(154.) There are some cases in which, although the place of the centre of gravity is not so obvious as in the examples just given, still it may be discovered without any mathematical process, which is not easily understood. Suppose A B C, _fig. 43._, to be a flat triangular plate of uniform thickness and density. Let it be imagined to be divided into narrow bars, by lines parallel to the side A C, as represented in the figure. Draw B D from the angle B to the middle point D of the side A C. It is not difficult to perceive, that B D will divide equally all the bars into which the triangle is conceived to be divided. Now if the flat triangular plate A B C be placed in a horizontal position on a straight edge coinciding with the line B D, it will be balanced: for the bars parallel to A C will be severally balanced by the edge immediately under their middle point; since that middle point is the centre of gravity of each bar. Since, then, the triangle is balanced on the edge, the centre of gravity must be somewhere immediately over it, and must, therefore, be within the plate at some point under the line B D. The same reasoning will prove that the centre of gravity of the plate is under the line A E, drawn from the angle A to the middle point E of the side B C. To perceive this, it is only necessary to consider the triangle divided into bars parallel to B C, and thence to show that it will be balanced on an edge placed under A E. Since then the centre of gravity of the plate is under the line B D, and also under A E, it must be under the point G, at which these lines cross each other; and it is accordingly at a depth beneath G, equal to half the thickness of the plate. This may be experimentally verified by taking a piece of tin or card, and cutting it into a triangular form. The point G being found by drawing B D and A E, which divide two sides equally, it will be balanced if placed upon the point of a pin at G. The centre of gravity of a triangle being thus determined, we shall be able to find the position of the centre of gravity of any plate of uniform thickness and density which is bounded by straight edges, as will be shown hereafter. (173.)