Archimedes lived centuries before Da Vinci. There is a water pump called Screw of Arquímedes. Basically it is to place inside a pipe a helical screw, whose diameter is slightly smaller than the inside diameter of the pipe. We assume the obligation to use only the vocabulary and practical notions that were available to an educated person in the Da Vinci era.
If the assembly prevents the screw from moving inside the pipe, then the water will move and the system will be used for pumping. If the screw were loose with the pipe full of water, instead of pumping it would travel inside the pipe when it started to turn.
Now think of an Archimedean screw submerged in the water and without the pipe. The screw rotates and the water is displaced perpendicular to the axis of the screw (radial direction). If the turn is fast enough, the volume of water water displaced radially in a time ##\Delta t## will be less than the volume of material of the propeller that tries to push the water to occupy a given region, in that same ##\Delta t##. This means that the screw encounters opposition when it rotates in the water, as does an oar. Turning quickly enough, the screw in the water will serve to propel a boat, as the paddle serves. Actually it will serve as a driver in any situation within the water. If the hull of a sunken ship, which is at the bottom of the sea, is fitted with a screw that turns rapidly, the ship could be brought to the surface.
Now let's recognize Da Vinci's mental breadth and knowledge. In the metal workshops of the time bellows were used to propel air and blow the fire of the forge, the furnace, etc. That same bellows propels water when it is submerged in water. The same bellows, used in the same way, do the same in the water and in the air. Could the same thing happen with the screw? That is, if it rotates in the air quickly enough, could it serve as an impeller, just as it does in the water? The air and water are fluid. They differ in density, in viscosity, etc. , but when they are displaced by the helix, they respond in the same way. A properly designed screw, rotating with the proper speed, should serve as an impeller in the open air. This way of reasoning was available to an educated person at that time. And much more within reach of someone as prepared as Da Vinci. To devise his screw-like helicopter did not need to raise what Newton raised later.