Under the head of “Mixed Mathematics,” applicable to both laws or abstract principles and facts, the discussion of things as actual and possible, we have first, mechanics, the science that treats of the various forces and their different effects. By force is meant any power that tends to prevent, produce, or modify motion. Three are recognized—(1) gravitation, or the attraction of bodies toward each other; (2) the cause, whatever it may be, of light, heat, and electricity; (3) life, an equally mysterious power producing the actions of animals and the growth of plants. These forces, though entirely unseen and their causes unknown, are definite quantities. We readily conceive of one force as equal to, or greater than another, and know that equal forces, applied in opposite directions, balance each other. To everything that moves there is force applied greater than the resistance to be overcome. A number of forces may act on an object at the same time, accelerating, retarding, or changing the direction of the motion given to it. When the forces are so balanced as to hold the body on which they act in a state of equilibrium, their action and consequent phenomena are investigated under the head of Statics, or the science which treats of bodies at rest. When motion is produced, Dynamics considers the laws that govern the moving bodies and the phenomena that result. These branches of mechanical science are of great practical importance, and a knowledge of them would save from many blunders and failures resulting from incompetence. The same laws govern in the movement of all bodies, whether solid or liquid. Hydrostatics, Hydrodynamics, Hydraulics, etc., are branches of the same science, and worthy of separate mention only because they apply the general principles of statics and dynamics to the phenomena of rest or motion in liquids. The foundation for all that is peculiar in these branches with the lengthened names, and that together may be called Hydro-mechanics, lies in the properties that distinguish the liquid from other states of material bodies, whether gaseous or solid, viz.: in the presence of cohesion, but with great mobility of parts and more or less elasticity. Some peculiarities are so noteworthy as to deserve mention even in this limited presentation. Because of the only slight cohesive attraction, and entire freedom of motion among the particles, liquid bodies possess no definite form of their own, but adapt themselves to the form of the excavations or vessels containing them. They, of course, vary much in their fluidity, the mobile liquids, as water and alcohol, flowing more readily than molasses, heavy oils, and tar. Fluids at rest press equally in all directions, upward, downward, and laterally. In this, also, they differ from solids that press only down, or in the direction of the center of gravitation. If not confined they can not be heaped up, but their particles seek a common level. An absolute water level is, of course, possible only when the area covered is so limited that lines joining all the points on the surface with the center of gravity are practically parallel, or their convergence an inappreciable quantity. In large bodies of water, as the ocean, the surface corresponds with the general rotundity of the earth.
The fact of the equal pressure of liquids in all directions, and with the same intensity, is found of great importance in practical mechanics. The strong pressure of a small column of water is finely illustrated by simple experiment with the water bellows, or hydraulic paradox, in which one pound of water in a tube lifts a hundred pounds on the top of the bellows, and the greater the disproportion between the diameter of the tube and that of the top of the bellows, the greater weight it will raise. More than two hundred years ago Pascal showed the enormous pressure exerted by a lofty column of water in a small tube. A strong cask was filled with water, and a small tube forty feet high closely fitted in its head, when a few pints of water poured into it burst the cask, and would have done so if it had been made of the strongest oaken staves and bound with hoops of iron. This is the power used in the hydraulic press, a very simple machine of much value in the industrial arts when there is a demand for great force that can be slowly and steadily applied, as in compressing cloth, oil cake, paper, gunpowder and numerous other things. Its parts are so few that it can be described without a model to represent it. A small, upright cylinder, with a closely fitting piston used as a pump to draw and force the water, and connected at the base by a tube with a much larger cylinder directly under the substance to be pressed, in which there is also a piston to be moved upward, though water tight. The whole is secured in position by powerful frame work. Beneath the piston the water is received. And knowing the principles of hydrostatics we can estimate its power. If the areas of the lower surfaces of the two pistons are to each other as one to four hundred square inches, one pound pressure on the small one will deliver to the lower surface of the large one a pressure equal to four hundred pounds weight. But let the arms of the lever used as the force pump handle be to each other as one to fifty. Then when a force of fifty pounds is applied at the end of the long arm of the lever it will descend with a force of 50×50=2,500, and there will be delivered on the lower surface of the large piston a power to raise it expressed by 50×50×400=1,000,000. Some allowance must be made for friction or other impediments, say one fourth, which is more than enough, and still a man or boy at the end of that pump handle would be able to lift at least three hundred and seventy-five tons.
