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Maths Help: What Am I Screwing Up?

elemtilas

Inkling
So, I am trying to sort out miles per degree of longitude for a planet. I can't seem to make sense of the ratio values (?) I'm getting when comparing Earth and Gea. There's probably something very simple that I'm not seeing, and am hoping someone can point me in the right direction!

What I know:

Earth circumference = 24,901 mi
Gea circumference = 34,851 mi

*Here* we use a 360 degree circle, so, at the equator, a degree of longitude is 69.17 miles.
*There* they use a 288 degree circle, so, at the equator, a degree of longitude is 121.01 miles.

I know that at 90deg north *here*, a degree of longitude is 0 miles; also, *there*, at 72deg north, a degree of longitude will be 0 miles.

When I used a ratio calculator to compare these numbers, of course I got different answers. (Calculators are dangerous toys!)

I've also got a list of Earth miles per degree longitude at varying latitudes that equate to the 72 degree system of Gea. (E.g., 18deg north on Gea is equivalent to 22 1/2 degrees north on Earth).

The calculator I used to get Terran measurements tells me that, at that latitude, a degree of longitude is 63.94 miles. I'm trying to go from there to Gean measurements. Is it just a matter of plugging in a number to multiply by? And if so, which one?

If I compare circumference measurements, I got a ratio of Earth:Gea :: 1:1.3995.
If I compare the miles/degree longitude at the equator, I got 1:1.749.
Shouldn't they be the same?

Help! What am I missing, apart from something like three or four years of high school maths?

Also, I have a notation that a Gean mile is 5760 feet (whereas a Terran mile is 5280 feet). Not sure how that fits or even if that fits into my confusion.
 

Vaporo

Inkling
You're overthinking this a bit.

Earth circumference = 24,901 mi
360 deg of rotation
24901 mi /360 deg= 69.1694 miles/degree at the equator

Gea circumference = 34,851 mi
288 deg of rotation
34,851 mi /288 deg= 121.0104 miles/degree at the equator

I believe that the formula for degrees between longitude lines at a particular latitude on earth would 69.1694*cos(latitude). This wouldn't be shortest-distance, though. It's the distance when traveling directly east-west along the latitude line. So, on Gea, it would be 121.0104*cos(360/288*Gean latitude).

Why 288 degrees? We use 360 because it's divisible by every number up to 10. 288 is only divisible by every number up to 9. If you're not familiar with the math, I'd just stick to a 360 degree circle. Makes the numbers a lot cleaner.
 
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ThinkerX

Myth Weaver
The way I work through this sort of thing (though I keep the 360 degree circle) :

At latitude +/- 45, the distance per degree of longitude will be half that for a degree at the equator.

At latitude +/- 60, the distance per degree of longitude will be 1/3rd that for a degree at the equator.
 
Hi,

I think what you're asking is is there a direct ratio for a degree of latitude in distance at different longitudes. There isn't. Consider that the Earth is roughly a sphere. Gaea too. At the equator you get according to your numbers roughly 70 miles travelled for every degree of horizontal rotation. At the north pole it's 0 miles. But at all the latitudes in between it isn't a linear relationship between 70 miles and 0 miles. The problem is that as you head north the change in the diameter and hence circumference of the circle / horizontal slice of the world changes. So say you go one degree north. The decrease in the size of the world's diameter / circumference at that point is minimal. On the other hand, say you're at the north pole and you go south one degree, the change in the world's diameter / circumference is much larger. There will be a formula to work it out - but alas I don't know what it is. Fair warning though, as it involves limits (maths term) it will be calculus based.

Cheers, Greg.
 

Viorp

Minstrel
First of all.
I'd use meters not miles.
Most math and caculations we made are done in meters.

Miles are not a very good unit when it comes to actual scientific stuff.
 

Vaporo

Inkling
First of all.
I'd use meters not miles.
Most math and caculations we made are done in meters.

