This was Euclid's first proposition on parallelograms, and Proclus speaks of it as the connecting link between the theory of parallels and that of parallelograms. The ancients, writing for mature students, did not add the words "and the figure is a parallelogram," because that follows at once from the first part and from the definition of "parallelogram," but it is helpful to younger students because it emphasizes the fact that here is a test for this kind of figure. THEOREM. _The diagonals of a parallelogram bisect each other._ This proposition was not given in Euclid, but it is usually required in American syllabi. There is often given in connection with it the exercise in which it is proved that the diagonals of a rectangle are equal. When this is taken, it is well to state to the class that carpenters and builders find this one of the best checks in laying out floors and other rectangles. It is frequently applied also in laying out tennis courts. If the class is doing any work in mensuration, such as finding the area of the school grounds, it is a good plan to check a few rectangles by this method. An interesting outdoor application of the theory of parallelograms is the following: [Illustration] Suppose you are on the side of this stream opposite to _XY_, and wish to measure the length of _XY_. Run a line _AB_ along the bank. Then take a carpenter's square, or even a large book, and walk along _AB_ until you reach _P_, a point from which you can just see _X_ and _B_ along two sides of the square. Do the same for _Y_, thus fixing _P_ and _Q_. Using the tape, bisect _PQ_ at _M_. Then walk along _YM_ produced until you reach a point _Y'_ that is exactly in line with _M_ and _Y_, and also with _P_ and _X_. Then walk along _XM_ produced until you reach a point _X'_ that is exactly in line with _M_ and _X_, and also with _Q_ and _Y_. Then measure _Y'X'_ and you have the length of _XY_. For since _YX'_ is [perp] to _PQ_, and _XY'_ is also [perp] to _PQ_, _YX'_ is || to _XY'_. And since _PM_ = _MQ_, therefore _XM_ = _MX'_ and _Y'M_ = _MY_. Therefore _Y'X'YX_ is a parallelogram. The properties of the parallelogram are often applied to proving figures of various kinds congruent, or to constructing them so that they will be congruent. [Illustration] For example, if we draw _A'B'_ equal and parallel to _AB_, _B'C'_ equal and parallel to _BC_, and so on, it is easily proved that _ABCD_ and _A'B'C'D'_ are congruent. This may be done by ordinary superposition, or by sliding _ABCD_ along the dotted parallels. There are many applications of this principle of parallel translation in practical construction work. The principle is more far-reaching than here intimated, however, and a few words as to its significance will now be in place.