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 schooling 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 accurately 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 remarkable 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 flute 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 watts, 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. laterally the extreme edge of dark drapery held by figure on right that blows towards venus is phi proportion with sides of picture.

"the rape 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 illustrating 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 minor 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 physiological reason underlying it.