I found these two items advertised on the Celestaire.com website:
--------------------
EMERGENCY NAVIGATION
by David Burch #5803 $16.95
A detailed account of how to find your position anywhere on the world's oceans after your electronics fail and you lose your sextant, watch, and almanac. The book is full of good sound navigational techniques and principles that will serve you well regardless of where and under what conditions you are sailing. You may never be confronted by a need for true emergency navigation, but even the most pleasurable sailing afternoon can be enlivened by a knowledge of the skills and methods taught in this book.

EMERGENCY NAVIGATION CARD
#4604 $12.95
A handy card to have on board in case your sight reduction system, sextant, or almanac becomes lost. The card is loose-leaf sized. You can punch holes in it if desired for insertion in a binder. It is plastic laminated for weatherproofing and durability. A wealth of information is provided on it in addition to: an emergency almanac for the sun, and declination tables for the stars; several methods for accomplishing sight reduction, including one that requires only the card itself; and angular markings on the perimeter of the card that allows approximate altitude measurements of celestial bodies.
--------------------

I don't plan to take up boating, let alone sail the Pacific single-handed using a sextant to navigate, but I am becoming increasingly fascinated by the human race's ability to navigate across thousands of miles of trackless ocean using mathematics (especially after reading Tami Oldham Ashcraft's "Red Sky in Mourning").

A Nautical Almanac and a set of Sight Reduction Tables is a bit heavy to take backpacking, but the above publications might well fit comfortably and not weigh too much. (Burch's book is listed at half a kilo; no weight is given for the Navigation Card, so I suspect it's probably negligible for anyone except the most fanatical ultra-lighters. <img src="/images/graemlins/laugh.gif" alt="" />

Has anyone used these or formed an (informed) opinion of them?

Are there any Formulas or Tables Giving Information about the Curvature of the Earth / Distances you Can See with your Eyes X Feet Above the Surface you Stand on? In either of these?

This is Something I've been Long Itching / Passionate!, -to Know! Of Just How Much the Earth Drops Away in Curvature!, -For Every Truly Straight and Level, -Horizontal Mile (or Kilometer) Outward!

I Have such Information out to 60 or so Miles, -from several Sources. I'd be Itching with Curiousity if I Didn't! But I am yet Itching to Get Out Hundreds and Thousands of Miles!

This is a Very Interesting Something!, -Which has Just Escaped Most People's Notice! Including Geographic and Sci-Tech Types! Astronomy and Science Books Abound with Planetary Information Tables, Geologic Era and Epoch Charts, the Periodic Table, Lists of Chemical Elements, Major Earthquakes and Volcanic Eruptions, Etc Etc Etc! But This One concerning the Very Curvature of our Earth!, -Nearly Always Escapes Notice!

Of Course with our Earth being about 8,000 Miles Wide, / 4,000 Miles from Surface to Center, -Then any Truly Horizontal Line Outward of 4,000 Miles, -Say from either one of the Poles, -Will End Just Above the Equator. A Right Angled Vertical Drop from there, -Will again 4,000 Miles Later, -Meet that Point on the Equator. So at Least Here, -I Know that it is a 4,000 Mile Drop. -For 4,000 Miles Outward. (All Figures are Round).

I'm Thinking of Constructing a Little Arm, -Which I'll Attach to my Globes North Pole. Extending Verically (And NON Saggingly) Outward. I'll Attach and Measure from Stings, from various Distances Outward! Thats How Much I'm Significantly Itching to Know!

So I'm just Wondering if the Two Resources that you Mention, -May Have Anything on This! And Hopefully Beyond just 60 Miles or so of Outward Distances. [color:"black"] [/color] [email]aardwolfe[/email]

I just found the Burch title at the bookstore today! Based on my brief skimming it is an excellent read. My sea daddie ( A senior coastie who smart E2's mentor under) taught us skills more in time with H.M.S. Pinafore than the glowing screens of our electronics array. Years later my boat took a 360 roll crossing the bar. The flying bridge ( and crew) were a shambles; no electronics, no radio, no searchlight, no nothing. We proceeded to assist a disabled fishing boat in little better shape than us. Return over that bar was suicidal, waiting out the worsening conditions more so. There was another refuge just north with a notorious entrance that ate the innattentive even in daylight. Remember the scene in Red October as they turn into the two pillars by Sean Connery's mental calculations? I took a final bearing on an all night Foster's freeze hamburger stand sign that would center itself on the bow at just the right moment, all else being correct DURING DAYLIGHT. We scraped some paint, but slipped in. This stuff is truly fascinating. I also recall a Free French tank commander entering Paris came up on a german tank directly across a famous landmark. He remembered the precise distance from some long ago boring school lesson. He destroyed the tank <img src="/images/graemlins/blush.gif" alt="" /> In Wind, Sand and Stars, Antoine de Saint- Exupery told of pilots marking their charts with " the little ditch unseen on an emergency landing field"

If you are interested in Navigation, emergency or otherwise, check out The American Practical Navigator. An epitome of Navigation originally by Nathaniel Bowditch.

