>That such a canon of form, and that the use of such geometrical +formula, entered into the education, and thereby facilitated +the practice of ancient Greek art, I have in a former +work expressed my firm belief, which is founded on the remarkable +fact, that for a period of nearly three centuries, and +throughout a whole country politically divided into states +often at war with each other, works of sculpture, architecture, +and ornamental design were executed, which surpass in symmetrical<span class="pagenum"><a id="Page_35"></a>[35]</span> +beauty any works of the kind produced during the +two thousand years that have since elapsed. So decided is +this superiority, that the artistic remains of the extraordinary +period I alluded to are, in all civilised nations, still held to be +the most perfect specimens of formative art in the world; and +even when so fragmentary as to be denuded of everything +that can convey an idea of expression, they still excite admiration +and wonder by the purity of their geometric beauty. +And so universal was this excellence, that it seems to have +characterised every production of formative art, however +humble the use to which it was applied.</p> + +<p>The common supposition, that this excellence was the result +of an extraordinary amount of genius existing among the +Greek people during that particular period, is not consistent +with what we know of the progress of mankind in any other +direction, and is, in the present state of art, calculated to +retard its progress, inasmuch as such an idea would suggest +that, instead of making any exertion to arrive at a like general +excellence, the world must wait for it until a similar supposed +psychological phenomenon shall occur.</p> + +<p>But history tends to prove that the long period of universal +artistic excellence throughout Greece could only be the result +of an early inculcation of some well-digested system of correct +elementary principles, by which the ordinary amount of genius +allotted to mankind in every age was properly nurtured and +cultivated; and by which, also, a correct knowledge and appreciation +of art were disseminated amongst the people generally. +Indeed, Müller, in his “Ancient Art and its Remains,” shews +clearly that some certain fixed principles, constituting a science +of proportions, were known in Greece, and that they formed +the basis of all artists’ education and practice during the +period referred to; also, that art began to decline, and its<span class="pagenum"><a id="Page_36"></a>[36]</span> +brightest period to close, as this science fell into disuse, and +the Greek artists, instead of working for an enlightened community, +who understood the nature of the principles which +guided them, were called upon to gratify the impatient whims +of pampered and tyrannical rulers.</p> + +<p>By being instructed in this science of proportion, the Greek +artists were enabled to impart to their representations of the +human figure a mathematically correct species of symmetrical +beauty; whether accompanying the slender and delicately +undulated form of the Venus,—its opposite, the massive and +powerful mould of the Hercules,—or the characteristic representation +of any other deity in the heathen mythology. And +this seems to have been done with equal ease in the minute figure +cut on a precious gem, and in the most colossal statue. The +same instruction likewise enabled the architects of Greece to +institute those varieties of proportions in structure called the +Classical Orders of Architecture; which are so perfect that, since +the science which gave them birth has been buried in oblivion, +classical architecture has been little more than an imitative art; +for all who have since written upon the subject, from Vitruvius +downwards, have arrived at nothing, in so far as the great +elementary principles in question are concerned, beyond the +most vague and unsatisfactory conjectures. For a more clear +understanding of the nature of this application of the Pythagorean +law of number to the harmony of form, it will be requisite +to repeat the fact, that modern science has shewn that +the cause of the impression, produced by external nature upon +the sensorium, called light, may be traced to a molecular or +ethereal action. This action is excited naturally by the sun, +artificially by the combustion of various substances, and sometimes +physically within the eye. Like the atmospheric pulsations +which produce sound, the action which produces light is<span class="pagenum"><a id="Page_37"></a>[37]</span> +capable, within a limited sphere, of being reflected from some +bodies and transmitted through others; and by this reflection +and transmission the visible nature of forms and figures is +communicated to the sensorium. The eye is the medium of +this communication; and its structural beauty, and perfect +adaptation to the purpose of conveying this action, must, like +those of the ear, be left to the anatomist fully to describe. It +is here only necessary to remark, that the optic nerve, like +the auditory nerve, ends in a carefully protected fluid, which +is the last of the media interposed between this peculiarly +subtle action and the nerve upon which it impresses the +presence of the object from which it is reflected or through +which it is transmitted, and the nature of such object made +perceptible to the mind. The eye and the ear are thus, in +one essential point, similar in their physiology, relatively to +the means provided for receiving impressions from external +nature; it is, therefore, but reasonable to believe that the eye +is capable of appreciating the exact subdivision of spaces, just +as the ear is capable of appreciating the exact subdivision of +intervals of time; so that the division of space into exact +numbers of equal parts will æsthetically affect the mind +through the medium of the eye.</p> + +<p>We assume, therefore, that the standard of symmetry, so +estimated, is deduced from the simplest law that could have +been conceived—the law that the angles of direction must all +bear to some fixed angle the same simple relations which the +different notes in a chord of music bear to the fundamental +note; that is, relations expressed arithmetically by the smallest +natural numbers. Thus the eye, being guided in its estimate +by direction rather than by distance, just as the ear is guided +by number of vibrations rather than by magnitude, both it +and the ear convey simplicity and harmony to the mind without<span class="pagenum"><a id="Page_38"></a>[38]</span> +effort, and the mind with equal facility receives and appreciates +them.</p> + +<h3><i>On the Rectilinear Forms and Proportions of Architecture.</i></h3> + +<p>As we are accustomed in all cases to refer direction to the +horizontal and vertical lines, and as the meeting of these +lines makes the right angle, it naturally constitutes the +fundamental angle, by the harmonic division of which a +system of proportion may be established, and the theory of +symmetrical beauty, like that of music, rendered susceptible of +exact reasoning.</p> + +<p>Let therefore the right angle be the fundamental angle, +and let it be divided upon the quadrant of a circle into the +harmonic parts already explained, thus:—</p> + +<table> + <tr> + <th></th> + <th>Right Angle.</th> + <th>Supertonic Angles.</th> + <th>Mediant Angles.</th> + <th>Subdominant Angles.</th> + <th>Dominant Angles.</th> + <th>Submediant Angles.</th> + <th>Subtonic Angles.</th> + <th>Semi-subtonic Angles.</th> + <th>Tonic Angles.</th> + </tr> + <tr> + <td class="tdr">I.</td> + <td>(1)</td> + <td>(⁸⁄₉)</td> + <td>(⁴⁄₅)</td> + <td>(³⁄₄)</td> + <td>(²⁄₃)</td> + <td>(³⁄₅)</td> + <td>(⁴⁄₇)</td> + <td>(⁸⁄₁₅)</td> + <td>(¹⁄₂)</td> + </tr> + <tr> + <td class="tdr">II.</td> + <td>(¹⁄₂)</td> + <td>(⁴⁄₉)</td> + <td>(²⁄₅)</td> + <td>(³⁄₈)</td> + <td>(¹⁄₃)</td> + <td>(³⁄₁₀)</td> + <td>(²⁄₇)</td> + <td>(⁴⁄₁₅)</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdr">III.</td> + <td>(¹⁄₄)</td> + <td>(²⁄₉)</td> + <td>(¹⁄₅)</td> + <td>(³⁄₁₆)</td> + <td>(¹⁄₆)</td> + <td>(³⁄₂₀)</td> + <td>(¹⁄₇)</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₈)</td> + </tr> + <tr> + <td class="tdr">IV.</td> + <td>(¹⁄₈)</td> + <td>(¹⁄₉)</td> + <td>(¹⁄₁₀)</td> + <td>(³⁄₃₂)</td> + <td>(¹⁄₁₂)</td> + <td>(³⁄₄₀)</td> + <td>(¹⁄₁₄)</td> + <td>(¹⁄₁₅)</td> + <td>(¹⁄₁₆)</td> + </tr> +</table> + +<p>In order that the analogy may be kept in view, I have +given to the parts of each of these four scales the appropriate +nomenclature of the notes which form the diatonic scale in +music.</p> + +<p>When a right angled triangle is constructed so that its +two smallest angles are equal, I term it simply the triangle +of (¹⁄₂), because the smaller angles are each one-half of the +right angle. But when the two angles are unequal, the +triangle may be named after the smallest. For instance, when +the smaller angle, which we shall here suppose to be one-third<span class="pagenum"><a id="Page_39"></a>[39]</span> +of the right angle, is made with the vertical line, the triangle +may be called the vertical scalene triangle of (¹⁄₃); and when +made with the horizontal line, the horizontal scalene triangle +of (¹⁄₃). As every rectangle is made up of two of these right +angled triangles, the same terminology may also be applied to +these figures. Thus, the equilateral rectangle or perfect +square is simply the rectangle of (¹⁄₂), being composed of two +similar right angled triangles of (¹⁄₂); and when two vertical +scalene triangles of (¹⁄₃), and of similar dimensions, are united +by their hypothenuses, they form the vertical rectangle of (¹⁄₃), +and in like manner the horizontal triangles of (¹⁄₃) similarly +united would form the horizontal rectangle of (¹⁄₃). As the +isosceles triangle is in like manner composed of two right +angled scalene triangles joined by one of their sides, the same +terminology may be applied to every variety of that figure. +All the angles of the first of the above scales, except that +of (¹⁄₂), give rectangles whose longest sides are in the horizontal +line, while the other three give rectangles whose longest sides +are in the vertical line. I have illustrated in <a href="#plate01">Plate I.</a> the +manner in which this harmonic law acts upon these elementary +rectilinear figures by constructing a series agreeably to the +angles of scales II., III., IV. Throughout this series <i>a b c</i> is +the primary scalene triangle, of which the rectangle <i>a b c e</i> +is composed; <i>d c e</i> the vertical isosceles triangle; and when +the plate is turned, <i>d e a</i> the horizontal isosceles triangle, both +of which are composed of the same primary scalene triangle.</p> + +<div class="sidenote"><a href="#plate01">Plate I.</a></div> + +<p>Thus the most simple elements of symmetry in rectilinear +forms are the three following figures:—</p> + +<ul> +<li>The equilateral rectangle or perfect square,</li> +<li>The oblong rectangle, and</li> +<li>The isosceles triangle.</li> +</ul> + +<p>It has been shewn that in harmonic combinations of<span class="pagenum"><a id="Page_40"></a>[40]</span> +musical sounds, the æsthetic feeling produced by their agreement +depends upon the relation they bear to each other with +reference to the number of pulsations produced in a given +time by the fundamental note of the scale to which they +belong; and that the more simply they relate to each other +in this way the more perfect the harmony, as in the common +chord of the first scale, the relations of whose parts are in the +simple ratios of 2:1, 3:2, and 5:4. It is equally consistent +with this law, that when applied to form in the composition of +an assortment of figures of any kind, their respective proportions +should bear a very simple ratio to each other in order +that a definite and pleasing harmony may be produced +amongst the various parts. Now, this is as effectually done +by forming them upon the harmonic divisions of the right +angle as musical harmony is produced by sounds resulting +from harmonic divisions of a vibratory body.</p> + +<p>Having in previous works<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a> given the requisite illustrations of +this fact in full detail, I shall here confine myself to the most +simple kind, taking for my first example one of the finest +specimens of classical architecture in the world—the front +portico of the Parthenon of Athens.</p> + +<p>The angles which govern the proportions of this beautiful +elevation are the following harmonic parts of the right angle—</p> + +<table> + <tr> + <th>Tonic Angles.</th> + <th>Dominant Angles.</th> + <th>Mediant Angles.</th> + <th>Subtonic Angle.</th> + <th>Supertonic Angles.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(¹⁄₇)</td> + <td>(¹⁄₉)</td> + </tr> + <tr> + <td>(¹⁄₄)</td> + <td>(¹⁄₆)</td> + <td>(¹⁄₁₀)</td> + <td></td> + <td>(¹⁄₁₈)</td> + </tr> + <tr> + <td>(¹⁄₈)</td> + <td></td> + <td></td> + <td></td> + <td></td> + </tr> + <tr> + <td>(¹⁄₁₆)</td> + <td></td> + <td></td> + <td></td> + <td></td> + </tr> +</table> + +<div class="sidenote"><a href="#plate02">Plate II.</a></div> + +<p>In <a href="#plate02">Plate II.</a> I give a diagram of its rectilinear orthography, +which is simply constructed by lines drawn, either horizontally,<span class="pagenum"><a id="Page_41"></a>[41]</span> +vertically, or obliquely, which latter make with either of the +former lines one or other of the harmonic angles in the above +series. For example, the horizontal line AB represents the +length of the base or surface of the upper step of the substructure +of the building. The line AE, which makes an +angle of (¹⁄₅) with the horizontal, determines the height of the +colonnade. The line AD, which makes an angle of (¹⁄₄) with +the horizontal, determines the height of the portico, exclusive +of the pediment. The line AC, which makes an angle of (¹⁄₃) +with the horizontal, determines the height of the portico, including +the pediment. The line GD, which makes an angle +of (¹⁄₇) with the horizontal, determines the form of the pediment. +The lines EZ and LY, which respectively make angles +of (¹⁄₁₆) and (¹⁄₁₈) with the horizontal, determine the breadth of +the architrave, frieze, and cornice. The line <i>v n u</i>, which makes +an angle of (¹⁄₃) with the vertical, determines the breadth of +the triglyphs. The line <i>t d</i>, which makes an angle of (¹⁄₂), +determines the breadth of the metops. The lines <i>c b r f</i>, and +<i>a i</i>, which make each an angle of (¹⁄₆) with the vertical, determine +the width of the five centre intercolumniations. The +line <i>z k</i>, which makes an angle of (¹⁄₈) with the vertical, determines +the width of the two remaining intercolumniations. The +lines <i>c s</i>, <i>q x</i>, and <i>y h</i>, each of which makes an angle of (¹⁄₁₀) +with the vertical, determine the diameters of the three columns +on each side of the centre. The line <i>w l</i>, which makes an +angle of (¹⁄₉) with the vertical, determines the diameter of +the two remaining or corner columns.</p> + +<p>In all this, the length and breadth of the parts are determined +by horizontal and vertical lines, which are necessarily +at right angles with each other, and the position of which are +determined by one or other of the lines making the harmonic +angles above enumerated.</p> + +<p><span class="pagenum"><a id="Page_42"></a>[42]</span></p> + +<p>Now, the lengths and breadths thus so simply determined +by these few angles, have been proved to be correct by their +agreement with the most careful measurements which could +possibly be made of this exquisite specimen of formative art. +These measurements were obtained by the “Society of Dilettanti,” +London, who, expressly for that purpose, sent Mr +F. C. Penrose, a highly educated architect, to Athens, where +he remained for about five months, engaged in the execution +of this interesting commission, the results of which are now +published in a magnificent volume by the Society.<a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a> The +agreement was so striking, that Mr Penrose has been publicly +thanked by an eminent man of science for bearing testimony +to the truth of my theory, who in doing so observes, “The +dimensions which he (Mr Penrose) gives are to me the surest +verification of the theory I could have desired. The minute +discrepancies form that very element of practical incertitude, +both as to execution and direct measurement, which always +prevails in materialising a mathematical calculation made +under such conditions.”<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a></p> + +<p>Although the measurements taken by Mr Penrose are +undeniably correct, as all who examine the great work +just referred to must acknowledge, and although they have +afforded me the best possible means of testing the accuracy of +my theory as applied to the Parthenon, yet the ideas of Mr +Penrose as to the principles they evolve are founded upon the +fallacious doctrine which has so long prevailed, and still prevails, +in the æsthetics of architecture, viz., that harmony may be +imparted by ratios between the lengths and breadths of parts.</p> + +<p>I have taken for my second example an elevation which, +although of smaller dimensions, is no less celebrated for the +beauty of its proportions than the Parthenon itself, viz., the<span class="pagenum"><a id="Page_43"></a>[43]</span> +front portico of the temple of Theseus, which has also been +measured by Mr Penrose.</p> + +<p>The angles which govern the proportions of this elevation +are the following harmonic parts of the right angle:—</p> + +<table> + <tr> + <th>Tonic Angles.</th> + <th>Dominant Angles.</th> + <th>Mediant Angles.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(²⁄₅)</td> + </tr> + <tr> + <td>(¹⁄₄)</td> + <td>(¹⁄₆)</td> + <td>(¹⁄₅)</td> + </tr> + <tr> + <td></td> + <td>(¹⁄₁₂)</td> + <td></td> + </tr> +</table> + +<div class="sidenote"><a href="#plate03">Plate III.</a></div> + +<p>A diagram of the rectilinear orthography of this portico is +given in <a href="#plate03">Plate III.</a> Its construction is similar to that of +the Parthenon in respect to the harmonic parts of the right +angle, and I have therefore only to observe, that the line A E +makes an angle of (¹⁄₄); the line A D an angle of (¹⁄₃); the +line A C an angle of (²⁄₅); the line G D an angle of (¹⁄₆); and +the lines E Z and L Y angles of (¹⁄₁₂) with the horizontal.</p> + +<p>As to the colonnade or vertical part, the line <i>a b</i>, which +determines the three middle intercolumniations, makes an +angle of (¹⁄₅); the line <i>c d</i>, which determines the two outer +intercolumniations, makes an angle of (¹⁄₆); and the line <i>e f</i>, +which determines the lesser diameter of the columns, makes an +angle of (¹⁄₁₂) with the vertical. I need give no further details +here, as my intention is to shew the simplicity of the method +by which this theory may be reduced to practice, and because I +have given in my other works ample details, in full illustration +of the orthography of these two structures, especially the first.<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a></p> + +<p>The foregoing examples being both horizontal rectangular +compositions, the proportions of their principal parts have +necessarily been determined by lines drawn from the extremities +of the base, making angles with the horizontal line, and forming<span class="pagenum"><a id="Page_44"></a>[44]</span> +thereby the diagonals of the various rectangles into which, in +their leading features, they are necessarily resolved. But the +example I am now about to give is of another character, +being a vertical pyramidal composition, and consequently the +proportions of its principal parts are determined by the angles +which the oblique lines make with the vertical line representing +the height of the elevation, and forming a series of isosceles +triangles; for the isosceles triangle is the type of all pyramidal +composition.</p> + +<p>This third example is the east end of Lincoln Cathedral, a +Gothic structure, which is acknowledged to be one of the +finest specimens of that style of architecture existing in this +country.</p> + +<p>The angles which govern the proportions of this elevation +are the following harmonic parts of the right angle:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Subtonic.</th> + <th>Supertonic.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(¹⁄₇)</td> + <td>(²⁄₉)</td> + </tr> + <tr> + <td>(¹⁄₄)</td> + <td>(¹⁄₆)</td> + <td>(¹⁄₁₀)</td> + <td></td> + <td>(¹⁄₉)</td> + </tr> + <tr> + <td></td> + <td>(¹⁄₁₂)</td> + <td></td> + <td></td> + <td></td> + </tr> +</table> + +<div class="sidenote"><a href="#plate04">Plate IV.</a></div> + +<p>In <a href="#plate04">Plate IV.</a> I give a diagram of the vertical, horizontal, and +oblique lines, which compose the orthography of this beautiful +elevation.</p> + +<p>The line A B represents the full height of this structure. +The line A C, which makes an angle of (²⁄₉) with the vertical, +determines the width of the design, the tops of the aisle windows, +and the bases of the pediments on the inner buttresses; +A G, (¹⁄₅) with the vertical, that of the outer buttress; A F, +(¹⁄₉) with the vertical, that of the space between the outer and +inner buttresses and the width of the great centre window; +and A E, (¹⁄₁₂) with vertical, that of both the inner buttresses +and the space between these. A H, which makes (¹⁄₄) with the +vertical, determines the form of the pediment of the centre,<span class="pagenum"><a id="Page_45"></a>[45]</span> +and the full height of the base and surbase. A I, which makes +(¹⁄₃) with the vertical, determines the form of the pediment of +the smaller gables, the base of the pediment on the outer +buttress, the base of the ornamental recess between the outer +and inner buttresses, the spring of the arch of the centre +window, the tops of the pediments on the inner buttresses, +and the spring of the arch of the upper window. A K, which +makes (¹⁄₂), determines the height of the outer buttress; and +A Z, which makes (¹⁄₆) with the horizontal, determines that of +the inner buttresses. For the reasons already given, I need not +here go into further detail.<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">[11]</a> It is, however, worthy of remark +in this place, that notwithstanding the great difference which +exists between the style of composition in this Gothic design, +and in that of the east end of the Parthenon, the harmonic +elements upon which the orthographic beauty of the one +depends, are almost identical with those of the other.</p> + +<h3><i>On the Curvilinear Forms and Proportions of Architecture.</i></h3> + +<p>Each regular rectilinear figure has a curvilinear figure that +exclusively belongs to it, and to which may be applied a corresponding +terminology. For instance, the circle belongs to +the equilateral rectangle; that is, the rectangle of (¹⁄₂), an +ellipse to every other rectangle, and a composite ellipse to +every isosceles triangle. Thus the most simple elements of +beauty in the curvilinear forms of architectural design are the +following three figures:—</p> + +<ul> +<li>The circle,</li> +<li>The ellipse, and</li> +<li>The composite ellipse.