Eddington number

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This article is about astrophysics. For the measure in cycling, see Arthur_Eddington#Eddington_secondary_number.

In astrophysics, the Eddington number, NEdd, is the number of protons in the observable universe. The name honors the British astrophysicist Arthur Eddington who, in 1938, was the first to propose a calculation of NEdd, and to explain why this number could be important for cosmology and the foundations of physics.

In the late 1930s, the best experimental value of the fine structure constant, α, was about 1/136. Eddington began by arguing, from aesthetic and numerological considerations, that α should be exactly 1/136. He then gave a "proof" that NEdd = 136×2256, or about 1.57×1079. In the 1938 Tarner Lecture at Trinity College, Cambridge, Eddington averred that:

I believe there are 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 protons in the universe and the same number of electrons.[1]

This large number was soon named the "Eddington number." Shortly thereafter, improved measurements of α yielded values closer to 1/137, whereupon Eddington changed his 'proof' to show that α had to be exactly 1/137[2] – a feat for which Punch dubbed him "Sir Arthur Adding-One."

The most precise value of α obtained experimentally (as of 2012):[3]

\alpha^{-1} = 137.035\,999\,174(35).

Hence no one maintains any longer that α is the reciprocal of an integer. Nor does anyone take seriously a mathematical connection between the value of α and NEdd. More defendable estimates of NEdd point to a value of about 1080.[citation needed] These estimates assume that all matter can be taken to be hydrogen, and require assumed values for the numbers and sizes of galaxies and stars in the universe.

On possible roles for NEdd in contemporary cosmology, especially its connection with the large number coincidences, see Barrow (2002) (easier) and Barrow and Tipler (1986: 224–31) (harder).


  1. ^ Eddington (1939), lecture titled "The Philosophy of Physical Science."
  2. ^ Eddington (1946)
  3. ^ Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, Makiko Nio (2012). "Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant". Physical Review Letters 109 (11): 111807. arXiv:1205.5368v2. Bibcode:2012PhRvL.109k1807A. doi:10.1103/PhysRevLett.109.111807. 

See also[edit]


  • John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
  • -------- & Frank J. Tipler (1986). The Anthropic Cosmological Principle. London: Oxford University Press. 
  • Dingle, H. (1954). The Sources of Eddington's Philosophy. London: Cambridge University Press. 
  • Arthur Eddington (1928). The Nature of the Physical World. London: Cambridge University Press. 
  • -------- (1935). New Pathways in Science. London: Cambridge University Press. 
  • -------- (1939). The Philosophy of Physical Science. London: Cambridge University Press. 
  • -------- (1946). Fundamental Theory. London: Cambridge University Press. 
  • Kilmister, C.W. & Tupper, B.O.J. (1962). Eddington's Statistical Theory. London: Oxford University Press. 
  • Mohr, P.J. & Taylor, B.N. (2005). "CODATA recommended values of the fundamental phh". Reviews of Modern Physics 77: 1–107. Bibcode:2005RvMP...77....1M. doi:10.1103/RevModPhys.77.1. 
  • Slater, N.B. (1957). Development and Meaning in Eddington's Fundamental Theory. London: Cambridge University Press. 
  • Whittaker, E.T. (1951). Eddington's Principle in the Philosophy of Science. London: Cambridge University Press. 
  • -------- (1958). From Euclid to Eddington. New York: Dover.