This is the standard navigation manual for any serious student. It also covers weather forcasting, maps, and a host of other interesting topics. A must read for those interested in navigation, ship operation or ocean studies.

For Scott
From Ch 22, pg 340, the distance (D) to the visible horizon in nautical miles is
D = 1.17 times the square root of the height of the observers eye in feet.

This book is free, on the web. It is about 27 megs and I don't have the url anymore as I have it on my hard drive. Google should find it.

I was trained in celestial nav. and learned a survival method using primitive tools (stick,string etc) and a memorized table of a few primary nav stars. It was pretty complex and took a bit of work to get proficient, but it did work and would plot you to within 25 miles or less, depending on a lot of variables. I have not seen the technique mentioned in anything for years, so one would have to check out some of the Air Force survival manuals from the late 50's or so.

It assumed you were a navigator and understood the complexities and also had a watch set to the correct time (GMT). We had periodic refreshers and practice sessions to stay current. A lot of work to keep current, but like most survival things, it only is worth it if you need it. Then of course it is too late to learn it.

If I remember correctly we made a fixture that would allow us to measure a stars zenith time with a great deal of accuracy. We would lay on our back under the fixture and wait for one of the memorized stars to cross the fixture and note its zenith time. Then there was a number (which changed over time) assigned to each star. We then did a memorized series of calculations (about 15) and resolved our location. It was suprisingly accurate.

We would have several guys do it independently and then compare our answers. Of course we were highly motivated. I doubt that I could do it now even if it were written down. It is amazing what you can do when you are scared.

Nomad, -Thanks for That! [color:"black"] [/color] [email]Nomad[/email]

It doesn't strike me as a particularly difficult math problem. Probably harder to describe that it would be to solve it if I could give you a diagram.

Draw a circle - this represents the earth. Then draw a line segment from the centre straight out to the height you want to measure from. (Let's say it's 1000 miles.) Then draw a tangent to the circle that just goes through the far end of the line. (A tangent is a line that just touches the circle at one point - think of laying a yardstick on top of a basketball, it's touching the ball but doesn't stick into the ball - that's a tangent.)

Anyway, these three points - T, where the tangent touches the earth; X, where the observer is above the earth, and C, the centre of the earth - form a right triangle.

So let's say the observer is 1,000 miles above the earth. Then the distance between him and the centre of the earth, XC, is 1,000 + r, where r is the radius of the earth. That's the hypotenuse of a right triangle. The other two sides of the right triangle are CT, which is a radius of the earth and so is of length r, and XT, which is the observer's straight line view to the horizon. (Hopefully, it'll make more sense if you draw a picture of it.)

The angular distance will be given by angle TCX, where:

cos(angle(TCX)) = r/(1000+r)

(Of course, the total coverage that the observer can see will be a circle on the surface of the earth.)

Multiply angle(TCX) (in degrees) by 60 to get the number of nautical miles (along the earth's surface).

The actual straight-line distance that the observer can see is the length of line XT, which is obtained using Pythagoras's theorem:

XT = sqrt((1000+r)^2 - r^2)

I hope this is what you were looking for, and Chris K. doesn't have to shut down this thread for "too much math" <img src="/images/graemlins/laugh.gif" alt="" />

Celestaire is selling Bowditch on CD-ROM for about $60 US. I decided to buy a copy of that at the same time - it's such a well-known publication that anyone who's really interested in this stuff can't go wrong. <img src="/images/graemlins/smile.gif" alt="" />

Chris K., thanks for the feedback on the Burch book. I just ordered it, and the Emergency Nav card, and the BOWDITCH Plus on CD-ROM. I reckon that'll be enough to get me started <img src="/images/graemlins/wink.gif" alt="" />

Thanks For That Too!, -Aardwolfe. I Knew it would be a Trigonometric Problem as Well. But my Half Year to a Year of Trig in Hi Skool, -is Rustier than Rusty! So I Wasn't Going to Explore there! But Thanks for your Invaluable Math and Formulae there Anyway! I'm Gonna Print your Post.

Another Problem that I Notice in some of the Formulas I've Seen, -is that their Outstretched Horizontal Level Line, "Stretches Out Beyond"!, -So that it's "Drop Down Point" to the Earth, is Perfectly on the Vertical. (If Extended Deeper into the Earth, -It would Reach the Center).

Let me Now Clarify Better.

As with you, -Here Too!, -a Picture would Do Better towards Explanation! But I Must Use Words. So Here Does Go!

In my Earlier Post, -I spoke of Drawing a Straight, Non-Sagging, Horizontal Line, Outward from Earth. One Endpoint of that Line, -Lets Call it it's Left End, -or "A", -Rests At and On the North Pole. It Extends Out 4,000 Miles Rightward (Lets Call that Eastward). It Ends at a Point in Outer Space. We'll Call that Point "B". This Line is Parrellel to the Plane of Earth's Equator.

Now MY Dropoff from that Point B, -Vertical *with Respect to Earth's Pole to Pole Axis!*, -of Which it is Parrellel to. *Would NOT be Vertical with Respect to a Dropoff Line from Point B Directly Towards Earth's Center!

In Order for the Above to Possibly Be the Case!, -the Outstretched Horizontal Line from the North Pole, -MUST Be a Good Bit Longer than the 4,000 Miles Originally Given!

I've Seen Formulas and Diagrams where that Line has been so Lengthened!

To Arrive at a Vertical Dropdown 8With Respect to Reaching the Earth's Center!*

That is Not What I am Interested In! / Seeking! / Nor Want!

I'm Speaking Rather of a Line A - B, -Which ONLY Goes Outward 4,000 Miles! (Or the Earth's Radius / Distance From Surface to it's Center).

I Do NOT then "Right Angle it to a Center Reaching Dropoff!* As I've Seen Others Do!

I Rather Drop Mine Down Right to Where it would Become a Tangent, -with a Point on Earth's Equator! This Drop Down Line from B of Line A - B, -We Will Now Call a New Line, -Line B - C.

This New Line B - C is Both Right Angled to the Original (and Still Present) Line A - B.

And It (B -C) is Parrallel to the Earth's Polar Axis. (4,000 Miles West of It.).

My Drop Down Line B - C *DOES NOT* Meet the Earth at a Purely Vertical, "Up and Down", Sense! But Rather at an Angle!, -with Respect to Such Verticality.

In Short, -Earth is in Round Figures 8,000 Miles in Diameter. Making of Course for an Approximately 4,000 Mile Radius. The Distance from Either Pole, -or for that Matter Any Point on Earth's Surface, -to Earth's Center, -is Also that Same Radius of 4,000 Miles.

I'm Simply Extending a Line, -Line A - B, -Horizontally Outward, / Parellel to the PLANE of Earth's Equator, / At Perpendicular Right Angles to the Line Coinciding with Earth's Axis.

For 4,000 Miles.

Here's Where I Now Depart from some of the Standard Formulas I've Seen!

When I now "Drop Off"!, -I DON'T now Drive Toward the Center of the Earth!

I Rather Drop to Meet a Point on Earth's Equator at a Tangent!

THIS is How I've Arrived at a 4,000 Mile Dropdown in 4,000 Miles Outward!, -There in my Earlier Post.

I'm NOT Interested in How Many Miles Directly Below me, -Towards Earth's Center, -the Surface of the Earth is!

(That, Incidentally, -is a Good Deal LESS than 4,000 Miles! WHILE an Extension All the Way to Earth's Center, -Would Be a Good Ways MORE than 4,000 Miles!)

Rather!, -I'm Simply Interested in Making MY Dropoff Line!, -Rather Meet a Point on the Earth's Equator, -at a Tangent!

All TOWARDS my Finding Out What I Want to Know!, -*How Much Distance the Curvature of the Earth Itself!, -Drops Down Away from you*, -*For Every TRULY Horizontal Mile Outward!*

You See, -There's a DIFFERENCE Between the Aims and Assumptions of Different Formulas!

MY Dropoff is NOT an EXTENDED Line A - B. That Others Use in Order to Drop Down Vertically toward the Center of the Earth!

MINE Rather Goes Only Horizontally Outward the Length of Earth's Radius / 4,000 Miles!

So as I Can TANGENTIALLY Meet the Equator!

For I Want to Know How Much the Earth Drops / Curves Down!, -in 4,000 Miles / It's Radius! Of Horizontal Line A - B Outward! And for Many Intermediate Values Between Zero and 4,000 Miles Outward!

That is the Difference!