Miles are not a very good unit when it comes to actual scientific stuff.

I'm afraid that I disagree. S/he should definitely be using furlongs as their primary unit of measure.

The math is exactly the same no matter what unit you use. From a scientific standpoint, there is little difference between the imperial and metric system, only a matter of convenience for some scientific calculations. If elemtilas wants to use miles, that's what they should use.
 

CupofJoe

Myth Weaver
Miles are not a very good unit when it comes to actual scientific stuff.
Oh... I dunno... The British did well enough for about 400 years :p
In real life the metric system does make the maths easier but also easier to get lost in the scale of things.
If you have to work out inches, feet, yard, and Miles you concentrate a lot more to get the magnitudes right.
And that is why a slide rule is superior to a calculator every time!:D
 

elemtilas

Inkling
You're overthinking this a bit.

That's what I was afraid of!

I believe that the formula for degrees between longitude lines at a particular latitude on earth would 69.1694*cos(latitude). This wouldn't be shortest-distance, though. It's the distance when traveling directly east-west along the latitude line. So, on Gea, it would be 121.0104*cos(360/288*Gean latitude).

Yay! There's a formula for that! Thanks for this! I think this will be very helpful.

Why 288 degrees? We use 360 because it's divisible by every number up to 10. 288 is only divisible by every number up to 9.

In this particular case, we're looking at differing cultures of mathematics. There are lands in the West that use a 360 degree circle --- Heropea, Demeteia, Atelante. In the North and East, they use a 288 degree circle. My area of greatest concentration is Narutanea, in the East, so it's a 288 degree circle for me! I'm not sure how it got that way, though.

If you're not familiar with the math, I'd just stick to a 360 degree circle. Makes the numbers a lot cleaner.

Would have been easier, perhaps. But there are so many other things, from our perspective *here*, that are "less convenient" *there* (in some places e.g. a 26 hour day, and it's not always just a matter of more divisions on the clock face) that this is just one more scoop of the gumbo.
 
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elemtilas

Inkling
First of all.
I'd use meters not miles.
Most math and caculations we made are done in meters.

Miles are not a very good unit when it comes to actual scientific stuff.

I don't think that'd be the best way for me to go. Cos I'd have to convert from miles to meters just to do the maths, then back to miles again, and then from decimal to dozenal. You're making me work too hard here! :eek:

Also, "scientific"?? Who said I'm looking for a(n Earth-friendly) scientific solution! Ultimately, I'm looking for an in-world scheme and while I appreciate the utility of the SI *here*, and use it all the time, it just doesn't port well. The SI-analogue of *there* is very much more like the old Roman & Imperial systems of *here*.
 

elemtilas

Inkling
I'm afraid that I disagree. S/he should definitely be using furlongs as their primary unit of measure.


Actually, leuyves / leagues. I just asked in miles because I figured that might be easier to handle!

The math is exactly the same no matter what unit you use. From a scientific standpoint, there is little difference between the imperial and metric system, only a matter of convenience for some scientific calculations. If elemtilas wants to use miles, that's what they should use.

Exactly. Six of one half a decimal dozen of the other.
 

skip.knox

toujours gai, archie
Moderator
I gotta ask: why 288? Why not 289, or 117? Just wondering why you chose that particular number. Our number of 360 actually has some basis, both cultural and mathematical. It's interesting that it appears the 360 figure was arrived at by separate cultures (Egypt Babylon, India).

It's fine if 288 is just a number, but I was curious as to your reasoning.
 

Simpson17866

Minstrel
I gotta ask: why 288? Why not 289, or 117? Just wondering why you chose that particular number. Our number of 360 actually has some basis, both cultural and mathematical. It's interesting that it appears the 360 figure was arrived at by separate cultures (Egypt Babylon, India).