</li> +</ul> + +<p>I find it necessary in this place to go into some details<span class="pagenum"><a id="Page_46"></a>[46]</span> +regarding the specific character of the two latter figures, +because the proper mode of describing these beautiful curves, +and their high value in the practice of the architectural +draughtsman and ornamental designer, seem as yet unknown. +In proof of this assertion, I must again refer to Mr Penrose’s +great work published by the “Society of Dilettanti.” At page +52 of that work it is observed, that “by whatever means an +ellipse is to be constructed mechanically, it is a work of time +(if not of absolute difficulty) so to arrange the foci, &c., as to +produce an ellipse of any exact length and breadth which may +be desired.” Now, this is far from being the case, for the +method of arranging the foci of an ellipse of any given length +and breadth is extremely simple, being as follows:—</p> + +<p>Let A B C (figure 1) be the length, and D B E the breadth +of the desired ellipse.</p> + +<figure class="figcenter illowp100" id="figure1" style="max-width: 37.5em;"> + <img class="w100" src="images/figure1.jpg" alt=""> + <figcaption class="caption"><p>Fig. 1.</p></figcaption> +</figure> + +<p>Take A B upon the compasses, and place the point of one +leg upon E and the point of the other upon the line A B, it +will meet it at F, which is one focus: keeping the point of +the one leg upon E, remove the point of the other to the line +B C, and it will meet it at G, which is the other focus. +But, when the proportions of an ellipse are to be imparted<span class="pagenum"><a id="Page_47"></a>[47]</span> +by means of one of the harmonic angles, suppose the angle +of (¹⁄₃), then the following is the process:—</p> + +<p>Let A B C (figure 2) represent the length of the intended +ellipse. Through B draw B <i>e</i> indefinitely, at right angles with +A B C; through C draw the line C <i>f</i> indefinitely, making, with +B C, an angle of (¹⁄₃).</p> + +<p>Take B C upon the compasses, and place the point of one +leg upon D where C f intersects B <i>e</i>, and the point of the +other upon the line A B, it will meet it at F, which is one +focus. Keeping the point of one leg still upon D, remove +the point of the other to the line B C, and it will meet it at +G, which is the other focus.</p> + +<figure class="figcenter illowp100" id="figure2" style="max-width: 34.375em;"> + <img class="w100" src="images/figure2.jpg" alt=""> + <figcaption class="caption"><p>Fig. 2.</p></figcaption> +</figure> + +<p>The foci being in either case thus simply ascertained, the +method of describing the curve on a small scale is equally +simple.</p> + +<div class="sidenote"><a href="#plate05">Plate V.</a></div> + +<p>A pin is fixed into each of the two foci, and another into +the point D. Around these three pins a waxed thread, +flexible but not elastic, is tied, care being taken that the knot +be of a kind that will not slip. The pin at D is now removed, +and a hard black lead pencil introduced within the thread +band. The pencil is then moved around the pins fixed in the +foci, keeping the thread band at a full and equal tension;<span class="pagenum"><a id="Page_48"></a>[48]</span> +thus simply the ellipse is described. When, however, the +governing angle is acute, say less than (¹⁄₆), it is requisite to +adopt a more accurate method of description,<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">[12]</a> as the architectural +examples which follow will shew. But architectural +draughtsmen and ornamental designers would do well to +supply themselves, for ordinary practice, with half a dozen +series of ellipses, varying in the proportions of their axes from +(⁴⁄₉) to (¹⁄₆) of the scale, and the length of their major axes +from 1 to 6 inches. These should be described by the above +simple process, upon very strong drawing paper, and carefully +cut out, the edge of the paper being kept smooth, +and each ellipse having its greater and lesser axes, its foci, +and the hypothenuse of its scalene triangle drawn upon +it. To exemplify this, I give <a href="#plate05">Plate V.</a>, which exhibits the +ellipses of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆), inscribed in their rectangles, +on which <i>a b</i> and <i>c d</i> are respectively the greater and lesser +axes, <i>o o</i> the foci, and <i>d b</i> the angle of each. Such a series +of these beautiful figures would be found particularly useful +in drawing the mouldings of Grecian architecture; for, to +describe the curvilinear contour of such mouldings from +single points, as has been done with those which embellish +even our most pretending attempts at the restoration of +that classical style of architecture, is to give the resemblance +of an external form without the harmony which constitutes +its real beauty.</p> + +<p>Mr Penrose, owing to the supposed difficulty regarding the +description of ellipses just alluded to, endeavours to shew +that the curves of all the mouldings throughout the Parthenon<span class="pagenum"><a id="Page_49"></a>[49]</span> +were either parabolic or hyperbolic; but I believe such curves +can have no connexion with the elementary forms of architecture, +for they are curves which represent motion, and do +not, by continued production, form closed figures.</p> + +<p>But I have shewn, in a former work,<a id="FNanchor_13" href="#Footnote_13" class="fnanchor">[13]</a> that the contours of +these mouldings are composed of curves of the composite +ellipse,—a figure which I so name because it is composed simply +of arcs of various ellipses harmonically flowing into each +other. The composite ellipse, when drawn systematically upon +the isosceles triangle, resembles closely parabolic and hyperbolic +curves—only differing from these inasmuch as it possesses +the essential quality of circumscribing harmonically one of the +elementary rectilinear figures employed in architecture, while +those of the parabola and hyperbola, as I have just observed, +are merely curves of motion, and, consequently, never can harmonically +circumscribe or be resolved into any regular figure.</p> + +<p>The composite ellipse may be thus described.</p> + +<div class="sidenote"><a href="#plate06">Plate VI.</a></div> + +<p>Let A B C (<a href="#plate06">Plate VI.</a>) be a vertical isosceles triangle of +(¹⁄₆), bisect A B in D, and through D draw indefinitely D <i>f</i> +perpendicular to A B, and through B draw indefinitely B <i>g</i>, +making the angle D B <i>g</i> (¹⁄₈), D <i>f</i> and B <i>g</i> intersecting each +other in M. Take B D and D M as semi-axes of an ellipse, +the foci of which will be at <i>p</i> and <i>q</i>, in each of these, and in +each of the foci <i>h t</i> and <i>k r</i> in the lines A C and B C, fix +a pin, and one also in the point M, tie a thread around these +pins, withdraw the pin from M, and trace the composite ellipse +in the manner already described with respect to the simple +ellipse.</p> + +<p>In some of my earlier works I described this figure by taking +the angles of the isosceles triangle as foci; but the above +method is much more correct. As the elementary angle of<span class="pagenum"><a id="Page_50"></a>[50]</span> +the triangle is (¹⁄₆), and that of the elliptic curve described +around it (¹⁄₈), I call it the composite ellipse of (¹⁄₆) and (¹⁄₈), +their harmonic ratio being 4:3; and so on of all others, according +to the difference that may thus exist between the elementary +angles.</p> + +<p>The visible curves which soften and beautify the melody of +the outline of the front of the Parthenon, as given in Mr +Penrose’s great work, I have carefully analysed, and have +found them in as perfect agreement with this system, as its +rectilinear harmony has been shewn to be. This I demonstrated +in the work just referred to<a id="FNanchor_14" href="#Footnote_14" class="fnanchor">[14]</a> by a series of twelve plates, +shewing that the entasis of the columns (a subject upon which +there has been much speculation) is simply an arc of an ellipse +of (¹⁄₄₈), whose greater axis makes with the vertical an angle of +(¹⁄₆₄); or simply, the form of one of these columns is the frustrum +of an elliptic-sided or prolate-spheroidal cone, whose +section is a composite ellipse of (¹⁄₄₈) and (¹⁄₆₄), the harmonic +ratio of these two angles being 4:3, the same as that of the +angles of the composite ellipse just exemplified.</p> + +<div class="sidenote"><a href="#plate07">Plate VII.</a></div> + +<div class="sidenote"><a href="#plate08">Plate VIII.</a></div> + +<p>In <a href="#plate07">Plate VII.</a> is represented the section of such a cone, of +which A B C is the isosceles triangle of (¹⁄₄₈), and B D and +D M the semi-axes of an ellipse of (¹⁄₆₄). M N and O P are +the entases of the column, and <i>d e f</i> the normal construction of +the capital. All these are fully illustrated in the work above +referred to,<a id="FNanchor_15" href="#Footnote_15" class="fnanchor">[15]</a> in which I have also shewn that the curve of the +neck of the column is that of an ellipse of (¹⁄₆); the curve of +the capital or echinus, that of an ellipse of (¹⁄₁₄); the curve of +the moulding under the cymatium of the pediment, that of an +ellipse of (¹⁄₃); and the curve of the bed-moulding of the cornice +of the pediment, that of an ellipse of (¹⁄₃). The curve of the +cavetto of the soffit of the corona is composed of ellipses of (¹⁄₆)<span class="pagenum"><a id="Page_51"></a>[51]</span> +and (¹⁄₁₄); the curve of the cymatium which surmounts the +corona, is that of an ellipse of (¹⁄₃); the curve of the moulding +of the capital of the antæ of the posticum, that of an +ellipse of (¹⁄₃); the curves of the lower moulding of the same +capital are composed of those of an ellipse of (¹⁄₃) and of the +circle (¹⁄₂); the curve of the moulding which is placed between +the two latter is that of an ellipse of (¹⁄₃); the curve of the upper +moulding of the band under the beams of the ceiling of the +peristyle, that of an ellipse of (¹⁄₃); the curve of the lower +moulding of the same band, that of an ellipse of (¹⁄₄); and the +curves of the moulding at the bottom of the small step or +podium between the columns, are those of the circle (¹⁄₂) and of +an ellipse of (¹⁄₃). I have also shewn the curve of the fluting +of the columns to be that of (¹⁄₁₄). The greater axis of each of +these ellipses, when not in the vertical or horizontal lines, +makes an harmonic angle with one or other of them. In +<a href="#plate08">Plate VIII.</a>, sections of the two last-named mouldings are +represented full size, which will give the reader an idea of the +simple manner in which the ellipses are employed in the +production of those harmonic curves.</p> + +<p>Thus we find that the system here adopted for applying +this law of nature to the production of beauty in the abstract +forms employed in architectural composition, so far from +involving us in anything complicated, is characterised by +extreme simplicity.</p> + +<p>In concluding this part of my treatise, I may here repeat +what I have advanced in a late work,<a id="FNanchor_16" href="#Footnote_16" class="fnanchor">[16]</a> viz., my conviction of +the probability that a system of applying this law of nature in +architectural construction was the only great practical secret +of the Freemasons, all their other secrets being connected, not +with their art, but with the social constitution of their society.<span class="pagenum"><a id="Page_52"></a>[52]</span> +This valuable secret, however, seems to have been lost, as its +practical application fell into disuse; but, as that ancient +society consisted of speculative as well as practical masons, +the secrets connected with their social union have still been +preserved, along with the excellent laws by which the brotherhood +is governed. It can scarcely be doubted that there was +some such practically useful secret amongst the Freemasons +or early Gothic architects; for we find in all the venerable +remains of their art which exist in this country, symmetrical +elegance of form pervading the general design, harmonious proportion +amongst all the parts, beautiful geometrical arrangements +throughout all the tracery, as well as in the elegantly +symmetrised foliated decorations which belong to that style of +architecture. But it is at the same time worthy of remark, +that whenever they diverged from architecture to sculpture +and painting, and attempted to represent the human figure, +or even any of the lower animals, their productions are such +as to convince us that in this country these arts were in a +very degraded state of barbarism—the figures are often much +disproportioned in their parts and distorted in their attitudes, +while their representations of animals and chimeras are whimsically +absurd. It would, therefore, appear that architecture, +as a fine art, must have been preserved by some peculiar +influence from partaking of the barbarism so apparent in the +sister arts of that period. Although its practical secrets have +been long lost, the Freemasons of the present day trace the +original possession of them to Moses, who, they say, “modelled +masonry into a perfect system, and circumscribed its mysteries +by <i>land-marks</i> significant and unalterable.” Now, as Moses +received his education in Egypt, where Pythagoras is said to +have acquired his first knowledge of the harmonic law of +numbers, it is highly probable that this perfect system of the<span class="pagenum"><a id="Page_53"></a>[53]</span> +great Jewish legislator was based upon the same law of nature +which constituted the foundation of the Pythagorean philosophy, +and ultimately led to that excellence in art which is +still the admiration of the world.</p> + +<p>Pythagoras, it would appear, formed a system much more +perfect and comprehensive than that practised by the Freemasons +in the middle ages of Christianity; for it was as +applicable to sculpture, painting, and music, as it was to +architecture. This perfection in architecture is strikingly +exemplified in the Parthenon, as compared with the Gothic +structures of the middle ages; for it will be found that the +whole six elementary figures I have enumerated as belonging +to architecture, are required in completing the orthographic +beauty of that noble structure. And amongst these, none +conduce more to that beauty than the simple and composite +ellipses. Now, in the architecture of the best periods of +Gothic, or, indeed, in that of any after period (Roman architecture +included), these beautiful curves seem to have been +ignored, and that of the circle alone employed.</p> + +<p>Be those matters as they may, however, the great law of +numerical harmonic ratio remains unalterable, and a proper +application of it in the science of art will never fail to be as +productive of effect, as its operation in nature is universal, certain, +and continual.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_54"></a>[54]</span></p> + +<h2 class="nobreak" id="AS_DEVELOPED_IN_THE_HUMAN_HEAD_AND_COUNTENANCE">THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE HUMAN HEAD AND COUNTENANCE.</h2> + +</div> + +<p>The most remarkable characteristics of the human head and +countenance are the globular form of the cranium, united as +it is with the prolate spheroidal form produced by the parts +which constitute the face, and the approximation of the profile +to the vertical; for in none of the lower animals does the skull +present so near a resemblance to a combination of these +geometric forms, nor the plane of the face to this direction. +We also find that although these peculiar characteristics are +variously modified among the numerous races of mankind, yet +one law appears to govern the beauty of the whole. The +highest and most cultivated of these races, however, present +only an approximation to the perfect development of those +distinguishing marks of humanity; and therefore the beauty +of form and proportion which in nature characterises the human +head and countenance, exhibits only a partial development of +the harmonic law of visible beauty. On the other hand, we +find that, in their sculpture, the ancient Greeks surpassed +ordinary nature, and produced in their beau ideal a species of +beauty free from the imperfections and peculiarities that constitute +the individuality by which the countenances of men are +distinguished from each other. It may be requisite here to<span class="pagenum"><a id="Page_55"></a>[55]</span> +remark, that this species of beauty is independent of the more +intellectual quality of expression. For as Sir Charles Bell has +said, “Beauty of countenance may be defined in words, as well +as demonstrated in art. A face may be beautiful in sleep, and +a statue without expression may be highly beautiful. But it +will be said there is expression in the sleeping figure or in the +statue. Is it not rather that we see in these the capacity for +expression?—that our minds are active in imagining what may +be the motions of these features when awake or animated? +Thus, we speak of an expressive face before we have seen a +movement grave or cheerful, or any indication in the features +of what prevails in the heart.”</p> + +<p>This capacity for expression certainly enhances our admiration +of the human countenance; but it is more a concomitant +of the primary cause of its beauty than the cause itself. This +cause rests on that simple and secure basis—the harmonic law +of nature; for the nearer the countenance approximates to +an harmonious combination of the most perfect figures in +geometry, or rather the more its general form and the relation +of its individual parts are arranged in obedience to that law, +the higher its degree of beauty, and the greater its capacity +for the expression of the passions.</p> + +<p>Various attempts have been made to define geometrically +the difference between the ordinary and the ideal beauty of the +human head and countenance, the most prominent of which is +that of Camper. He traced, upon a profile of the skull, a line +in a horizontal direction, passing through the foramen of the +ear and the exterior margin of the sockets of the front teeth of +the upper jaw, upon which he raised an oblique line, tangential +to the margin of these sockets, and to the most prominent part +of the forehead. Agreeably to the obliquity of this line, he +determined the relative proportion of the areas occupied by the<span class="pagenum"><a id="Page_56"></a>[56]</span> +brain and by the face, and hence inferred the degree of intellect. +When he applied this measurement to the heads of the +antique statues, he found the angle much greater than in +ordinary nature; but that this simple fact afforded no rule for +the reproduction of the ideal beauty of ancient Greek art, is +very evident from the heads and countenances by which his +treatise is illustrated. Sir Charles Bell justly remarks, that +although, by Camper’s method, the forehead may be thrown +forward, yet, while the features of common nature are preserved, +we refuse to acknowledge a similarity to the beautiful forms of +the antique marbles. “It is true,” he says, “that, by advancing +the forehead, it is raised, the face is shortened, and the eye +brought to the centre of the head. But with all this, there is +much wanting—that which measurement, or a mere line, will +not shew us.”—“The truth is, that we are more moved by the +features than by the form of the whole head. Unless there be +a conformity in every feature to the general shape of the head, +throwing the forehead forward on the face produces deformity; +and the question returns with full force—How is it that we are +led to concede that the antique head of the Apollo, or of the +Jupiter, is beautiful when the facial line makes a hundred degrees +with the horizontal line? In other words—How do we +admit that to be beautiful which is not natural? Simply for +the same reason that, if we discover a broken portion of an +antique, a nose, or a chin of marble, we can say, without deliberation—This +must have belonged to a work of antiquity; +which proves that the character is distinguishable in every part—in +each feature, as well as in the whole head.”</p> + +<p>Dr Oken says upon this subject:<a id="FNanchor_17" href="#Footnote_17" class="fnanchor">[17]</a>—“The face is beautiful +whose nose is parallel to the spine. No human face has grown<span class="pagenum"><a id="Page_57"></a>[57]</span> +into this estate; but every nose makes an acute angle with the +spine. The facial angle is, as is well known, 80°. What, as +yet, no man has remarked, and what is not to be remarked, +either, without our view of the cranial signification, the old +masters have felt through inspiration. They have not only +made the facial angle a right angle, but have even stepped +beyond this—the Romans going up to 96°, the Greeks even to +100°. Whence comes it that this unnatural face of the Grecian +works of art is still more beautiful than that of the Roman, +when the latter comes nearer to nature? The reason thereof +resides in the fact of the Grecian artistic face representing +nature’s design more than that of the Roman; for, in the +former, the nose is placed quite perpendicular, or parallel to +the spinal cord, and thus returns whither it has been derived.”</p> + +<p>Other various and conflicting opinions upon this subject have +been given to the world; but we find that the principle from +which arose the ideal beauty of the head and countenance, as +represented in works of ancient Greek art, is still a matter of +dispute. When, however, we examine carefully a fine specimen, +we find its beauty and grandeur to depend more upon the +degree of harmony amongst its parts, as to their relative proportions +and mode of arrangement, than upon their excellence +taken individually. It is, therefore, clear that those (and they +are many) who attribute the beauty of ancient Greek sculpture +merely to a selection of parts from various models, must be in +error. No assemblage of parts from ordinary nature could +have produced its principal characteristic, the excess in the +angle of the facial line, much less could it have led to that +exquisite harmony of parts by which it is so eminently distinguished; +neither can we reasonably agree with Dr Oken +and others, who assert that it was produced by an exclusive<span class="pagenum"><a id="Page_58"></a>[58]</span> +degree of the inspiration of genius amongst the Greek people +during a certain period.</p> + +<p>That the inspiration of genius, combined with a careful +study of nature, were essential elements in the production of +the great works which have been handed down to us, no one +will deny; but these elements have existed in all ages, whilst +the ideal head belongs exclusively to the Greeks during the +period in which the schools of Pythagoras and Plato were +open. Is it not, therefore, reasonable to suppose, that, besides +genius and the study of nature, another element was employed +in the production of this excellence, and that this +element arose from the precise mathematical doctrines taught +in the schools of these philosophers?