Then the angle DAB is one-third of the angle CAB. Mr. Byrne asserts that this problem is not capable of solution by the straight line and circle. Mathematicians have in vain attempted to solve it geometrically, that is, by the ruler and compasses only.
[Plate VI.—Figure 1.]
Perpetuum Mobile. Desaguliers demonstrated the absurdity of attempting to raise weights enclosed in a cellular wheel, simply by providing for their approach in succession nearer to the centre on the ascending side, while they should be projected further from the centre on the descending side. He remarks:—
Those who think the velocity of the weight is the line it describes, expect that that weight shall be overpoised, which describes the shortest line, and therefore contrive machines to cause the ascending weight to describe a shorter line than the descending weight.
For example, in the circle A B D a, the weights A and B being supposed equal, it is imagined that, if by any contrivance whatever, whilst the weight A describes the arc A a, the weight B is carried in any arc, as B b, so as to come nearer the centre in its rising, than if it went up the arc B D; the said weight shall be overpoised, and consequently, by a number of such weights, a perpetual motion produced.
Now the velocity of any weight is not the line which it describes in general, but the height that it rises up to, or falls from, with respect to its distance from the centre of the earth. So that when the weight describes the arc A a, its velocity is the line A C, which shows the perpendicular descent, and likewise the line B C denotes the velocity of the weight B, or the height that it rises to, when it ascends in any of the arcs B b, instead of the arc B D: so that, in this case, whether the weight B, in its ascent be brought nearer the centre or not, it loses no velocity, which it ought to do, in order to be raised up by the weight A.
Indeed, if the weight at B, could by any means spring as it were, or be lifted up to x, and move in the arc x b, the end would be answered, because then the velocity would be diminished, and become xC.
[Figure 2.]
In "The Life, Times, and Scientific Labours of the Marquis of Worcester," 1865, page 454, will be found a full account of the present diagram, which is intended to illustrate as far as possible, an approach to the probable construction of the wheel by the Marquis in the 56th article of his memorable "Century of Inventions."
If any likely-looking method, could, more than another, render hopelessness more hopeless, surely this mechanical demonstration must prove most efficient for that purpose. For here, we actually produce a wheel agreeing to the terms with which Desaguliers closes his demonstration, when he suggests the only likely method to effect the end proposed, namely, perpetual motion. We find the fallen weight is absolutely "lifted up" as he desires, and "moves in the arc" he describes, and yet although he declares that then "the end would be answered"—it absolutely is not answered in this instance.