# Straight-lined international and interior borders **URL:** \ **Category:** Factual Questions\ **Created:** [January 2, 2025, 3:43pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484 "2025-01-02T15:43:50Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [July 17, 2025, 10:57pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/41 "2025-07-17T22:57:09Z")
I assume no need to derive a formula, since formulas for the distance along a geodesic on whichever Earth spheroid you’re using have been published. In 1941 Lambert gave us formulas that are comparatively simple and are usually? correct within a few meters on lines thousands of kilometers long. Around 1955 Rainsford gave us longer formulas that were accurate to about a millimeter for lines up to 19000+ km. In 1975-76 Vincenty improved Rainsford’s formulas; Vincenty’s formulas are online if you want them. In 1981 Bowring gave us simpler formulas that were about-millimeter-accurate for lines up to 150 km, and in 1996 he gave us more long-line formulas. Of course the Coast and Geodetic Survey wasn’t using any of that when they triangulated the US, and their calculated distances weren’t accurate to a millimeter. NAD27 lat-lons are given to a thousandth of a second, around a tenth of a foot, and that’s probably just to reduce rounding errors. Anyway, the point is, the latitude and longitude of a point are defined by its distance and direction (on the spheroid) from a known point. So in the old days there had to be a starting point, which (in NAD27) was Meades Ranch, in Kansas. ---
**Author:** ![md-2000](https://avatars.discourse-cdn.com/v4/letter/m/9d8465/32.png) [@md-2000](https://boards.straightdope.com/u/md-2000)\ **Post date:** [July 18, 2025, 12:19pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/42 "2025-07-18T12:19:14Z")
> [@Freddy\_the\_Pig](#): > > Congress added extra bits of Montana (the squiggles in the northwest and northeast) to the park in 1929 and 1932. What’s with the small sqiggles/dips in the south border of Montana itself in the park? It’s more pronounced that the other irregularities in their south border… I can understand jogs in the north-south lines due to the earth’s shape, but why east-west? ---
**Author:** ![dtilque](https://avatars.discourse-cdn.com/v4/letter/d/d6d6ee/32.png) [@dtilque](https://boards.straightdope.com/u/dtilque)\ **Post date:** [July 18, 2025, 2:57pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/43 "2025-07-18T14:57:06Z")
> [@md-2000](#): > > I can understand jogs in the north-south lines due to the earth’s shape, but why east-west? Usually, irregularities in straight east-west borders are due to poor surveying. Of which there is a lot of in the borders of the states. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [July 18, 2025, 4:45pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/44 "2025-07-18T16:45:17Z")
> [@md-2000](#): > > I can understand jogs in the north-south lines due to the earth’s shape, but why east-west? No difference between north-south and east-west. You can’t expect either one to be a perfect straight line on a map, and it’s not because of “poor surveying”. Real-life surveying, more like. How would they go about surveying the US-Canadian border in 1800-whatever? They’ve been told to place markers each mile along the 49th parallel. So they’re in wilderness Montana, many miles from any point of known latitude-longitude. How do they decide where the 49th parallel is? Nothing to do but use the stars. The problem with that is, the resulting border won’t be a straight line on the map. I guess they must have had some way to estimate the deflection of the vertical, but they can’t expect to get it exactly right, so they’d be doing well to keep the markers within 200 feet of the line that’s now the 49th parallel on the topo maps. In any case, once the markers are placed, that’s it – the border is now defined (I assume) as a “straight line” from each marker to the next marker. (The straight line from marker to marker doesn’t exactly follow the 49th parallel, of course.) The same applies to state borders in the US. They can never be straight, whether east-west or north-south. (A diagonal border like California-Nevada is tougher. Just defining “straight” is a problem there.) ---
**Author:** ![Q.Q.Switcheroo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/q.q.switcheroo/32/5713_2.png) [@Q.Q.Switcheroo](https://boards.straightdope.com/u/Q.Q.Switcheroo)\ **Post date:** [July 18, 2025, 4:47pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/45 "2025-07-18T16:47:16Z")
> [@Timz](#): > > How would they go about surveying the US-Canadian border in 1800-whatever? They’ve been told to place markers each mile along the 49th parallel. So they’re in wilderness Montana, many miles from any point of known latitude-longitude. How do they decide where the 49th parallel is? If you’ve got a few months to spare, check out Pynchon’s _Mason and Dixon_. ---
