Aaardwolf´s math is correct if you want to know where the longest line of sight meets the earth surface (assuming earth is sphere).
The problem described in your last post is totally different. If I understood it properly here is a solution (again assuming earth is a sphere):
There are three parallel planes. One goes through the equator, one touches the pole and one goes through a point on the earth surface with a certain horizontal distance from the pole. The drop is the distance between the the latter two planes.

earth radius R
horizontal distance l
drop d

d = R - sqrt(R^2 - l^2)

It´s only valid for l <= R. Posting some graphic may help to understand the problem more clearly.

Well, if I've understood your problem correctly, this is what you're asking:

You're standing at the North Pole. You travel out horizontally a distance of I miles, where "horizontally" is defined as "parallel to the equator", and drop down vertically until you reach the earth (where "vertically" is defined as "parallel to the earth's axis").

Max is correct - this only makes sense if I is less than or equal to the earth's radius r. If I > r, then you'll miss the earth entirely and shoot off into infinity.

It's a fairly straightforward problem, although I don't see any useful application for it. Using the pythagorean formula, the answer is

drop = r - sqrt(r^2 - I^2)

From What you just said, It Seems that What you and Max are Saying, -May be the Same as my Saying "A 4,000 Mile Length or Less", (Meaning the Horizontal Line Going Out from the North Pole).

Indeed, Any Greater, and One Does Miss any Earth Tangent, -and Continues Off into Space Indefinitely! This is Something that I either said, -but More Likely Implied, -in my earlier Posts.

But Some People Do Draw the Horizontal Line Out Farther! NOT to Drop it Directly Downward as we've just Described. But to Go Outward some More, -So as to then Turn Back on a Line that would Take you Right to the Center of the Earth!

But in Our System of Just Directly Dropping Down, -the Center of the Earth is Missed too! Only at the Pole Itself, -Can Going Directly Downward (Thru the Earth), Take you to the Center of the Earth.

Of Course I'm Not Interested in the Center of the Earth! From your Description, -You might Not be either! Earlier I Didn't Know So. Whether you / your System, -was "Pro or Con", With Respect to Working With the Earth's Center.

Some Systems However Do! I Simply Do Not! I Don't Want to Come Back Down on the Earth from Above! (There you'd Never Have a Tangent!) I Just Want to Drop Down to Meet it at a Tangent to the Equator!

Of course I'll Measure Less Than 4,000 Mile Values too, -Towards Arriving at my Drop Off / Earth Curvature Figures.

You've Correctly Restated some of It. I'd like to Add Something that you Didn't Restate now, -And thats that you Drop Down Vertically Till you Reach the Earth *At an Equatorial Tangent"!*

That is Parrallel to the Earth's Axis, as you Mention. But it's Important to Note that our Construction is a Parralell to that Axis, -But One that is 4,000 Miles *Outward* from that Axis.

This could be One of those Things in Life, -that you *Don't* Do for a Practical Application!

Be it your Helpful Formulas, my Globe Measurements, or Both!, -I Will Get this Mystery and these Figures Solved! [color:"black"] [/color] [email]aardwolfe[/email]

Oh!, -His Math Itself is Correct! as is Your's and Nomad's too. What You and I also say in our Posts, -is Correct too. And That's Not Neccessarily at Variance with Aardwolfe and his Math Itself! At Least Not in All Ways! And He After All, -is Plenty More of a Mathematician than I! My Points and Reasonings Nevertheless Stands! But Once Again!, -At Least Much of That is Not Neccessarily At Variance! With His, Your's, and Nomad's Math! Thanks for Your Help, Contributions, and Input as Well! [color:"black"] [/color] [email]M_a_x[/email]

I think celestial navigation to get absolute position is not that useful for backpackers. (Of course it's nice to find north using the Big Dipper, and stuff like that). But even with a decent marine sextant, sight reduction tables, and a precise chronometer, the best position fix you can expect will be plus or minus several miles. That's accurate enough to steer across the ocean without going way off course, but not much help finding your way back to camp in the woods.

Much more useful are coastal navigation methods. Rather than sighting on distant astronomical objects, you use your sextant to sight on known landmarks like mountains and measure the angles between them, and then just use a protractor and ruler to find your location on a chart. A sextant is probably too bulky for backpacking, but fancy backpacking compasses have gun-sights for measuring angles like that, albeit with less accuracy than you could get with a sextant.

Finding position at sea was the great navigational problem of the early seafaring era. The British Admiralty offered a huge reward for a practical method in the 1600's, and many fancy astronomers tried to invent purely celestial techniques and failed. Only the development of the accurate marine chronometer (which most people back then thought was technologically impossible) made a practical solution. Dava Sobel's book "Longitude" tells the story and is fascinating. There was a PBS video made from it, which you might be able to find at the library or something.

Bowditch's book is downloadable for free (the US government bought the copyright from the author in the 1800's and has been updating the book since then) so paying $60 for a CD-ROM sounds like a rip-off to me.

Here's a real cool page someone here found recently, about how to make a sextant from a CD-ROM and case:

http://www.tecepe.com.br/nav/CDSextantProject.htm

Well, from what I've read, I think a good marine sextant is more accurate than you give it credit for. The pictures I've seen indicate they are calibrated to the tenth of an arc-minute, which is a tenth of a nautical mile on a great circle.

Having said that, I wasn't really thinking of taking a sextant on my next backpacking trip. But this forum isn't concerned solely with backpacking or getting lost in the woods, either - I'm also interested in surviving if I manage to get lost in bad weather in a light plane, for example. Not that I'd be likely to carry a sextant for that either <img src="/images/graemlins/wink.gif" alt="" />

The problem with traditional celestial navigation for most "survival" scenarios is that it requires several sets of printed documents (or a computer that will generate the equivalent information), none of which may be available.

Lofty Wiseman, in The SAS Survival Guide, gives a technique for finding True North using a vertical stick and the sun's shadow. I now realise that, basically, he's constructing a crude sextant and taking the Sun's Meridian Passage (I think). With a good wrist-watch and a measuring tape, and a knowledge of high-school geometry (one of the few subjects I ever scored a 100% in :nerd: ) and trigonometry, I think you should (theoretically) be able to measure your latitude and longitude. (If you can measure the tip of the sun's shadow at noon to within 1 cm - about the width of my index finger - using a yardstick 1 m high, that should give a True Altitude accurate to within +/- 5 arc-minutes - approximately.)

Does knowing your position within a radius of 5 nm help? It would probably depend on the circumstances. It might help you pinpoint your position on a map, for example (if you were hopelessly lost), or determine which side of a particular river you're on. If you were in dire circumstances (plane crash, search called off, and the only map available is a 1:500,000 VFR Sectional/VNC - or worse, a 1:1,000,000 WAC) I might well try to figure out my lat/long using any available technique - just for something to do, if nothing else.

But mostly, it's just something I would like to know how to do.

Sextants were used in aircraft during WW2 for navigation. The accuracy is there, depending on the skill of the navigator. A CD ROM of Bowditch? Thats an abomination against the natural order of the universe. Real navigators ( and would be navigators) lugged that massive blue tome everywhere with all the gravity ( literal and literary) of a preacher and his King James. <img src="/images/graemlins/grin.gif" alt="" />

Good sextants are as acurate as you state. The errors are introduced with the observation and time measurement. There are systematic errors like refraction and variation of the suns radius and random errors in the observation as you can´t hold the sextant really steady. You may not be able to adjust it with the precision of the calibration. Most literature states that you should expect a position error of about 5 nm even if you have a good sextant and apply all corrections.