How Far is the Horizon?

I'm not sure why this has become an interest for me in recent weeks. Partly, I think, because I've been (re-)reading Patrick O'Brian's Aubrey/Maturin series of historical novels about the English Navy during the first part of the 19^th century.

Anyway, it's fairly easy to work out with some basic maths if you know the radius of the earth.

Assume the earth is a sphere, with centre C; and imagine that you are looking at the horizon, H, from point E, which is a distance h above the earth:



The distance to the horizon is d and the radius of the earth is r – which NASA says is on average 6,371 km, approximately 3,959 miles.. Since ∠EHC is a right angle, we know from Pythagoras' Theorem that:


... which can be rearranged to find d:


That's all very well and good, but is not very useful. Let's define d_m to be the distance to the horizon in miles, and d_f, h_f and r_f to be the measurements in feet, respectively, of the distance to the horizon, our height above the earth and the radius of the earth. Then, because there are 5,280 (3 x 1,760) feet in a mile:


Substituting in the actual radius of the earth (3,959 x 5,280 feet) and rearranging the formula a bit, we get:


Now, it so happens that 3,959 is almost exactly three-quarters of 5,280, so:


And, because the first term is so small compared with the second, we can approximate even further, and say:


So, if you're 6 ft tall, the distance to the horizon when you're standing on a beach as the water's edge is about 3 miles. If you were on the deck of Captain Aubrey's Surprise, say 18 ft above sea level, you'd be able to see over 5 miles; but at the top of the mast, which was probably 80 ft above sea level, you'd be able to see over twice as far.

In fact, this approximate calculation is remarkable accurate. At altitudes up to about 20,000 ft the difference between the "exact" distance given by equation (A) above and the approximate equation (B) is only about 0.01% - just a few yards.

Don't Worry 12 - Get Your Thinking Straight (B)

Another way for Christians to avoid worry.

12. Get your thinking straight: get the facts right

We need to train our minds to think rationally. Sometimes we worry about impossibilities … things that can never happen. As a simple example, it’s not much use - at least in this country - worrying that our child will meet a stampeding elephant on the way to school. And even if things are not impossible, they’re often extremely improbable. We need to research the facts carefully. If it’s futile to worry - how much more futile is it to worry about something that will never happen or that is not based on facts.

I am an old man and have known a great many troubles, but most of them never happened. (Mark Twain)

How much pain they have cost us, the evils which have never happened. (Thomas Jefferson)

Remember the two on the road to Emmaus after Jesus’ death. They were depressed, worried and confused. When Jesus met with them they stood still, their faces downcast (Luke 24:18) ... and explained to him:

[Jesus] was a prophet, powerful in word and deed before God and all the people. The chief priests and our rulers handed him over to be sentenced to death, and they crucified him; but we had hoped that he was the one who was going to redeem Israel. And what is more, it is the third day since all this took place. (Luke 24:19-21) ... and now it’s all gone wrong!

Jesus had to take them back to first principles, show them the facts ... especially the fact of his resurrection. It was then that:

their eyes were opened and they recognized him, and he disappeared from their sight. They asked each other, "Were not our hearts burning within us while he talked with us on the road and opened the Scriptures to us?" (Luke 24:31:32)

Flat Earth? Never!

Part of a quote from David Warren by Post-Darwinist. In the same post, she also helpfully refers to Myth of the Flat Earth by Jeffrey Burton Russell.

One of the constants in the long clash between Scientism & Christianity has been the repetition & elaboration by the Scienticists of quite incredible myths & lies, that are still used in our public schools, media, &c, to mock & slander Christians & Christianity. I know about the potency of these myths & lies, for I myself was taught many of them, in school, saw them endlessly repeated in the press, heard them repeated by all liberal adults, & actually believed several of them until I came to riper years, & began to realize that public atheism requires the defence of a "bodyguard of lies." (And you will find them all repeated uncritically in Dawkins, Hitchens, Sam Harris, & all the current bestselling atheist tracts.)

...

Among the most universally taught & least subtle, lies taught to this day, is that Christians throughout the Middle Ages believed the world was flat; & that the Church taught this, & defended it as religious doctrine against Copernicus, Kepler, & Galileo. Yet the Church never taught the earth was flat, nor did any educated person, Christian or otherwise, ever believe it.

Indeed, the old Ptolemaic system -- universally accepted by intelligent Christians, Heretics, Atheists, Jews, Muslims, & even Hindus until the age of Copernicus -- was very clear on the fact that the earth was a sphere.