In astrophysics, the **Eddington number**, *N*_{Edd}, 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 *N*_{Edd}, 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 *N*_{Edd} = 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.

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 – a feat for which *Punch* dubbed him "Sir Arthur Adding-One."

The best present-day estimate (2008) of the value of the fine structure constant is:

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 *N*_{Edd}. More defendable estimates of *N*_{Edd} point to a value of about 1080. 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 *N*_{Edd} in contemporary cosmology, especially its connection with the large number coincidences, see Barrow (2002) (easier) and Barrow and Tipler (1986: 224–31) (harder).

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