It's fine if 288 is just a number, but I was curious as to your reasoning.
Prime factorization would be a... Factor :cool:

289 = 17*17, so you can look at the 17th fraction of a circle, or the 17th fraction of that fraction.
117 = 3*3*13, so you can divide the circle into 3; 9; 13; 39; or 117 pieces.

288 = 2*2*2*2*2*3*3, and can be divided into 2; 3; 4; 6; 8; 9; 12; 16; 18; 24; 32; 36; 48; 72; 96; 144; or 288

Maybe not quite as flexible as 360 = 2*2*2*3*3*5, but still pretty good ;)
 
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elemtilas

Inkling
I gotta ask: why 288? Why not 289, or 117? Just wondering why you chose that particular number.

In large part because it's two gross. That is, "200" in base-12.

Our number of 360 actually has some basis, both cultural and mathematical. It's interesting that it appears the 360 figure was arrived at by separate cultures (Egypt Babylon, India).

Sure. In The World, 360 also appears in a couple different places, notably the very ancient lands of Shumur (*there*'s Sumeria) and Atelante (Atlantis, precursor of the Misereans (Egyptians) and later people they influenced).

It's fine if 288 is just a number, but I was curious as to your reasoning.

I am sure there are some other rationales that I'm not even aware of. I can't be expected to know èverything about the place, right? ;)
 

elemtilas

Inkling
I'd just like to thank everyone for their input and perspectives! (And especially for the formula!)

The long and short of this exercise is that, as part of the Atlas I'm making, I wanted a chart that would show the distances between the degree markings on the map.

After some happy number crunching, I've determined, for example, that at 27 degrees north, each degree is l.33/1'117 or 33 leagues 1 and 117/288 miles.
 

pmmg

Myth Weaver
How exact do you want this math?

You should lose approx. 1.34 miles of longitudinal distance per degree for every 1 degree of latitude moving away from the equator towards the pole. Running the math I used, I was off by 1.1 miles at what would be your arctic circle (66 degrees) assuming other similar properties towards earth. This was calculated using the 288 degree compass. The error would likely increase in ratio (but not actual miles) as you go closer to 90 degrees (which would be zero miles of longitudinal distance). So the error would increase, but the distance of error would be negligible in actual footage. Probably enough to have a probe not land safely on the planets surface, but not enough that a reader would notice.

If you want the math, I will send it to you.

Also, I did round some numbers.


My fault, I did not see there was a page two. Seems you already got it.
 
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elemtilas

Inkling
How exact do you want this math?

You should lose approx. 1.34 miles of longitudinal distance per degree for every 1 degree of latitude moving away from the equator towards the pole. Running the math I used, I was off by 1.1 miles at what would be your arctic circle (66 degrees) assuming other similar properties towards earth. This was calculated using the 288 degree compass. The error would likely increase in ratio (but not actual miles) as you go closer to 90 degrees (which would be zero miles of longitudinal distance). So the error would increase, but the distance of error would be negligible in actual footage. Probably enough to have a probe not land safely on the planets surface, but not enough that a reader would notice.

If you want the math, I will send it to you.

Also, I did round some numbers.


My fault, I did not see there was a page two. Seems you already got it.

Am about three quarters the way through the chart now.

There's no real need for exactitude. It's not like the airship flyboys have GPS navigation or anything!, and no one's trying to land a probe anywhere.

The way I see it, if the map and chart can get them within visual distance of their destination, then thy're golden! (At 5000 feet you can see something like 80+ miles, so even if they're quite a bit off course, correction should be easy enough.) All flying is done by surface navigation rather than great circle navigation, so the need to know distances is more for planning stop overs and estimating travel time.

Arctic Circle is só far north that no one would ever have any reason go that far! Wark Island is the farthest north land known to exist at about 45 degrees north. No one sane would ever want to travel that far, let alone any further. Heck, even the Wark islanders can't stand being on Wark Island, what with the lousy weather and lousy food and lousy gods and lousy disposition of the natives, and did I mention the lousy food?
 
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