Wishing to measure the distance across a river, some boys sighted from A to a point P. They then turned and measured AB at right angles to AP. They placed a stake at O, halfway from A to B, and drew a perpendicular to AB at B. They placed a stake at C, on this perpendicular, and in line with O and P. They then found the width by measuring BC. Prove that they were right.
This involves the ranging of a line, and the running of a line at right angles to a given line, both of which have been described in [Chapter IX]. It is also fairly accurate to run a line at any angle to a given line by sighting along two pins stuck in a protractor.
Theorem. Two triangles are congruent if two angles and the included side of the one are equal respectively to two angles and the included side of the other.
Euclid combines this with his Proposition 26:
If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides, and the remaining angle to the remaining angle.
He proves this cumbersome statement without superposition, desiring to avoid this method, as already stated, whenever possible. The proof by superposition is old, however, for Al-Nairīzī[60] gives it and ascribes it to some earlier author whose name he did not know. Proclus tells us that "Eudemus in his geometrical history refers this theorem to Thales. For he says that in the method by which they say that Thales proved the distance of ships in the sea, it was necessary to make use of this theorem." How Thales did this is purely a matter of conjecture, but he might have stood on the top of a tower rising from the level shore, or of such headlands as abound near Miletus, and by some simple instrument sighted to the ship. Then, turning, he might have sighted along the shore to a point having the same angle of declination, and then have measured the distance from the tower to this point. This seems more reasonable than any of the various plans suggested, and it is found in so many practical geometries of the first century of printing that it seems to have long been a common expedient. The stone astrolabe from Mesopotamia, now preserved in the British Museum, shows that such instruments for the measuring of angles are very old, and for the purposes of Thales even a pair of large compasses would have answered very well. An illustration of the method is seen in Belli's work of 1569, as here shown. At the top of the picture a man is getting the angle by means of the visor of his cap; at the bottom of the picture a man is using a ruler screwed to a staff.[61] The story goes that one of Napoleon's engineers won the imperial favor by quickly measuring the width of a stream that blocked the progress of the army, using this very method.
This proposition is the reciprocal or dual of the preceding one. The relation between the two may be seen from the following arrangement: