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]