</p> + +<p>An application of the great harmonic law seems to prove +that there is no object in nature in which the science of +beauty is more clearly developed than in the human head +and countenance, nor to the representations of which the +same science is more easily applied; and it is to the mode in +which this is done that the varieties of sex and character +may be imparted to works of art. Having gone into full +detail, and given ample illustrations in a former work,<a id="FNanchor_18" href="#Footnote_18" class="fnanchor">[18]</a> it is +unnecessary for me to enter upon that part of the subject in +this <i>résumé</i>; but only to shew the typical structure of beauty +by which this noble work of creation is distinguished.</p> + +<p>The angles which govern the form and proportions of the +human head and countenance are, with the right angle, a +series of seven, which, from the simplicity of their ratios to +each other, are calculated to produce the most perfect concord. +It consists of the right angle and its following parts—</p> + +<p><span class="pagenum"><a id="Page_59"></a>[59]</span></p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Subtonic.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(¹⁄₇)</td> + </tr> + <tr> + <td>(¹⁄₄)</td> + <td>(¹⁄₆)</td> + <td></td> + <td></td> + </tr> +</table> + +<p>These angles, and the figures which belong to them, are +thus arranged:—</p> + +<div class="sidenote"><a href="#plate09">Plate IX.</a></div> + +<p>The vertical line A B (<a href="#plate09">Plate IX.</a> fig. 2) represents the +full length of the head and face. Taking this line as the +greater axis of an ellipse of (¹⁄₃), such an ellipse is described +around it. Through A the lines A G, A K, A L, A M, and +A N, are drawn on each side of the line A B, making, with +the vertical, respectively the angles of (¹⁄₃), (¹⁄₄), (¹⁄₅), (¹⁄₆), and +(¹⁄₇). Through the points G, K, L, M, and N, where these +straight lines meet the curved line of the ellipse, horizontal +lines are drawn by which the following isosceles triangles are +formed, A G G, A K K, A L L, A M M, and A N N. From +the centre X of the equilateral triangle A G G the curvilinear +figure of (¹⁄₂), viz., the circle, is described circumscribing that +triangle.</p> + +<p>The curvilinear plane figures of (¹⁄₂) and (¹⁄₃), respectively, +represent the solid bodies of which they are sections, viz., a +sphere and a prolate spheroid. These bodies, from the manner +in which they are here placed, are partially amalgamated, +as shewn in figures 1 and 3 of the same plate, thus representing +the form of the human head and countenance, both in +their external appearance and osseous structure, more correctly +than they could be represented by any other geometrical +figures. Thus, the angles of (¹⁄₂) and (¹⁄₃) determine the +typical form.</p> + +<p>From each of the points <i>u</i> and <i>n</i>, where A M cuts G G on +both sides of A B, a circle is described through the points <i>p</i> +and <i>q</i>, where A K cuts G G on both sides of A B, and with<span class="pagenum"><a id="Page_60"></a>[60]</span> +the same radius a circle is described from the point <i>a</i>, where +K K cuts A B.</p> + +<p>The circles <i>u</i> and <i>n</i> determine the position and size of the +eyeballs, and the circle <i>a</i> the width of the nose, as also the +horizontal width of the mouth.</p> + +<p>The lines G G and K K also determine the length of the +joinings of the ear to the head. The lines L L and M M determine +the vertical width of the mouth and lips when at +perfect repose, and the line N N the superior edge of the chin. +Thus simply are the features arranged and proportioned on +the facial surface.</p> + +<p>It must, however, be borne in mind, that in treating simply +of the æsthetic beauty of the human head and countenance, +we have only to do with the external appearance. In +this research, therefore, the system of Dr Camper, Dr Owen, +and others, whose investigations were more of a physiological +than an æsthetic character, can be of little service; because, +according to that system, the facial angle is determined by +drawing a line tangential to the exterior margin of the sockets +of the front teeth of the upper jaw, and the most prominent +part of the forehead. Now, as these sockets are, when the +skull is naturally clothed, and the features in repose, entirely +concealed by the upper lip, we must take the prominent part +of it, instead of the sockets under it, in order to determine +properly this distinguishing mark of humanity. And I believe +it will be found, that when the head is properly poised, +the nearer the angle which this line makes with the horizontal +approaches 90°, the more symmetrically beautiful will be the +general arrangement of the parts (see line <i>y z</i>, figure 3, +<a href="#plate09">Plate IX.</a>).</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_61"></a>[61]</span></p> + +<h2 class="nobreak" id="AS_DEVELOPED_IN_THE_FORM_OF_THE_HUMAN_FIGURE">THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE.</h2> + +</div> + +<p>The manner in which this science is developed in the symmetrical +proportions of the entire human figure, is as remarkable +for its simplicity as it has been shewn to be in those of the +head and countenance. Having gone into very full details, +and given ample illustration in two former works<a id="FNanchor_19" href="#Footnote_19" class="fnanchor">[19]</a> upon this +subject, I may here confine myself to the illustration of one +description of figure, and to a reiteration of some facts stated +in these works. These facts are, <i>1st</i>, That on a given line the +human figure is developed, as to its principal points, entirely +by lines drawn either from the extremities of this line, or +from some obvious or determined localities. <i>2d</i>, That the +angles which these lines make with the given line, are all +simple sub-multiples of some given fundamental angle, or +bear to it a proportion expressible under the most simple relations, +such as those which constitute the scale of music. +<i>3d</i>, That the contour is resolved into a series of ellipses of +the same simple angles. And, <i>4th</i>, That these ellipses, like +the lines, are inclined to the first given line by angles which +are simple sub-multiples of the given fundamental angle.<span class="pagenum"><a id="Page_62"></a>[62]</span> +From which four facts, and agreeably to the hypothesis I +have adopted, it results as a natural consequence that the +only effort which the mind exercises through the eye, in +order to put itself in possession of the data for forming its +judgment, is this, that it compares the angles about a point, +and thereby appreciates the simplicity of their relations. In +selecting the prominent features of a figure, the eye is not +seeking to compare their relative distances—it is occupied +solely with their relative positions. In tracing the contour, +in like manner, it is not left in vague uncertainty as to what +is the curve which is presented to it; unconsciously it feels +the complete ellipse developed before it; and if that ellipse +and its position are both formed by angles of the same +simple relative value as those which aided its determination +of the positions of the prominent features, it is satisfied, and +finds the symmetry perfect.</p> + +<p>Müller, and other investigators into the archæology of art, +refer to the great difficulty which exists in discovering the +principles which the ancients followed in regard to the proportions +of the human figure, from the different sexes and +characters to which they require to be applied. But in the +system thus founded upon the harmonic law of nature, no +such difficulty is felt, for it is as applicable to the massive +proportions which characterise the ancient representations of +the Hercules, as to the delicate and perfectly symmetrical +beauty of the Venus. This change is effected simply by an +increase in the fundamental angle. For instance, in the +construction of a figure of the exact proportions of the Venus, +the right angle is adopted. But in the construction of a +figure of the massive proportions of the Hercules, it is requisite +to adopt an angle which bears to the right angle the ratio +of 6:5. The adoption of this angle I have shewn in another<span class="pagenum"><a id="Page_63"></a>[63]</span> +work<a id="FNanchor_20" href="#Footnote_20" class="fnanchor">[20]</a> to produce in the Hercules those proportions which +are so characteristic of physical power. The ellipses which +govern the outline, being also formed upon the same larger +class of angles, give the contour of the muscles a more massive +character. In comparing the male and female forms thus +geometrically constructed, it will be found that that of the +female is more harmoniously symmetrical, because the right +angle is the fundamental angle for the trunk and the limbs +as well as for the head and countenance; while in that of the +male, the right angle is the fundamental angle for the head +only. It may also be observed, that, from the greater proportional +width of the pelvis of the female, the centres +of that motion which the heads of the thigh bones perform +in the cotyloid cavities, and the centres of that still +more extensive range of motion which the arm is capable +of performing at the shoulder joints, are nearly in the same +line which determines the central motion of the vertebral +column, while those of the male are not; consequently all +the motions of the female are more graceful than those of +the male.</p> + +<p>This difference between the fundamental angles, which +impart to the human figure, on the one hand, the beauty of +feminine proportion and contour, and on the other, the grandeur +of masculine strength, being in the ratio of 5:6, allows +ample latitude for those intermediate classes of proportions +which the ancients imparted to their various other deities in +which these two qualities were blended. I therefore confine +myself to an illustration of the external contour of the form, +and the relative proportions of all the parts of a female figure, +such as those of the statues of the Venus of Melos and Venus +of Medici.</p> + +<p><span class="pagenum"><a id="Page_64"></a>[64]</span></p> + +<p>The angles which govern the form and proportions of such +a figure are, with the right angle, a series of twelve, as +follows:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Subtonic.</th> + <th>Supertonic.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(¹⁄₇)</td> + <td>(¹⁄₉)</td> + </tr> + <tr> + <td>(¹⁄₄)</td> + <td>(¹⁄₆)</td> + <td>(¹⁄₁₀)</td> + <td>(¹⁄₁₄)</td> + <td></td> + </tr> + <tr> + <td>(¹⁄₈)</td> + <td>(¹⁄₁₂)</td> + <td></td> + <td></td> + <td></td> + </tr> +</table> + +<p>These angles are employed in the construction of a diagram, +which determines the proportions of the parts throughout the +whole figure. Thus:—</p> + +<div class="sidenote"><a href="#plate10">Plate X.</a></div> + +<p>Let the line A B (fig. 1, <a href="#plate10">plate X.</a>) represent the height of +the figure to be constructed. At the point A, make the +angles of C A D (¹⁄₃), F A G (¹⁄₄), H A I (¹⁄₅), K A L (¹⁄₆), and +M A N (¹⁄₇). At the point B, make the angles K B L (¹⁄₈), +U B A (¹⁄₁₂), and O B A (¹⁄₁₄).</p> + +<p>Through the point K, in which the lines A K and B K +intersect one another, draw P K O parallel to A B, and +through C F H and M, where this line meets A C, A F, A H, +and A M, draw C D, F G, H I, and M N, perpendicular to +A B; draw also K L perpendicular to A B; join B F and B H, +and through C draw C E, making with A B the angle (¹⁄₂), +which completes the arrangement of the eleven angles upon +A B; for F B G is very nearly (¹⁄₁₀), and H B I very +nearly (¹⁄₉).</p> + +<p>At the point <i>f</i>, where A C intersects O B, draw <i>f a</i> +perpendicular to A B; and through the point <i>i</i>, where B O +intersects M N, draw S <i>i</i> T parallel to A C.</p> + +<p>Through <i>m</i>, where S <i>i</i> T intersects F B, draw <i>m n</i>; through +<i>β</i>, where S <i>i</i> T intersects K B, draw <i>β w</i>; through T draw T <i>g</i>, +making an angle of (¹⁄₃) with O P. Join N P, M B, and <i>g</i> P,<span class="pagenum"><a id="Page_65"></a>[65]</span> +and where N P intersects K B, draw Q R perpendicular to +A B.</p> + +<p>On A E as a diameter, describe a circle cutting A C in <i>r</i>, +and draw <i>r o</i> perpendicular to A B.</p> + +<p>With A <i>o</i> and <i>o r</i> as semi-axes, describe the ellipse A <i>r e</i>, +cutting A H in <i>t</i>; and draw <i>t u</i> perpendicular to A B. +With A <i>u</i> and <i>t u</i>, as semi-axes describe the ellipse A <i>t d</i>. +On <i>a</i> L, as major axis, describe the ellipse of (¹⁄₃).</p> + +<p>For the side aspect or profile of the figure the diagram is +thus constructed—</p> + +<p>On one side of a line A B (fig. 2, <a href="#plate10">Plate X.</a>) construct the +rectilinear portion of a diagram the same as fig. 1. Through <i>i</i> +draw W Y parallel to A B, and draw A <i>z</i> perpendicular to A B. +Make W <i>a</i> equal to A <i>a</i> (fig. 1), and on <i>a l</i>, as major axis, +describe the ellipse of (¹⁄₄). Through <i>a</i> draw <i>a p</i> parallel to +A F, and through <i>p</i> draw <i>p t</i> perpendicular to W Y. Through +<i>a</i> draw <i>f a u</i> perpendicular to W Y.</p> + +<p>Upon a diameter equal to A E describe a circle whose +circumference shall touch A B and A <i>z</i>. With semi-axes +equal to A <i>o</i> and <i>o r</i> (fig. 1), describe an ellipse with its major +axis parallel to A B, and its circumference touching O P and +<i>z</i> A.</p> + +<div class="sidenote"><a href="#plate11">Plate XI.</a></div> + +<p>Thus simply are the diagrams of the general proportions of +the human figure, as viewed in front and in profile, constructed; +and <a href="#plate11">Plate XI.</a> gives the contour in both points of view, as +composed entirely of the curvilinear figures of (¹⁄₂), (¹⁄₃), (¹⁄₄), +(¹⁄₅), and (¹⁄₆).</p> + +<p>Further detail here would be out of place, and I shall +therefore refer those who require it to the Appendix, or the +more elaborate works to which I have already referred.</p> + +<p>The beauty derived from proportion, imparted by the +system here pointed out, and from a contour of curves derived<span class="pagenum"><a id="Page_66"></a>[66]</span> +from the same harmonic angles, is not confined to the human +figure, but is found in various minor degrees of perfection in +all the organic forms of nature, whether animate or inanimate, +of which I have in other works given many examples.<a id="FNanchor_21" href="#Footnote_21" class="fnanchor">[21]</a></p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_67"></a>[67]</span></p> + +<h2 class="nobreak" id="AS_DEVELOPED_IN_COLOURS">THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS.</h2> + +</div> + +<p>There is not amongst the various phenomena of nature one that +more readily excites our admiration, or makes on the mind a +more vivid impression of the order, variety, and harmonious +beauty of the creation, than that of colour. On the general +landscape this phenomenon is displayed in the production of +that species of harmony in which colours are so variously blended, +and in which they are by light, shade, and distance modified +in such an infinity of gradation and hue. Although genius +is continually struggling, with but partial success, to imitate +those effects, yet, through the Divine beneficence, all whose +organs of sight are in an ordinary degree of perfection can +appreciate and enjoy them. In winter this pleasure is often +to a certain extent withdrawn, when the colourless snow alone +clothes the surface of the earth. But this is only a pause in +the general harmony, which, as the spring returns, addresses +itself the more pleasingly to our perception in its vernal +melody, which, gradually resolving itself into the full rich hues +of luxuriant beauty exhibited in the foliage and flowers of +summer, subsequently rises into the more vivid and powerful +harmonies of autumn’s colouring. Thus the eye is prepared +again to enjoy that rest which such exciting causes may be +said to have rendered necessary.</p> + +<p><span class="pagenum"><a id="Page_68"></a>[68]</span></p> + +<p>When we pass from the general colouring of nature to that +of particular objects, we are again wrapt in wonder and admiration +by the beauty and harmony which so constantly, and in +such infinite variety, present themselves to our view, and +which are so often found combined in the most minute objects. +And the systematic order and uniformity perceptible amidst +this endless variety in the colouring of animate and inanimate +nature is thus another characteristic of beauty equally prevalent +throughout creation.</p> + +<p>By this uniformity in colour, various species of animals are +often distinguished; and in each individual of most of these +species, how much is this beauty enhanced when the uniformity +prevails in the resemblance of their lateral halves! The +human countenance exemplifies this in a striking manner; the +slightest variety of colour between one and another of the +double parts is at once destructive of its symmetrical beauty. +Many of the lower animals, whether inhabitants of the earth, +the air, or the water, owe much of their beauty to this kind of +uniformity in the colour of the furs, feathers, scales, or shells, +with which they are clothed.</p> + +<p>In the vegetable kingdom, we find a great degree of uniformity +of colour in the leaves, flowers, and fruit of the same +plant, combined with all the harmonious beauty of variety +which a little careful examination develops.</p> + +<p>In the colours of minerals, too, the same may be observed. +In short, in the beauty of colouring, as in every other species +of beauty, uniformity and variety are found to combine.</p> + +<p>An appreciation of colour depends, in the first place, as +much upon the physical powers of the eye in conveying a proper +impression to the mind, as that of music on those of the +ear. But an ear for music, or an eye for colour, are, in so far +as beauty is concerned, erroneous expressions; because they<span class="pagenum"><a id="Page_69"></a>[69]</span> +are merely applicable to the impression made upon the senses, +and do not refer to the æsthetical principles of harmony, by +which beauty can alone be understood.</p> + +<p>A good eye, combined with experience, may enable us to +form a correct idea as to the purity of an individual colour, or +of the relative difference existing between two separate hues; +but this sort of discrimination does not constitute that kind of +appreciation of the harmony of colour by which we admire +and enjoy its development in nature and art. The power of +perceiving and appreciating beauty of any kind, is a principle +inherent in the human mind, which may be improved by cultivation +in the degree of the perfection of the art senses. +Great pains have been bestowed on the education of the ear, +in assisting it to appreciate the melody and harmony of sound; +but still much remains to be done in regard to the cultivation +of the eye, in appreciating colour as well as form.</p> + +<p>It is true, that there are individuals whose powers of vision +are perfect, in so far as regards the appreciation of light, shade, +and configuration, but who are totally incapable of perceiving +effects produced by the intermediate phenomenon of colour, +every object appearing to them either white, black, or neutral +gray; others, who are equally blind as to the effect of one of +the three primary colours, but see the other two perfectly, either +singly or combined; while there are many who, having the full +physical power of perceiving all the varieties of the phenomenon, +and who are even capable of making nice distinctions +amongst a variety of various colours, are yet incapable of appreciating +the æsthetic quality of harmony which exists in their +proper combination. It is the same with respect to the effects +of sounds upon the ear—some have organs so constituted, that +notes above or below a certain pitch are to them inaudible; while +others, with physical powers otherwise perfect, are incapable<span class="pagenum"><a id="Page_70"></a>[70]</span> +of appreciating either melody or harmony in musical composition. +But perceptions so imperfectly constituted are, by the +goodness of the Creator, of very rare occurrence; therefore all +attempts at improvement in the science of æsthetics must be +suited to the capacities of the generality of mankind, amongst +whom the perception of colour exists in a variety as great as +that by which their countenances are distinguished. Artists +now and then appear who have this intuitive perception in +such perfection, that they are capable of transferring to their +works the most beautiful harmonies and most delicate gradations +of colours, in a manner that no acquired knowledge +could have enabled them to impart. To those who possess +such a gift, as well as to those to whom the ordinary powers +of perception are denied, it would be equally useless to offer +an explanation of the various modes in which the harmony of +colour develops itself, or to attempt a definition of the +many various colours, hues, tints, and shades, arising out of +the simple elements of this phenomenon. But to those whose +powers lie between these extremes, being neither above nor +below cultivation, such an explanation and definition must +form a step towards the improvement of that inherent principle +which constitutes the basis of æsthetical science.</p> + +<p>Although the variety and harmony of colour which nature +is continually presenting to our view, are apparent to all whose +visual organs are in a natural state, and thus to the generality +of mankind; yet a knowledge of the simplicity by which this +variety and beauty are produced, is, after ages of philosophic +research and experimental inquiry, only beginning to be properly +understood.</p> + +<p>Light may be considered as an active, and darkness a +passive principle in the economy of Nature, and colour an +intermediate phenomenon arising from their joint influence;<span class="pagenum"><a id="Page_71"></a>[71]</span> +and it is in the ratios in which these primary principles act +upon each other, by which I here intend to explain the science +of beauty as evolved in colour. It has been usual to consider +colour as an inherent quality in light, and to suppose that +coloured bodies absorb certain classes of its rays, and reflect or +transmit the remainder; but it appears to me that colour is +more probably the result of certain modes in which the opposite +principles of motion and rest, or force and resistance, operate +in the production, refraction, and reflection of light, and that +each colour is mutually related, although in different degrees, +to these active and passive principles.</p> + +<p>White and black are the representatives of light and darkness, +or activity and rest, and are therefore calculated as +pigments to reduce colours and hues to tints and shades.</p> + +<p>Having, however, fully illustrated the nature of tints and +shades in a former work,<a id="FNanchor_22" href="#Footnote_22" class="fnanchor">[22]</a> I shall here confine myself to +colours in their full intensity—shewing the various modifications +which their union with each other produce, along with +the harmonic relations which these modifications bear to the +primaries, and to each other in respect to warmth and coolness +of tone, as well as to light and shade.