Book IV treats of the area of polygons, and offers a large number of practical applications. Since the number of applications to the measuring of areas of various kinds of polygons is unlimited, while in the first three books these applications are not so obvious, less effort is made in this chapter to suggest practical problems to the teachers. The survey of the school grounds or of vacant lots in the vicinity offers all the outdoor work that is needed to make Book IV seem very important. THEOREM. _Two rectangles having equal altitudes are to each other as their bases._ Euclid's statement (Book VI, Proposition 1) was as follows: _Triangles and parallelograms which are under the same height are to one another as their bases_. Our plan of treating the two figures separately is manifestly better from the educational standpoint. In the modern treatment by limits the proof is divided into two parts: first, for commensurable bases; and second, for incommensurable ones. Of these the second may well be omitted, or merely be read over by the teacher and class and the reasons explained. In general, it is doubtful if the majority of an American class in geometry get much out of the incommensurable case. Of course, with a bright class a teacher may well afford to take it as it is given in the textbook, but the important thing is that the commensurable case should be proved and the incommensurable one recognized. Euclid's treatment of proportion was so rigorous that no special treatment of the incommensurable was necessary. The French geometer, Legendre, gave a rigorous proof by _reductio ad absurdum_. In America the pupils are hardly ready for these proofs, and so our treatment by limits is less rigorous than these earlier ones. THEOREM. _The area of a rectangle is equal to the product of its base by its altitude._ The easiest way to introduce this is to mark a rectangle, with commensurable sides, on squared paper, and count up the squares; or, what is more convenient, to draw the rectangle and mark the area off in squares.