Sorry for the delay, Chris, but you've already had a few good answers. The "How Sails Work" is by Paul Bogotaj appears good, though I've only skimmed it so far. I do like the two line explanation of Upwash, which is generally misunderstood and ignored, whereas a good foil shape will enhance it and put it to good use.
In an effort to simplify things, think of anything the sticks out in an airflow to be an airfoil, in that it foils the wind. The wing of an aeroplane is pushed through the air by its engine, a glider is pulled down by gravity, but due to the shape of its airfoils slices forward as it descends. When the wind blows the sail of a boat produces a force which drives it and adding the wind speed to the boat speed results in an apparent wind blowing past the rig. A round post planted in the ground also 'foils' the air, and when the wind blows there is a resulting force pushing it straight downwind. We call that drag.
Now if we alter the cross sectional shape of our pole cleverly and angle it so that it deflects the airflow sideways the force produced will angle off a bit to one side, giving a total force at an angle to the wind flow, which will have a measurable strength and direction. It therefore is a vector. For convenience we split this vector into two components, the one angled straight down wind, which we call drag, and the other at right angles to the apparent wind, which we call lift. The ratio of the lift to the drag can vary from zero for our round post to about 60 for a glider wing stuck in the ground at the right angle and the right wind speed. A sailing rig will be in between these extremes as the wind varies in strength from naught to gale force and the angle can vary by 360°.
It matters not whether we are talking about wings, sails or poles as they all 'foil' the wind and produce more or less lift and drag. The clever trick is for us to get the best lift and drag figures to suit our particular requirements. A zero L/D is fine on the dead run which is why the flat junk rigs will go downwind, but by introducing camber and a slot the split junk still has some airflow deflection on the dead run and will produce a usable L/D and therefore a greater total force for better performance, eve on this point of sail.
It is easy to see that an aeroplane in steady level flight requires the lift to equal the weight and the engine thrust to equal the drag. The boat sailing to windward is not so clear, but if you think of the boat sailing at 45° to the wind with the sail set out at 20° to the centre line then the apparent wind hits the sail at about 25° with the drag going down wind and the lift at right angle to the apparent wind. Now if we add the two vectors to get the total wind force again and then split it again but with one component along the boats direction of sailing and the other at right angles we end up with a small driving force which pushes the boat forward (and therefore up wind) and a bigger force pushing it sideways forming leeway and making it heel.
Now to get the best windward performance we want the biggest driving force and the smallest sideways force. Working back through the vectors, this means we want the biggest lift and the smallest drag which means the best Lift/Drag ratio.
Getting lots of lift is easy as you can increase the area and put lots of camber in the sail, but the drag will also go up and you won't gain. The key is to get good lift and minimise the drag which is not so easy. The total drag is made up of a number of separate types, form drag, skin friction and others, but the one that I have been talking about is induced drag which is the main problem and is a by product of making lift. You can't have one without the other, unfortunately.
The commercial aviation industry has spent millions on this problem and now we have airliners with a glide performance comparable with early gliders! We used to practice total engine failures in the Boeing 737 simulator, typically losing both engines at 30,000 feet over Clacton-on-Sea and gliding to land at Heathrow. We always ended up high and had to fly two or three orbits to lose the height and judge our final approach. That calculated out at about a 1 in 20 glide angle, or 3° slope. The two seat Slingsby T21 glider used by the ATC in the early days had a glide angle of 1 in 21, about the same, but the difference being that it had a best glide angel at about 70 knots whereas the B737 was best at 250 knots.
The same type of improvement is being made in sailing rigs for racing, but the cruising boats are away behind. The flat junk rig was at the bottom of the pile when it came to L/D ratios, but the later cambered rigs are seriously better. The fact that the split junk rig in Poppy can heel the boat to windward is a clear indication how far forward the total force vector is and just how good the L/D ratio is. In the right hands this rig will embarrass the pointy headed brigade. Where I could outperform most similar boats I never keep it up from beginning to end of any race. Edward is up against hot race tuned boats, which ain't fair, in my opinion, and which does not show off his real speed.
I hope that covers the first and second bullet points in the first mail in this thread.
For the third point I agree that the dynamic range of an aeroplane can be low about two or three to one and a sailing boat's can be infinite, but that makes the challenge more interesting, but not any different.
The difference in apparent wind at top and bottom of the rig has been discussed and when actually calculated does not make that much angular difference, in practice being only a few degrees! The Bermudan rig needs twist because the sail chord is reducing with height so it needs to twist to present the correct entry angle with change of height. The mast also ruins the L/D as the chord shortens. Poppy, with the parallel luff and leech shape sail performs well with negligible twist, and Edward has arrived at the same conclusion. Don't be misled by the inefficient Bermudan rig.
Chris, you ask, “Why then do aircraft fall out of the sky fairly catastrophically when all engine power is lost?” They don't. Unfortunately modern pilots have become so dependent on avionics and autopilots that they have forgotten their basic training. It scares me rigid. Every commercial pilot must practice engine failures on take off every 6 months, and we used to practice failures on approach, but few airlines will do more than the financial minimum and include failures in cruise and other interesting tricks. Pity, because it was great fun (and often quite exciting!)
Sorry about the long-winded simplified expatiation, but some might find it helpful.
Cheers, Slieve.