in general. See Encycl.

Arithmetical Mean, is half the sum of the extremes. So 4 is an arithmetical mean between 2 and 6, or be- tween 3 and 5, or between 1 and 7; also an arithmeti- cal mean between \(a\) and \(b\) is \(\frac{a + b}{2}\), or \(\frac{1}{2}a + \frac{1}{2}b\).

Geometrical Mean, commonly called a mean propor- tional, is the square root of the product of the two ex- tremes; so that, to find a mean proportional between two given extremes, multiply these together, and ex- tract the square root of the product. Thus, a mean pro- portional between 1 and 9, is \(\sqrt{1 \times 9} = \sqrt{9} = 3\); a mean between 2 and 4 is \(\sqrt{2 \times 4} = \sqrt{8} = 2\); also, the mean between 4 and 6 is \(\sqrt{4 \times 6} = \sqrt{24}\); and the mean between \(a\) and \(b\) is \(\sqrt{ab}\).

Harmonical Mean. See Harmonical Proportion, Encycl.

Mean and Extreme Proportion, or Extreme and Mean Proportion, is when a line or any quantity is so divided that the less part is to the greater, as the greater is to the whole.

Mean Anomaly of a Planet, is an angle which is al- ways proportional to the time of the planet's motion from the aphelion or perihelion, or proportional to the area described by the radius vector; that is, as the whole periodic time is one revolution of the planet, is to the time past the aphelion or perihelion, so is 360° to the mean anomaly. See ANOMALY, Encycl.

Mean Conjunction or Opposition, is when the mean place of the sun is in conjunction, or opposition, with the mean place of the moon in the ecliptic.

Mean Distance of a Planet from the Sun, is an arith- metic mean between the planet's greatest and least dis- tances.

Mean Motion, is that by which a planet is supposed to move equably in its orbit; and it is always propor- tional to the time.

Mean Time, or Equal Time, is that which is mea- sured by an equable motion, as a clock; as distinguis- hed from apparent time, arising from the unequal motion of the earth or sun.

Universal or Perpetual MEASURE, is a kind of MECHANICS.—Our readers will recollect that in the article Physics, Encycl. we proposed to distinguish by the term Mechanical Philosophy that part of natural science which treats of the local motions of bodies, and the causes of those phenomena. And, although all the changes which we observe in material nature are accompanied by local motion, and, when completely explained, are the effects (perhaps very remote) of those powers of matter which we call moving forces, and of those alone, yet, in many cases, this local motion is not observed, and we only perceive certain ultimate results of those changes of place. This is the case (for example) in the solution of a grain of silver in a phial of aquafortis. In the beginning of the experiment, the particles of silver are contained in a small space at the bottom of the phial; but they are finally raised from the bottom, and uniformly disseminated over the whole fluid. If we fix our attention steadily on one particle, and trace it in its whole progress, we contemplate nothing but a particle of matter acted on by moving forces, and yielding to their action. Could we state, for every situation of the particle, the direction and intensity of the moving force by which it is impelled, we could construct a figure, or a formula, which would tell us the precise direction and velocity with which it changes its place, and we could delineate its path, and tell the time when it will arrive at that part of the vessel where it finally rests in perfect equilibrium. Newton having done all this in the case of bodies acted on by the moving force called gravity, has given us a complete system of mechanical astronomy. The philosopher who shall be as fortunate in ascertaining the paths and motions of the particles of silver, till the end of this experiment, will establish a system of the mechanical solution of silver in aquafortis; and the theorems and formulæ which characterize this particular moving force, or this modification of force, stating the laws of variation by a change of distance, will be the complete theory of this chemical fact. It is this modification of moving force which is usually (but most vaguely) called the chemical affinity, or the elective attraction of silver and aquafortis.

But, alas! we are, as yet, far from having attained this perfection of chemical knowledge. All that we have yet discovered is, that the putting the bit of silver into the spirit of salt will not give occasion to the exertion of this moving force; and we express this observation, by calling that unknown force (unknown, because we are ignorant of the law of its action) an affinity, an elective attraction. And we have observed many such elections, and have been able to clasps them, and to tell on what occasions they will or will not be exerted; and this scrap of the complete theory becomes a most valuable acquisition, and the classification of those scraps a most curious, and extensive, and important science. The chemical philosopher has also the pleasure of seeing gradual approaches made by ingenious men to the complete mechanical explanation of these unseen motions and their causes, of which he has arranged the ultimate results.

