of attack constant. Their blades were only 5% less efficient than those used 100 years later. Understanding of low-speed propeller aerodynamics was complete by the 1920s, although increased power and smaller diameters added design constraints. Alberto Santos Dumont, another early pioneer, applied the knowledge he gained from experiences with airships to make a propeller with a steel shaft and aluminium blades for his 14 bis biplane. Some of his designs used a bent aluminium sheet for blades, thus creating an airfoil shape. They were heavily undercambered, and this plus the absence of lengthwise twist made them less efficient than the Wright propellers. Even so, this may have been the first use of aluminium in the construction of an airscrew. Theory In the nineteenth century, several theories concerning propellers were proposed. The momentum theory or disk actuator theory – a theory describing a mathematical model of an ideal propeller – was developed by W.J.M. Rankine (1865), A.G. Greenhill (1888) and R.E. Froude (1889). The propeller is modelled as an infinitely thin disc, inducing a constant velocity along the axis of rotation and creating a flow around the propeller. A screw turning through a solid will have zero "slip"; but as a propeller screw operates in a fluid (either air or water), there will be some losses. The most efficient propellers are large-diameter, slow-turning screws, such as on large ships; the least efficient are small-diameter and fast-turning (such as on an outboard motor). Using Newton's laws of motion, one may usefully think of a propeller's forward thrust as being a reaction proportionate to the mass of fluid sent backward per time and the speed the propeller adds to that mass, and in practice there is more loss associated with producing a fast jet than with creating a heavier, slower jet. (The same applies in aircraft, in which larger-diameter turbofan engines tend to be more efficient than earlier, smaller-diameter turbofans, and even smaller turbojets, which eject less mass at greater speeds.) Propeller geometry The geometry of a marine screw propeller is based on a helicoidal surface. This may form the face of the blade, or the faces of the blades may be described by offsets from this surface. The back of the blade is described by offsets from the helicoid surface in the same way that an aerofoil may be described by offsets from the chord line. The pitch surface may be a