Vick,

There are two types of answers to your question.

ANSWER 1.

Geometry is concerned with ideal objects. An ideal line has only one dimension -- length with no thickness; an ideal point has no dimensions at all, just position. Thus a tangent line touches a sphere at exactly one point, namely that point where the radius is perpendicular to the tangent. Go a fraction of a micrometer along the tangent line away from the tangency point and you will no longer be in contact with the sphere; neither the size of the sphere nor the length of the line makes any difference. No items in the physical world can have the properties of an ideal line -- not even a thread pulled tight nor a laser beam, because these both have a thickness. Because an ideal line has no thickness, we would not be able to see it; of course, we would never be able to see a point. The effectiveness of the formulas of physics depends on how closely the objects being measured approximate their ideal forms.

ANSWER 2.

There is a neat formula for the approximate distance in miles to the horizon from a point f feet above the earth:

distance = $\sqrt{\frac{3f}{2}}\mbox{ miles.}$

Thus, a person whose eyes are 6 feet above the earth would be able to see for 3 miles (because 3 is the square root of $3 \times \frac62 = 9$). For a discussion of this formula see the entry "line of sight" in our Quandaries and Queries file. This assumes, of course, that the earth is a perfectly round sphere whose radius is about 3960 miles; moreover, your long flat ruler would be 8 inches above the earth at a distance of 1 mile from the spot it is set down, and still only 6 feet above the earth when 3 miles away. Of course, the earth is not perfectly round, and there is nothing in the physical world that is perfectly flat. Imagine a long bridge across a wide body of water -- it would have to follow the curvature of the earth, but the difference between the high point in the middle and the low point at the two ends would be imperceptible, no doubt less than differences caused by imperfections in the building materials. Bridges appear flat. So do railroad tracks across the flat prairies that stretch in a straight line from horizon to horizon.

The real answer to your query seems to be that there is no such thing as a flat ruler in the physical world.

Chris