To prove this demonstratively would require some abstruse mathematical investigation. It may, however, be rendered in some degree apparent by reasoning of a looser and more popular nature. As the point B is raised to E, it is also drawn aside towards the right. At the same time the other extremity D of the rod B D is raised to E′, and is drawn aside towards the left. The ends of the rod B D being thus at the same time drawn equally towards opposite sides, its middle point P will suffer no lateral derangement, and will move directly upwards. On the other hand, if B be moved downwards to F, it will be drawn laterally to the right, while D being moved to F′ will be drawn to the left. Hence, as before, the middle point P sustains no lateral derangement, but merely descends. Thus, as the extremities B and D move upwards and downwards in circles, the middle point P moves upwards and downwards in a straight line.[6] [6] In a strictly mathematical sense, the path of the point P is a curve, and not a straight line; but in the play given to it in its application to the steam-engine, it moves through a part only of its entire locus, and this part extending equally on each side of a point of inflection, the radius of curvature is infinite, so that in practice the deviation from a straight line, when proper proportions are observed in the rods, is imperceptible. The application of this geometrical principle in the steam-engine evinces much ingenuity. The same arm of the beam usually works two pistons, that of the cylinder and that of the _air-pump_. The apparatus is represented on the arm of the beam in _fig. 175._ The beam moves alternately upwards and downwards on its axis A. Every point of it, therefore, describes a part of a circle of which A is the centre. Let B be the point which divides the arm A G into two equal parts A B and B G; and let C D be a straight rod, equal in length to G B, and fixed on a centre or pivot C on which it is at liberty to play. The end D of this rod is connected by a straight bar with the point B, by pivots on which the rod B D turns freely. If the beam be now supposed to rise and fall alternately, the points B and D will move upwards and downwards in circular arcs, and, as already explained with respect to the points B D, _fig. 174._, the middle point P of the connecting rod B D will move upwards and downwards without lateral deflection. To this point one of the piston rods which are to be worked is attached.