(120.) Although the guidance of the air-pump rod in a true vertical line is not so necessary as that of the steam piston, [Pg199] and as the air-pump piston is always brought down by its own weight and that of its rod, the connection of the air-pump piston-rod with the beam, by any contrivance of the kind now described, was not so necessary. Nevertheless, by a slight addition to the mechanical contrivance which has been just described, Watt obtained the means of at once preserving the true rectilinear motion of both piston-rods. [Illustration: _Fig._ 37.] Let the lever represented by O P in _fig._ 36. be conceived to be prolonged to twice its length, as represented in _fig._ 37., so that O P′ shall be twice O P. Let the points P _p_ be connected by a link as before. Let a link P′ _x′_, equal in length to the link P _p_ be attached to the point P′, and let the extremity _x′_ of this link be connected with the point _p_ by another link, equal in length to P P′, by pivots at _x′_ and _p_, so that the figure P P′ _x′ p_ shall be a jointed parallelogram, the angles of which will be capable of altering their magnitude with every change of position of the rods _o p_ and O P. Thus, when the rod O P descends, the angles of the parallelogram at P and _x′_ will be diminished in magnitude, while the angles at P′ and _p_ will be increased in magnitude. Now, let a line be conceived to be drawn from O to _x′_. It is evident that that line will pass through the middle point of the link _p_ P, for the triangle O P _x_ is in all respects similar to the greater triangle O P′ _x′_ only on half the scale, so that every side of the one is [Pg200] half the corresponding side of the other. Therefore P _x_ is half the length of P′ _x′_; but P′ _x′_ was made equal to P _p_, and therefore _p x_ is half of P _p_, that is to say, _x_ is the middle point of P _p_. It has been already shown, that in the alternate motion of the rods _o p_, O P in ascending and descending, the point _x_ is moved upwards and downwards in a true vertical line. Now since the triangle O P _x_ is in all respects similar to O P′ _x′_, and subject to a similar motion during the ascent and descent of the rods, it is apparent that the point _x′_ must be subject to a motion in all respects similar to that which affects the points _x_, except that the point _x′_ will move through double the space. In fact, the principle of the mechanism is precisely similar to that of the common pantograph, where two rods are so connected as that the motion of the one governs the motion of the other, so that whatever line or figure may be described by one, a similar line or figure must be described by the other. Since, then, the point _x_ is moved upwards and downwards in a vertical straight line, the point _x′_ will also be moved in a vertical straight line of double the length.