</p> + +<p>The primary colours are red, yellow, and blue. Of these, +yellow is most allied to light, and blue to shade, while red is +neutral in these respects, being equally allied to both. In +respect to tone, that of red is warm, and that of blue cool, +while the tone of yellow is neutral. The ratios of their relations +to each other in these respects will appear in the harmonic +scales to which, for the first time, I am about to subject colours, +and to systematise their various simple and compound relations, +which are as follow:—</p> + +<p><span class="pagenum"><a id="Page_72"></a>[72]</span></p> + +<p>From the binary union of the primary colours, the +secondary colours arise—</p> + +<p>Orange colour, from the union of yellow and red.</p> + +<p>Green, from the union of yellow and blue.</p> + +<p>Purple, from the union of red and blue.</p> + +<p>From the binary union of the secondary colours, the primary +hues arise—</p> + +<p>Yellow-hue, from the union of orange and green.</p> + +<p>Red-hue, from the union of orange and purple.</p> + +<p>Blue-hue, from the union of purple and green.</p> + +<p>From the binary union of the primary hues, the secondary +hues arise—</p> + +<p>Orange-hue, from the union of yellow-hue and red-hue.</p> + +<p>Green-hue, from the union of yellow-hue and blue-hue.</p> + +<p>Purple-hue, from the union of red-hue and blue-hue.</p> + +<p>Each hue owes its characteristic distinction to the proportionate +predominance or subordination of one or other of the +three primary colours in its composition.</p> + +<p>It follows, that in every hue of <i>red</i>, yellow and blue are subordinate; +in every hue of <i>yellow</i>, red and blue are subordinate; +and in every hue of <i>blue</i>, red and yellow are subordinate. In +like manner, in every hue of <i>green</i>, red is subordinate; in +every hue of <i>orange</i>, blue is subordinate; and in every hue of +<i>purple</i>, yellow is subordinate.</p> + +<p>By the union of two primary colours, in the production of +a secondary colour, the nature of both primaries is altered; and +as there are only three primary or simple colours in the scale, +the two that are united harmonically in a compound colour, +form the natural contrast to the remaining simple colour.</p> + +<p>Notwithstanding all the variety that extends beyond the +six positive colours, it may be said that there are only three<span class="pagenum"><a id="Page_73"></a>[73]</span> +proper contrasts of colour in nature, and that all others are +simply modifications of these.</p> + +<p>Pure red is the most perfect contrast to pure green; because +it is characterised amongst the primary colours by warmth of +tone, while amongst the secondary colours green is distinguished +by coolness of tone, both being equally related to the +primary elements of light and shade.</p> + +<p>Pure yellow is the most perfect contrast to pure purple; +because it is characterised amongst the primary colours as most +allied to light, whilst pure purple is characterised amongst +the secondaries as most allied to shade, both being equally +neutral as to tone.</p> + +<p>Pure blue is the most perfect contrast to pure orange; +because it is characterised amongst the primary colours as +not only the most allied to shade, but as being the coolest in +tone, whilst pure orange is characterised amongst the secondaries +as being the most allied to light and the warmest in +tone. The same principle operates throughout all the modifications +of these primary and secondary colours.</p> + +<p>Such is the simple nature of contrast upon which the beauty +of colouring mainly depends.</p> + +<p>It being now established as a scientific fact, that the effect +of light upon the eye is the result of an ethereal action, similar +to the atmospheric action by which the effect of sound is +produced upon the ear; also, that the various colours which +light assumes are the effect of certain modifications in this +ethereal action;—just as the various sounds, which constitute +the scale of musical notes, are known to be the effect of certain +modifications in the atmospheric action by which sounds in +general are produced:</p> + +<p>Therefore, as harmony may thus be impressed upon the +mind through either of these two art senses—hearing and<span class="pagenum"><a id="Page_74"></a>[74]</span> +seeing—the principles which govern the modifications in the +ethereal action of light, so as to produce through the eye the +effect of harmony, cannot be supposed to differ from those +principles which we know govern the modifications of the +atmospheric action of sound, in producing through the ear a +like effect. I shall therefore endeavour to illustrate the +science of beauty as evolved in colours, by forming scales of +their various modifications agreeably to the same Pythagorean +system of numerical ratio from which the harmonic elements +of beauty in sounds were originally evolved, and by which +I have endeavoured in this, as in previous works, to systematise +the harmonic beauty of forms.</p> + +<figure class="figcenter illowp80" id="illus2" style="max-width: 43.75em;"> + <img class="w100" src="images/illus2.jpg" alt=""> +</figure> + +<p>It will be observed, that with a view to avoid complexity<span class="pagenum"><a id="Page_75"></a>[75]</span> +as much as possible, I have, in the arrangement of the above +series of scales, not only confined myself to the merely elementary +parts of the Pythagorean system, but have left out +the harmonic modifications upon (¹⁄₁₁) and (¹⁄₁₃), in order that +the arithmetical progression might not be interrupted.<a id="FNanchor_23" href="#Footnote_23" class="fnanchor">[23]</a></p> + +<p>The above elementary process will, I trust, be found sufficient +to explain the progress, by harmonic union, of a primary +colour to a toned gray, and how the simple and compound +colours naturally arrange themselves into the elements of five +scales, the parts of which continue from primary to secondary +colour; from secondary colour to primary hue; from primary +hue to secondary hue; from secondary hue to primary-toned +gray; and from primary-toned gray to secondary-toned gray +in the simple ratio of 2:1; thereby producing a series of the +most beautiful and perfect contrasts.</p> + +<p>The natural arrangement of the primary colours upon the +solar spectrum is red, yellow, blue, and I have therefore +adopted the same arrangement on the present occasion. Red +being, consequently, the first tonic, and blue the second, the +divisions express the numerical ratios which the colours bear +to one another, in respect to that colourific power for which +red is pre-eminent. Thus, yellow is to red, as 2:3; blue +to yellow, as 3:4; purple to orange, as 5:6; and green to +purple, as 6:7.</p> + +<p>The following series of completed scales are arranged upon +the foregoing principle, with the natural connecting links of +red-orange, yellow-orange, yellow-green, and blue-green, introduced +in their proper places.</p> + +<p>The appropriate terminology of musical notes has been +adopted, and the scales are composed as follows:—</p> + +<p><span class="pagenum"><a id="Page_76"></a>[76]</span></p> + +<ul> +<li>Scale I. consists of primary and secondary colours;</li> +<li>Scale II. of secondary colours and primary hues;</li> +<li>Scale III. of primary and secondary hues;</li> +<li>Scale IV. of secondary hues and primary-toned grays; and</li> +<li>Scale V. of primary and secondary-toned grays;</li> +</ul> + +<p>All the parts in each of these scales, from the first tonic to +the second, relate to the same parts of the scale below them +in the simple ratio of 2:1; and serially to the first tonic in +the following ratios:—</p> + +<p class="center">8:9, 4:5, 3:4, 2:3, 3:5, 4:7, 8:15, 1:2.</p> + +<h3><i>First Series of Scales.</i></h3> + +<table class="borders"> + <tr> + <th></th> + <th>Tonic.</th> + <th>Supertonic.</th> + <th>Mediant.</th> + <th>Subdominant.</th> + <th>Dominant.</th> + <th>Submediant.</th> + <th>Subtonic.</th> + <th>Semi-Subtonic.</th> + <th>Tonic.</th> + </tr> + <tr> + <td rowspan="2" class="tdr valign">I.</td> + <td>(¹⁄₂)</td> + <td>(⁴⁄₉)</td> + <td>(²⁄₅)</td> + <td>(³⁄₈)</td> + <td>(¹⁄₃)</td> + <td>(³⁄₁₀)</td> + <td>(²⁄₇)</td> + <td>(⁴⁄₁₅)</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td>Red.</td> + <td>Red-orange.</td> + <td>Orange.</td> + <td>Yellow-orange.</td> + <td>Yellow.</td> + <td>Yellow-green.</td> + <td>Green.</td> + <td>Blue-green.</td> + <td>Blue.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">II.</td> + <td>(¹⁄₄)</td> + <td>(²⁄₉)</td> + <td>(¹⁄₅)</td> + <td>(³⁄₁₆)</td> + <td>(¹⁄₆)</td> + <td>(³⁄₂₀)</td> + <td>(¹⁄₇)</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₈)</td> + </tr> + <tr> + <td>Green.</td> + <td>Blue-green hue.</td> + <td>Blue hue.</td> + <td>Blue-purple hue.</td> + <td>Purple hue.</td> + <td>Red-purple hue.</td> + <td>Red hue.</td> + <td>Red-orange hue.</td> + <td>Orange.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">III.</td> + <td>(¹⁄₈)</td> + <td>(¹⁄₉)</td> + <td>(¹⁄₁₀)</td> + <td>(³⁄₃₂)</td> + <td>(¹⁄₁₂)</td> + <td>(³⁄₄₀)</td> + <td>(¹⁄₁₄)</td> + <td>(¹⁄₁₅)</td> + <td>(¹⁄₁₆)</td> + </tr> + <tr> + <td>Red hue.</td> + <td>Red-orange hue.</td> + <td>Orange hue.</td> + <td>Yellow-orange hue.</td> + <td>Yellow hue.</td> + <td>Yellow-green hue.</td> + <td>Green hue.</td> + <td>Blue-green hue.</td> + <td>Blue hue.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">IV.</td> + <td>(¹⁄₁₆)</td> + <td>(¹⁄₁₈)</td> + <td>(¹⁄₂₀)</td> + <td>(³⁄₆₄)</td> + <td>(¹⁄₂₄)</td> + <td>(³⁄₈₀)</td> + <td>(¹⁄₂₈)</td> + <td>(¹⁄₃₀)</td> + <td>(¹⁄₃₂)</td> + </tr> + <tr> + <td>Green hue.</td> + <td>Blue-green-toned gray.</td> + <td>Blue-toned gray.</td> + <td>Blue-purple-toned gray.</td> + <td>Purple hue.</td> + <td>Red-purple-toned gray.</td> + <td>Red-toned gray.</td> + <td>Red-orange-toned gray.</td> + <td>Orange hue.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">V.</td> + <td>(¹⁄₃₂)</td> + <td>(¹⁄₃₆)</td> + <td>(¹⁄₄₀)</td> + <td>(³⁄₁₂₈)</td> + <td>(¹⁄₄₈)</td> + <td>(³⁄₁₆₀)</td> + <td>(¹⁄₅₆)</td> + <td>(¹⁄₆₀)</td> + <td>(¹⁄₆₄)</td> + </tr> + <tr> + <td>Red-toned gray.</td> + <td>Red-orange-toned gray.</td> + <td>Orange-toned gray.</td> + <td>Yellow-orange-toned gray.</td> + <td>Yellow-toned gray.</td> + <td>Yellow-green-toned gray.</td> + <td>Green-toned gray.</td> + <td>Blue-green-toned gray.</td> + <td>Blue-toned gray.</td> + </tr> +</table> + +<p><span class="pagenum"><a id="Page_77"></a>[77]</span></p> + +<p>To the scales of chromatic power I add another series of +scales, in which yellow, being the first tonic, and blue the +second, the numerical divisions express the ratios which the +colours in each scale bear to one another in respect to light +and shade. Thus red is to yellow, in respect to light, as 2:3; +blue to red, as 3:4; green to orange, as 5:6, and purple to +green, as 6:7.</p> + +<p>These scales may therefore be termed scales for the colour-blind, +because, in comparing colours, those whose sight is thus +defective, naturally compare the ratios of the light and shade +of which different colours are primarily constituted.</p> + +<figure class="figcenter illowp80" id="illus3" style="max-width: 43.75em;"> + <img class="w100" src="images/illus3.jpg" alt=""> +</figure> + +<p>The following is a series of five complete scales of the harmonic<span class="pagenum"><a id="Page_78"></a>[78]</span> +parts into which the light and shade in colours may be +divided in each scale according to the above arrangement:—</p> + +<h3><i>Second Series of Scales.</i></h3> + +<table class="borders"> + <tr> + <th></th> + <th>Tonic.</th> + <th>Supertonic.</th> + <th>Mediant.</th> + <th>Subdominant.</th> + <th>Dominant.</th> + <th>Submediant.</th> + <th>Subtonic.</th> + <th>Semi-Subtonic.</th> + <th>Tonic.</th> + </tr> + <tr> + <td rowspan="2" class="tdr valign">I.</td> + <td>(¹⁄₂)</td> + <td>(⁴⁄₉)</td> + <td>(²⁄₅)</td> + <td>(³⁄₈)</td> + <td>(¹⁄₃)</td> + <td>(³⁄₁₀)</td> + <td>(²⁄₇)</td> + <td>(⁴⁄₁₅)</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td>Yellow.</td> + <td>Yellow-orange.</td> + <td>Orange.</td> + <td>Red-orange.</td> + <td>Red.</td> + <td>Red-purple.</td> + <td>Purple.</td> + <td>Blue-purple.</td> + <td>Blue.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">II.</td> + <td>(¹⁄₄)</td> + <td>(²⁄₉)</td> + <td>(¹⁄₅)</td> + <td>(³⁄₁₆)</td> + <td>(¹⁄₆)</td> + <td>(³⁄₂₀)</td> + <td>(¹⁄₇)</td> + <td>(²⁄₁₅)</td> + <td>(¹⁄₈)</td> + </tr> + <tr> + <td>Purple.</td> + <td>Blue-purple hue.</td> + <td>Blue hue.</td> + <td>Blue-green hue.</td> + <td>Green.</td> + <td>Yellow-green hue.</td> + <td>Yellow hue.</td> + <td>Yellow-orange hue.</td> + <td>Orange.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">III.</td> + <td>(¹⁄₈)</td> + <td>(¹⁄₉)</td> + <td>(¹⁄₁₀)</td> + <td>(³⁄₃₂)</td> + <td>(¹⁄₁₂)</td> + <td>(³⁄₄₀)</td> + <td>(¹⁄₁₄)</td> + <td>(¹⁄₁₅)</td> + <td>(¹⁄₁₆)</td> + </tr> + <tr> + <td>Yellow hue.</td> + <td>Yellow-orange hue.</td> + <td>Orange hue.</td> + <td>Red-orange hue.</td> + <td>Red hue.</td> + <td>Red-purple hue.</td> + <td>Purple hue.</td> + <td>Blue-purple hue.</td> + <td>Blue hue.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">IV.</td> + <td>(¹⁄₁₆)</td> + <td>(¹⁄₁₈)</td> + <td>(¹⁄₂₀)</td> + <td>(³⁄₆₄)</td> + <td>(¹⁄₂₄)</td> + <td>(³⁄₈₀)</td> + <td>(¹⁄₂₈)</td> + <td>(¹⁄₃₀)</td> + <td>(¹⁄₃₂)</td> + </tr> + <tr> + <td>Purple hue.</td> + <td>Blue-purple-toned gray.</td> + <td>Blue-toned gray.</td> + <td>Blue-green-toned gray.</td> + <td>Green hue.</td> + <td>Yellow-green-toned gray.</td> + <td>Yellow-toned gray.</td> + <td>Yellow-orange-toned gray.</td> + <td>Orange hue.</td> + </tr> + <tr> + <td rowspan="2" class="tdr valign">V.</td> + <td>(¹⁄₃₂)</td> + <td>(¹⁄₃₆)</td> + <td>(¹⁄₄₀)</td> + <td>(³⁄₁₂₈)</td> + <td>(¹⁄₄₈)</td> + <td>(³⁄₁₆₀)</td> + <td>(¹⁄₅₆)</td> + <td>(¹⁄₆₀)</td> + <td>(¹⁄₆₄)</td> + </tr> + <tr> + <td>Yellow-toned gray.</td> + <td>Yellow-orange-toned gray.</td> + <td>Orange-toned gray.</td> + <td>Red-orange-toned gray.</td> + <td>Red-toned gray.</td> + <td>Red-purple-toned gray.</td> + <td>Purple-toned gray.</td> + <td>Blue-green-toned gray.</td> + <td>Blue-toned gray.</td> + </tr> +</table> + +<p>Should I be correct in arranging colours upon scales identical +with those upon which musical notes have been arranged, +and in assuming that colours have the same ratios to +each other, in respect to their harmonic power upon the eye, +which musical notes have in respect to their harmonic power +upon the ear, the colourist may yet be enabled to impart +harmonic beauty to his works with as much certainty and +ease, as the musician imparts the same quality to his compositions: +for the colourist has no more right to trust exclusively<span class="pagenum"><a id="Page_79"></a>[79]</span> +to his eye in the arrangement of colours, than the +musician has to trust exclusively to his ear in the arrangement +of sounds.</p> + +<p>We find, in comparing the dominant parts in the first +and second scales of the second series, that they are equal +as to light and shade, so that their relative powers of contrast +depend entirely upon colour. Hence it is that red +and green are the two colours, the difference between which +the colour-blind are least able to appreciate. Professor +George Wilson, in his excellent work, “Researches on +Colour-Blindness,” mentions the case of an engraver, which +proves the power of the eye in being able to appreciate +these original constituents of colour, irrespective of the intermediate +phenomenon of tone. This engraver, instead of +expressing regret on account of his being colour-blind, +observed to the professor, “My defective vision is, to a certain +extent, a useful and valuable quality. Thus, an engraver has +two negatives to deal with, <i>i.e.</i>, white and black. Now, +when I look at a picture, I see it only in white and black, +or light and shade, or, as artists term it, the effect. I find +at times many of my brother engravers in doubt how to +translate certain colours of pictures, which to me are matters +of decided certainty and ease. Thus to me it is valuable.”</p> + +<p>The colour-blind are therefore as incapable of receiving +pleasure from the harmonious union of various colours, as +those who, to use a common term, have no ear for music, are +of being gratified by the “melody of sweet sounds.”</p> + +<p>The generality of mankind are, however, capable of appreciating +the harmony of colour which, like that of both sound +and form, arises from the simultaneous exhibition of opposite +principles having a ratio to each other. These principles are in +continual operation throughout nature, and from them we<span class="pagenum"><a id="Page_80"></a>[80]</span> +often derive pleasure without being conscious of the cause. +All who are not colour-blind must have felt themselves struck +with the harmonic beauty of a cloudless sky, although in it +there is no configuration, and at first sight apparently but one +colour. Now, as we know that there can be no more impression +of harmony made upon the mind by looking upon a +single colour, than there could be by listening to a single +continued musical note, however sweet its tone, we are apt +at first to imagine that the organ of vision has, in some +measure, conveyed a false impression to the mind. But it +has not done so; for light, when reflected from the atmosphere, +produces those cool tones of blue, gray, and purple, +which seem to clothe the distant mountains; but, when +transmitted through the same atmosphere, it produces those +numerous warm tints, the most intense of which give the +gorgeous effects which so often accompany the setting sun. +We have, therefore, in the upper part of a clear sky, where +the atmosphere may be said to be illuminated principally +by reflection from the surface of the earth, a comparatively +cool tone of blue, the result of reflection, which gradually +blends into the warm tints, the result of transmission through +the same atmosphere. Such a composition of harmonious +colouring is to the eye what the voice of the soft breath of +summer amongst the trees, the hum of insects on a sultry +day, or the simple harmony of the Æolian harp, is to the ear. +To such a composition of chromatic harmony must also be +referred the universal concurrence of mankind in appreciating +the peculiar beauty of white marble statuary. That the +principal constituent of beauty in such works ought to be +harmony of form, no one will deny; but this is not the only +element, as appears from the fact, that a cast in plaster of +Paris, of a fine white marble statue, although identical in<span class="pagenum"><a id="Page_81"></a>[81]</span> +form, is far less beautiful than the original. Now this undoubtedly +must be the consequence of its having been changed +from a semi-translucent substance, which, like the atmosphere, +can transmit as well as reflect light, to an opaque substance, +which can only reflect it. Thus the opposite principles of +chromatic warmth and coolness are equally balanced in white +marble—the one being the natural result of the partial transmission +of light, and the other that of its reflection.</p> + +<p>As a series of coloured illustrations would be beyond the +scope of this <i>résumé</i>, I may refer those who wish to prosecute +the inquiry, with the assistance of such a series, to my published +works upon the subject.<a id="FNanchor_24" href="#Footnote_24" class="fnanchor">[24]</a></p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_82"></a>[82]</span></p> + +<h2 class="nobreak" id="APPLIED_TO_THE_FORMS_AND_PROPORTIONS">THE SCIENCE OF BEAUTY, APPLIED TO THE FORMS AND PROPORTIONS OF ANCIENT +GRECIAN VASES AND ORNAMENTS.</h2> + +</div> + +<p>In examining the remains of the ornamental works of the +ancient Greek artists, it appears highly probable that the harmony +of their proportions and melody of their contour are +equally the result of a systematised application of the same +harmonic law. This probability not being fully elucidated in +any of my former works, I will require to go into some detail +on the present occasion. I take for my first illustration an +unexceptionable example, viz.:—</p> + +<h3><i>The Portland Vase.</i></h3> + +<p>Although this beautiful specimen of ancient art was found +about the middle of the sixteenth century, inclosed in a marble +sarcophagus within a sepulchral chamber under the Monte del +Grano, near Rome, and although the date of its production +is unknown, yet its being a work of ancient Grecian art is +undoubted; and the exquisite beauty of its form has been +universally acknowledged, both during the time it remained +in the palace of the Barberini family at Rome, and since it +was added to the treasures of the British Museum. The<span class="pagenum"><a id="Page_83"></a>[83]</span> +forms and proportions of this gem of art appear to me to +yield an obedience to the great harmonic law of nature, similar +to that which I have instanced in the proportions and contour +of the best specimens of ancient Grecian architecture.</p> + +<div class="sidenote"><a href="#plate12">Plate XII.</a></div> + +<p>Let the line A B (<a href="#plate12">Plate XII.</a>) represent the full height +of the vase. Through A draw A <i>a</i>, and through B draw B <i>b</i> +indefinitely, A <i>a</i> making an angle of (¹⁄₂), and B <i>b</i> an angle of +(¹⁄₃), with the vertical. Through the point C, where A <i>a</i> and +B <i>b</i> intersect one another, draw D C E vertical. Through +A C and B respectively, draw A D, C F, and B E horizontal. +Draw similar lines on the other side of A B, and the rectilinear +portion of the diagram is complete.</p> + +<p>The curvilinear contour may be thus added:—</p> + +<p>Take a cut-out ellipse of (¹⁄₄), whose greater axis is equal +to the line A B, and</p> + +<p><i>1st.</i> Place it upon the diagram, so that its circumference +may be tangential to the lines C E and C F, and its greater +axis <i>m n</i> may make an angle of (¹⁄₅) with the vertical, and trace +its circumference.</p> + +<p><i>2d.</i> Place it with its circumference tangential to that of the +first at the point m, while its greater axis (of which <i>o p</i> is a +part) is in the horizontal, and trace the portion of its circumference +<i>q o r</i>.</p> + +<p><i>3d.</i> Place it with its circumference tangential to that of the +above at <i>v</i>, while its greater axis (of which <i>u v</i> is a part) makes +an angle of (³⁄₁₀) with the vertical, and trace the portion of its +circumference <i>s v t</i>.</p> + +<p>Thus the curvilinear contour of the body and neck are +harmonically determined.</p> + +<p>The curve of the handle may be determined by the same +ellipse placed so that its greater axis (of which <i>i k</i> is a part) +makes an angle of (¹⁄₆) with the vertical.</p> + +<p><span class="pagenum"><a id="Page_84"></a>[84]</span></p> + +<p>Make similar tracings on the other side of A B, and the +diagram is complete. The inscribing rectangle D G E K is +that of (²⁄₅).</p> + +<p>The outline resulting from this diagram, not only is in perfect +agreement with my recollection of the form, but with +the measurements of the original given in the “Penny Cyclopædia;” +of the accuracy of which there can be no doubt. +They are stated thus:—“It is about ten inches in height, and +beautifully curved from the top downwards; the diameter at +the top being about three inches and a-half; at the neck or +smallest part, two inches; at the largest (mid-height), seven +inches; and at the bottom, five inches.”