First of all, that's a book on flight dynamics, not a pilot's study guide. Second, right before those two statements it says "simplifying assumptions". And right after, it also says that the aircraft is a rigid body (not true), it has constant mass (also not true), it has no turbines (again, not true), and that there is constant wind with no turbulence (almost never true). 

Simplifying assumptions are not things we assume are actually true. They are things we pretend are true because it makes the math easier, and they are close enough to true that the math still works.

Here's the relevant bit:
https://imgur.com/KMjimev

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You guys must enjoy spending overtime searching for oddball cases.
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Are you aware that lacking air navigation certification is a major cause of aviation accidents? 
IOW flat-earthers should be disqualified from any aircraft certification requiring navigation!
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Pilots and aeronautical engineers concerned with aircraft stability don't have to think about the shape of the earth.
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But when you get into world-wide navigation, they most certainly do.
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Key sentence in quoted article below: 
For short distances on Earth (around 20 km or less), it is acceptable to neglect Earth’s spheroidal shape and utilize planar-trigonometry. 
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Regarding the question of flight stability, all concerns are what is happening RIGHT NOW with the aircraft, which means that 20 kilometers is enormously larger than the dimensions of the plane, consequently no concern for the curvature of the earth is relevant for such studies of flight stability!
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As the long-range and ultra-long range non-stop commercial flights slowly turn into reality, the accurate characterization and optimization of flight trajectories becomes even more essential. An airplane flying non-stop between two antipodal points on spherical Earth along the Great Circle (GC) route is covering distance of about 10,800 NM (20,000 km) over-the-ground. Taking into consideration winds (Daidzic, 2014; Daidzic; 2016a) and the flight level (FL), the required air range may exceed 12,500 NM for antipodal ultra-long flights. No civilian or military airplane today (without inflight refueling capability) is capable of such ranges. About 40-60% increase in air range performance will be required from the future airplanes to achieve truly global range (GR). 

There is no need to elaborate on the economic aspects of finding the shortest trajectories between two points on Earth. However, many other factors may cause perturbations of such trajectories when considering the minimum-cost, minimum-fuel, or any other goal-function in complex optimizations. The Earth is not a perfect sphere and due to rotational and gravitational effects the shape is more of an oblate spheroid. Actually, Earth’s shape is even more complicated and more appropriately treated in terms of tesseral surface harmonics (Tikhonov and Samarskii, 1990). Earth’s Polar radius is about 21 km shorter than the Equatorial. 

Several Earth’s shape approximations are used: 
 Idealized spherical Earth of equivalent volume (implicitly used in International Standard Atmosphere or ISA definition). 
 Reference mathematical ellipsoid of revolution (WGS-84, IERS/ITRS). Smooth and oblate. 
 Geoid or particular equipotential surface that approximates mean sea-level (MSL). Irregular and locally smooth. Physically the most important measure of Earth’s shape. 

An example of Geoid in use is WGS-84 (revision 2004) EGM96 Geoid. 
 Actual or physical Earth surface with all terrain details. This is fractal dimension, scale dependent and mathematically intractable. Using spherical Earth approximation is sufficient for the majority of longrange air navigation problems. Due to economy of flight, we are particularly interested in the shortest distances between two arbitrary points on Earth. Differential geometry classifies such lines on smooth surfaces as geodesic lines. On the spherical Earth model a geodesic is a Great Circle (GC) or Orthodrome segment. GCs distances are not necessarily always shortest with respect to time as atmospheric wind plays significant role in distance-time optimization problems. 

The optimization of flight trajectories taking into account atmospheric factors, extended operations (ETOPS) procedures (De Florio, 2016; FAA, 2008 ), airspace restriction, etc., is a difficult task. Finding geodesics on smooth ellipsoidal Earth has been solved. However, finding geodesic between two arbitrary points on the actual Earth surface considering all the vertical terrain features is practically impossible. Although, it has been with us for many years, the theory of GC and rhumb-line navigation has not been presented clearly and comprehensively for air navigation practitioners, operators, and students. 
One of the stated purposes of this article is to review and summarize differential geometry and calculus of variation theories as applied to spheres. That will relive readers from searching and consulting multiple sources using different and often confusing terminology. We are only considering spherical Earth approximation and present theory of GC (Orthodrome geodesic) and rhumb-line (Loxodrome) navigation. 

For short distances over certain terrestrial regions we also provide some simplified approximate formulas and define their limits of use. Several ultra-long-range navigational problems utilizing existing major international airports are fully solved using Orthodromes and Loxodromes. Graphic representation utilizing Mercator (cylindrical) and azimuthal (planar) projections is presented. The longest commercial non-stop flights today are reaching 8,000 NM (Daidzic, 2014). Increase in air range of, at least, 40% is required to achieve full GR connecting any two airports on the Earth (Daidzic, 2014). 
Practically, due to airspace restrictions, ETOPS procedures, and other considerations, the range of existing long-range subsonic airplanes may need to increase by at least 50%. The main purpose of this article is to give complete and comprehensive consideration of short and long (geodesic) lines on spherical Earth for the purpose of air navigation. While clearly professional navigation planning software is available to major airlines/operators and ATC system it is mostly used as a blackbox. The objective is also to remove mysteries behind the long-range navigational calculations and provide working equations. Spherical approximation is satisfactory for overwhelming number of long-range air navigation problems. In particular, Orthodromes and Loxodromes are typically considered the two most important curves for air navigation. For that purpose we coded working equations into several software platforms (Basic, Fortran, IDL, and Matlab). 

