Draw a straight line, as shown in the diagram, from the milking-stool perpendicular to the near bank of the river, and continue it to the point A, which is the same distance from that bank as the stool. 
If you now draw the straight line from A to the door of the dairy, it will cut the river at B. 
Then the shortest route will be from the stool to B and thence to the door. 
Obviously the shortest distance from A to the door is the straight line, and as the distance from the stool to any point of the river is the same as from A to that point, the correctness of the solution will probably appeal to every reader without any acquaintance with geometry. .

Math Genius