width of a stream that blocked the progress of the army, using this very method. [Illustration: SIXTEENTH-CENTURY MENSURATION Belli's "Del Misurar con la Vista," Venice, 1569] This proposition is the reciprocal or dual of the preceding one. The relation between the two may be seen from the following arrangement: Two triangles are congruent if two _sides_ and the included _angle_ of the one are equal respectively to two _sides_ and the included _angle_ of the other. 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. In general, to every proposition involving _points_ and _lines_ there is a reciprocal proposition involving _lines_ and _points_ respectively that is often true,--indeed, that is always true in a certain line of propositions. This relation is known as the Principle of Reciprocity or of Duality. Instead of points and lines we have here angles (suggested by the vertex points) and lines. It is interesting to a class to have attention called to such relations, but it is not of sufficient importance in elementary geometry to justify more than a reference here and there. There are other dual features that are seen in geometry besides those given above. THEOREM. _In an isosceles triangle the angles opposite the equal sides are equal._ This is Euclid's Proposition 5, the second of his theorems, but he adds, "and if the equal straight lines be produced further, the angles under the base will be equal to one another." Since, however, he does not use this second part, its genuineness is doubted. He would not admit the common proof of to-day of supposing the vertical angle bisected, because the problem about bisecting an angle does not precede this proposition, and therefore his proof is much more involved than ours. He makes _CX_ = _CY_, and proves [triangles]_XBC_ and _YAC_ congruent,[62] and also [triangles]_XBA_ and _YAB_ congruent. Then from [L]_YAC_ he takes [L]_YAB_, leaving [L]_BAC_, and so on the other side, leaving [L]_CBA_, these therefore being equal. [Illustration] This proposition has long been called the _pons asinorum_, or bridge of asses, but no one knows where or when the name arose. It is usually stated that it came from the fact that fools could not cross this bridge, and it is a fact that in the Middle Ages this was often the limit of the student's progress in geometry. It has however been suggested that the name came from Euclid's figure, which resembles the simplest type of a wooden truss bridge. The name is applied by the French to the Pythagorean Theorem. Proclus attributes the discovery of this proposition to Thales. He also says that Pappus (third century A.D.), a Greek commentator on Euclid, proved the proposition as follows: