Why does this calculation show Gustave Whitehead's propellers were more than 100% efficient? I have tried to calculate the efficiency of Gustave Whitehead's propellers using the information he provided to the American Inventor Magazine in 1902 (see the citation below).
Number of propellers = 2
Engine power per propeller, $P$ = 20 hp
Propeller diameter, $D$ = 6 ft
Static thrust per propeller, $T$ = 254 lbf
Propeller efficiency, $\eta_{prop}$
Gear efficiency, $\eta_{gears}$ &lt; 100%
Air density, $\rho=1.2 \frac{kg}{m^3}$
Doing the calculations: $\eta_{prop} &gt; \sqrt{\frac{T^3}{(P^2 \cdot \pi \cdot \eta_{gears} \cdot \frac{D^2}{2} \cdot \rho )}} = 101.4\%$ which is impossible.
Is there a mistake? If yes, where?
Note: I used the formula that relates static thrust to efficiency, power and propeller diameter.

The Whitehead Flying Machine
Has the End been Finally Attained, and is the Dirigible Balloon to Go?
Editor, American Inventor
Dear Sir: Replying to your recent letter, I take pleasure in sending you the following description of my flying machine No. 22, the latest that I have constructed:
This machine was built in four months with the aid of 14 skilled mechanics and cost about $1,700 to build. It is run by a 40 horse-power kerosene motor of my own design, especially constructed for strength, power and lightness, weighing but 120 pounds complete. It will run for a week at a time if required, without running hot, stopping, or in any possible manner troubling the operator. No electrical apparatus is required for ignition purposes. Ignition is accomplished by its own heat and compression; it runs about 800 revolutions per minute, has five cylinders and no fly-wheel is used. It requires a space 10 inches wide, 4 feet long and 10 inches high. ...
The propellers are 6 feet in diameter and have a projecting blade-surface of 4 square feet each. They are made of wood and are covered with very thin aluminum sheeting. The propellers run about 600 revolutions per minute under full power and turn in opposite directions. When running at full speed they will exert a thrust of 508 pounds. I measured this thrust by attaching the machine to a post by means of a dynamometer and running the engines at full speed. ...
I have no photographs taken yet of No. 22, but send you some of No. 21, as these machines are exactly alike, except the details mentioned. No. 21 has made four trips, the longest one and a half miles, on August 14, 1901.   The wings of both machines measure 30 feet from tip to tip, and the length of the entire machine is 32 feet.   It will run on the ground 50 miles an hour, and in air travel at about 70 miles.  I believe that if wanted it would fly 100 miles an hour. The power carried is considerably more than necessary. ...
Trusting this will interest your readers, I remain, Very truly yours,
GUSTAVE WHITEHEAD
Bridgeport, Conn.

Source: The American Inventor Magazine
, 1 April 1902
.
The equation you referenced shows prop diameter squared, divided by 2, not 4.

Even then, the calculation shows that marketing is to experimentation as about 2:1 :)