a square sail is a very simple device. it catches a following wind and pushes the vessel forward. the lateen sail is more complex. it is set at an angle to the wind. it works by creating a difference in air pressure between the two sides (concave and convex) of the sail. this is the same way a fore-and-aft sail works. this is why the lateen sail is considered the ancestor of the fore-and-aft sail. the advantages of the lateen sail is that it is effective in lighter winds. it causes less drag and so is more efficient. it allows the vessel to sail much "closer to the wind" (meaning it can sail at up to about 45 degrees into the wind). this made the lateen sail a very important invention.