The ordinary chemist, however, and even many most acute and penetrating enquirers, do not think of all these motions. Familiarly conversant with the results, they consider them as principles, and as topics to reason from. They think a chemical phenomenon insufficiently explained, when they have pointed out the affinity under which it is arranged. Thus they ascribe the propagation of heat to the expansive nature of fire, and imagine that they conceive clearly how the effect is produced. But if a mathematical philosopher should say, "What is this which you call an expansive fluid? Explain to me distinctly, in what manner this property which you call expansiveness operates in producing the propagation of heat?"—We imagine that the chemist would find himself put to a stand. He will then, perhaps, for the first time, try to form a distinct conception of an expansive fluid, and its manner of operation. He will naturally think of air, and will reflect on the manner in which air actually expands or occupies more room; and he will thus contemplate local motion and mechanical pressure. He will find, too late, that this gives him no affluence; because the phenomena which he has been accustomed to explain by the expansiveness of fluids have no resemblance whatever to what we see result from the actual expansion of air. Experience has made him acquainted with many effects which the air produces during its expansion; but they are of a totally different kind from those which he thought that he had sufficiently explained by the expansiveness of fire. The only resemblance he observes is, that the air and the heat, which were formerly perceived only in a small space, now appear in a much larger space. The mathematician now desires him to tell in what manner he conceives this expansiveness, or this actual expansion of air or gas. The chemist is then obliged to consider the air or gas as consisting of atoms or particles, which must be kept in their present situation by an external force, the most familiar of all to his imagination, namely, pressure; and all pressures are equally fit. Pressure is a moving force, and can only be opposed to such another moving force; therefore expansiveness supposes, that the particles are under the influence of something which would separate them from each other, if it were not opposed by something perfectly of the same kind. It cannot be opposed by greenness, nor by loudness, nor by fear, but only by what is competent to the production of motion; and it may be opposed by any such natural power; therefore by gravity, or by magnetism, or electricity, or corporeal attraction, or by an elective attraction. The chemist, being thus led to the contemplation of the phenomenon in its most simple state, can now judge with some distinctness, what is the nature of those powers with which expansiveness can- Mechanics be brought to co-operate or combine. And only now will he be able to speculate on the means for explaining the propagation of heat; and he will perceive, that the general laws of motion, and of the action of moving forces (doctrines which we comprehended under the title of Dynamics, Suppl.), must be referred to for a complete explanation of all chemical phenomena. The same may be said of the phenomena perceived in the growth of vegetables and animals. All of them lead us ultimately to the contemplation of an atom, which is characterized by being susceptible of local motion, and requires for this purpose the agency of what we call a moving force.

We would distinguish this particular object of our contemplation (consisting of two constituent parts, the atom and the force, related, in fact, to each other by constant conjunction) by the term MECHANISM. We conceive it to be the characteristic of what we call MATTER; and we would consider it as the most simple MECHANICAL PHENOMENON. We are disposed to think, that this moving force is as simple and uniform as the atom to which it is related; and we would ascribe the inconceivable diversity of the moving forces which we see around us to combinations of this universal force exerted by many atoms at once; and therefore modified by this combination, in the very same manner as we frequently see those seemingly different moving forces combine their influence on sensible mass of tangible matter, giving it a sensible local motion. Having formed such notions, we would say that we do not conceive either the atom or the force as being matter, but the two thus related. And we would then say, that whatever object of contemplation does not ultimately lead us to this complex notion is IMMATERIAL; meaning by the epithet nothing more than the negation of this particular character of the object. It is equivalent to saying, that the phenomenon does not lead the mind to the consideration of an atom actuated by a moving force; that is, moved, or prevented from moving, by an opposite pressure or force.

Such is the extension which the discoveries of last century have enabled us to give to the use of the term mechanism, mechanical action, mechanical cause, &c.

The Greeks, from whom we have borrowed the term, gave it a much more limited meaning; confining it to those motions which are produced by the intervention of machines. Even many of the naturalists of the present day limit the term to those motions which are the immediate consequences of impulse, and which are cases of sensible motion. Thus the chemist says, that printers' ink is a mechanical fluid, but that ink for writing is a chemical fluid. We make no objection to the distinction, because chemistry is really a vast body of real and important science, although we have, as yet, been able to class only very complicated phenomena, and are far from the knowledge of its elements. This distinction made by the chemists is very clear, and very proper to be kept in view; but we should be at a loss for a term to express the analogy which is perceivable between these sensible motions and the hidden motions which obtain even in the chemical phenomena, unless we give mechanism a still greater extension than the effects of percussion or impulsion.

Mechanics, in the ancient sense of the word, considers only the energy of organs, machines. The authors who have treated the subject systematically, have observed, that all machines derive their efficacy from a few simple forms and dispositions, which may be given to that piece of matter called the tool, 'οπανος, or machine, which is interposed between the workman or natural agent, and the task to be performed, which is always something to be moved, in opposition to resisting pressures. To those simple forms they have given the name of MECHANICAL POWERS, simple powers, simple machines.

The machine is interposed for various reasons.

1. In order to enable a natural power, having a certain determinate intensity, which cannot be increased, to balance or overcome another natural power, acting with a greater intensity. For this purpose, a piece of solid matter is interposed, connected in such a manner with firm supports, that the pressure exerted on the impelled point by the power occasions the excitement of a pressure at the working point, which is equal or inferior to the resistance, arising from the work, to the motion of that point. Thus, if a rod three feet long be supported at one foot from the end to which the resistance of two pounds is applied, and if a pressure of one pound be applied to the other end of the rod, perpendicular to its length, the cohesive forces which connect the particles of the rod will all be excited, in certain proportions, according to their situation, and the supported point will be made to press on its support, as much as three pounds would press on it; and a pressure in the opposite direction will be excited at the working point, equal to the pressure of two pounds. The resistance will therefore be balanced, and it will be overcome by increasing the natural power acting on the long division of the rod. This is called a LEVER. Toothed wheels and pinions are a perpetual succession of levers in one machine or mechanical power.

