APPENDIX

If you add a line of 5 inches to one of 8 inches you produce one 13 inches long, and if you proceed by always adding the last two you arrive at a series of lengths, 5, 8, 13, 21, 34, 55 inches, &c. Mr. William Schooling¹ tells me that any two of these lines adjoining one another are practically in the same proportion to each other; that is to say, one 8 inches is 1.600 times the size of one 5 inches, and the 13-inch line is 1.625 the size of the 8-inch, and the 21-inch line being 1.615 times the 13-inch line, and so on. With the mathematician's love of accuracy, Mr. Schooling has worked out the exact proportion that should exist between a series of quantities for them to be in the same proportion to their neighbours, and in which any two added together would produce the next. There is only one proportion that will do this, and although very formidable, stated exactly, for practical purposes, it is that between 5 and a fraction over 8. Stated accurately² to eleven places of decimals it is (1 + sqrt(5))/2 = 1.61803398875 (nearly).

We have evidently here a very unique proportion. Mr. Schooling has called this the Phi proportion, and it will be convenient to refer to it by this name.



THE PHI PROPORTION
EC is 1.618033, &c., times size of     AB,
CD     BC,
DE     CD, &c.,
AC=CD
BD=DE, &c.

290Testing this proportion on the reproductions of pictures in this book in the order of their appearing, we find the following remarkable³ results:

"Los Meninas," Velazquez, page 60 [Transcribers Note: Plate IX].—The right-hand side of light opening of door at the end of the room is exactly Phi proportion with the two sides of picture; and further, the bottom of this opening is exactly Phi proportion with the top and bottom of canvas.

It will be noticed that this is a very important point in the "placing" of the composition.

"Fête Champêtre," Giorgione, page 151 [Transcribers Note: Plate XXXIII].—Lower end of flute⁴ held by seated female figure exactly Phi proportion with sides of picture, and lower side of hand holding it (a point slightly above the end of flute) exactly Phi proportion with top and bottom of canvas. This is also an important centre in the construction of the composition.

"Bacchus and Ariadne," Titian, page 154 [Transcribers Note: Plate XXXIV].—The proportion in this picture both with top and bottom and sides of canvas comes in the shadow under chin of Bacchus; the most important point in the composition being the placing of this head.

"Love and Death," by Watts⁵, page 158 [Transcribers Note: Plate XXXV].—Point from which drapery radiates on figure of Death exactly Phi proportion with top and bottom of picture.

Point where right-hand side of right leg of Love cuts dark edge of steps exactly Phi proportion with sides of picture.

"Surrender of Breda," by Velazquez, page 161 [Transcribers Note: Plate XXXVI].—First spear in upright row on the right top of picture, exactly Phi proportion with sides of canvas. Height of gun carried horizontally by man in middle distance above central group, exactly Phi proportion 291with top and bottom of picture. This line gives height of group of figures on left, and is the most important horizontal line in the picture.

"Birth of Venus," Botticelli, page 166 [Transcribers Note: Plate XXXVII].—Height of horizon line Phi proportion with top and bottom of picture. Height of shell on which Venus stands Phi proportion with top and bottom of picture, the smaller quantity being below this time. Laterally⁷ the extreme edge of dark drapery held by figure on right that blows towards Venus is Phi proportion with sides of picture.

"The Rape⁶ of Europa," by Paolo Veronese, page 168 [Transcribers Note: Plate XXXVIII].—Top of head of Europa exactly Phi proportion with top and bottom of picture. Right-hand side of same head slightly to left of Phi proportion with sides of picture (unless in the reproduction a part of the picture on the left has been trimmed away, as is likely, in which case it would be exactly Phi proportion).

I have taken the first seven pictures reproduced in this book that were not selected with any idea of illustrating⁸ this point, and I think you will admit that in each some very important quantity has been placed in this proportion. One could go on through all the illustrations were it not for the fear of becoming wearisome; and also, one could go on through some of the minor⁹ relationships, and point out how often this proportion turns up in compositions. But enough has been said to show that the eye evidently takes some especial pleasure in it, whatever may eventually be found to be the physiological¹⁰ reason underlying¹¹ it.

The End

1 schooling   

n.教育；正规学校教育

参考例句：

• A child's access to schooling varies greatly from area to area.孩子获得学校教育的机会因地区不同而大相径庭。
• Backward children need a special kind of schooling.天赋差的孩子需要特殊的教育。

2 accurately   

adv.准确地，精确地

参考例句：

• It is hard to hit the ball accurately.准确地击中球很难。
• Now scientists can forecast the weather accurately.现在科学家们能准确地预报天气。

3 remarkable   

adj.显著的，异常的，非凡的，值得注意的

参考例句：

• She has made remarkable headway in her writing skills.她在写作技巧方面有了长足进步。
• These cars are remarkable for the quietness of their engines.这些汽车因发动机没有噪音而不同凡响。

4 flute   

n.长笛；v.吹笛

参考例句：

• He took out his flute, and blew at it.他拿出笛子吹了起来。
• There is an extensive repertoire of music written for the flute.有很多供长笛演奏的曲目。

5 watts   

（电力计量单位）瓦，瓦特( watt的名词复数 )

参考例句：

• My lamp uses 60 watts; my toaster uses 600 watts. 我的灯用60瓦，我的烤面包器用600瓦。
• My lamp uses 40 watts. 我的灯40瓦。

6 rape   

n.抢夺，掠夺，强奸；vt.掠夺，抢夺，强奸

参考例句：

• The rape of the countryside had a profound ravage on them.对乡村的掠夺给他们造成严重创伤。
• He was brought to court and charged with rape.他被带到法庭并被指控犯有强奸罪。

7 laterally   

ad.横向地；侧面地；旁边地

参考例句：

• Shafts were sunk, with tunnels dug laterally. 竖井已经打下，并且挖有横向矿道。
• When the plate becomes unstable, it buckles laterally. 当板失去稳定时，就发生横向屈曲。

8 illustrating   

给…加插图( illustrate的现在分词 )； 说明； 表明； （用示例、图画等）说明

参考例句：

• He upstaged the other speakers by illustrating his talk with slides. 他演讲中配上幻灯片，比其他演讲人更吸引听众。
• Material illustrating detailed structure of graptolites has been etched from limestone by means of hydrofluoric acid. 表明笔石详细构造的物质是利用氢氟酸从石灰岩中侵蚀出来。

9 minor   

adj.较小(少)的，较次要的；n.辅修学科；vi.辅修

参考例句：

• The young actor was given a minor part in the new play.年轻的男演员在这出新戏里被分派担任一个小角色。
• I gave him a minor share of my wealth.我把小部分财产给了他。

10 physiological   

adj.生理学的，生理学上的

参考例句：

• He bought a physiological book.他买了一本生理学方面的书。
• Every individual has a physiological requirement for each nutrient.每个人对每种营养成分都有一种生理上的需要。

11 underlying   

adj.在下面的，含蓄的，潜在的

参考例句：

• The underlying theme of the novel is very serious.小说隐含的主题是十分严肃的。
• This word has its underlying meaning.这个单词有它潜在的含义。