It is interesting and valuable to a class to have its attention called to the fact that the perimeter of a rectangle is no criterion as to the area. Thus, if a rectangle has an area of 1 square foot and is only 1/440 of an inch high, the perimeter is over 2 miles. The story of how Indians were induced to sell their land by measuring the perimeter is a very old one. Proclus speaks of travelers who described the size of cities by the perimeters, and of men who cheated others by pretending to give them as much land as they themselves had, when really they made only the perimeters equal. Thucydides estimated the size of Sicily by the time it took to sail round it. Pupils will be interested to know in this connection that of polygons having the same perimeter and the same number of sides, the one having equal sides and equal angles is the greatest, and that of plane figures having the same perimeter, the circle is the greatest. These facts were known to the Greek writers, Zenodorus (_ca._ 150 B.C.) and Proclus (410-485 A.D.). The surfaces of rectangular solids may now be found, there being an advantage in thus incidentally connecting plane and solid geometry wherever it is natural to do so. THEOREM. _The area of a parallelogram is equal to the product of its base by its altitude._ The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, separate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the measurement appear to be merely an approximation. The cutting of the paper is in every way more satisfactory. THEOREM. _The area of a triangle is equal to half the product of its base by its altitude._ Of course, the Greeks would never have used the wording of either of these two propositions. Euclid, for example, gives this one as follows: _If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle._ As to the parallelogram, he simply says it is equal to a parallelogram of equal base and "in the same parallels," which makes it equal to a rectangle of the same base and the same altitude. The number of applications of these two theorems is so great that the teacher will not be at a loss to find genuine ones that appeal to the class. Teachers may now introduce pyramids, requiring the areas of the triangular faces to be found. The Ahmes papyrus (_ca._ 1700 B.C.) gives the area of an isosceles triangle as 1/2 _bs_, where _s_ is one of the equal sides, thus taking _s_ for the altitude. This shows the primitive state of geometry at that time. THEOREM. _The area of a trapezoid is equal to half the sum of its bases multiplied by the altitude._ [Illustration]