2. The natural power may act with a certain velocity which cannot be changed, and the work requires to be performed with a greater velocity. A machine is interposed, moveable round a fixed support, and the distances of the impelled and working points are taken in the proportion of the two velocities. Then are we certain, that when the power acts with its natural velocity, the working point is moving with the velocity we desire.

3. The power may act only in one unchangeable direction, and the resistance must be overcome in another direction. As when a quantity of coals must be brought from the bottom of a pit, and we have no power at command but the weight of a quantity of water. We let the water pull down one end of a lever, either immediately or by a rope, and we hang the coals on the other end, while the middle point is firmly supported. This lever may be made perpetual, by lapping the ropes round a cylinder which turns round an axis firmly supported. This is a FIXED FULLEY. We can let unequal powers in opposition, by lapping each rope round a different cylinder, having the same axis. This is a WIND-LASS or GIN. All these forms derive their energy from the lever virtually contained in them.

Any of these three purposes may be gained by the interposition of a solid body in another way. Instead of being supported in one point, round which it is moveable, it may be supported by a solid path, along which it is impelled, and by its shape it thrusts the resisting body This is the case with the wedge when it is employed to force up a swaying joint, or press things strongly together. If this wedge be lapped or formed round an axis, it becomes a screw or spiral wiper. This is also the operation of the balance wheel of a horizontal or cylinder watch. The oblique face of the tooth is a wedge, which thrusts the edge of the cylinder out of its way. The pallet of a clock or watch is also a wedge, acted on in the opposite direction.

There are different forms in which a solid body is interposed as a mechanic power. All are reducible to the lever and the wedge.

But there are other mechanic powers besides those now mentioned. The carmen have a way of lowering a cask of liquor into a cellar, by passing a rope under it, making the end fast to some stake close to the ground, and bringing the other end of the rope round the cask, and thus letting it slip down in the sight of the rope. In this process they feel but half of its weight, the other half being supported by the end of the rope that is fastened to the stake. This is called a parabola by the seamen. A hanging pulley is quite the same with this more artless method. The weight hangs by the axis of the pulley, and each half of the hanging rope carries half of the weight, and the person who pulls one of them upwards acts only against half of the weight, the other being carried by the hook to which the standing rope is fastened. This mechanical power does not (as is commonly imagined) derive its efficacy from the pulley's turning round an axis. If it were made fast, or if the tackle rope merely passed through a loop of the rope which carries the weight, it would still require only half of the weight acting on the running rope to balance it. The use of the motion round an axis is merely to avoid a very great friction.

When the two hanging parts of the rope are not parallel, but inclined at any angle, the force necessary for balancing the weight is to the weight as the side is to the diagonal of the parallelogram formed by the directions of the three ropes. Varignon calls this the funicular machine or power. Our sailors call it the swigg.

We may employ the guayau versum pressure of fluidity with great effect as a mechanic power. Thus, in the hydrostatic bellows described by Gravendeel, § 1451, and by Desaguilliers, the weight of a few ounces of water is made to raise several hundred pounds. In like manner, Dr Wallis of Oxford, by blowing with a pipe into a bladder, raised 64 pounds lying on it. Otto Guericke of Magdeburgh made a child balance, and even overcome, the pull exerted by the emperor's six coach horses, by merely sucking the air from below a piston. Mr Bramah, ironmonger in Piccadilly, London, has lately obtained a patent for a machine acting on this principle as a press*. A piston of one-fourth of an inch in diameter, forces water into a cylinder of 12 inches diameter, and by this intervention raises the piston of the cylinder. A boy, acting with the fourth part of his strength on the small piston by means of a lever, raises 42 tons, or 94,080 lbs, pressing on the great piston. It is very surprising, that this application of the guayau versum pressure of fluids has been overlooked for more than a century, although the principle has been inculcated and lectured on by every itinerant teacher, and illustrated by the above mentioned experiments of Gravendeel and Wallis; nay, it has been expressly taught as a mechanic power of great efficacy by the Professor of Natural Philosophy at Edinburgh every session of the college for these twenty years past, but he never thought of putting it in practice. It forms a most compendious machine of prodigious power, and is susceptible of the greatest strength. If the same multiplication of power be attempted by toothed wheels, pinions, and racks, it is scarcely possible to give strength enough to the teeth of the racks, and the machine becomes very cumbersome and of great expense. But Mr Bramah's machine may be made abundantly strong in very small compass. It only requires very accurate execution. We give it all praise; but Mr Bramah is mistaken when he publishes it as the invention or discovery of a new mechanic power; for it has been familiar to every student of mechanics and hydraulics ever since Boyle's first publication of his hydrostatic paradoxes.