</p> + +<p>The harmonic elements of this beautiful form, therefore, +appear to be the following parts of the right angle:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Submediant.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(³⁄₁₀)</td> + </tr> + <tr> + <td>(¹⁄₄)</td> + <td>(¹⁄₆)</td> + <td></td> + <td></td> + </tr> +</table> + +<p>When we reflect upon the variety of harmonic ellipses that +may be described, and the innumerable positions in which +they may be harmonically placed with respect to the horizontal +and vertical lines, as well as upon the various modes in which +their circumferences may be combined, the variety which may +be introduced amongst such forms as the foregoing appears +almost endless. My second example is that of—</p> + +<h3><i>An Ancient Grecian Marble Vase of a Vertical Composition.</i></h3> + +<p>I shall now proceed to another class of the ancient Greek +vase, the form of which is of a more complex character. The<span class="pagenum"><a id="Page_85"></a>[85]</span> +specimen I have chosen for the first example of this class is +one of those so correctly measured and beautifully delineated +by Tatham in his unequalled work.<a id="FNanchor_25" href="#Footnote_25" class="fnanchor">[25]</a> This vase is a work of +ancient Grecian art in Parian marble, which he met with in +the collection at the Villa Albani, near Rome. Its height is +4 ft. 4¹⁄₂ in.</p> + +<div class="sidenote"><a href="#plate13">Plate XIII.</a></div> + +<p>The following is the formula by which I endeavour to develop +its harmonic elements:—</p> + +<p>Let A B (<a href="#plate13">Plate XIII.</a>) represent the full height of this vase. +Through B draw B D, making an angle of (¹⁄₅) with the vertical. +Through D draw D O vertical, through A draw A C, +making an angle of (²⁄₅); through B draw B L, making an +angle of (¹⁄₂), and B S, making an angle of (³⁄₁₀), each with the +vertical. Through A draw A D, through B draw B O, through +L draw L N, through C draw C F, and through S draw S P, +all horizontal. Through A draw A H, making an angle of +(¹⁄₁₀) with the vertical, and through H draw H M vertical. +Draw similar lines on the other side of A B, and the rectilinear +portion of the diagram is complete, and its inscribing +rectangle that of (³⁄₈).</p> + +<p>The curvilinear portion may thus be added—</p> + +<p>Take a cut-out ellipse of (¹⁄₃), whose greater axis is about +the length of the body of the intended vase, place it with its +lesser axis upon the line S P, and its greater axis upon the +line D O, and trace the part <i>a b</i> of its circumference upon the +diagram. Place the same ellipse with one of its foci upon C, +and its greater axis upon C F, and trace its circumference +upon the diagram. Take a cut-out ellipse of (¹⁄₅), whose +greater axis is nearly equal to that of the ellipse already used;<span class="pagenum"><a id="Page_86"></a>[86]</span> +place it with its greater axis upon M H, and its lesser axis +upon L N, and trace its circumference upon the diagram. +Make similar tracings upon the other side of A B, and the +diagram is complete. In this, as in the other diagrams, +the strong portions of the lines give the contour of the vase. +The harmonic elements of this classical form, therefore, appear +to be the right angle and its following parts:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Submediant.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(²⁄₅)</td> + <td>(³⁄₁₀)</td> + </tr> + <tr> + <td></td> + <td></td> + <td>(¹⁄₅)</td> + <td></td> + </tr> + <tr> + <td></td> + <td></td> + <td>(¹⁄₁₀)</td> + <td></td> + </tr> +</table> + +<p>My third example is that of—</p> + +<h3><i>An Ancient Grecian Vase of a Horizontal Composition.</i></h3> + +<p>This example belongs to the same class as the last, but it +is of a horizontal composition. It was carefully drawn from +the original in the museum of the Vatican by Tatham, in +whose etchings it will be found with its ornamental decorations. +The diagram of its harmonic elements may be constructed +as follows:—</p> + +<div class="sidenote"><a href="#plate14">Plate XIV.</a></div> + +<p>Let A B (<a href="#plate14">Plate XIV.</a>) represent the full height of the vase. +Through B draw B D, making an angle of (²⁄₅) with the vertical. +Through A draw A H, A L, and A C, making respectively +the following angles, (¹⁄₅) with the vertical, (⁴⁄₉) with the +vertical, and (³⁄₁₀) with the horizontal. These angles determine +the horizontal lines H B, L N, and C F, which divide the vase +into its parts, and the inscribing rectangle D G K O is (³⁄₈). +This completes the rectilinear portion of the diagram. The +ellipse by which the curvilinear portion is added is one of (¹⁄₅), +the greater axis of which, at <i>a b</i>, as also at <i>c d</i>, makes an angle<span class="pagenum"><a id="Page_87"></a>[87]</span> +of (¹⁄₁₂) with the vertical, and the same axis at <i>e f</i> an angle of +(¹⁄₁₂) with the horizontal.</p> + +<p>The harmonic elements of this vase, therefore, appear to +be:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Submediant.</th> + <th>Supertonic.</th> + </tr> + <tr> + <td>The Right Angle.</td> + <td>(¹⁄₁₂)</td> + <td>(²⁄₅)</td> + <td>(³⁄₁₀)</td> + <td>(⁴⁄₉)</td> + </tr> + <tr> + <td></td> + <td></td> + <td>(¹⁄₅)</td> + <td></td> + <td></td> + </tr> +</table> + +<p>My remaining examples are those of—</p> + +<h3><i>Etruscan Vases.</i></h3> + +<p>Of these vases I give four examples, by which the simplicity +of the method employed in applying the harmonic law +will be apparent.</p> + +<div class="sidenote"><a href="#plate15">Plate XV.</a></div> + +<p>The inscribing rectangle D G E K of fig. 1, <a href="#plate15">Plate XV.</a>, +is one of (³⁄₈), within which are arranged tracings from an +ellipse of (³⁄₁₀), whose greater axis at <i>a b</i> makes an angle of +(¹⁄₁₂), at <i>c d</i> an angle of (³⁄₁₀), and at <i>e f</i> an angle of (³⁄₄), with +the vertical. The harmonic elements of the contour of this +vase, therefore, appear to be:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Subdominants.</th> + <th>Submediant.</th> + </tr> + <tr> + <td>The Right Angle.</td> + <td>(¹⁄₁₂)</td> + <td>(³⁄₄)</td> + <td>(³⁄₁₀)</td> + </tr> + <tr> + <td></td> + <td></td> + <td>(³⁄₈)</td> + <td></td> + </tr> +</table> + +<p>The inscribing rectangle L M N O of fig. 2 is that of (¹⁄₂), +within which are arranged tracings from an ellipse of (¹⁄₃), +whose greater axis, at <i>a b</i> and <i>c d</i> respectively, makes angles of +(¹⁄₂) and (⁴⁄₉) with the horizontal, while that at <i>e f</i> is in the +horizontal line. The harmonic elements of the contour of +this vase, therefore, appear to be:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Subtonic.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(⁴⁄₉)</td> + </tr> +</table> + +<p><span class="pagenum"><a id="Page_88"></a>[88]</span></p> + +<div class="sidenote"><a href="#plate16">Plate XVI.</a></div> + +<p>The inscribing rectangle P Q R S of fig. 1, <a href="#plate16">Plate XVI.</a>, +is one of (⁴⁄₉), within which are arranged tracings from an +ellipse of (³⁄₈), whose greater axis, at <i>a b</i>, <i>c d</i>, and <i>e f</i>, makes respectively +angles of (¹⁄₆) with the horizontal, (³⁄₅) and (⁴⁄₅) with +the vertical. Its harmonic elements, therefore, appear to be:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Supertonic.</th> + <th>Subdominant.</th> + <th>Submediant.</th> + </tr> + <tr> + <td>The Right Angle.</td> + <td>(¹⁄₆)</td> + <td>(⁴⁄₅)</td> + <td>(⁴⁄₉)</td> + <td>(³⁄₈)</td> + <td>(³⁄₅)</td> + </tr> +</table> + +<p>The inscribing rectangle T U V X of fig. 2 is one of (⁴⁄₉), +within which are arranged tracings from an ellipse of (³⁄₈) +whose greater axis at <i>a b</i> is in the vertical line, and at <i>c d</i> +makes an angle of (¹⁄₂). The harmonic elements of the contour +of this vase, therefore, appear to be:—</p> + +<table> + <tr> + <th>Tonic.</th> + <th>Submediant.</th> + <th>Supertonic.</th> + </tr> + <tr> + <td>(¹⁄₂)</td> + <td>(³⁄₈)</td> + <td>(⁴⁄₉)</td> + </tr> +</table> + +<p>These four Etruscan vases, the contours of which are thus +reduced to the harmonic law of nature, are in the British +Museum, and engravings of them are to be found in the well-known +work of Mr Henry Moses, Plates 4, 6, 14, and 7, respectively, +where they are represented with their appropriate +decorations and colours.</p> + +<p>To these, I add two examples of—</p> + +<h3><i>Ancient Grecian Ornament.</i></h3> + +<p>I have elsewhere shewn<a id="FNanchor_26" href="#Footnote_26" class="fnanchor">[26]</a> that the elliptic curve pervades +the Parthenon from the entases of the column to the smallest +moulding, and we need not, therefore, be surprised to find it<span class="pagenum"><a id="Page_89"></a>[89]</span> +employed in the construction of the only two ornaments +belonging to that great work.</p> + +<div class="sidenote"><a href="#plate17">Plate XVII.</a></div> + +<p>In the diagram (<a href="#plate17">Plate XVII.</a>), I endeavour to exhibit the +geometric construction of the upper part of one of the ornamental +apices, termed antefixæ, which surmounted the cornice +of the Parthenon.</p> + +<p>The first ellipse employed is that of (¹⁄₃), whose greater axis +<i>a b</i> is in the vertical line; the second is also that of (¹⁄₃), whose +greater axis <i>c d</i> makes, with the vertical, an angle of (¹⁄₁₂); +the third ellipse is the same with its major axis <i>e f</i> in the +vertical line. Through one of the foci of this ellipse at A the +line A C is drawn, and upon the part of the circumference C <i>e</i>, +the number of parts, 1, 2, 3, 4, 5, 6, 7, of which the surmounting +part of this ornament is to consist, are set off. +That part of the circumference of the ellipse whose larger axis +is <i>c d</i> is divided from <i>g</i> to <i>c</i> into a like number of parts. The +third ellipse employed is one of (¹⁄₄).</p> + +<p>Take a cut-out ellipse of this kind, whose larger axis is +equal in length to the inscribing rectangle. Place it with its +vertex upon the same ellipse at <i>g</i>, so that its circumference +will pass through C, and trace it; remove its apix first to <i>p</i>, +then to <i>q</i>, and proceed in the same way to <i>q</i>, <i>r</i>, <i>s</i>, +<i>t</i>, <i>u</i>, and <i>v</i>, +so that its circumference will pass through the seven divisions +on <i>c g</i> and <i>e</i> C: <i>v o</i>, <i>u n</i>, <i>t m</i>, +<i>s i</i>, <i>r k</i>, <i>q j</i>, <i>p l</i>, and <i>g x</i>, are parts +of the larger axes of the ellipses from which the curves are +traced. The small ellipse of which the ends of the parts are +formed is that of (¹⁄₃).</p> + +<div class="sidenote"><a href="#plate18">Plate XVIII.</a></div> + +<p>In the diagram (<a href="#plate18">Plate XVIII.</a>), I endeavour to exhibit the +geometric construction of the ancient Grecian ornament, commonly +called the <i>Honeysuckle</i>, from its resemblance to the flower +of that name. The first part of the process is similar to that +just explained with reference to the antefixæ of the Parthenon,<span class="pagenum"><a id="Page_90"></a>[90]</span> +although the angles in some parts differ. The contour is +determined by the circumference of an ellipse of (¹⁄₃), whose +major axis A B makes an angle of (¹⁄₉) with the vertical, and the +leaves or petals are arranged upon a portion of the perimeter +of a similar ellipse whose larger axis E F is in the vertical +line, and these parts are again arranged upon a similar ellipse +whose larger axis C D makes an angle of (¹⁄₁₂) with the vertical. +The first series of curved lines proceeding from 1, 2, 3, 4, 5, +6, 7, and 8, are between K E and H C, part of the circumference +of an ellipse of (¹⁄₃); and those between C H and A G +are parts of the circumference of four ellipses, each of (¹⁄₃), but +varying as to the lengths of their larger axes from 5 to 3 inches. +The change from the convex to the concave, which produces +the ogie forms of which this ornament is composed, takes +place upon the line C H, and the lines <i>a b</i>, <i>c d</i>, +<i>e f</i>, <i>g h</i>, <i>i k</i>, <i>l m</i>, +<i>n o</i>, and <i>p q</i>, are parts of the larger axis of the four ellipses +the circumference of which give the upper parts of the petals +or leaves.</p> + +<p>This peculiar Grecian ornament is often, like the antefixæ +of the Parthenon, combined with the curve of the spiral scroll. +But the volute is so well understood that I have not rendered +my diagrams more complex by adding that figure. Many +varieties of this union are to be found in Tatham’s etchings, +already referred to. The antefixæ of the Parthenon, and its +only other ornament the honeysuckle, as represented on the +soffit of the cornice, are to be found in Stewart’s “Athens.”</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<p><span class="pagenum"><a id="Page_91"></a>[91]</span></p> + +<h2 class="nobreak" id="APPENDIX">APPENDIX.</h2> + +</div> + +<h3 id="APPENDIX_I">No. I.</h3> + +<p>In pages <a href="#Page_34">34</a>, <a href="#Page_35">35</a>, and +<a href="#Page_58">58</a>, I have reiterated an opinion advanced in several +of my former works, viz., that, besides genius, and the study of nature, an +additional cause must be assigned for the general excellence which characterises +such works of Grecian art as were executed during a period commencing +about 500 <span class="allsmcap">B.C.</span>, and ending about 200 <span class="allsmcap">B.C.</span> And that this cause +most probably was, that the artists of that period were instructed in a +system of fixed principles, based upon the doctrines of Pythagoras and +Plato. This opinion has not been objected to by the generality of those +critics who have reviewed my works; but has, however, met with an +opponent, whose recondite researches and learned observations are worthy +of particular attention. These are given in an essay by Mr C. Knight +Watson, “On the Classical Authorities for Ancient Art,” which appeared in +the <i>Cambridge Journal of Classical and Sacred Philology</i> in June 1854. As +this essay is not otherwise likely to meet the eyes of the generality of my +readers, and as the objections he raises to my opinion only occupy two out +of the sixteen ample paragraphs which constitute the first part of the essay, +I shall quote them fully:—</p> + +<div class="blockquote"> + +<p>“The next name on our list is that of the famous Euphranor (<span class="allsmcap">B.C.</span> 362). +For the fact that to the practice of sculpture and of painting he added an +exposition of the theory, we are indebted to Pliny, who says (xxxv. 11, 40), +‘Volumina quoque composuit de symmetria et coloribus.’ When we reflect +on the <i>critical</i> position occupied by Euphranor in the history of Greek art, +as a connecting link between the idealism of Pheidias and the naturalism +of Lysippus, we can scarcely overestimate the value of a treatise on art +proceeding from such a quarter. This is especially the case with the first of +the two works here assigned to Euphranor. The inquiries which of late +years have been instituted by Mr D. R. Hay of Edinburgh, on the proportions +of the human figure, and on the natural principles of beauty as illustrated<span class="pagenum"><a id="Page_92"></a>[92]</span> +by works of Greek art, constitute an epoch in the study of æsthetics +and the philosophy of form. Now, in the presence of these inquiries, or of +such less solid results as Mr Hay’s predecessors in the same field have +elicited, it naturally becomes an object of considerable interest to ascertain +how far these laws of form and principles of beauty were consciously +developed in the mind, and by the chisel, of the sculptor: how far any +such system of curves and proportions as Mr Hay’s was used by the Greek +as a practical manual of his craft. Without in the least wishing to impugn +the accuracy of that gentleman’s results—a piece of presumption I should +do well to avoid—I must be permitted to doubt whether the ‘Symmetria’ +of Euphranor contained anything analogous to them in kind, or indeed equal +in value. It must not be forgotten that the truth of Mr Hay’s theory is +perfectly compatible with the fact, that of such theory the Greek may have +been utterly ignorant. It is on this fact I insist: it is here that I join +issue with Mr Hay, and with his reviewer in a recent number of <i>Blackwood’s +Magazine</i>. Or, to speak more accurately,—while I am quite prepared to +find that the Elgin marbles will best of all stand the test which Mr Hay has +hitherto applied, I believe, to works of a later age, I am none the less convinced +that it is precisely that golden age of Hellenic art to which they +belong, precisely that first and chief of Hellenic artists by whom they were +executed, to which and to whom any such line of research on the laws of +form would have been pre-eminently alien. Pheidias, remember, by the +right of primogeniture, is the ruling spirit of idealism in art. Of spontaneity +was that idealism begotten and nurtured: by any such system as +Mr Hay’s, that spontaneity would be smothered and paralysed. Pheidias +copied an idea in his own mind—‘Ipsius in mente insidebat species pulchritudinis +eximia quædam’ (<i>Cic.</i>);—later ages copied <i>him</i>. He created: they +criticised. He was the author of Iliads: they the authors of Poetics. Doubtless, +if you unsphere the <i>spirit</i> of Mr Hay’s theories, you will find nothing +discordant with what I have here said. That is a sound view of Beauty +which makes it consist in that due subordination of the parts to the whole, +that due relation of the parts to each other, which Mendelssohn had in his +mind when he said that the essence of beauty was ‘unity in variety’—variety +beguiling the imagination, the perception of unity exercising the +thewes and sinews of the intellect. On such a view of beauty, Mr Hay’s +theory may, <i>in spirit</i>, be said to rest. But here, as in higher things, it is the +letter that killeth, while the spirit giveth life. And accordingly I must +enter a protest against any endeavour to foist upon the palmy days of +Hellenic art systems of geometrical proportions incompatible, as I believe, +with those higher and broader principles by which the progress of ancient +sculpture was ordered and governed—systems which will bear nothing of +that ‘felicity and chance by which’—and not by rule—‘Lord Bacon believed +that a painter may make a better face than ever was:’ systems which +take no account of that fundamental distinction between the schools of<span class="pagenum"><a id="Page_93"></a>[93]</span> +Athens and of Argos, and their respective disciples and descendants, without +which you will make nonsense of the pages of Pliny, and—what is worse—sense +of the pages of his commentators;—systems, in short, which may have +their value as instruments for the education of the eye, and for instructions +in the arts of design, but must be cast aside as matters of learned trifling +and curious disputation, where they profess to be royal roads to art, and to +map the mighty maze of a creative mind. And even as regards the application +of such a system of proportions to those works of sculpture which are +posterior to the Pheidian age, only partial can have been the prevalence +which it or any other <i>one</i> system can have obtained. The discrepancies +of different artists in the treatment of what was called, technically called, +<i>Symmetria</i> (as in the title of Euphranor’s work) were, by the concurrent +testimony of all ancient writers, far too salient and important to warrant the +supposition of any uniform scale of proportions, as advocated by Mr Hay. +Even in Egypt, where one might surely have expected that such uniformity +would have been observed with far greater rigour than in Greece, the discoveries +of Dr Lepsius (<i>Vorläufige Nachricht</i>, Berlin, 1849) have elicited +three totally different κανόνες, one of which is identical with the system of +proportions of the human figure detailed in Diodorus. While we thus venture +to differ from Mr Hay on the historical data he has mixed up with his +inquiries, we feel bound to pay him a large and glad tribute of praise for +having devised a system of proportions which rises superior to the idiosyncracies +of different artists, which brings back to one common type the +sensations of eye and ear, and so makes a giant stride towards that <i>codification</i>, +if I may so speak, of the laws of the universe which it is the business +of the science to effect. I have no hesitation in saying, that, for scientific +precision of method and importance of results, Albert Durer, Da Vinci, and +Hogarth, not to mention less noteworthy writers, must all yield the palm +to Mr Hay.</p> + +<p>“I am quite aware that in the digression I have here allowed myself, on +systems of proportions prevalent among ancient artists, and on the probable +contents of such treatises as that of Euphranor, <i>De Symmetria</i>, I have laid +myself open to the charge of treating an intricate question in a very perfunctory +way. At present the exigencies of the subject more immediately +in hand allow me only to urge in reply, that, as regards the point at issue—I +mean the ‘solidarité’ between theories such as Mr Hay’s and the practice +of Pheidias—the <i>onus probandi</i> rests with my adversaries.”</p> +</div> + +<p>I am bound, in the first place, gratefully to acknowledge the kind and +complimentary notice which, notwithstanding our difference of opinion, this +author has been pleased to take of my works; and, in the second, to assure +him that if any of them profess to be “royal roads to art,” or to “map +the mighty maze of a creative mind,” they certainly profess to do more +than I ever meant they should; for I never entertained the idea that a<span class="pagenum"><a id="Page_94"></a>[94]</span> +system of æsthetic culture, even when based upon a law of nature, was +capable of effecting any such object. But I doubt not that this too common +misapprehension of the scope and tendency of my works must arise from a +want of perspicuity in my style.</p> + +<p>I have certainly, on one occasion,<a id="FNanchor_27" href="#Footnote_27" class="fnanchor">[27]</a> gone the length of stating that +as poetic genius must yield obedience to the rules of rhythmical measure, +even in the highest flights of her inspirations; and musical genius must, in +like manner, be subject to the strictly defined laws of harmony in the most +delicate, as well as in the most powerfully grand of her compositions; so +must genius, in the formative arts, either consciously or unconsciously have +clothed her creations of ideal beauty with proportions strictly in accordance +with the laws which nature has set up as inflexible standards. If, therefore, +the laws of proportion, in their relation to the arts of design, constitute +the harmony of geometry, as definitely as those that are applicable to poetry +and music produce the harmony of acoustics; the former ought, certainly, +to hold the same relative position in those arts which are addressed to the +eye, that is accorded to the latter in those which are addressed to the ear. +Until so much science be brought to bear upon the arts of design, the +student must continue to copy from individual and imperfect objects in +nature, or from the few existing remains of ancient Greek art, in total ignorance +of the laws by which their proportions are produced, and, what is equally +detrimental to art, the accuracy of all criticism must continue to rest upon +the indefinite and variable basis of mere opinion.