Thursday 4 February 2016

39) “Practical distance measurements taken from “The Australian Handbook, Almanack, Shippers’ and Importers’ Directory” state that the straight line distance between Sydney and Nelson is 1550 statute miles. Their given difference in longitude is 22 degrees 2’14”. Therefore if 22 degrees 2’14” out of 360 is 1550 miles, the entirety would measure 25,182 miles. This is not only larger than the ball-Earth is said to be at the equator, but a whole 4262 miles greater than it would be at Sydney’s southern latitude on a globe of said proportions.”

For the next few points I will simply quote these two sources:

 “False claim (incomprehension of the model).
The distance between Sydney and Nelson is roundabout 2100 km or 1300 miles or 1135 sm or 18.9°. And 360°*2100km/18.9° is how much? Right, 40,000 km.
I suspect that the cited 1877 "The Australian handbook and almanac, and shippers' and importers' directory" was either not stating a perfectly straight line or that is was just off by a certain amount; unfortunately I don't have a copy of my own. Needless to say, these distances and durations of travel don't work on the flat earth azimuthal map.
I also just found out how to mistakenly get the value of 22°. It is the difference in longitude between these two places. There are two big problems in taking this value as a reference for distance:
1st Sydney and Nelson are not on the same latitude. That means you have to go southeast from Sydney to Nelson and not just east. It's just the wrong line to begin with.
2nd Even if they were on the same latitude, the shortest distance between them wouldn't follow the latitude and the true angular distance wouldn't be 22°. That is because no latitude (except for 0° at the equator) represents the true diameter of earth. 
If you were to calculate the distance between these two places following the latitude, you would have to use the diameter of your latitude and not the diameter of earth. Anyway, take a piece of string and try to connect two places on the same latitude (except 0°). You will see that the line will NOT follow the latitude, but actually spans an arc around it. It only works for places that lie exactly on the equator.”

“They state here that the straight line distance between Sydney and Nelson is 1550 statute miles (2494.483 km). And, the longitude difference is 22 degrees 2'14'' (22.0372 degrees).
They say that the entirety would measure 25182 miles (40526 km). So, that gives a radius of 6449 km, versus the 6371 km average radius. 
To calculate the distance, we must first convert to radians. We find that 22.0372⋅π180≈0.3846. If we assume the average radius is okay (6371 km), we find a distance ds=rdϕ=0.3846∗6371 =2450.40:[km] - that's pretty nice. 
In fact, Nelson lies to the north of Sydney: 
Sydney at 33.8650° S, 151.2094° E [Google/Sydney Longitude Latitude] 
·                     Nelson at 41°18' S, 173°16' E [MapsOfWorld/Latitude Longitude / New Zealand] 

This gives a difference of 7.435°, 22.0573° - so the longitude difference agrees. We just calculate the latitude difference too, assume the radius is constant (the ~7 km difference won't matter much) and use the Pythagorean Theorem to calculate the distance. The result is 2588 km. Again, not too bad.

So our result is 2588 km assuming a perfect sphere (it isn't) versus the distance as they tell it being 2494 km. Of course, we can't check the longitude/latitude they took for the 1550 statute miles. Additionally, we are assuming a straight line instead of the correct Great Circle (shortest difference over a spherical surface) - it's not weird that we don't get the exact result. 
I'm going to say that these calculations match the given straight line distance.” 

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