# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Bowditch: Distance to visible horizon**

**From:**Frank Reed

**Date:**2012 Dec 6, 11:13 -0800

Robin:

Good catch! In the print edition from 1984, the factors are correct: 1.17 multiplies the square root of the height in feet, 2.11 multiplies when the height is in meters (a nitpick in the PS below). But as you note, in the later pdf of the 1995 edition, the second factor is wrong. It's now listed as 2.07. I also checked the print version, which by that date was probably produced from the digital copy, and the incorrect value is there, too. It's only a few percent difference, but it's a surprising error.

What does this tell us? How could a glaring error in a formula go un-noticed for so many years? Well, first of all, we're seeing the slow decline of traditional navigation methods. Who needs a formula when the computer does all the work? There are actually plenty of little errors in these more recent editions of Bowditch. And I think there's another factor at work. Bowditch is an American reference work. A great many American navigators and students of navigation have been told that Bowditch, either today's or the first edition from 1802, is a globally important work --the "navigator's bible". But it's not. It is a reflection of American and, more specifically in late 20th century editions, USN navigational practice and culture. And Americans don't use meters. As we all know, only Canadians, Europeans, and various antipodean peoples use the heathen meter. :)

For the rest of you folks worrying about refraction factors and such, sure, those can change the value of the factor out front, and the extent of terrestrial refraction in the formulas has indeed changed over the years (and even within individual editions of Bowditch, but the ratio between the factors in imperial versus metric units of the same formula in the same edition should be the same.

The distance to the horizon and the dip of the horizon and various other calculations look like geometry problems. But they're not. They're physics problems. Light rays travelling nearly parallel to the Earth's surface are bent by the density gradient in the atmosphere. The paths of light rays skimming the surface of the Earth bend downward about one-sixth of a minute of arc for every minute of arc (nautical mile) travelled. This is known as "terrestrial refraction". And unlike astronomical refraction, which we need to account for when measuring the altitudes of celestial objects, the terrestrial refraction is rather variable --simply because the density gradient is a function of the weather. The effect of terrestrial refraction is remarkably simple. It increase the effective size of the Earth; photons can go farther because the atmosphere lets them follow the curve of the Earth. So you can work out a formula for the dip or the distance to the horizon using simple geometry and pretending that light rays are perfect straight lines --ignoring refraction. When you're done, you replace the radius of the Earth by a larger number. Do that in every formula, and you completely account for terrestrial refraction. In the literature of terrestrial refraction, this change in the effective radius of the Earth is handled by a factor "beta0" which divides the radius. That is, R = Re/beta0 where Re is the actual radius of the Earth (in whatever units you like). The value of beta0 is variable but 0.83 is a reasonable average choice. Amazingly, if you read through the explanations of the tables in modern editions of Bowditch, you will find that various different values of beta0 were used to generate the tables, reflecting the organic growth of the document in earlier decades. These values are usually listed in Bowditch to four digits, e.g. beta0=0.8279, but that extra precision is seriously misleading. Those are meaningless extra digits, and daily variability in the terrestrial refraction swamps such small differences.

-FER

PS: If you're checking numbers carefully, 2.12 in the metric formula would be closer if the feet factor is exactly 1.17. But this is just rounding. If the latter factor is really, let's say, 1.167 rounded up, then 2.11 would be closer. Not a real difference. The factor Robin discovered, 2.07, is simply inconsistent. It's wrong.

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