Such has been the term applied to the enunciation of the truth, that any column of water, however small, may be made to raise any weight, however large, experimentally shown in the familiar piece of apparatus known as the water bellows. This proposition is theoretically correct, although there are practical limits to its application. Why it should be considered paradoxical,however,any more than the action of a lever, has always been a puzzle to us. Theoretically, it is just as true of the lever, that any weight, however small, may be made by its means to raise any weight, however large, as of the water bellows, or the hydrostatic press. In either case, on the principle of “ virtual velocities,” the weight of the body which raises, multiplied into the distance it moves, will always equal the weight of the body raised multiplied into the distance it moves, friction being supposed to be nothing. And, practically,in all cases,the weight which raises must be enough heavier than would be found by this equation, to overcome the friction of the apparatus, whether bellows or lever. Some of our correspondents are puzzling their heads over the theory of hydrostatic pressure as applied to the press of Brahma, and we are in receipt of not less than a dozen inquiries in regard to this subject. We will endeavor to answer these inquiries definitely in this article. The subject only becomes obscure, when we attempt to get back of nature's laws, to find out why things are as they are. We shall confine ourselves to the simple question of 7iOW they are. The equilibrium of fluids was ascribed by Pascal to the principle of virtual velocities above mentioned. This principle or law of nature has been thus enunciated: “Forces in equilibrium must be to each other as their velocities.” It may be added, that when any two forces are so related to each other that the motion which each tends to produce is in an opposite direction to that of the other, and so that the distances through which each would move, if an additional force were made to aid either, would he inversely as the forces themselves, then unless an additional force be made to aid one or the other of the two forces thus related, neither will produce motion. An example of two forces thus related would be two springs, one having a strength equal to the support of two pounds, the other a strength equal to the support of four pounds, attached to fixed supports, and acting upon the ends of a lever gix feet long, resting upon a fulcrum placed two feet from one end and four feet from the other—the two-pound spring acting upon the longer arm, and the four-pound spring upon the shorter. In this case, no motion would take place unless ose of the springs were assisted by an additional force. The two forces would be in equilibrium. Now, when a small column of water supports a larger column, their weights are two forces, exactly so related. Neither column can descend without the other ascends,i.e., moves in an opposite direction, and the distances through which the columns would move would be inversely, as their weights. That either may move, an additional force must be applied to at least one of them, which will cause a motion in both. But an infinitesimal additional force applied to one column would be sufficient to destroy the equilibrium,unless some resistance or counteracting force should immediately impede the motion of the other column. Moreover, the properties of fluids are such,that the weights of any two columns of fluids, connected at their bases by a fluid medium, invariably sustain the relation we have described, unless some other force acts upon one or both columns. It is unnecessary for our present purpose to complicate the question by a consideration of columns of unequal diameters in different parts, the columns here spoken of being those of uniform diameter throughout. Further, although this law of virtual velocities has been the subject of many explanatory efforts, we know no more about it to-day than we do about the nature of gravity. All we can do is to recognize its existence as we do that of gravity, all else must be merely fruitless speculation. The hydrostatic press of Brahma,applies an additional force to one of two fluid columns in equilibrium, to not only destroy the equilibrium, but, also, to overcome a counteracting force or resistance opposed to the motion of the opposite column. We have said the two forces in two such columns when no additional force is applied,are the weights of the columns ; but as the weights of the columns are to each other as their sectional areas.these areas may be used as the representatives of the two forces, and it will be more convenient to so consider them. But as these areas, when geometrically similar, are to each other as the squares of their diameters, we may operate still more conveniently by making these the representatives of the two forces. Let the small column of a hydrostatic press be one inch in diameter, and the large column be two inches in diameter. When these columns are in equilibrium, the weights will be to each other as their sectional areas, which are to each other as the squares of their diameters, or as one is to four. Here we have a force of one balancing a force of four, simply because they are so related, that if motion should take place by the action of an additional force on either column, one must move in an opposite direction four times as far as the other. It follows that, as the motion produced by this force must be transmitted through the fluid medium connecting the two columns at their bases, and as this medium is the condition which establishes the peculiar relation between the two forces, the ratio between the force applied and the resistance it will overcome must be exactly the same as existed at first between the two columns, so that if a force of six pounds be applied through a piston resting on the top of the smaller column, it will balance a weight of twenty-four pounds applied through a piston resting on the top of the larger column ; and any less force than twenty-four pounds, applied through a piston, to the top of the larger column, would be raised one inch for every four inches the smaller piston descends. It also follows, that the quantity of fluid displaced from under the smaller piston is exactly equal to that injected into the larger cylinder,andthat the stroke of the small piston must always be through a greater distance than the movement of larger piston in the same time, the distances being inversely as the forces. The principle which underlies the action of of this machine, namely, the principle of virtual velocities, is as immutable and as inscrutable as the existence of matter and force. We have here, also, a reason why great hydrostatic power, generated by a small column of water in such'a press cannot be made to generate a motion any more rapid than could be produced by the motion of the small column itself, and as a further and final deduction,the greater the difference between the diameters of the pistons, and the greater the consequent power of the press,the slower will be the motion of the larger piston. All of these facts have been proved by experiment, and we have shown that the law of virtual velocities is sufficient to account for them.
This article was originally published with the title "The Hydrostatic Paradox" in Scientific American 21, 16, 249 (October 1869)