The main goal of this article was to provide the fundamental theory and understanding, while the computations can be executed in any high-level programming language. Detailed mathematical derivations are presented in several appendices as to relive a reader interested only in the final results from the heavy mathematical interpretations involving differential geometry, variational calculus, topology and  other mathematical fields. Indeed, a knowledge of differential geometry, plane and space vectors and vector calculus, and calculus of variation (variational calculus) is required for in-depth understanding of the subject matters. 

The most important geometric and topological properties of spheres have been reviewed and working equations provided. An added benefit of presented long-range navigation solutions is relatively easy implementation of the actual Point-of-Equal-Time (PET), Point-of-NoReturn (PNR) and ETOPS limitations for given wind conditions (Daidzic, 2016a) and one-engine-inoperative (OEI) cruising speeds (Daidzic, 2016b). In a future contribution, and for the academic completeness an ellipsoidal Earth model will be introduced with the Great Ellipse (GE) substituting Great Circle (GC). True geodesic computations are complicated even for a smooth ellipsoid of revolution requiring iterative solvers. Generally, GC calculations on spherical Earth are sufficient for reliable air navigation flight planning purposes, considering all other uncertainties involved and mandatory fuel reserves. 

Air navigation should be a mandatory course in every professional pilot curriculum and especially so in aviation university education. Unfortunately, it often is not, which results in operational safety degradations. Too much and/or uneducated reliance on sophisticated electronic navigation technology did and certainly will continue to cause aviation accidents and incidents.
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Short distance Great Circle navigation on a perfect sphere
As mentioned previously the cosine-formula for GC (Equation C8 ) fails for very short arcs. While mathematically exact, the problem occurs with the finite number of digits representation as was illustrated by Sinnott (1984). Essentially for angular distances of less than 1 arc-minute it is better to use haversine formula (Equations C14 and C15) although modern 64-bit floating point arithmetic’s can push the envelope to less than 1 arc-second. Such small angular distances have important astronomical, but practically no air navigation applications. For short distances on Earth (around 20 km or less), it is acceptable to neglect Earth’s spheroidal shape and utilize planar-trigonometry. 

Here, we will distinguish two special cases (1) Polar-coordinates flat-Earth formula used with polar azimuthal projections for high latitudes, and (2) planar projections using simple Pythagoras theorem (Euclidian space). Close to the geographic poles (NP or SP), the polar projections (De Remer and McLean, 1998; Jeppesen, 2007; Struik, 1988; Underdown and Palmer, 2001) will result in meridians being represented as radials coming out of poles, while the lines of latitude are concentric circles with separation distance between them changing depending on the type of projection. The polar stereographic azimuthal projections is illustrated in Figure D1. Polar gnomonic would be very similar, but the distance between the lines of latitude will be increasing rapidly (Equator is in infinity). Orthographic projection has center of projection in infinity (Figure D2). The lines of latitude are decreasingly spaced concentric circles. Conjugate hemisphere is not represented. 

The irony of Neil's nonsense math post is not lost on anyone. 

It says there are "several" theories on spherical math, and it is SO complicated to navigate because of the spherical "assumption."


Seems to me a flat earth map solves all navigatiom problems. 

Neil bolded this from his cut and paste: "For short distances on Earth (around 20 km or less), it is acceptable to neglect Earth’s spheroidal shape and utilize planar-trigonometry."

Yeah, I don't know of many flights that are less than 20km (12.4 miles).  It would be a challenge to find two airports that close to each other ... even in the same city.  I'd hop in a car first or take the bus ... or ride my bike.

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Flat-earthers don't need any help making fools of themselves.
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I find this to be very persuasive.

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Embry-Riddle is in the business of preparing pilots and navigators for doing a great job.
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They have no vested interest in promoting ignorance.
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But they bemoan the preponderance of unqualified applicants to positions of responsibility in aeronotics:
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Air navigation should be a mandatory course in every professional pilot curriculum and especially so in aviation university education. Unfortunately, it often is not, which results in operational safety degradations. Too much and/or uneducated reliance on sophisticated electronic navigation technology did and certainly will continue to cause aviation accidents and incidents.
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Found under "things flat-earthers can't stand to see repeated."
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IOW just as a flat-earther can deny the existence of satellites while continuing to use his GPS (which relies on satellite data transmission to operate AT ALL), so too an unqualified aviator can think the earth is "flat" while continuing to use his electronic navigation technology (which includes consideration of the spheroid shape of the earth). But when his technology fails to operate properly for whatever reason he finds himself at a total loss to make do safely without it, just like a car driver who suffers the loss of GPS guidance may be unable to get out a Thomas Guide and find his way without the GPS.
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What happened to all the satellites in Elon Musk's Tesla view of earth from outer space?  Technology for tv and cell phones is ground based.  That's why people in the hills cannot get good reception even though they have access to the sky.  No tower, no service.  "Satellite" service is merely tower/repeater service localized for hilly areas, and even then, some areas STILL are without service.