The sciences we have been considering under the general name of mechanics, which is derived from a Greek word that means to contrive, invent, construct, have much to do with machinery, with the methods of construction, the propelling forces, and the phenomena produced. There were machinists and some simple machines propelled by human or brute force, by weights and springs, by falling or running water, and air in motion before the laws of motion and forces were understood, or the rude mechanic arts began to assume the character of a science. The machines were, of course, imperfect, and lacked efficiency, while many of those now in use seem nearly perfect and adapted to the work expected of them. But notwithstanding the marvelous advance that has been made in the manufacture of machinery, and the intelligent application of mechanical powers, we look for still greater things as possible in the future.
It is well, however, never to forget that whatever the seeming may be, the most perfect machine of human invention does not create force. That is as impossible for man as it is to give life or create matter. All he can do is to collect, concentrate and use, to the best advantage, the forces that exist. He may by skillful appliances gain a great mechanic advantage, and overcome very formidable resistance, but he must be content to do it very slowly; and it has been often said that “what he gains in power he loses in speed.” In many cases this seems a necessity, and he must submit to it. His simplest machine, if the fulcrum is placed very near the weight, gives a man tremendous power gained by his position at the long arm of the machine. But the point at which he applies the force must move much faster and a greater distance than the object against which it is directed. So when a man with a system of pulleys raises to the top of a tower a block of granite that four men might lift from the ground he sacrifices in speed what he gains in the new way of applying the force he has for the purpose.
You visit a large manufacturing establishment or the mechanical department of a great national or international industrial exposition and see a whole acre of machinery of all kinds, shafts, wheels, saws, lathes, and spindles in rapid motion, and, astonished at the complications, inquire for the power that carries the whole. You will possibly find it is in some remote part of the premises, and shut up in the motionless boiler where the steam is said to be generated, which only means that the water heated expands and struggles to escape from its confinement, while man understanding the laws of its action manages to liberate the force under conditions that make it his servant.
The science of numbers and magnitudes, useful in discussing the distances, measurements, and motions of terrestrial bodies, is especially so in its application to astronomy.
Astronomy as a physical science will receive consideration in the next number; here only the mathematical elements are noticed, and they are everywhere manifest. The same general laws control all material bodies, those near to us, and those seen at a distance. So the science of the stars is not now mere theory, but has all the elements of mathematical certainty. When dealing with such vast numbers and magnitudes as engage the astronomer’s attention, with a few known principles or laws, and abundant recorded telescopic observations for the basis of their work, men can calculate even more accurately than they can count or measure. Having once prepared their theorem, aided by the logarithms of Napier[1] that simplify and shorten the more difficult arithmetical calculations, they can readily determine the distance, magnitude and motions of a planet, and know that it is done with sufficient exactness. The distances of the heavenly bodies are generally determined by their parallax, that is the difference between the directions of the bodies as seen from two different points. The inclination of the lines thus drawn is the angle of parallax. By supposing the lines prolonged to the sun, and other lines drawn through the points selected to the center of the earth a quadrangle is formed, all the angles and sides of which are easily found. In measuring very minute parallaxes it may not be possible to determine the exact position of the body as projected on the celestial sphere, but in that case recourse can be had to relative parallax, or the difference between the parallaxes of two bodies lying nearly in the same direction. The best opportunity for this is afforded by the transit of Venus, and on this account great interest is felt in that phenomenon, and extensive preparations are made for taking accurate observations.
The figure, size and density of the celestial bodies have all been calculated with approximate certainty. The orbits, through which they pass in their revolutions, described, and their velocities ascertained.
There is a solar system of which the sun is the center, and in its relation to the planets stationary, though really moving on through infinite space; the orbits through which planets move are not circles, but more or less elliptic, having the sun at one focus of the ellipse.
That planets move in ellipses was announced by Kepler[2] as the first law governing their motions, and a second deduced from this and confirmed by observations, is that they do not move with equal velocity in all parts of their orbits; and that a line drawn from the center of the earth to the center of the sun passes over equal spaces in equal times. He also found as a third law that the squares of the times of the revolutions of the planets are proportional to the cubes of their mean distances from the sun.