34 “Ship captains in navigating great distances at sea never need to factor the supposed curvature of the Earth into their calculations. Both Plane Sailing and Great Circle Sailing, the most popular navigation methods, use plane, not spherical trigonometry, making all mathematical calculations on the assumption that the Earth is perfectly flat. If the Earth were in fact a sphere, such an errant assumption would lead to constant glaring inaccuracies. Plane Sailing has worked perfectly fine in both theory and practice for thousands of years, however, and plane trigonometry has time and again proven more accurate than spherical trigonometry in determining distances across the oceans.”

Yes, ship’s navigators do take account of curvature and use spherical geometry; they use latitude and longitude, which are just angles in a spherical coordinate system - so why use that instead of just cartesian coordinates? They use sextants and almanacs to determine their position on the globe.  

It is important to note that it is the angle of the horizon with the celestial object that is important. The celestial object is far enough away that the same results couldn't be reproduced on a flat surface - the fact that celestial navigation works is testimony to the earth being an oblate spheroid. For more information, start here and scroll down to Point 34.

By the way, I mentioned before that my dad was a navigator on Lancaster bombers in the RAF. He always worked on the assumption that he was flying over the surface of a sphere. If he had been wrong, he would have died, and I wouldn’t be here now.

Please note that a 2D surface is not the same as a flat surface . Run your finger all the way round the outside of a cylindrical glass. The path it took is two dimensional, but it is still curved.  Try the same with a ball. Same thing.

If you want more read this extract from this source;

"There are 2D map projections of earth that preserve angles, like the Mercator projection. This one was a great invention for sailors as the angles they measured on earth were indeed consistent with the angles on the map. However, this projection is neither equal-area nor equidistant. 

Anybody dealing with maps should at some point come across the fact, that there is not ONE single 2D-model of earth that can do all three things at the same time (preserving angles, equidistant and equal-area). The standard flatearth map for instance is an azimuthal projection which is equidistant, but ONLY for distances measured through the center of the map. All other distances plus angles and surface areas are being distorted.

https://en.wikipedia.org/wiki/List_of_map_projections

Again, this is perfectly understood:

https://en.wikipedia.org/wiki/Theorema_Egregium
That is why every flat map is a compromise, getting one thing right, but many things wrong because you just can't display a 3D-sphere on a 2D-map without any distortion.

That is also why a conformal projection (preserving angles) like Mercator's ONLY works for terrestrial navigation. If you used a flat earth model for celestial navigation (with a sextant for instance), you would be completely lost because it just doesn't work (we'll have more on that later). 

All of the impressive feats in sailing history that relied on celestial navigation, like Shackleton's lifeboat journey to South Georgia, just wouldn't have worked if they had navigated on a flat earth model.

https://en.wikipedia.org/wiki/Voyage_of_the_James_Caird#The_open-boat_journey"

And have a look at: All The Ships At Sea: A Pragmatic Test Of Earth's Shape. In short, none of the records of sea voyages over the last two centuries make sense on a flat earth. Every recorded course would be impossible.

https://www.metabunk.org/all-the-ships-at-sea-a-pragmatic-test-of-earths-shape.t8954/#post-208543