Goals: The student will|
- 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).
Start by bringing up the story of Columbus:
Probably everyone here has heard about Christopher Columbus. In 1492 he sailed westward from Spain with 3 little ships, trying--so he said--to reach India, from which all sorts of valuable spices could be brought.
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
-- If we stand on the seashore and watch a ship sailing away, how do we know that its disappearance is due to the curvature of the Earth and not just because of distance?
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).
--(Section 8a) If we stand at an elevation h kilometers above sea level, how far (at sea or on flat sea-level land) is the horizon--not allowing for the bending of light by the atmosphere?
The distance is equal to 112.88 kilometers times the square root (SQRT) of (h).
D (kilometers) = 112.88*SQRT h .
[Optional, tricky question--may be presented on the board by the teacher.]
What is the formula in miles, with h in feet? 1 mile = 1609 meters, 1 foot = 30.5 cm = 0.000305 kilometers.
This is best solved in stages. Let H be the height in feet. Then
h = 0.000305 H
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
D (kilometers) = 1.9714*SQRT(H)
To convert kilometers to miles, we divide by 1.609, so:
D (miles) = 1.2252*SQRT(H)
--(Section 8a) In a boat on the ocean at night, how far can we see the light of a distant lighthouse, whose lantern is at height h above the water?
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.
--(Section 8a) How did the Colorado mountain known as Pike's Peak get its name? (see story, sect. 8a)
Back to section (8) on Eratosthenes, Columbus, etc. The questions below can be used in the lesson or in reviewing it later.
-- What information did Eratosthenes use to estimate the size of the Earth?
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.
-- How did Eratosthenes use this information to estimate the size of the Earth?
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.
-- Who else in the age before Columbus estimated the size of the Earth?
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.
-- Who first suggested that if the Earth were round, one could reach India by sailing westward?
The Roman writer Strabo suggested India could be reached by sailing westward.
-- Why did Spanish experts reject the proposal of Columbus, to sail west to India?
Because from what they knew about the size of the Earth, they felt the distance was too great.
--How did Columbus respond to this?
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.
-- How was the unit of distance known as the "meter" originally based on the size of the Earth?
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.]
Optional concluding discussion: Why is the Earth round? Why, in fact, do all planets and the Moon appear round?
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.)