A navigator uses the same principle when he "doubles the angle on the bow" to find his distance from a lighthouse or other object.
If he is sailing on the course ABC and notes a lighthouse L when he is at A, and takes the angle A, and if he notices when the angle that the lighthouse makes with his course is just twice the angle noted at A, then BL = AB. He has AB from his log (an instrument that tells how far a ship goes in a given time), so he knows BL. He has "doubled the angle on the bow" to get this distance.
It would have been possible for Thales, if he knew this proposition, to have measured the distance of the ship at sea by some such device as this:
Make a large isosceles triangle out of wood, and, standing at T, sight to the ship and along the shore on a line TA, using the vertical angle of the triangle. Then go along TA until a point P is reached, from which T and S can be seen along the sides of a base angle of the triangle. Then TP = TS. By measuring TB, BS can then be found.
Theorem. The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side.
If the postulate is assumed that a straight line is the shortest path between two points, then the first part of this theorem requires no further proof, and the second part follows at once from the axiom of inequalities. This seems the better plan for beginners, and the proposition may be considered as semiobvious. Euclid proved the first part, not having assumed the postulate. Proclus tells us that the Epicureans (the followers of Epicurus, the Greek philosopher, 342-270 B.C.) used to ridicule this theorem, saying that even an ass knew it, for if he wished to get food, he walked in a straight line and not along two sides of a triangle. Proclus replied that it was one thing to know the truth and another thing to prove it, meaning that the value of geometry lay in the proof rather than in the mere facts, a thing that all who seek to reform the teaching of geometry would do well to keep in mind. The theorem might simply appear as a corollary under the postulate if it were of any importance to reduce the number of propositions one more.
If the proposition is postponed until after those concerning the inequalities of angles and sides of a triangle, there are several good proofs.
