"TheDistance to the Horizon ", section #8a, on disk Shorizon.htm, on the web
          http://www.phy6.org/stargaze/Shorizon.htm

"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web
          http://www.phy6.org/stargaze/Sintro.htm

• Understand the concept of the horizon at sea as evidence that the Earth is round: the gradual disappearance of distant objects, bottom before top, and their reappearance if we climb higher.

• Calculate the way Erathosthenes estimated the size of the Earth.

• Know of other estimates of the size of the Earth before Columbus, and the origin of the idea of "sailing west to India. "

• Appreciate that the opposition to Columbus did not come from "flat earth" supporters, but from people who objected to his underestimate of the Earth's size (and who were right).

• Recognize the origin of the unit of distance "meter. "

• [Understand why the Earth is round (see optional discussion at the end of this lesson plan).]

Terms: Horizon, meter.

Stories and extras: The story of measurements of the Earth's size, involving Columbus, Eratosthenes, Posidonius, El-Ma'mun, Strabo and the French Academy of the late 1700's.

(8a) Distance to the Horizon

Goals:The student will learn

• To derive and apply a formula for the distance of the horizon from a given height (neglecting light bending in the atmosphere). The derivation uses (a) The theorem of Pythagoras (b) the notion that in a calculation we can sometimes neglect small quantities without incurring a great error.

• About the 1806 sighting of Pike's Peak (US History).

    Before Columbus, everyone knew India was far to the east of Spain, but the way east over land was blocked by the Turks and Arabs, while the Portugese led in exploring the sea-route around Africa. Columbus argued that since the Earth was round, one could also reach India by sailing westward, and he proposed to do so, by crossing the ocean west of Spain. No one knew what was on the other side of that ocean, only some islands were known, fairly close to Europe.

    In the end he sailed west with his 3 ships and discovered a "new world." The story is told that many people opposed the idea, and that only at the last moment Isabella and Ferdinand, queen and king of Spain, changed their minds and supported him. That seems true.

    It is also told that Queen Isabella sold her jewels to finance the journey, and that is doubtful. And many people believe that those who opposed Columbus claimed his scheme would not work, because "everyone knew the Earth was flat" and that if he persisted sailing west, his ships may fall off the edge. It is a cute story, but is quite false.

    The real story is longer and much more interesting. Even the ancient Greeks knew the Earth was round, because they traveled widely by sea, and wherever they went, they observed that their view was limited by a horizon.

    Start here with the first part of section #8 (up to the first sub-headline) then present section #8a on the horizon and review it. Guiding questions and additional tidbits
    The questions below may be used in the presentation, in the review afterwards, or in both

Reason #1: The bottom of the ship disappears first, the top last.

Reason #2: After a ship has completely disappeared over the horizon, if we climb to the top of a nearby tower or hill, it may become visible again

--On the board, go over the derivation of the distance of the horizon in section (8a).

The distance is equal to 112.88 kilometers times the square root (SQRT) of (h).
    D (kilometers) = 112.88*SQRT h .

This is best solved in stages. Let H be the height in feet. Then

        h = 0.000305 H        
and
            SQRT(h) = SQRT(0.000305) * SQRT(H) = 0.017464 * SQRT(H)

(Surprised that the square root is bigger than the number? It is, for all positive numbers less than 1. For instance SQRT(1/4) = 1/2, because 0.5*0.5=0.25
We get
               D (kilometers) = 1.9714*SQRT(H)

To convert kilometers to miles, we divide by 1.609, so:

                 D (miles) = 1.2252*SQRT(H)

Same formula as above.

    However, on a really dark night, we will see the "loom" of light, the brightening of the horizon by the light of the distant lighthouse, a bit before we see the light itself.

See story, sect. 8a.

On the day when in Syene in southern Egypt, he observed the noontime Sun was directly overhead ("at zenith"), as evidenced by its reflection in a deep well. The Sun in Alexandria, some 500 miles to the north, was not overhead, as shown by the fact that a vertical column cast a shadow at noon.

The shadow of the column suggested that the displacement from zenith of the Sun in Alexandria was 1/50 of a circle.

If the Earth is a sphere, the "straight up" directions in Syene and Alexandria must differ by 1/50 of a circle.

If Alexandria is exactly north of Syene, the two are on the same line of longitude. Let us complete that line into a circle around the Earth. Then [diagram on the board] the line between the two points, which is 500 miles long, must be 1/50 of the circle.

The full circle around the Earth is then 25,000 miles long, and the radius of the Earth is 25,000/(2π ) = 25,000/6.2832 = 3979 miles, close to the correct value of 3960 miles.

A number of other estimates exist for the size of the Earth, including those by Posidonius and by Arab scientists under Caliph El-Mamun, who ruled in Baghdad around the year 825.

The Roman writer Strabo suggested India could be reached by sailing westward.

Because from what they knew about the size of the Earth, they felt the distance was too great.

Columbus responded with a different, smaller estimate of the size of the Earth (one which we know was wrong). That made the circumference of the Earth (the distance around it) about 2/3 of what was generally believed. Subtracting from that the (known) distant from Spain to India going eastward, what remained, the distance from Spain westward to India, was not too large.

The meter was defined as 1/10,000,000 of the distance from the pole to the equator.

[Based on astronomical observations, a bar of metal was marked by two scratches, and the distance between them was taken as the "standard meter." By modern observations, this is not exactly 1/10,000,000 of the pole-equator distance. But it is still the standard meter, now redefined in units of wavelengths of light, which provide an accurate standard not related to any material object.]

The reason is gravity. If the Earth were made of water, a spherical shape would be stable. In that case, if we took a wedge of the Earth, starting with one square mile on the surface and tapering down to the middle of the Earth (sketch on the board), all such wedges would have the same height and therefore the same weight.

If one wedge were by any reason higher, it would be heavier, gravity would pull it down with greater force and it would sag down, until it was at the same height as the rest of the surface.

The Earth is not fluid like water, and indeed, it has mountains which stick up above the surface. But the rocks of which it is made up also yield to pressure, although not as easily as water. The highest mountain, Mt. Everest, is nearly 9 kilometers high, and physicists have calculated that is about as high as a mountain can get on Earth, before its weight pushes away the ground under it and makes it sink.

Mars has a giant volcano, Olympus Mons, about 25 km high. Mars rocks can support a bigger mountain, because Mars is less massive and its gravity is weaker. Ganymede is a big moon of Jupiter (slightly larger than ours) covered with ice. The "Voyager" mission has taken pictures of round marks on it--like the craters of our Moon, and probably caused like them by meteorites hitting the surface. However, they are completely flat--no raised ridges--because ice, unlike rock, yields easily to the pull of gravity. Over long time-spans, ice flows like a liquid as it does in glaciers. (see section (7) on precession.)