School: a monthly record of educational thought and progress . in area to the sum of thesquares, or like regular polygons constructed on the othertwo sides of the triangle. The truth of the statement, The semicircle on the hypotenuse is equal in areato the sum of the semicircles on the other two sides ofthe right-angled triangle (XII. 2) is demonstrablewith equal facility, and since the doubles of equalsare equal, the circle on the hypotenuse as diameteris equal to the sum of the circles with the other sidesas diameters. When, however, the students are askedto construct triangles or oblongs on
School: a monthly record of educational thought and progress . in area to the sum of thesquares, or like regular polygons constructed on the othertwo sides of the triangle. The truth of the statement, The semicircle on the hypotenuse is equal in areato the sum of the semicircles on the other two sides ofthe right-angled triangle (XII. 2) is demonstrablewith equal facility, and since the doubles of equalsare equal, the circle on the hypotenuse as diameteris equal to the sum of the circles with the other sidesas diameters. When, however, the students are askedto construct triangles or oblongs on the sides of a right-angled triangle some difficulty is sure to be met idea that the proposition has limitations will nowbegin to assert itself, but by the aid of a few moreconstructions—under the guidance of the teacher—and suggestive questions, the general principle will beestablished, viz., The area of a figure on the hypo-tenuse of a right-angled triangle is equal to the sum of theareas of similar figures constructed on the other two ?4— I am equal to theother twofellows. Fig. I. The special significance of the word similar inthis case is realised, as well as a lasting impression createdby drawingon the blackboard some such diagram as is 40 SCHOOL: A MONTHLY RECORD OF shown in Fig. i, where the figures on the sides of thetriangle are really irregular but similar polygons. Volumes. Sight and experiment are admittedly valuableaids to students in helping them to form correct andfixed mathematical conceptions. Such being the caseno apology is needed for the introduction of the apparatusreferred to in this section. A very thorough grasp ofElementary Volume Mensuration is easily and quicklygiven to students by means of the following apparatus : * (i) Prisms.—Larger cubes and oblong prisms shouldbe built up from small black and white unit cubes(Fig. 2) to enable the pupil to obtain a correct ideaof the measurement of volume, and to realise that the volume
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