An experiment to test the hypothesis that the Earth is flat. (Spoiler alert. It s not.)

Transcription

1 An experiment to test the hypothesis that the Earth is flat. (Spoiler alert. It s not.)

2 We are going to do a thought experiment, something that has served a lot of famous scientists well. Ours has to be a thought experiment because we can t afford the time or cost to do it in fact. There are too many of us and it would require at least two or three days to accomplish, never mind the logistics and the expense. Even though this is just a thought experiment, I am keeping it accurate. When I say that the results at some place would look like so-and-so, they would look like that. If you wish to check my assertions, on your own time and at your own expense, be my guest! That is the nature of scientific skepticism the reason that we demand that our experiments and those of others produce reproducible results.

3 We take a standard scientific approach to this problem: propose a hypothesis and test it. We select the hypothesis that the Earth is flat and seek evidence that disproves it. The main point is to see how this hypothetico-deductive process works. Proof that Earth is not flat is simply a bonus. The class will divide into thirds. I will accompany one group to their station and a research assistant will accompany each of the other groups to theirs.

4 N Latitude 45 N Chippewa Falls, WI Latitude 30 N New Iberia, LA We will be comparing observations at 3 locations along the 92 nd meridian west. These places are pretty close to directly north/south of each other, so the distances between are easy to estimate or measure and our observations can all be made at essentially the same time. Chippewa Falls, WI is very nearly at 45 N latitude. ~Longitude 92 W Latitude 0 (equator) North peak, Albemarle I., Galapagos New Iberia, LA sits at a latitude very near 30 N. The northern of two volcanic peaks on Albemarle, Galapagos Islands, lies on the Equator.

5 The setup at each location is very simple. We drive a stake into level ground, taking care that it is exactly plumb (perfectly vertical). We also take care that exactly 1m of the stick projects above the ground. We also make sure the ground is perfectly level. The sun is part of our experiment, so it is relevant to note that it is above us, but not necessarily directly overhead! We do the experiment at local high noon on the fall or the spring equinox. This is when the sun has reached zenith (local noon) and when it passes directly overhead at the equator (the equinox). We do this at exactly this time so that the setup at Albemarle will have a peculiar geometry, which will become obvious pretty soon. (locally) level ground The sun is somewhere that-a-way N Stick, 1m long

6 Albemarle (I ve never been, so this is where my group goes.) Because Albemarle is on the equator, and because we are observing on the equinox (currently about September or March 21) at local noon, the sun will be directly overhead here. Paths of solar rays This means that at exactly noon the stick will cast no visible shadow. More precisely, the shadow will be directly at the base of the stick, and will be exactly the same size as the stick s diameter. The sun is exactly that-a-way Both the stick and the path of incoming solar rays are perpendicular to the ground. Map view N Stick, 1m long Stick N

7 Chippewa Falls The group at Chippewa Falls, with the exact same setup and at the exact same time, will see something completely different. The shadow they see on the ground indicates that the sun is NOT directly overhead, but is some distance to the south. The solar rays are coming in at an angle A with respect to the ground. This is called the angle of incidence. The rays, the shadow, and the stick form a right triangle. B The sun is exactly that-a-way Map view N Stick A Shadow

8 Chippewa Falls Notice that the stick and its shadow are the two legs of a right triangle, and the ray path is the hypotenuse. This group has made sure the stick is 1m long. It is a simple matter for them to measure the shadow, but they must measure very, very precisely! At Chippewa Falls the shadow would be exactly 1.00m long, the same length as the stick. B Stick 1m high A Shadow 1m long

9 Chippewa Falls Now remember basic trig: Tan(A) = length of opposite leg/length of adjacent leg = stick length/shadow length. In this particular case the stick and the shadow are both 1m, so tan(a) = 1m/1m In other words, the tangent of A is 1. B We can use the arctan (or tan -1 ) function of a calculator, or check a table in a book, to see which angle corresponds to a tangent of 1. Stick 1m high A Shadow 1m long

10 The page at left is copied from the CRC Handbook of Standard Mathematical Tables and shows a table of trig functions like you d find in any trig book. Notice first that the numbers in the Tan column (or any of the others) are all different. One Tan value corresponds to exactly one angle. With Tan = 1, the corresponding angle is 45, which is the value for angle A in the diagram.

11 Chippewa Falls This small triangle that we have constructed on the ground is easily measured and its angles determined. Just for practice, what is angle B? We have made this little triangle because it is similar to another, much larger (hypothetical) triangle that exists in the solar system similar in the geometric sense. B This real triangle would be very difficult to measure even if it actually exists. We are hypothesizing its existence when we hypothesize a flat Earth but if the hypothesis fails the existence of this triangle will also fail. Stick 1m high A=45 Shadow 1m long

12 Remember that we are hypothesizing a flat Earth, so the drawing shows flat ground between Chippewa Falls and Albemarle. What we are about to do is true if and only if this hypothesis is correct! If the Earth is not flat, what we are about to do, and to conclude from it, is utter nonsense. Just be forewarned. Ray path to Albemarle A B Stick 1m high

