I was looking for WW2 aircraft indicators and I found one navigation equipment called API (air position indicator). But I couldn't find how it works exactly????
The API uses not only heading and airspeed but they are fed to mechanisms within the instrument that solve the navigation problem using sine, cosine and secant functions. The output was latitude and longitude from the departure or a known fix (geographical position) to give the no wind position of the airplane. This was then used by the navigator to give a dedreconing position by plotting the last known wind vector from the no wind position.
These links (to download documents) give interesting information about a "Ground Position Indicator" from 1953, which sounds like an "Air Position Indicator" with added capability to correct for assumed wind speed and direction. The device took into account the spherical shape of the earth, but apparently did not incorporate acceleration sensors as per a true Inertial Navigation System.
(Click on all 3 links under "Ford Instrument Company")
And here is a link for information about the "Air Position Indicator":
Apparently the only inputs to the "Air Position Indicator" were true airspeed, gyro-magnetic compass heading, and initial lat/lon. It automated the ded. reckoning process, taking into account the spherical shape of the earth, and magnetic variation, but assuming no wind. Apparently the pilot or navigator then corrected the result for assumed wind speed and direction.