In Fig. 1, in the annexed diagram, we show the distance forward and upward, _a_ to _b_, the rider must be thrown before he gets beyond the gravity line, _g_, in the Ordinary; Fig. 2 shows the distance when the wheel will not revolve backward through the fork. In either case, the header is supposed to be taken on a smooth road and not against an obstruction; this can easily occur in vaulting into the saddle or in leaning too far forward. It will be noticed that the distance the rider is elevated, or, in other words, the amount of work done against gravity, is in both cases the same, but the distance forward he must be thrown is considerably greater in Fig. 2. This is for the reason that while the point of contact, _h_, with the ground remains the same in Fig. 1, in Fig. 2 the point rolls on to _i_. For more accurate illustration of the work to be done against gravity, and the distance forward the rider must be thrown, see the header curves in Figs. 4 and 5, farther on. We see, then, that the advantage which the anti-header (No. 2) has over the Ordinary machine (No. 1) is not so very great when in both cases a smooth road is considered; when, however, we consider the element of an obstacle in the path, the difference is much more in favor of No. 2. Let us compare the action of both classes of machines against a four-inch obstruction. In all cases the action of No. 1 machine will be the same,—that is, the wheel will remain in contact at _h_, Fig. 1, and the saddle will go on over, just as it does in the case of no obstruction at all. But in No. 2 the very act of taking the header must raise the entire weight and roll the system upon the obstacle, as shown in Fig. 3. [Illustration: Anti-header wheel action on obstruction.] The point of contact, _h_, over and beyond which the centre of gravity must be thrown, will not only move forward, as shown in Fig. 2, but will move to the top of the obstacle _i_, Fig. 3. Or, if the question is one of a rut or indentation in the surface of the roadway, No. 2 will be caused to roll partially or altogether out of the rut. Now, since the rider, by the action of his momentum and that of the machine, is rolled upon the obstacle or out of the rut, it is easily seen that if he is attending strictly to his work and is at all a skilful rider, he can, by a lively thrust upon the pedal at the opportune time, right himself and keep the drive-wheel rolling on, in which case the rear part of the machine will, in all ordinary cases, drop back upon the ground, from which, of course, it will have raised. [Illustration: Ordinary header curve, any obstruction.] [Illustration: Anti-header attachment, smooth road.] [Illustration: Anti-header, four-inch obstruction.] [Illustration: Anti-header, eight-inch obstruction.] In the diagrams herewith annexed, Fig. 4 shows the curve of the saddle of a No. 1 (Ordinary) machine on a level road, and which would be the same against any obstruction. Scale, one-sixteenth.