Like With what you Said, -A Picture or Diagram should Explain It Better and Easier, -than my Verbal Explanation here!

I'm Not Sure that Your's or Nomad's Formulas, -Comport with the Specific, Different Need, -that I am Talking About Here!

But Thanks a Bunch to Both of You for It Anyway!

I'll Have to Get Better at Trig Anyway! Before I could Use It!

So I'll Still Construct my Physical Measuring Setup on my Globe! Towards These Ends. Sometime Anyway!

I've just Seen Too Many Formulas which Extend Themself Outward!, -BEYOND Earth's 4,000 Mile Radius! In Order to Go Back Down Vertically Toward Earth's Center!

Such Center Heading Verticality is NOT What I'm Interested In or Want! But Rather What I've now Described!

I Cannot Know or Be Sure that Your's or Nomad's Formula Satisfies Such! Plus my Not now Being Up to Trig!

So I'll Still Pursue this my "Physical Measuring Way"! But Thanks Kindly and a Lot Anyway to Both of You!, -For your Such!

ADDED EDIT NOTE, - -I Speak of a Tangentality to the Equator!, -Whereas I Think you Spoke of One with the Pole! (I Also, Incidentally Speak of One with the Pole.) It's 5 AM here in the East! Bedtime!, -Rather than Digging Back into your Post Time!, -Right Now! [color:"black"] [/color] [email]aardwolfe[/email]

Aard, -I've just Re-Read your 07-02-04 / 9 05 PM Post, -and Understand your Reasoning Even Clearer, -Here in the Freshness of a New Day!

But it Appears that your Formula *Does* have the "Horizontal Measuring Line" Extending Out Too Far!

We Both Use Tangents to the Surface of the Earth. But it Appears that we Use Different Tangents! At Least in Part.

Also, -Differences perhaps in the Lengths of our Line Extensions!

Let me Try to Basically and Verbally Outline these Procedural Differences of Our's Better!

Just Imagine a Circle on Paper Before you! That is the Earth.

Next, Draw a Square around that Circle. So as it Tangents the Circle in Four Places. Two of those Tangents will be Each of Earth's Two Poles! The North and South Pole.

The Other Two such Tangents will be Two Separate, (and Incidentally Opposite), Points On and Tangent with Earth's Equator.

Now We'll Arbitrarily Choose the Upper Right Hand Corner of Such Square. And Work with That. (Any Corner Will Do! Though on Two of Them, -Earth's *South* Pole is the Reference Point!, -Rather than the North for the Other Two Corners.). On the Arbitrary Upper Right Corner of the Square we Choose to Work with, -We are Referencing from the North Pole.

That Corner, as are All Other Corners, -is a Right Angle.

I Simply Want to Go Horizontally Outward for 4,000 Miles, -Beginning *From* Earth's North Pole. To That 4,000 Miles Away Point in Space! / *To* that Particular Right Angle and Corner of the Square.

Where if you Then Drop a Vertical Line, -*Vertical as Determined by What is Vertical on the Circle / Square Diagram Before you!*. And *NOT* on a Drop Down Line Towards the Center of the Earth!, -as So Many Formulas Seem to Do! Thats!, -the Difference Between the Two Systems and Methods!, -That I am Talking About here!

If and When One Drops Down in the Kind of Verticality that I am Speaking About, -You Will Again Intersect the Earth Tangentially, -at a Point on it's Equator! Which Just Happens to Also Be, -4,000 Miles Later!

*Thats* How I Get a 4,000 Mile Deep Dropoff, -For 4,000 Miles Horizontally Outward!

But if you were Standing at the Surface, Right By the Equator, -You would See that Incoming Drop Down Line, -Approaching and Tangenting / Intersecting, *At an ANGLE!* NOT Directly From Above You!

The ONLY Way a Earth Centered Vertical Dropdown, -Can Seem to Come at you from Directly Above!, -Is IF the Original Horizontally Outward Line from the Poles, -*Went Out FURTHER than 4,000 Miles!!!!!*

In Fact, -and This is Another Very Important Point!, -For Right Angles such as This, -The "Other System's" Horizontal Extension, -Can Never Successfully Suffice!

For There that Line Extending Horizontally Outward from the Pole, -and Another Line Originating at Earth's Center, -and Then Passing Thru and Directly Above such Equatorial Observer, -and Onward into Space, -*These Two Lines will Always Be Parrellel! And Thus Never Meet! It Goes Back to Having Started from a Right Angle!

But with any Angles Less than Right Angles, -an Eventual, -but *Extended*, -Intersection of those Two Lines, -Will Eventually Occur!