13 We construct this larger triangle involving not sticks and shadows, but the ground over long distances (if it is flat) and the solar ray paths that cast the shadows. The path from the sun to Albemarle (which we know is at a right angle to the ground because of the shadow it doesn t create) is one leg of the large triangle and the distance the sun sits above Albemarle is its length. The hypothetically flat Earth between the locations is the other leg and the distance between Albemarle and Chippewa Falls is its length. The ray path to Chippewa Falls is the hypotenuse. Work out which parts of the small triangle correspond to the parts of the big one. A Shadow 1m long B Stick 1m high NOT TO SCALE!!! Millions of meters Ray path to Albemarle and distance to sun

14 If necessary, take a moment to review the notion of similar triangles in this diagram. The sides indicated are in reference to angle A. Those in parentheses are for the small triangle. (b) (opposite) B opposite A (adjacent) (c) C adjacent

15 Note that the bottom right-hand angles of both the large (sun/shadow/ground) triangle and the small (stick/shadow/ground) triangle at Chippewa Falls are right angles because both sticks are vertical. B Also note that angle A is shared by both triangles. (Angle of incidence.) If two angles of a triangle are equal, the third labelled B ) Must also be equal. Therefore the two triangles are similar!. Ray path to Albemarle A B The rays leaving the sun for the two locations must be diverging at 45 because A=45, and therefore B must be 45 in both cases. IFF THE GROUND IS FLAT BETWEEN THEM!

16 The ground distance from Albemarle to Chippewa Falls can be carefully measured from a map and found to be ~5000km. B Remember tan(a) = opposite leg/adjacent leg, regardless of the scale of the triangle. In this case (since we already know that A = 45 ) [tan(45)=x/5000km], where X is the elevation of the sun above Albemarle. Remember that tan(45) = 1 We can rewrite the equation as: 1=(X/5000km) Ray path to Albemarle X (= ~5000km) Rearranging we find X = ~5000km. A B (The 1m shadow is insignificant in comparison to the 5000km [5,000,000m] between the locations, so we can safely ignore it in our calculations). ~ km

17 WE NEED TO STOP NOW AND BE SURE WE UNDERSTAND WHAT WE HAVE JUST DONE We have used our flat-earth hypothesis to calculate a specific value for a specific physical distance -- the sun sits 5000km above the Earth s surface! Our math is perfectly correct. If our assumptions are all correct then the sun has just been proven to be 5000km up. Forget that you know that this value is wrong (it s really ~150 million km), we are interested in testing our hypothesis purely by its own internal logic. (And, after all, how do you know how far it is to the sun?) If our flat-earth hypothesis fails, there will be no remaining reason to take the 5000km height of the sun seriously. In fact, as we will see at the end of the experiment we will have no evidence bearing on the exact elevation of the sun whatsoever.

18 We can now treat this proven value as a new hypothesis or sub-hypothesis, and propose a test for it. If the sun sits 5000km above Albemarle when we view it from Chippewa falls, it should also sit 5000km above Albemarle if we view it from anywhere else. If that does not happen, what do we conclude? There is another hypothesis that we must necessarily be making in our model that should be obvious by simple reasoning. If the sun is only 5000 km away then it must not be very big. In fact, we should be able to measure how big it looks and from that determine how big it would actually have to be to look that size at 5000 km distance. Let s move on.

19 New Iberia When we set up our experiment in Louisiana we see something a little different from what we saw in Wisconsin. Careful measurement of the shadow reveals it to be 57.74cm (which we have to convert to m), just a little over half as long as it was farther north. Because this shadow leg is shorter than at Chippewa Falls, the angle of solar incidence must be different. That is, angles A and B here are not the same size as in Wisconsin. B We get the size of the angle A as we always have: tan(a) = 1m/0.5774m = and arctan (1.7319) = 60. Stick 1m high A Shadow m long

20 New Iberia Again we have two similar triangles the small (stick/shadow/ray-path) one and the (Sun/Albemarle/New Iberia) one. B The measured ground distance between Albemarle and New Iberia is ~3335km. Solving the tangent function again gives us a value for X, just as it did for the Chippewa Falls triangle. Tan(A) = (solar elevation/ground distance) or tan(60) = (X/3335km) or = (X/3335km) Path of solar rays X Rearranging, we get: X = (3335km*1.7319) = ~5775km A B Stick 3335km New Iberia Albemarle

21 Both New Iberia and Chippewa Falls? When we put all the places on one diagram, with their distances drawn to scale, we see clearly that there s a problem. Our math indicates that the sun must be 5000km above the ground if we take one triangle seriously and 5775km if we take the other seriously. This is a 15.5% discrepancy, and this is substantial. It seems that two of our groups will be arguing over who has the height correct, and we need some clear thinking to help settle the matter. Height of sun is not predictable at Albemarle I. Height of sun is 5775km according to New Iberia. Height of sun is 5000km according to Chippewa Falls. 5000km 3335km

22 It s time to slow down again and figure out what we ve done. The triangular relationships at Chippewa Falls proved that if the Earth is flat, the sun is ~5000km directly above Albemarle at noon on the equinox. However, the triangular relationships at New Iberia now prove that if the Earth is flat, the sun is ~5775km directly above Albemarle at noon on the equinox. There are 3 possible explanations, which we will examine individually: 1) There is some problem with our measurements or our mathematics. 2) How high the sun is depends on where (or who) you happen to be. 3) There is some problem with our assumptions.

