This so-called problem is of doubtful classification. The author of the problem did not claim that the discovery of the problem discloses any means for attaining Perpetual Motion, and, yet, it is apparent that if the author of the problem was correct in his solution of it, Perpetual Motion was thereby already within his grasp. The difficulty about it all is that while the problem is quite interesting, the author's solution shows that he was not familiar with even fundamental mechanics. The name of the author was J. T. Desagulier, LL.D., F. R. S. He was a minister of the gospel, but evidently gave considerable attention to mechanical questions. He is mentioned in chapter X of this work.
Rev. Desagulier presented two problems of the balance. One he calls "A Proposition on the Balance, not taken notice of by Mechanical Writers, explained and confirmed by an Experiment." The article under this heading is as follows:
In the last papers I published in "Philosophical Transaction" against this perpetual motion, described in No. 177, I intreated the author to permit me to say nothing as to what alterations he might make in his engine, resolving to leave it to others to show him that upon that principle all he can do signifies nothing. But I find since, in the "Nouvelles de la Republique" for December last, that he still persists to urge some new contrivances, which being added, he conceives his engine must succeed. To this I answer, that I undertook only to shew that his first device would faile, which yet I should scarce have done if I had thought a dispute of this nature could have lasted so long. To come, therefore, to the point where he saith that this engine may well succeed without alteration, because he hath tryed with liquors put into bellows immersed in water; I again say that I grant him the truth of the experiments, but deny the consequences he would draw from them. I have already given the reasons of my dissent, which this gentleman is not pleased to understand. But to end all controversies, he may please to consult Mr. Perrault, De la Hire, or any other at Paris well known to be skilled in hydraulicks, and I doubt not but he will find them of the same opinion with Mr. Boyle, Mr. Hook, and other knowing persons here, who all agree that our author is in this matter under a mistake.
A Proposition on the Balance, not taken notice of by Mechanical Writers, explained and confirmed by an Experiment.
A B is a balance, on which is supposed to hang at one end, B, the scale E, with a man in it, who is counterpoised by the weight W hanging at A, the other end of the balance. I say, that if such a man, with a cane or any rigid straight body, pushes upwards against the beam anywhere between the points C and B (provided he does not push directly against B), he will thereby make himself heavier, or overpoise the weight W, though the stop G G hinders the scale E from being thrust outwards from C towards G G. I say likewise, that if the scale and man should hang from D, the man, by pushing upwards against B, or anywhere between B and D (provided he does not push directly against D), will make himself lighter, or be overpoised by the weight W, which before did only counterpoise the weight of his body and the scale.
If the common center of gravity of the scale E, and the man supposed to stand in it, be at k, and the man, by thrusting against any part of the beam, cause the scale to move outwards so as to carry the said common center of gravity to k x, then, instead of B E, L l will become the line of direction of the compound weight, whose action will be increased in the ratio of L C to B C. This is what has been explained by several writers of mechanics; but no one, that I know of, has considered the case when the scale is kept from flying out, as here by the post G G, which keeps it in its place, as if the strings of the scale were become inflexible. Now, to explain this case, let us suppose the length B D of half of the brachium B C to be equal to 3 feet, the line B E to 4 feet, the line E D of 5 feet to be the direction in which the man pushes, D F and F E to be respectively equal and parallel to B E and B D, and the whole or absolute force with which the man pushes equal to (or able to rise) 10 stone. Let the oblique force E D (= 10 stone) be resolved into the two E F and E B (or its equal F D) whose directions are at right angles to each other, and whose respective quantities (or intensities) are as 6 and 8, because E F and B E are in that proportion to each other and to E D. Now, since E F is parallel to B D C A, the beam, it does no way affect the beam to move it upwards; and therefore there is only the force represented by F D, or 8 stone, to push the beam upwards at D. For the same reason, and because action and reaction are equal, the scale will be pushed down at E with the force of 8 stone also. Now, since the force at E pulls the beam perpendicularly downwards from the point B, distant from C the whole length of the brachium B C, its action downwards will not be diminished, but may be expressed by 8 × B C; whereas the action upwards against D will be half lost, by reason of the diminished distance from the center, and is only to be expressed by 8 × B C/2; and when the action upwards to raise the beam is subtracted from the action downwards to depress it, there will still remain 4 stone to push down the scale; because 8 × B C - 8 × B C/2 = 4 B C. Consequently, a weight of 4 stone must be added at the end A to restore the æquilibrium. Therefore a man, &c., pushing upwards under the beam between B and D, becomes heavier. Q. E. D.
On the contrary, if the scale should hang at F, from the point D, only 3 feet from the center of motion C, and a post G G hinders the scale from being pushed inwards towards C, then, if a man in this scale F pushes obliquely against B with the oblique force above mentioned, the whole force, for the reasons before given (in resolving the oblique force into two others acting in lines perpendicular to each other) will be reduced to 8 stone, which pushes the beam directly upwards at B, while the same force of 8 stone draws it directly down at D towards F. But as C D is only equal to half of C B, the force at D, compared with that at B, loses half its action, and therefore can only take off the force of 4 stone from the push upwards at B; and consequently the weight W at A will preponderate, unless an additional weight of 4 stone be hanged at B. Therefore, a man, &c., pushing upwards under the beam between B and D, becomes lighter.