</p> + +<p>It cannot be denied that men of great artistic genius are possessed of an +intuitive feeling of appreciation for what is beautiful, by means of which +they impart to their works the most perfect proportions, independently of +any knowledge of the definite laws which govern that species of beauty. +But they often do so at the expense of much labour, making many trials +before they can satisfy themselves in imparting to them the true proportions +which their minds can conceive, and which, along with those other qualities of +expression, action, or attitude, which belong more exclusively to the province +of genius. In such cases, an acquaintance with the rules which constitute +the science of proportion, instead of proving fetters to genius, would doubtless +afford her such a vantage ground as would promote the more free exercise of +her powers, and give confidence and precision in the embodiment of her +inspirations; qualities which, although quite compatible with genius, are not +always intuitively developed along with that gift.</p> + +<p>It is also true that the operations of the conceptive faculty of the mind +are uncontrolled by definite laws, and that, therefore, there cannot exist any +rules by the inculcation of which an ordinary mind can be imbued with +genius sufficient to produce works of high art. Nevertheless, such a mind may +be improved in its perceptive faculty by instruction in the science of proportion, +so as to be enabled to exercise as correct and just an appreciation of<span class="pagenum"><a id="Page_95"></a>[95]</span> +the conceptions of others, in works of plastic art, as that manifested by the +educated portion of mankind in respect to poetry and music. In short, it +appears that, in those arts which are addressed to the ear, men of genius +communicate the original conceptions of their minds under the control of +certain scientific laws, by means of which the educated easily distinguish the +true from the false, and by which the works of the poet and musical composer +may be placed above mere imitations of nature, or of the works of +others; while, in those arts that are addressed to the eye in their own peculiar +language, such as sculpture, architecture, painting, and ornamental +design, no such laws are as yet acknowledged.</p> + +<p>Although I am, and ever have been, far from endeavouring “to foist upon +the palmy days of Hellenic art” any system incompatible with those higher +and more intellectual qualities which genius alone can impart; yet, from +what has been handed down to us by writers on the subject, meagre as it is, +I cannot help continuing to believe that, along with the physical and metaphysical +sciences, æsthetic science was taught in the early schools of Greece.</p> + +<p>I shall here take the liberty to repeat the proofs I advanced in a former +work as the ground of this belief, and to which the author, from whose +essay I have quoted, undoubtedly refers. It is well known that, in the time +of Pythagoras, the treasures of science were veiled in mystery to all but the +properly initiated, and the results of its various branches only given to the +world in the works of those who had acquired this knowledge. So strictly +was this secresy maintained amongst the disciples and pupils of Pythagoras, +that any one divulging the sacred doctrines to the profane, was expelled the +community, and none of his former associates allowed to hold further intercourse +with him; it is even said, that one of his pupils incurred the displeasure +of the philosopher for having published the solution of a problem +in geometry.<a id="FNanchor_28" href="#Footnote_28" class="fnanchor">[28]</a> The difficulty, therefore, which is expressed by writers, +shortly after the period in which Pythagoras lived, regarding a precise +knowledge of his theories, is not to be wondered at, more especially when it +is considered that he never committed them to writing. It would appear, +however, that he proceeded upon the principle, that the order and beauty so +apparent throughout the whole universe, must compel men to believe in, +and refer them to, an intelligible cause. Pythagoras and his disciples sought +for properties in the science of numbers, by the knowledge of which they +might attain to that of nature; and they conceived those properties to be +indicated in the phenomena of sonorous bodies. Observing that Nature +herself had thus irrevocably fixed the numerical value of the intervals of +musical tones, they justly concluded that, as she is always uniform in her +works, the same laws must regulate the general system of the universe.<a id="FNanchor_29" href="#Footnote_29" class="fnanchor">[29]</a><span class="pagenum"><a id="Page_96"></a>[96]</span> +Pythagoras, therefore, considered numerical proportion as the great principle +inherent in all things, and traced the various forms and phenomena of the +world to numbers as their basis and essence.</p> + +<p>How the principles of numbers were applied in the arts is not recorded, farther +than what transpires in the works of Plato, whose doctrines were from the +school of Pythagoras. In explaining the principle of beauty, as developed in +the elements of the material world, he commences in the following words:—“But +when the Artificer began to adorn the universe, he first of all figured +with forms and numbers, fire and earth, water and air—which possessed, +indeed, certain traces of the true elements, but were in every respect so constituted +as it becomes anything to be from which Deity is absent. But we +should always persevere in asserting that Divinity rendered them, as much +as possible, the most beautiful and the best, when they were in a state of +existence opposite to such a condition.” Plato goes on further to say, that +these elementary bodies must have forms; and as it is necessary that every +depth should comprehend the nature of a plane, and as of plane figures the +triangle is the most elementary, he adopts two triangles as the originals or +representatives of the isosceles and the scalene kinds. The first triangle of +Plato is that which forms the half of the square, and is regulated by the +number, 2; and the second, that which forms the half of the equilateral +triangle, which is regulated by the number, 3; from various combinations +of these, he formed the bodies of which he considered the elements to be +composed. To these elementary figures I have already referred.</p> + +<p>Vitruvius, who studied architecture ages after the arts of Greece had been +buried in the oblivion which succeeded her conquest, gives the measurements +of various details of monuments of Greek art then existing. But he +seems to have had but a vague traditionary knowledge of the principle of +harmony and proportion from which these measurements resulted. He says—“The +several parts which constitute a temple ought to be subject to the +laws of symmetry; the principles of which should be familiar to all who +profess the science of architecture. Symmetry results from proportion, +which, in the Greek language, is termed analogy. Proportion is the commensuration +of the various constituent parts with the whole; in the +existence of which symmetry is found to consist. For no building can +possess the attributes of composition in which symmetry and proportion are +disregarded; nor unless there exist that perfect conformation of parts which +may be observed in a well-formed human being.” After going at some +length into details, he adds—“Since, therefore, the human figure appears to +have been formed with such propriety, that the several members are commensurate +with the whole, the artists of antiquity (meaning those of Greece +at the period of her highest refinement) must be allowed to have followed +the dictates of a judgment the most rational, when, transferring to works of +art principles derived from nature, every part was so regulated as to bear +a just proportion to the whole. Now, although the principles were universally<span class="pagenum"><a id="Page_97"></a>[97]</span> +acted upon, yet they were more particularly attended to in the construction +of temples and sacred edifices, the beauties or defects of which +were destined to remain as a perpetual testimony of their skill or of their +inability.”</p> + +<p>Vitruvius, however, gives no explanation of this ancient principle of proportion, +as derived from the human form; but plainly shews his uncertainty +upon the subject, by concluding this part of his essay in the following words: +“If it be true, therefore, that the decenary notation was suggested by the +members of man, and that the laws of proportion arose from the relative +measures existing between certain parts of each member and the whole body, +it will follow, that those are entitled to our commendation who, in building +temples to their deities, proportioned the edifices, so that the several parts +of them might be commensurate with the whole.” +It thus appears certain that the Grecians, at the period of their highest +excellence, had arrived at a knowledge of some definite mathematical law +of proportion, which formed a standard of perfectly symmetrical beauty, +not only in the representation of the human figure in sculpture and painting, +but in architectural design, and indeed in all works where beauty of +form and harmony of proportion constituted excellence. That this law was +not deduced from the proportions of the human figure, as supposed by +Vitruvius, but had its origin in mathematical science, seems equally certain; +for in no other way can we satisfactorily account for the proportions of the +beau ideal forms of the ancient Greek deities, or of those of their architectural +structures, such as the Parthenon, the temple of Theseus, &c., or for +the beauty that pervades all the other formative art of the period.</p> + +<p>This system of geometrical harmony, founded, as I have shewn it to be, +upon numerical relations, must consequently have formed part of the Greek +philosophy of the period, by means of which the arts began to progress +towards that great excellence which they soon after attained. A little +further investigation will shew, that immediately after this period a theory +connected with art was acknowledged and taught, and also that there existed +a Science of Proportion.</p> + +<p>Pamphilus, the celebrated painter, who flourished about four hundred +years before the Christian era, from whom Apelles received the rudiments +of his art, and whose school was distinguished for scientific cultivation, +artistic knowledge, and the greatest accuracy in drawing, would admit no +pupil unacquainted with geometry.<a id="FNanchor_30" href="#Footnote_30" class="fnanchor">[30]</a> The terms upon which he engaged +with his students were, that each should pay him one talent (£225 sterling) +previous to receiving his instructions; for this he engaged “to give them, +<i>for ten years</i>, lessons founded on an excellent theory.”<a id="FNanchor_31" href="#Footnote_31" class="fnanchor">[31]</a></p> + +<p>It was by the advice of Pamphilus that the magistrates of Sicyon ordained +that the study of drawing should constitute part of the education of the<span class="pagenum"><a id="Page_98"></a>[98]</span> +citizens—“a law,” says the Abbé Barthélémie, “which rescued the fine arts +from servile hands.”</p> + +<p>It is stated of Parrhasius, the rival of Zeuxis, who flourished about the +same period as Pamphilus, that he accelerated the progress of art by purity +and correctness of design; “for he was acquainted with the science of +Proportions. Those he gave his gods and heroes were so happy, that artists +did not hesitate to adopt them.” Parrhasius, it is also stated, was so +admired by his contemporaries, that they decreed him the name of Legislator.<a id="FNanchor_32" href="#Footnote_32" class="fnanchor">[32]</a> +The whole history of the arts in Egypt and Greece concurs to prove +that they were based on geometric precision, and were perfected by a continued +application of the same science; while in all other countries we find +them originating in rude and misshapen imitations of nature.</p> + +<p>In the earliest stages of Greek art, the gods—then the only statues—were +represented in a tranquil and fixed posture, with the features exhibiting +a stiff inflexible earnestness, their only claim to excellence being symmetrical +proportion; and this attention to geometric precision continued as art +advanced towards its culminating point, and was thereafter still exhibited +in the neatly and regularly folded drapery, and in the curiously braided and +symmetrically arranged hair.<a id="FNanchor_33" href="#Footnote_33" class="fnanchor">[33]</a></p> + +<p>These researches, imperfect as they are, cannot fail to exhibit the great +contrast that exists between the system of elementary education in art practised +in ancient Greece, and that adopted in this country at the present +period. But it would be of very little service to point out this contrast, +were it not accompanied by some attempt to develop the principles which +seem to have formed the basis of this excellence amongst the Greeks.</p> + +<p>But in making such an attempt, I cannot accuse myself of assuming anything +incompatible with the free exercise of that spontaneity of genius which +the learned essayist says is the parent and nurse of idealism. For it is in +no way more incompatible with the free exercise of artistic genius, that +those who are so gifted should have the advantage of an elementary education +in the science of æsthetics, than it is incompatible with the free exercise +of literary or poetic genius, that those who possess it should have the +advantage of such an elementary education in the science of philology as +our literary schools and colleges so amply afford.</p> + +<p><span class="pagenum"><a id="Page_99"></a>[99]</span></p> + +<h3 id="APPENDIX_II">No. II.</h3> + +<p>The letter from which I have made a quotation at page <a href="#Page_42">42</a>, arose out of +the following circumstance:—In order that my theory, as applied to the +orthographic beauty of the Parthenon, might be brought before the highest +tribunal which this country afforded, I sent a paper upon the subject, +accompanied by ample illustrations, to the Royal Institute of British Architects, +and it was read at a meeting of that learned body on the 7th of +February 1853; at the conclusion of which, Mr Penrose kindly undertook +to examine my theoretical views, in connexion with the measurements he +had taken of that beautiful structure by order of the Dilettanti Society, and +report upon the subject to the Royal Institute. This report was published +by Mr Penrose, vol. xi., No. 539 of <i>The Builder</i>, and the letter from which +I have quoted appeared in No. 542 of the same journal. It was as +follows:—</p> + +<div class="blockquote"> + +<p class="center">“GEOMETRICAL RELATIONS IN ARCHITECTURE.</p> + +<p>“Will you allow me, through the medium of your columns, to thank Mr +Penrose for his testimony to the truth of Mr Hay’s revival of Pythagoras? +The dimensions which he gives are to me the surest verification of the +theory that I could have desired. The minute discrepancies form that very +element of practical incertitude, both as to execution and direct measurement, +which always prevails in materialising a mathematical calculation under +such conditions.</p> + +<p>“It is time that the scattered computations by which architecture has +been analysed—more than enough—be synthetised into those formulæ +which, as Mrs Somerville tells us, ‘are emblematic of omniscience.’ The +young architects of our day feel trembling beneath their feet the ground +whence men are about to evoke the great and slumbering corpse of art. +Sir, it is food of this kind a reviving poetry demands.</p> + +<div class="poetry-container"> +<div class="poetry"> + <div class="stanza"> + <div class="verse indent8">——‘Give us truths,</div> + <div class="verse indent0">For we are weary of the surfaces,</div> + <div class="verse indent0">And die of inanition.’</div> + </div> +</div> +</div> + +<p>“I, for one, as I listen to such demonstrations, whose scope extends with +every research into them, feel as if listening to those words of Pythagoras, +which sowed in the mind of Greece the poetry whose manifestation in beauty +has enchained the world in worship ever since its birth. And I am sure +that in such a quarter, and in such thoughts, <i>we</i> must look for the first +shining of that lamp of art, which even now is prepared to burn.</p> + +<p>“I know that this all sounds rhapsodical; but I know also that until the +architect becomes a poet, and not a tradesman, we may look in vain for<span class="pagenum"><a id="Page_100"></a>[100]</span> +architecture: and I know that valuable as isolated and detailed investigations +are in their proper bearings, yet that such purposes and bearings are to +be found in the enunciation of principles sublime as the generalities of +‘mathematical beauty.’</p> + +<p class="right">“<span class="smcap">Autocthon.</span>”</p> + +</div> + +<h3 id="APPENDIX_III">No. III.</h3> + +<p>Of the work alluded to at page <a href="#Page_58">58</a> I was favoured with two opinions—the +one referring to the theory it propounds, and the other to its anatomical +accuracy—both of which I have been kindly permitted to publish.</p> + +<p>The first is from Sir <span class="smcap">William Hamilton</span>, Bart., professor of logic and +metaphysics in the University of Edinburgh, and is as follows:—</p> + +<div class="blockquote"> + +<p>“Your very elegant volume is to me extremely interesting, as affording +an able contribution to what is the ancient, and, I conceive, the true theory +of the Beautiful. But though your doctrine coincides with the one prevalent +through all antiquity, it appears to me quite independent and original +in you; and I esteem it the more, that it stands opposed to the hundred +one-sided and exclusive views prevalent in modern times.—<i>16 Great King +Street, March 5, 1849.</i>”</p> +</div> + +<p>The second is from <span class="smcap">John Goodsir</span>, Esq., professor of anatomy in the +University of Edinburgh, and is as follows:—</p> + +<div class="blockquote"> + +<p>“I have examined the plates in your work on the proportions of the +human head and countenance, and find the head you have given as typical +of human beauty to be anatomically correct in its structure, only differing +from ordinary nature in its proportions being more mathematically precise, +and consequently more symmetrically beautiful.—<i>College, Edinburgh, 17th +April 1849.</i>”</p> +</div> + +<h3 id="APPENDIX_IV">No. IV.</h3> + +<p>I shall here shew, as I have done in a former work, how the curvilinear +outline of the figure is traced upon the rectilinear diagrams by portions of +the ellipse of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).</p> + +<div class="sidenote"><a href="#plate19">Plate XIX.</a></div> + +<p>The outline of the head and face, from points (1) to (3) (fig. 1, <a href="#plate19">Plate XIX.</a>), +takes the direction of the two first curves of the diagram. From point (3),<span class="pagenum"><a id="Page_101"></a>[101]</span> +the outline of the sterno-mastoid muscle continues to (4), where, joining the +outline of the trapezius muscle, at first concave, it becomes convex after +passing through (5), reaches the point (6), where the convex outline of the +deltoid muscle commences, and, passing through (7), takes the outline of the +arm as far as (8). The outline of the muscles on the side, the latissimus +dorsi and serratus magnus, commences under the arm at the point (9), and +joins the outline of the oblique muscle of the abdomen by a concave curve +at (10), which, rising into convexity as it passes through the points (11) and +(12), ends at (13), where it joins the outline of the gluteus medius muscle. +The outline of this latter muscle passes convexly through the point (14), and +ends at (15), where the outline of the tensor vaginæ femoris and vastus +externus muscle of the thigh commences. This convex outline joins the +concave outline of the biceps of the thigh at (16), which ends in that of the +slight convexity of the condyles of the thigh-bone at (17). From this point, +the outline of the outer surface of the leg, which includes the biceps, peroneus +longus, and soleus muscles, after passing through the point (18), continues +convexly to (19), where the concave outline of the tendons of the peroneus +longus continues to (20), whence the outline of the outer ankle and foot +commences.</p> + +<p>The outline of the mamma and fold of the arm-pit commences at (21), +and passes through the points (22) and (23). The circle at (24) is the outline +of the areola, in the centre of which the nipple is placed.</p> + +<p>The outline of the pubes commences at (25), and ends at the point (26), +from which the outline of the inner surface of the thigh proceeds over the +gracilis, the sartorius, and vastus internus muscles, until it meets the internal +face of the knee-joint at (27), the outline of which ends at (28). The outline +of the inside of the leg commences by proceeding over the gastrocnemius +muscle as far as (29), where it meets that of the soleus muscle, and continues +over the tendons of the heel until it meets the outline of the inner +ankle and foot at (30).</p> + +<p>The outline of the outer surface of the arm, as viewed in front, proceeds +from (8) over the remainder of the deltoid, in which there is here a slight +concavity, and, next, from (31) over the biceps muscle till (32), where it +takes the line of the long supinator, and passing concavely, and almost +imperceptibly, into the long and short radial extensor muscles, reaches the +wrist at (33). The outline of the inner surface of the arm from opposite +(9) commences by passing over the triceps extensor, which ends at (34), +then over the gentle convexity of the condyles of the bone of the arm at +(35), and, lastly, over the flexor sublimis which ends at the wrist-joint (36).</p> + +<p>The outline of the front of the figure commences at the point (1), (fig. 2, +<a href="#plate02">Plate II.</a>), and, passing almost vertically over the platzsma-myoidis muscles, +reaches the point (2), where the neck ends by a concave curve. From (2) +the outline rises convexly over the ends of the clavicles, and continues so +over the pectoral muscle till it reaches (3), where the mamma swells out<span class="pagenum"><a id="Page_102"></a>[102]</span> +convexly to (4), and returns convexly towards (5), where the curve becomes +concave. From (5) the outline follows the undulations of the rectus muscle +of the abdomen, concave at the points (6) and (7), and having its greatest +convexity at (8). This series of curves ends with a slight concavity at the +point (9), where the horizontal branch of the pubes is situated, over which +the outline is convex and ends at (10).