**Author:** ![Freddy\_the\_Pig](https://avatars.discourse-cdn.com/v4/letter/f/a587f6/32.png) [@Freddy\_the\_Pig](https://boards.straightdope.com/u/Freddy_the_Pig)\ **Post date:** [July 18, 2025, 6:37pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/46 "2025-07-18T18:37:54Z")
> [@md-2000](#): > > Fun facts - > Northwest Angle, a surveying error which results in the again isolated northern-most part of the continental USA just before the 49th parallel starts. I get a little prickly when this is described as “surveying error”, because the surveyors involved did their jobs perfectly. The “error” was on the part of politicians and diplomats working with inadequate information. The US-Canada boundary as far as the Northwest Angle was defined by the US and Great Britain in 1783; the boundary along the 49th parallel (as far as the Rockies) was defined by the Convention of 1818. It was only later, when they got boots (surveyors) on the ground, that they discovered that the two piece-parts fit together in such an illogical way, with northern Minnesota having a waterlogged northern nub. At that point the logical thing would have been for the US to cede the Northwest Angle to Canada. But countries have an aversion to giving up sovereignty unless there is a compelling reason to do so, and in this case there wasn’t. ---
**Author:** ![dtilque](https://avatars.discourse-cdn.com/v4/letter/d/d6d6ee/32.png) [@dtilque](https://boards.straightdope.com/u/dtilque)\ **Post date:** [July 18, 2025, 6:54pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/47 "2025-07-18T18:54:08Z")
> [@Timz](#): > > In any case, once the markers are placed, that’s it – the border is now defined (I assume) Not quite true. The boundary thus marked has to be accepted by the authorities that sent the surveyors out. There have been cases where a surveyed boundary was not accepted. IIRC, the N-S border between California and Nevada was surveyed three times before it was accepted. ---
**Author:** ![JRDelirious](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jrdelirious/32/9531_2.png) [@JRDelirious](https://boards.straightdope.com/u/JRDelirious)\ **Post date:** [July 19, 2025, 1:25pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/48 "2025-07-19T13:25:55Z")
In the book of _How the States Got Their Shapes_ they go over many such examples of a border being defined theoretically based on a latitude or longitude or a straight line from Point A to Point B, but the start (or end) point was yet unsurveyed. ---
**Author:** ![suranyi](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@suranyi](https://boards.straightdope.com/u/suranyi)\ **Post date:** [July 19, 2025, 9:16pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/49 "2025-07-19T21:16:24Z")
Yes, if there’s one thing that book (fascinating, I must say) makes clear, it’s that there’s a big difference between marking a line on a map and marking a line in the real world. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [July 19, 2025, 10:21pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/50 "2025-07-19T22:21:00Z")
At least as far as US state borders are concerned, I wonder if there is any value in resurveying the borders defined by lat and long using GPS and other modern methods. Of course all states affected would need to agree to the new borders. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [December 21, 2025, 10:51pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/51 "2025-12-21T22:51:26Z")
> [@Mark\_Finn](#): > > Bad surveying 200 or so years ago have led to parts of the straight-lined Canada-US border segments to be off by as much as a couple of hundred yards/meters The surveying wasn’t “bad”; doing better would have taken too long, and cost too much. Dunno when they started placing border markers – before 1900, I assume? So say you’re in Montana circa 1890, having been hired to place markers at one-mile intervals along the 49th parallel. How will you do it? Maybe a convenient star exists at declination 49.000000 degrees north, so you can use a zenith tube to find the point on the surface where the star passes directly overhead. You place your first marker, walk east for a mile, set up the zenith tube again and wait for the star to pass overhead the next night. Keep going for a few hundred miles. When someone comes along years later to map the territory, they find that the marker-to-marker-to-marker angles vary – in other words, they don’t plot at the same latitude on the map. Many of them are 200 meters north or south of the 49th parallel that‘s drawn on the map, even tho the star did pass directly above the marker. Drawing that map is what took too long, and cost too much for people that wanted the border marked. The original surveyor didn’t do anything wrong; he knew his markers weren’t going to form a smooth arc on the globe, due to the deflection of the vertical. He knew the only way to get a smooth arc was to triangulate the territory, and he didn’t have the time or the crew to do that. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [December 21, 2025, 10:52pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/52 "2025-12-21T22:52:55Z")
