How did Wilbur Wright create and use the formulas he mentions in his letter to Octave Chanute? In a letter written by Wilbur Wright and addressed to Octave Chanute, dated 1903/06/18, I have found two things that I do not quite understand.


  Our engine develops at the brake 15.6 horse power and we are convinced that this is very close to what we will be able to reach as a maximum. It furnishes in foot pounds at the brake, energy equivalent to 23 percent of the heat units
  contained in the fuel, so its efficiency is some twenty percent higher than the usual efficiency of gasoline engines which ranges from 14 to 18 percent of the total power contained in the fuel.


How did the two brothers make such a revolutionary improvement?
Modern gasoline engines have a maximum thermal efficiency of about 25% to 30%. The Wright brother's motor had an efficiency of 23% in 1903.


  The Dumont motor may develop more than 10 horsepower but it is with me a case of seeing before believing. The screw of the airship was 4 meters in diameter and had a pitch of 4 meters, and the engine at 1200 revolutions (which is the speed of maximum power of motors of this size) drove it 200 turns per minute and gave a thrust of 154 to 165 lbs. Now if you multiply the thrust, by the pitch divided by the circumference of the screw, and multiply this by the travel per minute of the center of pressure (which is about 5/6 of the circumference of the screw) you will find that the result is about 10 horse power.” 


It is evident that the Wrights used the relation:

Power = Thrust x (5/6) x pitch x (RPM/60 sec)

Where does this formula come from?

What exactly is that "center of pressure (which is about 5/6 of the circumference of the screw)"?
I do not have first-hand knowledge of this formula, so what I write below is speculation. But the Wrights used sound physical reasoning, and I will try do do the same.

Power in physics is the product of force and speed. The formula does indeed contain a force (thrust) and a speed (pitch times rotation speed) if the terms used by Wilbur are interpreted in a certain way. Pitch today is an angle, but I would think Wilbur meant with pitch what is termed distance D in the picture below.



Propeller advance ratio (picture source)

This is consistent with his earlier sentence where he gives a length for the term pitch. The term pitch * RPM/60 is actually the flight speed in m/s.

Now to the term 5/6: I would expect that Wilbur calculated the pitch from the incidence of his propeller. Since the propeller blade needs some angle of attack to produce thrust, the actual distance traveled through air is a little smaller than what Wilbur got. Therefore, I think the term 5/6 accounts for that slippage and had been determined empirically.

Now for the revolutionary increase in efficiency: I do not know what engines Wilbur used for comparison, but the low number he cites suggests that he used exceptionally poor specimens .