</p> + +<p>The outline of the thigh commences at the point (11) with a slight concave +curve, and then swells out convexly over the extensors of the leg, and, +reaching (12), becomes gently concave, and, passing over the patella at (13), +becomes again convex until it reaches the ligament of that bone, where it +becomes gently concave towards the point (14), whence it follows the slightly +convex curve of the shin-bone, and then, becoming as slightly concave, ends +with the muscles in front of the leg at (15).</p> + +<p>The outline of the back commences at the point (16), and, following with +a concave curve the muscles of the neck as far as (17), swells into a convex +curve over the trapezius muscle towards the point (18); passing through +which point, it continues to swell outward until it reaches (19), half way +between (18) and (20); whence the convexity, becoming less and less, falls +into the concave curve of the muscles of the loins at (21), and passing +through the point (22), it rises into convexity. It then passes through the +point (23), follows the outline of the gluteus maximus, the convex curve of +which rises to the point (24), and then returns inwards to that of (25), +where it ends in the fold of the hip. From this point the outline follows +the curve of the hamstring muscles by a slight concavity as far as (26), and +then, becoming gently convex, it reaches (27); whence it becomes again +gently concave, with a slight indication of the condyle of the thigh-bone at +(28), and, reaching (29), follows the convex curve of the gastrocnemius +muscle through the point (30), and falling into the convex curve of the +tendo Achilles at (31), ends in the concavity over the heel at (32).</p> + +<p>The outline of the front of the arm commences at the point (33), by a +gentle concavity at the arm-pit, and then swells out in a convex curve +over the biceps, reaching (34), where it becomes concave, and passing +through (35), again becomes convex in passing over the long supinator, and, +becoming gently concave as it passes the radial extensors, rises slightly at +(36), and ends at (37), where the outline of the wrist commences. The +outline of the back of the arm commences with a concave curve at (38), +which becomes convex as it passes from the deltoid to the long extensor +and ends at the elbow (39), from below which the outline follows the convex +curve of the extensor ulnaris, reaching the wrist at the point (40).</p> + +<p>It will be seen that the various undulations of the outline are regulated +by points which are determined generally by the intersections and sometimes +by directions and extensions of the lines of the diagram, in the same +manner in which I shewed proportion to be imparted, in a late work, to the +osseous structure. The mode in which the curves of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and<span class="pagenum"><a id="Page_103"></a>[103]</span> +(¹⁄₆) are thus so harmoniously blended in the outline of the female figure, +only remains to be explained.</p> + +<p>The curves which compose the outline of the female form are therefore +simply those of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).</p> + +<p>Manner in which these curves are disposed in the lateral outline (figure 1, +<a href="#plate19">Plate XIX.</a>):—</p> + +<table> + <tr> + <th></th> + <th></th> + <th colspan="3">Points.</th> + <th>Curves.</th> + </tr> + <tr> + <td class="tdl">Head</td> + <td>from</td> + <td class="tdr">1</td> + <td>to</td> + <td class="tdr">2</td> + <td>(¹⁄₂)</td> + </tr> + <tr> + <td class="tdl">Face</td> + <td>”</td> + <td class="tdr">2</td> + <td>”</td> + <td class="tdr">3</td> + <td>(¹⁄₃)</td> + </tr> + <tr> + <td class="tdl">Neck</td> + <td>”</td> + <td class="tdr">3</td> + <td>”</td> + <td class="tdr">4</td> + <td>(¹⁄₅)</td> + </tr> + <tr> + <td class="tdl">Shoulder</td> + <td>”</td> + <td class="tdr">4</td> + <td>”</td> + <td class="tdr">6</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl"><span class="ditto">”</span></td> + <td>”</td> + <td class="tdr">6</td> + <td>”</td> + <td class="tdr">8</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdl">Trunk</td> + <td>”</td> + <td class="tdr">9</td> + <td>”</td> + <td class="tdr">15</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdl"><span class="ditto">”</span></td> + <td>”</td> + <td class="tdr">21</td> + <td>”</td> + <td class="tdr">24</td> + <td>(¹⁄₂)</td> + </tr> + <tr> + <td class="tdl">Outer surface of thigh and leg</td> + <td>”</td> + <td class="tdr">15</td> + <td>”</td> + <td class="tdr">20</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl">Inner surface of thigh and leg</td> + <td>”</td> + <td class="tdr">25</td> + <td>”</td> + <td class="tdr">30</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl">Outer surface of the arm</td> + <td>”</td> + <td class="tdr">8</td> + <td>”</td> + <td class="tdr">33</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl">Inner surface of the arm</td> + <td>”</td> + <td class="tdr">9</td> + <td>”</td> + <td class="tdr">36</td> + <td>(¹⁄₆)</td> + </tr> +</table> + +<p>Manner in which they are disposed in the outline (figure 2, <a href="#plate19">Plate XIX.</a>):—</p> + +<table> + <tr> + <th></th> + <th></th> + <th colspan="3">Points.</th> + <th>Curves.</th> + </tr> + <tr> + <td class="tdl">Front of neck</td> + <td>from</td> + <td class="tdr">1</td> + <td>to</td> + <td class="tdr">2</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl"><span class="ditto">”</span> trunk</td> + <td>”</td> + <td class="tdr">2</td> + <td>”</td> + <td class="tdr">10</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdl">Back of neck</td> + <td>”</td> + <td class="tdr">16</td> + <td>”</td> + <td class="tdr">18</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl"><span class="ditto">”</span> trunk</td> + <td>”</td> + <td class="tdr">18</td> + <td>”</td> + <td class="tdr">23</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdl"><span class="ditto">”</span> ”</td> + <td>”</td> + <td class="tdr">23</td> + <td>”</td> + <td class="tdr">25</td> + <td>(¹⁄₃)</td> + </tr> + <tr> + <td class="tdl">Front of thigh and leg</td> + <td>”</td> + <td class="tdr">11</td> + <td>”</td> + <td class="tdr">13</td> + <td>(¹⁄₄)</td> + </tr> + <tr> + <td class="tdl"><span class="ditto">”</span> ” <span class="ditto">”</span></td> + <td>”</td> + <td class="tdr">13</td> + <td>”</td> + <td class="tdr">15</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl">Back of thigh and leg</td> + <td>”</td> + <td class="tdr">25</td> + <td>”</td> + <td class="tdr">32</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl">Front of the arm</td> + <td>”</td> + <td class="tdr">33</td> + <td>”</td> + <td class="tdr">37</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl">Back of the arm</td> + <td>”</td> + <td class="tdr">38</td> + <td>”</td> + <td class="tdr">40</td> + <td>(¹⁄₆)</td> + </tr> + <tr> + <td class="tdl">Foot</td> + <td>”</td> + <td class="tdr">0</td> + <td>”</td> + <td class="tdr">0</td> + <td>(¹⁄₆)</td> + </tr> +</table> + +<div class="sidenote"><a href="#plate20">Plate XX.</a></div> + +<p>In order to exemplify more clearly the manner in which these various +curves appear in the outline of the figure, I give in <a href="#plate20">Plate XX.</a> the whole +curvilinear figures, complete, to which these portions belong that form the +outline of the sides of the head, neck, and trunk, and of the outer surface of +the thighs and legs.</p> + +<p>The various angles which the axes of these ellipses form with the vertical, +will be found amongst other details in the works I have just referred to.</p> + +<p><span class="pagenum"><a id="Page_104"></a>[104]</span></p> + +<h3 id="APPENDIX_V">No. V.</h3> + +<p>At page <a href="#Page_85">85</a> I have remarked upon the variety that may be introduced +into any particular form of vase; and, in order to give the reader an idea +of the ease with which this may be done without violating the harmonic +law, I shall here give three examples:—</p> + +<div class="sidenote"><a href="#plate21">Plate XXI.</a></div> + +<p>The first of these (<a href="#plate21">Plate XXI.</a>) differs from the Portland vase, in the concave +curve of the neck flowing more gradually into the convex curve of the +body.</p> + +<div class="sidenote"><a href="#plate22">Plate XXII.</a></div> + +<p>The second (<a href="#plate22">Plate XXII.</a>) differs from the same vase in the same change +of contour, as also in being of a smaller diameter at the top and at the +bottom.</p> + +<div class="sidenote"><a href="#plate23">Plate XXIII.</a></div> + +<p>The third (<a href="#plate23">Plate XXIII.</a>) is the most simple arrangement of the elliptic +curve by which this kind of form may be produced; and it differs from the +Portland vase in the relative proportions of height and diameter, and in +having a fuller curve of contour.</p> + +<p>The following comparison of the angles employed in these examples, with +the angles employed in the original, will shew that the changes of contour +in these forms, arise more from the mode in which the angles are arranged +than in a change of the angles themselves:—</p> + +<table> + <tr> + <th></th> + <th>Line</th> + <th></th> + <th>Line</th> + <th></th> + <th>Line</th> + <th></th> + <th>Line</th> + <th></th> + <th>Line</th> + <th></th> + <th>Line</th> + <th></th> + <th></th> + <th></th> + <th></th> + <th></th> + </tr> + <tr> + <td class="nw">Plate VIII.</td> + <td><i>A C</i></td> + <td>(¹⁄₂)</td> + <td><i>B C</i></td> + <td>(¹⁄₃)</td> + <td><i>o p</i></td> + <td>(H)</td> + <td><i>v u</i></td> + <td>(³⁄₁₀)</td> + <td><i>m n</i></td> + <td>(¹⁄₃)</td> + <td><i>i k</i></td> + <td>(¹⁄₅)</td> + <td>ellipse</td> + <td>(¹⁄₄)</td> + <td>rectangle</td> + <td>(²⁄₅)</td> + </tr> + <tr> + <td class="nw">Plate XXI.</td> + <td></td> + <td>(¹⁄₂)</td> + <td></td> + <td>(¹⁄₃)</td> + <td></td> + <td>(²⁄₉)</td> + <td></td> + <td>(¹⁄₄)</td> + <td></td> + <td>(²⁄₉)</td> + <td></td> + <td>(¹⁄₅)</td> + <td></td> + <td>(¹⁄₄)</td> + <td></td> + <td>(²⁄₅)</td> + </tr> + <tr> + <td class="nw">Plate XXII.</td> + <td></td> + <td>(¹⁄₂)</td> + <td></td> + <td>(¹⁄₃)</td> + <td></td> + <td>(¹⁄₈)</td> + <td></td> + <td>(⁴⁄₉)</td> + <td></td> + <td>(¹⁄₃)</td> + <td></td> + <td>(¹⁄₅)</td> + <td></td> + <td>(¹⁄₄)</td> + <td></td> + <td>(²⁄₅)</td> + </tr> + <tr> + <td rowspan="2" class="nw valign">Plate XXIII.</td> + <td rowspan="2" class="valign"></td> + <td rowspan="2" class="valign">(¹⁄₂)</td> + <td rowspan="2" class="valign"></td> + <td rowspan="2" class="valign">(¹⁄₄)</td> + <td rowspan="2" class="valign"></td> + <td rowspan="2" class="valign">(H)</td> + <td rowspan="2" class="valign"></td> + <td rowspan="2" class="valign">(-)</td> + <td rowspan="2" class="valign"></td> + <td rowspan="2" class="valign">(¹⁄₅)</td> + <td rowspan="2" class="valign"></td> + <td rowspan="2" class="valign">(¹⁄₅)</td> + <td rowspan="2" class="valign">ellipses</td> + <td>{ (¹⁄₃) }</td> + <td rowspan="2" class="valign"></td> + <td rowspan="2" class="valign">(¹⁄₃)</td> + </tr> + <tr> + <td>{ (¹⁄₄) }</td> + </tr> +</table> + +<p>The harmonic elements of each are therefore simply the following parts +of the right angle:—</p> + +<table> + <tr> + <th></th> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Submediant.</th> + </tr> + <tr> + <td class="nw">Plate VIII.</td> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(³⁄₁₀)</td> + </tr> + <tr> + <td></td> + <td>(¹⁄₄)</td> + <td></td> + <td></td> + <td></td> + </tr> + <tr> + <th></th> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Supertonic.</th> + </tr> + <tr> + <td class="nw">Plate XXI.</td> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(²⁄₉)</td> + </tr> + <tr> + <td></td> + <td>(¹⁄₄)</td> + <td></td> + <td></td> + <td><span class="pagenum"><a id="Page_105"></a>[105]</span></td> + </tr> + <tr> + <th></th> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th>Supertonic.</th> + </tr> + <tr> + <td class="nw">Plate XXII.</td> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td>(⁴⁄₉)</td> + </tr> + <tr> + <td></td> + <td>(¹⁄₄)</td> + <td></td> + <td></td> + <td></td> + </tr> + <tr> + <td></td> + <td>(¹⁄₈)</td> + <td></td> + <td></td> + <td></td> + </tr> + <tr> + <th></th> + <th>Tonic.</th> + <th>Dominant.</th> + <th>Mediant.</th> + <th></th> + </tr> + <tr> + <td class="nw">Plate XXIII.</td> + <td>(¹⁄₂)</td> + <td>(¹⁄₃)</td> + <td>(¹⁄₅)</td> + <td></td> + </tr> + <tr> + <td></td> + <td>(¹⁄₄)</td> + <td></td> + <td></td> + <td></td> + </tr> +</table> + +<h3 id="APPENDIX_VI">No. VI.</h3> + +<p>So far as I know, there has been only one attempt in modern times, +besides my own, to establish a universal system of proportion, based +on a law of nature, and applicable to art. This attempt consists of a +work of 457 pages, with 166 engraved illustrations, by Dr Zeising, a professor +in Leipzic, where it was published in 1854.</p> + +<p>One of the most learned and talented professors in our Edinburgh University +has reviewed that work as follows:—</p> + +<p>“It has been rather cleverly said that the intellectual distinction between an +Englishman and a Scotchman is this—‘Give an Englishman two facts, and +he looks out for a third; give a Scotchman two facts, and he looks out for a +theory.’ Neither of these tests distinguishes the German; he is as likely to +seek for a third fact as for a theory, and as likely to build a theory on two facts +as to look abroad for further information. But once let him have a theory +in his mind, and he will ransack heaven and earth until he almost buries it +under the weight of accumulated facts. This remark applies with more than +common force to a treatise published last year by Dr Zeising, a professor in +Leipsic, ‘On a law of proportion which rules all nature.’ The ingenious +author, after proving from the writings of ancient and modern philosophers +that there always existed the belief (whence derived it is difficult to say), that +some law does bind into one formula all the visible works of God, proceeds +to criticise the opinions of individual writers respecting that connecting law. +It is not our purpose to follow him through his lengthy examination. +Suffice it to say that he believes he has found the lost treasure in the +<i>Timæus</i> of Plato, c. 31. The passage is confessedly an obscure one, and will +not bear a literal translation. The interpretation which Dr Zeising puts on +it is certainly a little strained, but we are disposed to admit that he does it +with considerable reason. Agreeably to him, the passage runs thus:—‘That +bond is the most beautiful which binds the things as much as possible +into one; and proportion effects this most perfectly when three things are<span class="pagenum"><a id="Page_106"></a>[106]</span> +so united that the greater bears to the middle the same ratio that the middle +bears to the less.’</p> + +<p>“We must do Dr Zeising the justice to say that he has not made more than +a legitimate use of the materials which were presented to him in the writings +of the ancients, in his endeavour to establish the fact of the existence of this +law amongst them. The canon of Polycletes, the tradition of Varro mentioned +by Pliny relative to that canon, the writings of Galen and others, +are all brought to bear on the same point with more or less force. The sum +of this portion of the argument is fairly this,—that the ancient sculptors had +<i>some</i> law of proportion—some authorised examplar to which they referred as +their work proceeded. That it was the law here attributed to Plato is by no +means made out; but, considering the incidental manner in which that law +is referred to, and the obscurity of the passages as they exist, it is, perhaps, +too much to expect more than this broad feature of coincidence—the +fact that some law was known and appealed to. Dr Zeising now proceeds to +examine modern theories, and it is fair to state that he appears generally to +take a very just view of them.</p> + +<p>“Let us now turn to Dr Zeising’s own theory. It is this—that in every +beautiful form lines are divided in extreme and mean ratio; or, that any +line considered as a whole, bears to its larger part the same proportion that +the larger bears to the smaller—thus, a line of 5 inches will be divided into +parts which are very nearly 2 and 3 inches respectively (1·9 and 3·1 inches). +This is a well-known division of a line, and has been called the <span class="smcap">golden</span> rule, +but when or why, it is not easy to ascertain. With this rule in his hand, +Dr Zeising proceeds to examine all nature and art; nay, he even ventures +beyond the threshold of nature to scan Deity. We will not follow him +so far. Let us turn over the pages of his carefully illustrated work, and +see how he applies his line. We meet first with the Apollo Belvidere—the +golden line divides him happily. We cannot say the same of the division +of a handsome face which occurs a little further on. Our preconceived +notions have made the face terminate with the chin, and not with the centre +of the throat. It is evident that, with such a rule as this, a little latitude +as to the extreme point of the object to be measured, relieves its inventor from +a world of perplexities. This remark is equally applicable to the <i>arm</i> which +follows, to which the rule appears to apply admirably, yet we have tried it +on sundry plates of arms, both fleshy and bony, without a shadow of success. +Whether the rule was made for the arm or the arm for the rule, we do not +pretend to decide. But let us pass hastily on to page 284, where the Venus +de Medicis and Raphael’s Eve are presented to us. They bear the application +of the line right well. It might, perhaps, be objected that it is remarkable +that the same rule applies so exactly to the existing position +of the figures, such as the Apollo and the Venus, the one of which is +upright, and the other crouching. But let that pass. We find Dr +Zeising next endeavouring to square his theory with the distances of the<span class="pagenum"><a id="Page_107"></a>[107]</span> +planets, with wofully scanty success. Descending from his lofty position, he +spans the earth from corner to corner, at which occupation we will leave him +for a moment, whilst we offer a suggestion which is equally applicable to +poets, painters, novelists, and theorisers. Never err in excess—defect is the +safe side—it is seldom a fault, often a real merit. Leave something for the student +of your works to do—don’t chew the cud for him. Be assured he will not +omit to pay you for every little thing which you have enabled him to discover. +Poor Professor Zeising! he is far too German to leave any field of discovery +open for his readers. But let us return to him; we left him on his +back, lost for a time in a hopeless attempt to double Cape Horn. We +will be kind to him, as the child is to his man in the Noah’s ark, and +set him on his legs amongst his toys again. He is now in the vegetable +kingdom, amidst oak leaves and sections of the stems of divers plants. +He is in his element once more, and it were ungenerous not to admit +the merit of his endeavours, and the success which now and then attends +it. We will pass over his horses and their riders, together with that +portly personage, the Durham ox, for we have caught a glimpse of a form +familiar to our eyes, the ever-to-be-admired Parthenon. This is the true +test of a theory. Unlike the Durham ox just passed before us, the Parthenon +will stand still to be measured. It has so stood for twenty centuries, +and every one that has scanned its proportions has pronounced them exquisite. +Beauty is not an adaptation to the acquired taste of a single nation, +or the conventionality of a single generation. It emanates from a deep-rooted +principle in nature, and appeals to the verdict of our whole humanity. +We don’t find fault with the Durham ox—his proportions are probably +good, though they be the result of breeding and cross-breeding; still we are +not sure whether, in the march of agriculture, our grandchildren may not +think him a very wretched beast. But there is no mistake about the Parthenon; +as a type of proportion it stands, has stood, and shall stand. Well, +then, let us see how Dr Zeising succeeds with his rule here. Alas! not a +single point comes right. The Parthenon is condemned, or its condemnation +condemns the theory. Choose your part. We choose the +latter alternative; and now, our choice being made, we need proceed no +further. But a question or two have presented themselves as we went +along, which demand an answer. It may be asked—How do you account +for the esteem in which this law of the section in extreme and mean ratio +was held? We reply—That it was esteemed just in the same way that a tree +is esteemed for its fruit. To divide a right angle into two or three, four or +six, equal parts was easy enough. But to divide it into five or ten such +parts was a real difficulty. And how was the difficulty got over? It was +effected by means of this golden rule. This is its great, its ruling application; +and if we adopt the notion that the ancients were possessed with the idea of +the existence of angular symmetry, we shall have no difficulty in accounting +for their appreciation of this problem. Nay, we may even go further, and<span class="pagenum"><a id="Page_108"></a>[108]</span> +admit, with Dr Zeising, the interpretation of the passage of Plato,—only +with this limitation, that Plato, as a geometer, was carried away by the +geometry of æsthetics from the thing itself. It may be asked again—Is it +not probable that <i>some</i> proportionality does exist amongst the parts of +natural objects? We reply—That, <i>à priori</i>, we expect some such system to +exist, but that it is inconsistent with the scheme of <i>least effort</i>, which pervades +and characterises all natural succession in space or in time, that that +system should be a complicated one. Whatever it is, its essence must be +simplicity. And no system of simple linear proportion is found in nature; +quite the contrary. We are, therefore, driven to another hypothesis, viz.—that +the simplicity is one of angles, not of lines; that the eye estimates by +search round a point, not by ascending and descending, going to the right +and to the left,—a theory which we conceive all nature conspires to prove. +Beauty was not created for the eye of man, but the eye of man and his +mental eye were created for the appreciation of beauty. Examine the forms +of animals and plants so minute that nothing short of the most recent +improvements in the microscope can succeed in detecting their symmetry; +or examine the forms of those little silicious creations which grew thousands +of years before Man was placed on the earth, and, with forms of marvellous +and varied beauty, they all point to its source in angular symmetry. This +is the keystone of formal beauty, alike in the minutest animalcule, and in +the noblest of God’s works, his own image—Man.”</p> + +<p class="titlepage">THE END.</p> + +<p class="center smaller">BALLANTYNE AND COMPANY, PRINTERS, EDINBURGH.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="footnotes"> + +<div class="chapter"> + +<h2 class="nobreak" id="FOOTNOTES">FOOTNOTES</h2> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> Sir David Brewster.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> No. CLVIII., October 1843.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> Diogenes Laërtius’s “Lives of the Philosophers,” literally translated. Bohn: +London.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> Ibid.