¶ Spherical Triangle: Equator As Square (3)
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Index Entry
Spherical Triangle: Equator As Square:
"If you make your small square a little bigger and your bigger one a little smaller by increasing the little one’s edges to one mile each, you will have a local one square mile-- a customary unit of western United States’ ranches-- and the big square will be approximately 24,999,999 square miles. As you do so, using great circle lines, which are the shortest distances on a sphere between any two points, to draw the squares’ edges, you will find the small square’s corner angles are increasing and the big one’s corners are decreasing. If you now make your square so that it’s area is one-half that of Earth 25 million square miles, in order to have all your edges the same and all your angles the same, you will find that each of your edges is approximately six thousand miles long and that each of the corners of both squares are 180 degrees each. That is to say that the edges of both squares lie along the Earth’s equator so that the areas of both are approximately 12,500,000 square miles.
“And so it would go with a triangle, a pentagon, an octagon, or any other equi-edged, closed line figure which you may draw on any system’s surface. The closed line surface figure will always and only divide the whole area into two complementary areas.”
