LESSONS ON FORM, BEING AN INTRODUCTION TO GEOMETRY. LESSON I. THE master places before his pupils a variety of objects, among which there should be (the following solids, viz.) the five regular solids, viz. several of the prisms and pyramids, the cylinder, the cone, and the sphere. Master.-I have set these objects before you, that you may find out some properties common to them all. Endeavour to discover them. (The answer of each pupil should be subjected to the consideration of the class, and be tried if in reality it equally applies to all objects.) Pupils. These and all other objects occupy a space. M.-In how many directions does each extend? P. In three directions: in length, in breadth, and in depth. One of the pupils said, "and in thickness." M. (holding up a book.)—Which would you call the length of this book?-Which is its breadth? Does B it extend in another direction?-By what word will you describe it? P.-Thickness. M.-Name an object of which it would be proper to say depth instead of thickness. P.-A well extends in length, in breadth, and in depth: so does the sea, a pond, a lake, a river. M.-Objects considered with reference to these three dimensions only are called solids. What other property have all solids in common? P.-They are all bounded by a surface. M. (holding up a sphere and a prism.)—In what does the surface of one of these objects chiefly differ from the surface of the other? P.—The one is composed of several surfaces, and the other is bounded only by one curved surface. M.-In what does a surface consist? P.-In extension of length and breadth: a surface is the boundary of anything. M.-What happens if a surface be removed from an object? P.-A part of the object is likewise removed by removing a surface. M. Is the object, by doing so, increased or decreased? P.-It is decreased. M.-In how many directions is it decreased? P. It is decreased either in length, or in breadth, or in thickness. M.-Can a surface exist without the object of which it is a surface? Can you hold a surface in your hand without holding the object itself? P.-No. On this question being asked, one of the pupils said, a shadow is a surface existing without a concomitant solid. The master will of course convince his pupils of the error, should a similar answer be given. M. (holding up a prism.)—What is meant by the surface of this object? P.-The assemblage of the several surfaces which bound it. M.-If we wish to distinguish one of these several surfaces from the total number of surfaces, it is usual to call it one of its faces. Now state what may be said in general of the number of faces by which all objects are bounded. P.-All objects are bounded either by one face only, or by several faces. M.-Now examine more minutely the faces of these objects, and class those together which you think to have similar faces. The master should allow the pupils some time for arranging the objects before him into groups, until they have perceived that they may be classed properly into three distinct groups;-the one comprehending those which are bounded by plane faces; the next, those that are bounded by plane and curved faces; and lastly, those that have only one curved surface. P.-All these objects are either bounded by plane |