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> Rose’s “Biographical Dictionary.”</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> Professor Laycock, now of the University of Edinburgh.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> “The Geometric Beauty of the Human Figure Defined,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a> Longman and Co., London.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a> See <a href="#APPENDIX">Appendix</a>.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a> “The Orthographic Beauty of the Parthenon,” &c., and “The Harmonic Law +of Nature applied to Architectural Design.”</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_11" href="#FNanchor_11" class="label">[11]</a> For further details, see “Harmonic Law of Nature,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_12" href="#FNanchor_12" class="label">[12]</a> By a very simple machine, which I have lately invented, an ellipse of any +given proportions, even to those of (¹⁄₆₄), which is the curve of the entases of the +columns of the Parthenon (see <a href="#plate07">Plate VII.</a>), and of any length, from half an inch +to fifty feet or upwards, may be easily and correctly described; the length and +angle of the required ellipse being all that need be given.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_13" href="#FNanchor_13" class="label">[13]</a> “The Orthographic Beauty of the Parthenon,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_14" href="#FNanchor_14" class="label">[14]</a> “The Orthographic Beauty of the Parthenon,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_15" href="#FNanchor_15" class="label">[15]</a> Ibid.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_16" href="#FNanchor_16" class="label">[16]</a> “The Harmonic Law of Nature applied to Architectural Design.”</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_17" href="#FNanchor_17" class="label">[17]</a> “Physio-philosophy.” By Dr Oken. Translated by Talk; and published by +the Ray Society. London, 1848.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_18" href="#FNanchor_18" class="label">[18]</a> “The Science of those Proportions by which the Human Head and Countenance, +as represented in Works of ancient Greek Art, are distinguished from those +of ordinary Nature.”</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_19" href="#FNanchor_19" class="label">[19]</a> “The Geometric Beauty of the Human Figure Defined,” &c., and “The +Natural Principles of Beauty Developed in the Human Figure.”</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_20" href="#FNanchor_20" class="label">[20]</a> “The Geometric Beauty of the Human Figure Defined,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_21" href="#FNanchor_21" class="label">[21]</a> “Essay on Ornamental Design,” &c., and “The Geometric Beauty of the Human +Figure,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_22" href="#FNanchor_22" class="label">[22]</a> “A Nomenclature of Colours, applicable to the Arts and Natural Sciences,” +&c., &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_23" href="#FNanchor_23" class="label">[23]</a> See <a href="#Page_24">pp. 24 and 25</a>.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_24" href="#FNanchor_24" class="label">[24]</a> “The Principles of Beauty in Colouring Systematised,” Fourteen Diagrams, +each containing Six Colours and Hues.</p> + +<p>“A Nomenclature of Colours,” &c., Forty Diagrams, each containing Twelve +Examples of Colours, Hues, Tints, and Shades.</p> + +<p>“The Laws of Harmonious Colouring,” &c., One Diagram, containing Eighteen +Colours and Hues.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_25" href="#FNanchor_25" class="label">[25]</a> “Etchings Representing the Best Examples of Grecian and Roman Architectural +Ornament, drawn from the Originals,” &c. By Charles Heathcote Tatham, +Architect. London: Priestly and Weale. 1826.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_26" href="#FNanchor_26" class="label">[26]</a> “The Orthographic Beauty of the Parthenon,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_27" href="#FNanchor_27" class="label">[27]</a> “Science of those Proportions,” &c.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_28" href="#FNanchor_28" class="label">[28]</a> Abbé Barthélémie’s “Travels of Anacharsis in Greece,” vol iv., pp. 193, 195.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_29" href="#FNanchor_29" class="label">[29]</a> Abbé Barthélémie (vol. ii., pp. 168, 169), who cites as his authorities, Cicer. De Nat. +Deor., lib. i., cap. ii., t. 2, p. 405; Justin Mart., Ovat. ad Gent., p. 10; Aristot. Metaph., +lib. i., cap. v., t. 2, p. 845.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_30" href="#FNanchor_30" class="label">[30]</a> Müller’s “Ancient Art and its Remains.”</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_31" href="#FNanchor_31" class="label">[31]</a> “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. ii., p. 325.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_32" href="#FNanchor_32" class="label">[32]</a> “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. vi., p. 225. The +authorities the Abbé quotes are—Quintil., lib. xii., cap. x., p. 744; Plin., lib. xxxv., cap. ix., +p. 691.</p> + +</div> + +<div class="footnote"> + +<p><a id="Footnote_33" href="#FNanchor_33" class="label">[33]</a> Müller’s “Archæology of Art,” &c.</p> + +</div> + +</div> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> + +<h2 class="nobreak" id="Works_by_the_Same_Author">Works by the Same Author.</h2> + +</div> + +<h3>I.<br> +<span class="smaller">In royal 8vo, with Copperplate Illustrations, price 2s. 6d.,</span><br> +THE HARMONIC LAW OF NATURE APPLIED TO ARCHITECTURAL DESIGN.</h3> + +<p class="center"><i>From the Athenæum.</i></p> + +<p>The beauty of the theory is its universality, and its simplicity. In +nature, the Creator accomplished his purposes by the simplest means—the +harmony of nature is indestructible and self-restoring. Mr Hay’s book on +the “Parthenon,” on the “Natural Principles of Beauty as developed in the +Human Figure,” his “Principles of Symmetrical Beauty,” his “Principles of +Colouring, and Nomenclature of Colours,” his “Science of Proportion,” and +“Essay on Ornamental Design,” we have already noticed with praise as the +results of philosophical and original thought.</p> + +<p class="center"><i>From the Daily News.</i></p> + +<p>This essay is a new application to Lincoln cathedral in Gothic +architecture, and to the Temple of Theseus in Greek architecture, of the +principles of symmetrical beauty already so profusely illustrated and +demonstrated by Mr Hay. The theory which Mr Hay has propounded in so many +volumes is not only a splendid contribution towards a science of æsthetic +proportions, but, for the first time in the history of art, proves the +possibility, and lays the foundations of such a science. To those who are +not acquainted with the facts, these expressions will sound hyperbolical, +but they are most true.</p> + +<h3>II.<br> +<span class="smaller">In royal 8vo, with Copperplate Illustrations, price 5s.,</span><br> +THE NATURAL PRINCIPLES OF BEAUTY, AS DEVELOPED IN THE HUMAN FIGURE.</h3> + +<p class="center"><i>From the Spectator.</i></p> + +<p>We cannot refuse to entertain Mr Hay’s system as of singular intrinsic +excellence. The simplicity of his law and its generality impress +themselves more deeply on the conviction with each time of enforcement. +His theory proceeds from the idea, that in nature every thing is effected +by means more simple than any other that could have been conceived,—an +idea certainly consistent with whatever we can trace out or imagine of +the all-wise framing of the universe.</p> + +<p class="center"><i>From the Sun.</i></p> + +<p>By founding (if we may so phrase it) this noble theory, Mr Hay has +covered his name with distinction, and has laid the basis, we conceive, +of no ephemeral reputation. By illustrating it anew, through the +medium of this graceful treatise, he has conferred a real boon upon +the community, for he has afforded the public another opportunity of +following the golden rule of the poet—by looking through the holy and +awful mystery of creation to the holier and yet more awful mystery of +Omnipotence.</p> + +<p class="center"><i>From the Cambridge Journal of Classical and Sacred Philology.</i></p> + +<p>The inquiries which of late years have been instituted by Mr D. R. Hay +of Edinburgh, on the proportions of the human figure, and on the natural +principles of beauty, as illustrated by works of Greek art, constitute an +epoch in the study of æsthetics and the philosophy of form.</p> + +<h3>III.<br> +<span class="smaller">In royal 8vo, with Copperplate Illustrations, price 5s.,</span><br> +THE ORTHOGRAPHIC BEAUTY OF THE PARTHENON REFERRED TO A LAW OF NATURE.</h3> + +<p class="center">To which are prefixed, a few Observations on the Importance of Æsthetic +Science as an Element in Architectural Education.</p> + +<p class="center"><i>From the Scottish Literary Gazette.</i></p> + +<p>We think this work will satisfy every impartial mind that Mr Hay has +developed the true theory of the Parthenon—that he has, in fact, to +use a kindred phraseology, both <i>parsed</i> and <i>scanned</i> this +exquisitely beautiful piece of architectural composition, and that, in +doing so, he has provided the true key by which the treasures of Greek +art may be further unlocked, and rendered the means of correcting, +improving, and elevating modern practice.</p> + +<p class="center"><i>From the Edinburgh Guardian.</i></p> + +<p>Again and again the attempt has been made to detect harmonic ratios in +the measurement of Athenian architecture, but ever without reward. Mr Hay +has, however, made the discovery, and to an extent of which no one had +previously dreamt.</p> + +<h3>IV.<br> +<span class="smaller">In 8vo, 100 Plates, price 6s.,</span><br> +FIRST PRINCIPLES OF SYMMETRICAL BEAUTY.</h3> + +<p class="center"><i>From the Spectator.</i></p> + +<p>This is a grammar of pure form, in which the elements of symmetrical, as +distinguished from picturesque beauty, are demonstrated, by reducing the +outlines or planes of curvilinear and rectilinear forms to their origin +in the principles of geometrical proportion. In thus analysing symmetry +of outline in natural and artificial objects, Mr Hay determines the fixed +principles of beauty in positive shape, and shews how beautiful forms may +be reproduced and infinitely varied with mathematical precision. Hitherto +the originating and copying of beautiful forms have been alike empirical; +the production of a new design for a vase or a jug has been a matter of +chance between the eye and the hand; and the copying of a Greek moulding +or ornament, a merely mechanical process. By a series of problems, Mr +Hay places both the invention and imitation of beautiful forms on a sure +basis of science, giving to the fancy of original minds a clue to the +evolving of new and elegant shapes, in which the infinite resources of +nature are made subservient to the uses of art.</p> + +<p>The volume is illustrated by one hundred diagrams beautifully executed, +that serve to explain the text, and suggest new ideas of beauty of +contour in common objects. To designers of pottery, hardware, and +architectural ornaments, this work is particularly valuable; but artists +of every kind, and workmen of intelligence, will find it of great utility.</p> + +<p class="center"><i>From the Athenæum.</i></p> + +<p>The volume before us is the seventh of Mr Hay’s works. It is the most +practical and systematic, and likely to be one of the most useful. It +is, in short, a grammar of form, or a spelling-book of beauty. This is +beginning at the right end of the matter; and the necessity for this kind +of knowledge will inevitably, though gradually, be felt. The work will, +therefore, be ultimately appreciated and adopted as an introduction to +the study of beautiful forms.</p> + +<p>The third part of the work treats of the Greek oval or composite ellipse, +as Mr Hay calls it. It is an ellipse of three foci, and gives practical +forms for vases and architectural mouldings, which are curious to the +architect, and will be very useful both to the potter, the moulder, +and the pattern-drawer. A fourth part contains applications of this to +practice. Of the details worked out with so much judgment and ingenuity +by Mr Hay, we should in vain attempt to communicate just notions without +the engravings of which his book is full. We must, therefore, refer to +the work itself. The forms there given are full of beauty, and so far +tend to prove the system.</p> + +<h3>V.<br> +<span class="smaller">In 8vo, 14 Coloured Diagrams, Second Edition, price 15s.,</span><br> +THE PRINCIPLES OF BEAUTY IN COLOURING SYSTEMATISED.</h3> + +<p class="center"><i>From the Spectator.</i></p> + +<p>In this new analysis of the harmonies of colour, Mr Hay has performed the +useful service of tracing to the operation of certain fixed principles +the sources of beauty in particular combinations of hues and tints; so +that artists may, by the aid of this book, produce, with mathematical +certainty, the richest effects, hitherto attainable by genius alone. Mr +Hay has reduced this branch of art to a perfect system, and proved that +an offence against good taste in respect to combinations of colour is, in +effect, a violation of natural laws.</p> + +<h3>VI.<br> +<span class="smaller">In 8vo, 228 Examples of Colours, Hues, Tints, and Shades, price 63s.,</span><br> +A NOMENCLATURE OF COLOURS, APPLICABLE TO THE ARTS AND NATURAL SCIENCES.</h3> + +<p class="center"><i>From the Daily News.</i></p> + +<p>In this work Mr Hay has brought a larger amount of practical knowledge +to bear on the subject of colour than any other writer with whom we are +acquainted, and in proportion to this practical knowledge is, as might be +expected, the excellence of his treatise. There is much in this volume +which we would most earnestly recommend to the notice of artists, house +decorators, and, indeed, to all whose business or profession requires a +knowledge of the management of colour. The work is replete with hints +which they might turn to profitable account, and which they will find +nowhere else.</p> + +<p class="center"><i>From the Athenæum.</i></p> + +<p>We have formerly stated the high opinion we entertain of Mr Hay’s +previous exertions for the improvement of decorative art in this country. +We have already awarded him the merit of invention and creation of the +new and the beautiful in form. In his former treatises he furnished a +theory of definite proportions for the creation of the beautiful in form. +In the present work he proposes to supply a scale of definite proportions +for chromatic beauty. For this purpose he sets out very properly with a +precise nomenclature of colour. In this he has constructed a vocabulary +for the artist—an alphabet for the artizan. He has gone further—he +constructs words for three syllables. From this time, it will be possible +to write a letter in Edinburgh about a coloured composition, which shall +be read off in London, Paris, St Petersburg, or Pekin, and shall so +express its nature that it can be reproduced in perfect identity. This Mr +Hay has done, or at least so nearly, as to deserve our thanks on behalf +of art, and artists of all grades, even to the decorative artizan—not +one of whom, be he house-painter, china pattern-drawer, or calico +printer, should be without the simple manual of “words for colours.”</p> + +<h3>VII.<br> +<span class="smaller">In post 8vo, with a Coloured Diagram, Sixth Edition, price 7s. 6d.,</span><br> +THE LAWS OF HARMONIOUS COLOURING ADAPTED TO INTERIOR DECORATIONS.</h3> + +<p class="center"><i>From the Atlas.</i></p> + +<p>Every line of this useful book shews that the author has contrived to +intellectualise his subject in a very interesting manner. The principles +of harmony in colour as applied to decorative purposes, are explained and +enforced in a lucid and practical style, and the relations of the various +tints and shades to each other, so as to produce a harmonious result, are +descanted upon most satisfactorily and originally.</p> + +<p class="center"><i>From the Edinburgh Review.</i></p> + +<p>In so far as we know, Mr Hay is the first and only modern artist who +has entered upon the study of these subjects without the trammels of +prejudice and authority. Setting aside the ordinances of fashion, as +well as the dicta of speculation, he has sought the foundation of +his profession in the properties of light, and in the laws of visual +sensation, by which these properties are recognised and modified. The +truths to which he has appealed are fundamental and irrefragable.</p> + +<p class="center"><i>From the Athenæum.</i></p> + +<p>We have regarded, and do still regard, the production of Mr Hay’s works +as a remarkable psychological phenomenon—one which is instructive both +for the philosopher and the critic to study with care and interest, not +unmingled with respect. We see how his mind has been gradually guided +by Nature herself out of one track, and into another, and ever and anon +leading him to some vein of the beautiful and true, hitherto unworked.</p> + +<h3>VIII.<br> +<span class="smaller">In 4to, 25 Plates, price 36s.,</span><br> +ON THE SCIENCE OF THOSE PROPORTIONS BY WHICH THE HUMAN HEAD AND +COUNTENANCE, AS REPRESENTED IN ANCIENT GREEK ART, ARE DISTINGUISHED FROM +THOSE OF ORDINARY NATURE.</h3> + +<p class="center smaller">(PRINTED BY PERMISSION.)</p> + +<p class="center"><i>From a Letter to the Author by Sir William Hamilton, Bart., Professor +of Logic and Metaphysics in the Edinburgh University.</i></p> + +<p>Your very elegant volume, “Science of those Proportions,” &c., is to me +extremely interesting, as affording an able contribution to what is the +ancient, and, I conceive, the true theory of the beautiful. But though +your doctrine coincides with the one prevalent through all antiquity, it +appears to me quite independent and original in you; and I esteem it the +more that it stands opposed to the hundred one-sided and exclusive views +prevalent in modern times.</p> + +<p class="center"><i>From Chambers’s Edinburgh Journal.</i></p> + +<p>We now come to another, and much more remarkable corroboration, which +calls upon us to introduce to our readers one of the most valuable and +original contributions that have ever been made to the Philosophy of Art, +viz., Mr Hay’s work “On the Science of those Proportions,” &c. Mr Hay’s +plan is simply to form a scale composed of the well-known vibrations of +the monochord, which are the alphabet of music, and then to draw upon +the quadrant of a circle angles <i>answering to these vibrations</i>. +With the series of triangles thus obtained he combines a circle and +an ellipse, the proportions of which are derived from the triangles +themselves; and thus he obtains an infallible rule for the composition of +the head of ideal beauty.</p> + +<h3>IX.<br> +<span class="smaller">In 4to, 16 Plates, price 30s.,</span><br> +THE GEOMETRIC BEAUTY OF THE HUMAN FIGURE DEFINED.</h3> + +<p class="center">To which is prefixed, a SYSTEM of ÆSTHETIC PROPORTION applicable to +ARCHITECTURE and the other FORMATIVE ARTS.</p> + +<p class="center"><i>From the Cambridge Journal of Classical and Sacred Philology.</i></p> + +<p>We feel bound to pay Mr Hay a large and glad tribute of praise for +having devised a system of proportions which rises superior to the +idiosyncrasies of different artists, which brings back to one common +type the sensations of Eye and Ear, and so makes a giant stride towards +that <i>codification</i> of the laws of the universe which it is the +business of science to effect. We have no hesitation in saying that, for +scientific precision of method and importance of results, Albert Durer, +Da Vinci, and Hogarth—not to mention less noteworthy writers—must all +yield the palm to Mr Hay.</p> + +<h3>X.<br> +<span class="smaller">In oblong folio, 57 Plates and numerous Woodcuts, price 42s.,</span><br> +AN ESSAY ON ORNAMENTAL DESIGN, IN WHICH ITS TRUE PRINCIPLES ARE DEVELOPED +AND ELUCIDATED, &c.</h3> + +<p class="center"><i>From the Athenæum.</i></p> + +<p>In conclusion, Mr Hay’s book goes forth with our best wishes. It must be +good. It must be prolific of thought—stimulant of invention. It is to be +acknowledged as a benefit of an unusual character conferred on the arts +of ornamental design.</p> + +<p class="center"><i>From the Spectator.</i></p> + +<p>Mr Hay has studied the subject deeply and scientifically. In this +treatise on ornamental design, the student will find a clue to the +discovery of the source of an endless variety of beautiful forms and +combinations of lines, in the application of certain fixed laws of +harmonious proportion to the purposes of art. Mr Hay also exemplifies +the application of his theory of linear harmony to the production of +beautiful forms generally, testing its soundness by applying it to the +human figure, and the purest creations of Greek art.</p> + +<p class="center"><i>From Fraser’s Magazine.</i></p> + +<p>Each part of this work is enriched by diagrams of great beauty, direct +emanations of principle, and, consequently, presenting entirely new +combinations of form. Had our space permitted, we should have made some +extracts from this “Essay on Ornamental Design;” and we would have done +so, because of the discriminating taste by which it is pervaded, and the +forcible observations which it contains; but we cannot venture on the +indulgence.</p> + +<h3>XI.<br> +<span class="smaller">In 4to, 17 Plates and 38 Woodcuts, price 25s.,</span><br> +PROPORTION, OR THE GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED.</h3> + +<h3>XII.<br> +<span class="smaller">In 4to, 18 Plates and numerous Woodcuts, price 15s.,</span><br> +THE NATURAL PRINCIPLES AND ANALOGY OF THE HARMONY OF FORM.</h3> + +<p class="center"><i>From the Edinburgh Review.</i></p> + +<p>Notwithstanding some trivial points of difference between Mr Hay’s views +and our own, we have derived the greatest pleasure from the perusal of +these works. They are all composed with accuracy and even elegance. His +opinions and views are distinctly brought before the reader, and stated +with that modesty which characterises genius, and that firmness which +indicates truth.</p> + +<p class="center"><i>From Blackwood’s Magazine.</i></p> + +<p>We have no doubt that when Mr Hay’s Art-discovery is duly developed and +taught, as it should be, in our schools, it will do more to improve the +general taste than anything which has yet been devised.</p> + +<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 75399 ***</div> +</body> +</html> + diff --git a/75399-h/images/cover.jpg b/75399-h/images/cover.jpg Binary files differnew file mode 100644 index 0000000..5f7ac15 --- /dev/null +++ b/75399-h/images/cover.jpg diff --git a/75399-h/images/figure1.jpg b/75399-h/images/figure1.jpg Binary files differnew file mode 100644 index 0000000..5cc5d05 --- /dev/null +++ b/75399-h/images/figure1.jpg diff --git a/75399-h/images/figure2.jpg b/75399-h/images/figure2.jpg Binary files differnew file 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