The apparatus adopted for carrying it into effect is represented on the arm which works the piston in fig. 15. The beam, moving on its axis C, every point in its arm moves in the arc of a circle of which C is the centre. Let B be the point which divides the arm A C into equal parts, A B and B C; and let D E be a straight rod equal in length to C B, and playing on the fixed centre or pivot D. The end E of this rod is connected by a straight bar, E B, with the point B, by pivots at B and E on which the rod B E plays freely. If the beam be supposed to move alternately on its axis C, the point B will move up and down in a circular arc, of which C is the centre, and at the same time the point E will move in an equal circular arc round the point D as a centre. According to what we have just explained, the middle point F of the rod B E will move up and down in a straight line. Also, let a rod, A G, equal in length to B E, be attached to the end A of the beam by a pivot on which it moves freely, and let its extremity G be connected with E by a rod, G E, equal in length to A B, and playing on pivots at G and E. By this arrangement the joint A G being always parallel to B E, the three points C, A, and G will be in circumstances precisely similar to the points C, B, and F, except that the system C A G will be on a scale of double the magnitude of C B F: C A being twice C B, and A G twice B F, it is clear, then, that whatever course the point F may follow, the point G must follow a similar line,[18] but will move twice as fast. But, since the point F has been already shown to move up and down in a straight line, the point G must also move up and down in a straight line, but of double the length.[19] [Footnote 18: It is, in fact, the principle of the pantograph. The points C, F, and G evidently lie in the same straight line, since C B : C A :: B F : A G, and the latter lines are parallel. Taking C as the common _pole_ of the _loci_ of the points F G, the _radius vector_ of the one will always be twice the corresponding _radius vector_ of the other; and therefore these curves are similar, similarly placed, and parallel. Hence, by the last note, the point G must move in a line differing imperceptibly from a right line.] [Footnote 19: It is not necessary that the rods, forming the parallel motion, should have the proportions which we have assigned to them. There are various proportions which answer the purpose, and which will be seen by reference to practical works on the steam engine.] By this arrangement the pistons of both the steam cylinder and air-pump are worked; the rod of the latter being attached to the point F, and that of the former to the point G.