In the illustrations of the Gothic windows given in Chapter XV only the square and circle were generally involved. Teachers who feel it necessary or advisable to go outside the regular work of geometry for the purpose of increasing the pupil's interest or of training his hand in the drawing of figures will find plenty of designs given in any pictures of Gothic cathedrals. For example, this picture of the noble window in the choir of Lincoln Cathedral shows the use of the square, hexagon, and pentagon. In the porch of the same cathedral, shown in the next illustration, the architect has made use of the triangle, square, and pentagon in planning his ornamental stonework. It is possible to add to the work in pure geometry some work in the mensuration of the curvilinear figures shown in these designs. This form of mensuration is not of much value, however, since it places before the pupil a problem that he sees at once is fictitious, and that has no human interest. [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND SQUARES] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE EQUILATERAL TRIANGLE] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE REGULAR HEXAGON] The designs given on page 283 involve chiefly the square as a basis, but it will be seen from one of the figures that the equilateral triangle and the hexagon also enter. The possibilities of endless variation of a single design are shown in the illustration on page 284, the basis in this case being the square. The variations in the use of the triangle and hexagon have been the object of study of many designers of Gothic windows, and some examples of these forms are shown on page 285. In more simple form this ringing of the changes on elementary figures is shown on page 286. Some teachers have used color work with such designs for the purpose of increasing the interest of their pupils, but the danger of thus using the time with no serious end in view will be apparent. [Illustration] In the matter of the mensuration of the circle the annexed design has some interest. The figure is not uncommon in decoration, and it is interesting to show, as a matter of pure geometry, that the area of the circle is divided into three equal portions by means of the four interior semicircles. [Illustration] An important application of the formula _a_ = [pi]_r_^2 is seen in the area of the annulus, or ring, the formula being _a_ = [pi]_r_^2 - [pi]_r'_^2 = [pi](_r_^2 - _r'_^2) = [pi](_r_ + _r'_)(_r_ - _r'_). It is used in finding the area of the cross section of pipes, and this is needed when we wish to compute the volume of the iron used. Another excellent application is that of finding the area of the surface of a cylinder, there being no reason why such simple cases from solid geometry should not furnish working material for plane geometry, particularly as they have already been met by the pupils in arithmetic.