> [@Hari\_Seldon](#): > > On account of the oblateness of the earth, the distance from 45 to the pole is less than that between 45 and the equator. Other way around. What if Earth were a pancake, ten times as wide as it is thick, but still with an elliptical cross-section? Where would 45 degrees latitude be? Out near the rim of the pancake. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [December 22, 2025, 12:10am UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/53 "2025-12-22T00:10:02Z")
> [@Timz](#): > > > [@Mark\_Finn](#): > > > > Bad surveying 200 or so years ago have led to parts of the straight-lined Canada-US border segments to be off by as much as a couple of hundred yards/meters > > The surveying wasn’t “bad”; doing better would have taken too long, and cost too much. There’s “bad” as in primitive state of the art, and “bad” as in poorly executes at the time, For sure better art means better results can be had more cheaply. I agree they did a reasonably workmanlike job for the tools, conditions, and era. But by modern standards it was still low accuracy and low precision. Which triggers issues now. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 22, 2025, 1:06am UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/54 "2025-12-22T01:06:14Z")
> [@Timz](#): > > What if Earth were a pancake, ten times as wide as it is thick, but still with an elliptical cross-section? Where would 45 degrees latitude be? Out near the rim of the pancake. That depends on how you define latitude, on a non-spherical Earth. There are multiple possible definitions, and so for anything that matters, you have to be clear which one you’re using. Which probably also isn’t the same for all applications. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [December 22, 2025, 1:59am UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/55 "2025-12-22T01:59:42Z")
Of course you are right. And I knew that too. I once read a book about two expeditions sent to Equador And northern Sweden to measure the length of a degree of latitude in each place and the latter was about a km longer. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [December 22, 2025, 11:06pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/56 "2025-12-22T23:06:20Z")
> [@Chronos](#): > > That depends on how you define latitude, on a non-spherical Earth. There are multiple possible definitions … Yes, there’s geocentric latitude, and parametric latitude, authalic latitude and blah blah blah. But when a reader sees “latitude” with no adjective, he rightly assumes it’s none of those. (Could possibly be astronomic, but 90% it’s geodetic, the kind of latitude that’s on the map.) ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [December 22, 2025, 11:20pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/57 "2025-12-22T23:20:57Z")
> [@LSLGuy](#): > > I agree they did a reasonably workmanlike job for the tools, conditions, and era. But by modern standards it was still low accuracy and low precision. Which triggers issues now. What issues now? People want to move markers? If someone had gone out in 1960 (or 1980, I suspect) to place markers along the 49th parallel, with modern equipment but no access to triangulation data or any other measurements done by then, lots of their markers would have been 100+ meters off too, however careful they were. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 23, 2025, 12:29am UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/58 "2025-12-23T00:29:19Z")
> [@Timz](#): > > Yes, there’s geocentric latitude, and parametric latitude, authalic latitude and blah blah blah. But when a reader sees “latitude” with no adjective, he rightly assumes it’s none of those. I think that the vast majority of readers, when they see “latitude”, don’t even stop to think about the fact that the precise definition matters on a nonspherical Earth. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [December 23, 2025, 1:37am UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/59 "2025-12-23T01:37:05Z")
Yup. “Latitude” has a “lies to children” non-rigorous definition on a spherical homogenous infinitely rigid and perpetually unchanging Earth and that’s the one pretty much every non-specialist uses. With no awareness there might even be other definitions. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 23, 2025, 2:45pm UTC](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484/60 "2025-12-23T14:45:34Z")
And, heck, I’m a lot more knowledgeable about the issue than most people, in that I know that the definitions differ, and I could come up with at least three possible definitions off the top of my head, all of which agree in the spherical limiting case… but I still wasn’t sure which one is actually the most commonly-used standard. [Previous page](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484.md?page=2) [Next page](https://boards.straightdope.com/t/straight-lined-international-and-interior-borders/1012484.md?page=4)