Flat Earth Claim
Mr. Thrive & Survive has a YouTube video "MAJOR LIES EXPOSED By TRIANGE RESEARCH" (sic) about how Great Circles are a lie and he used to sail and they never sailed a Great Circle path and his triangle research somehow proves all this.
As per usual, it is an unnecessarily long and rambling video (I might ramble but I try to stay on point) that ultimately fails when it tries to make its point. Let's cover a few of the more obvious misconceptions from the video...
Great Circles vs Rhumb Lines
A Great Circle path is the 'straight line' of a sphere. It doesn't mean that you go in a circle on the surface, you go in a radial circle around the Globe itself. The length of the Great Circle is the circumference. That's all it is.
If you put a string between any two points on a globe and pull it taunt it will mark out the Great Circle path. That's all the more "math" you need to plot one on a globe and it works for every pair of points. There is NO FLAT MAP in existence where you can draw a straight line between two arbitrary points and have that be the shortest path - period. You can make special maps where SOME straight lines are the shortest distance; for example the Gleason Map is actually the azimuthal equidistant projection where every point on the map to the center AND ONLY THE CENTER is accurate, every other path is distorted to make that fit. This is why this map utterly fails when you try to look at flight paths in the Southern Hemisphere.
The Equator is a Great Circle for example, it takes you all the way around. Every line of longitude is a Great Circle - but the only latitude line that is is the Equator -- all the other lines of latitude get shorter and shorter. It's called a Great Circle because it always goes all the way around the globe, never shorter. So the Longest possible route around the Globe is also the shortest path between two points that fall along that line.
Before we had satellite navigation and computers ships traveling shorter distances (maybe 1000 miles) would often take a path called a loxodrome (or Rhumb Line) because this is a path with a constant compass bearing so it is easier to manually navigate this way. However, over greater distances these paths become significantly longer than the Great Circle paths and it becomes very inefficient to continue to navigate using a Rhumb line route.
All modern, long haul shipping now uses Great Circle routes. You can see this in how the routes that are further north/south of the Equator appear as arcs on the Mercator projection.
Here is an example of a Great Circle (in Red) route using Google Earth showing why this is so:
This diagram shows the issue with the Rhumb line distances compared to the Great Circle path. The Rhumb line path turns out to be a spiral towards the pole.
The nice thing about Rhumb lines is that if you are navigating using a Mercator Projection they are the straight lines. [more info]
So my question to Mr. Thrive & Survive -- do you have any evidence for your claims about what routes you sailed and why?
He then argues that since non-Euclidean geometry wasn't invented until the 1800's ancient navigators couldn't POSSIBLY have used spherical trigonometry and offers this as "proof":
The problem is that isn't the WHOLE story.
Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, retrieved 2008-01-23
Shows that Ibn al-Haytham contributed to alternative geometries as far back as the 11th century and...
Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p. 470, in Roshdi Rashed & Régis Morelon (1996), Encyclopedia of the History of Arabic Science, Vol. 2, pp. 447–494, Routledge, London and New York:
"Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the nineteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the thirteenth century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
The Greek astronomer and mathematician Hipparchus produced the first known table of chords in 140 BCE. His work was further developed by astronomers Menelaus (ca. CE 100) and Ptolemy (ca. CE 100). The average sailor used tables and things like the ASTROLABE (often attributed to Hipparchus) - they didn't do the math by hand.
[See also: The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam]
But the deeper answer here is simple, Spherical & Conical geometry are NOT the same as formal non-Euclidean geometry -- non-euclidean geometry is a very abstract form as opposed to simply understanding spheres and basic trigonometry.
It takes about 5 minutes to research Hipparchus, Menelaus, & Ptolemy sufficiently to see that he is simply mistaken in his assessment.
This argument is the same as saying "Quantum Mechanics uses division so they couldn't have been using division 2000 years because that's Quantum Mechanics which was invented in the 20th century!" He confuses one component with the whole.
Horizon Rises To Eye-Level?
This is the simplest claim to completely debunk (as I have in this post which goes into more detail and has some video examples as well). I have personally taken images from airplanes that disprove this absurd claim that is just so blatantly false it's hard to believe that even Flat Earthers keep repeating it.
This first image is to show the Theodolite application works and that it works well. You can see that CAL is not activated so I have NOT recalibrated it, I'm using the default in all images. But, by all means, don't trust me, get the Theodolite app (or some professional instrument) and make your own observations.
To explain this a bit, the red square+cross reticle here marks out LEVEL from my position and you can see that, despite my purposefully tilting the phone, it nails the actual horizon here from ground level. The white cross-hairs just mark dead center.
This next image is from about 25,000 feet and I've pointed the 'dead center' cross hairs at the horizon and you can see this is marked as being 2.8° below level. This is exactly where it should be on a Globe of 3959 miles radius.
And as we go up to 38,000 feet the horizon drops to about 3.4° - again, exactly where it should be on a Globe of 3959 miles radius.
He doesn't prove anything in this video except that he doesn't actually understand sailing, spherical geometry, trigonometry, history, what the horizon actually looks like, or anything else that I can tell.