But we're Not Talking About Such in our Example! But are Rather Talking About a Right Angle!

However, -Should the Horizontally Outward from the Pole Line be any LESS than 4,000 Rounded Miles!, -You Can *Then* Deal with Less than Right Angles!

Along with "Greater Extensions" Outward, -as I Earlier Talked on here.

The Above Two Paragraphs, -are Typically What the "Other System" Does!

*But I'm NOT Asking How Far Directly Above Earth I Am, -Relative to it's Center!*

*I'm Asking How Far I Am Above Earth, -with Respect to Such "Tangential Equatorial Intersection Point", -That a *Direct!*, -NOT Extended, -Drop Down Line, -From the 4,000 Mile Off TIP!, -of Horizontal Line Outward from the Pole, -Would Take me!*

Other Conventional Systems, -Again for Less Than Right Angled Measurements, -Typically Extend Further Outward, -in *Order* to Later and Eventually, -Be Able to Drop Down to the Center of the Earth! THAT IS PRECISELY WHAT I AM Not INTERESTED IN! I Just Want to Know, How Far a Dropdown it is! / How Much the Earth Curves!, -From a Reference Horizontal Line of *4,000 Miles or Less!*, -NOT "or More!"

You'd Notice that in Many Systems, -including perhaps Your's, -that Your Final Drop Down Line toward the Center of the Earth, -*Is NOT Vertical* as Seen on the Diagram Right Before You!

THIS is the Crucial Difference Between the Two Methods!, -That I am Talking About!

In my Method, -I DON'T Employ or Reference From the Center of the Earth! For I Don't Need To! The Center of the Earth here has Nothing to Do With It!

When I Am 4,000 Miles Rightward of the North Pole, -Out in Space, -There on our Diagram, -I Then Make a Direct, Right Angled Turn, Downwards! Of Another 4,000 Miles to Meet the Earth at the Equator. I Do NOt so Drop Down on the Equator like a Rock, Directly from Above! But I Still Drop Directly Down!, -From an Outward Line from the Pole! That Right Angled Dropdown from Such Horizontal Polar Line, -IS How Much Earth Curves Downward! IN Those 4,000 Miles an Outward Going!, -From our North Polar Reference Point!

*Thats* the Kind of Curvature of the Earth I'm Interested in! NOT in the Other System!, -Which for my Interests and Purposes here, -Must Extend Out Too Far! Thats the Other System now! Which I Think Most to All Formulas typically Go By! Towards their Utilizing the Earth's Center! I Do NOT Need or Want the Earth's Center as a Reference Point!, in my System! I Entirely DON'T Need it! That, and it's Extended Line vs my NON-Extended One!, -is the Crucial Difference Between our Two Systems!

I Get the Feeling that Your Method is "Other / Conventional",of a System!

With Such Conventional Systems as I Often See, -One Sees "Vertical", -as Coming Up from the Center of the Earth!, -Wherever on Earth they are! MY Vertical, however, is Measured Along the Earth's Axis, and either the Right or Left Sides of the Diagrammed Square Before Us!

It's Important to Know, -that we are Coming From Two Different Bases / Premises / Starting Points, -to Begin With!

What Each of Us Considers to be Vertical or Curvature!

Thanks For your Formula! But it is Eminently Successful for *Your / Many Other's Systems*. But I'm Afraid Non-Applicable for Mine! Plus I'm Presently Too Rusty on Trig Anyway!

I'll Just Sometime Do my Physical Stringed Measurements, on my Globe! As in our Diagram, -A From the Pole Horizontal Extension, (But NOT Overextension!), -and a Vertical (on the Diagram) Dropdown Line to Tangent the Equator, -These Will Form such Upper Right Quadrant of Such Square! I'll Hang Strings at Least Every 500 Miles Outward from the North Pole. Mark Where they Touch the Surface of my Globe. And Take Appropriate Dropdown Length Measurements, in Miles! I may even Do This foe Every 100 Miles Outward from the Pole! Then my Somewhat Itching and Acheing Curiosity Will be Satisfied!

But I am NOT Interested in Extending Such a Line *Further* Outward from the Pole!, -Just so that I can Later Meet a Line, that can Intersect the Center of the Earth! Thats NOT the Kind of Vertical I'm Talking About!!!!!

I'm ONLY Interested in the *Diagram Vertical* Drop Down / Curvature of the Earth, -From 4,000 Miles and Less Out! Out From the Pole!, that is!

Thank You in Any Event! [color:"black"] [/color] [email]ScottRezaLogan[/email]