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.

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