23 1) There is some problem with our measurements or our mathematics. This is simply not the case. If you are appropriately skeptical, the measurements would be the first place to check. Get a map, measure the distances, use the map scale to convert them to kilometers, and you will find the we have measured them correctly. We are using the correct ground distance in our calculations to within about 1km, actually. (I didn t get these off a map, I got them in a much more precise way). Our solutions are also mathematically correct. Take them to a mathematician if you want to check. The mathematics of triangles is so straightforward that any trig solution like this is as good as a direct observation. Any scientist would take our mathematical solution here as being as factually true as if we had directly observed the entire system at once, say by standing on Saturn. Mathematical proofs establish factual truth, so within the model we have constructed (with it s included assumptions) the sun is both 5000 and 5775km above the ground.

24 2) How high the sun is depends on where you happen to be. Neither group wins the argument about who s right. The mathematical solutions of both groups are correct. Within our model, the sun is both 5000 and 5775km above the ground, at the same time. Anyone who needs an explanation for why the sun can t be in two places at once, go and ask a first grader.

25 3) There is a problem with our assumptions. As Sherlock Holmes tells Watson, probably at least once in every story and novel, when you have eliminated all the possible explanations but one, that one, no matter how bizarre, must be correct. In most cases scientists can t be sure they have all the possible explanations in front of them, but in this case we do. Either the conclusion is true, or it s not. If it s not, either it is a problem in our solution (our math) or a problem in our model. The conclusion is not true and there s no problem with our math, therefore there must be some problem with our model. Our model only has two assumptions in it: a flat earth, and a small, nearby sun. Which is wrong? Which is responsible for our absurd conclusions?

26 A Can we salvage a flat Earth if the sun is not small and close? Huge distance Earth Sun B

27 A Huge distance Sun If the sun is huge and very far away, and the flat Earth is tilted with respect to it (in this case at 45 ), then the angle of incidence everywhere would be the same as the tilt angle (45 ). Clearly this is only the case at Chippewa Falls, so we cannot salvage the flat Earth hypothesis with a big sun, no matter how that flat Earth is oriented. B

28 Let s have one more recap to see where we stand. We cannot hypothesize a flat Earth with a small, nearby sun because our triangular functions give absurd results. We cannot hypothesize a flat Earth with a huge distant sun because in that case the sun would shine everywhere from the same angle. We know that it does not. This means that the error in our assumption is not related to the size or position of the sun, but rather with the flat Earth.

29 Instead of a assuming that the sun s rays approach a flat surface at different angles, let s play with the idea that the Earth s surface is tilted by differing amounts in different places. We will stick with the idea of a huge, distant sun because this allows the rays to approach Earth in parallel paths. In this model, the paths don t change orientation from one part of the Earth to another, the ground does! Ray path Ray path Ray path (A=45 ) Chippewa Falls, WI (1m shadow) (A=60 ) New Iberia, LA (.5775m shadow) (A=90 ) Albemarle Is. (no shadow) The next slide simply arranges the three diagrams in a way that should make sense.

30 Ray path Ray path Ray path New Iberia, LA (.5775m shadow) Albemarle Is. (no shadow) Chippewa Falls, WI (1m shadow) 45 Equator (0 ) 30 Center of Earth

31 The larger in each pair of the triangles we ve been solving aren t really triangles. A triangle is constructed by drawing straight lines between three points. A straight line distance between Albemarle and Chippewa Falls would go through the curved Earth. A If we did this for these three points, the triangle would not be a right triangle and so the tangent function would not work anyway. The small triangles were okay, the flaw in our model was to assume we could scale up because of a flat Earth. B Albemarle Island Chippewa Falls

32 Be sure you understand that we cannot conclude from our experiment that the Earth is spherical, we only have enough information to conclude that it is not flat. Another way of saying this is that the alternative hypothesis to the flat Earth hypothesis is a not-flat Earth hypothesis. Similar experiments to this one, arrayed across both longitude and latitude in various ways would consistently be in agreement with a spherical Earth.

33 The spherical Earth model solves all our problems. At any point on Earth the trig functions give sensible patterns because the sun s rays come in from the same direction everywhere and the curvature of the Earth ensures that the ground, and therefore the stick, orientations are correct. The shadow length depends on latitude, or distance in terms of degrees of arc along the surface rather than the solar position. No matter what the distance to the sun, as long as it is big enough that the rays strike Earth along parallel paths, the triangle solutions are sensible. Though we haven t examined the idea, new locations come directly beneath the sun as the Earth turns (or, in an old rejected model, the Sun revolves), so you should be able to see that it must also be round in the east/west direction. A flat Earth is geometrically absurd. Therefore we reject the flat-earth hypothesis because of our tests. Acceptance the spherical Earth is really a separate issue. However, other experiments similar to ours would easily suggest that no other shape would give consistently sensible trigonometric results.

34 Of course, there are other ways of knowing the Earth is spherical.