The other problem presented by Rev. Desagulier is denominated by him "An Experiment explaining a Mechanical Paradox, that two bodies of equal weight suspended on a certain sort of balance do not lose their equilibrium by being removed, one farther from, the other nearer to, the center."
The article concerning this problem is as follows:
If the two weights P W hangs at the ends of the balance A B, whose center of motion is C, those weights will act against each other (because their directions are contrary) with forces made up of the quantity of matter in each multiplied by its velocity; that is, by the velocity which the motion of the balance turning about C will give to the body suspended. Now, the velocity of a heavy body is its perpendicular ascent or descent, as will appear by moving the balance into the position a b, which shews the velocity of P to be the perpendicular line e a, and the velocity of B will be the perpendicular line b g; for if the weights P and W are equal, and also the lines e a and b g, their momenta, made up of e a multiplied into W, and b g multiplied into P, will be equal, as will appear by their destroying one another in making an equilibrium. But if the body W was removed to M, and suspended at the point D, then, its velocity being only f d, it would be overbalanced by the body P, because f d multiplied into M would produce a less momentum than P multiplied into b g.
As the arcs A a, B b, and D d, described by the ends of the balance or points of suspension, are proportionable to their sines e a, g b, and d f, as also the radii or distances C A, C B, and C D; in the case of this common sort of balance, the arcs described by the weights, or their points of suspension, or the distances from the center, may be taken for velocities of the weights hanging at A, B, or D, and, therefore, the acting force of the weights will be reciprocally as their distances from the center.
Scholium.—The distances from the center are taken here for the velocities of the bodies, only because they are proportionable to the lines e a, b g, and f d, which are the true velocities; for there are a great many cases wherein the velocities are neither proportionable to the distances from the center of motion of a machine, nor to the arcs described by the weights or their points of suspension. Therefore, it is not a general rule that weights act in proportion to their distances from the center of motion; but a corollary of the general rule that weights act in proportion to their velocities, which is only true in some cases. Therefore, we must not take this case as a principle, which most workmen do, and all those people who make attempts to find the perpetual motion, as I have more amply shewn in the Phil. Trans., No. 369.
But to make this evident even in the balance, we need only take notice of the following experiment:—A C B E K D is a balance in the form of a parallelogram passing through a slit in the upright piece N O standing on the pedestal M, so as to be moveable upon the center pins C and K. To the upright pieces A D and B E of this balance are fixed at right angles the horizontal pieces F G and H I. That the equal weights P W must keep each other in æquilibrio, is evident; but it does not at first appear so plainly, that if W be removed to V, being suspended at 6, yet it shall still keep P in æquilibrio, though the experiment shews it. Nay, if W be successively moved to any of the points 1, 2, 3, E, 4, 5, or 6, the æquilibrium will be continued; or if, W hanging at any of those points, P be successively moved to D, or any of the points of suspension on the cross-piece F G, P will at any of those places make an æquilibrium with W. Now, when the weights are at P and V, if the least weight that is capable to overcome the friction at the points of suspension C and K be added to V, as u, the weight V will overpower, and that as much at V as if it was at W.
From what we have said above, the reason of this experiment will be very plain.
As the lines A C and K D, C B and K E, always continue of the same length in any position of the machine, the pieces A D and B E will always continue parallel to one another, and perpendicular to the horizon. However, the whole machine turns upon the points C and K, as appears by bringing the balance to any other position, as a b e d; and therefore, as the weights applied to any part of the pieces F G and H I can only bring down the pieces A D and B E perpendicularly, in the same manner as if they were applied to the hooks D and E, or to X and Y, the centers of gravity of A D and B E, the force of the weights (if their quantity of matter is equal) will be equal, because their velocities will be their perpendicular ascent or descent, which will always be as the equal lines 4 l and 4 L, whatever part of the pieces F G and H I the weights are applied to. But if to the weight at V be added the little weight u, those two weights will overpower, because in this case the momentum is made up of the sum of V and u multiplied by the common velocity 4 L.
Hence follows, that it is not the distance C 6 multiplied into the weight V which makes its momentum, but its perpendicular velocity L 4 multiplied into its mass. Q. E. D.
This is still further evident by taking out the pin at K; for then the weight P will overbalance the other weight at V, because then their perpendicular ascent and descent will not be equal.
The Rev. Dr. Desagulier was evidently a man of scientific turn and capacity. It is unusual to find ministers deeply interested in scientific matters, and yet, he seems to have been. The net result of his experiments can be succinctly stated as follows:
In the first problem there is no change in the distance of the center of gravity from the support, and, therefore, there could be no disturbance of the equilibrium.
In the second problem there is a change in the distance in the center of gravity from the support, and there must have been a disturbance of the equilibrium.