Science means classified knowledge. There may be much general knowledge that is not science. It attains to that dignity only when the particular facts known are generalized, and arranged in some order, instead of being jumbled together, and lying about loosely in the memory, to be taken up at random. Especially must the basal facts of the science be verified, not assumed.

Information that is general and assured, though as yet lacking system and a proper ordering of the elementary facts, may, and usually will in time advance to the dignity of science. History warrants this expectation. Only let not the boast be made, or the honor conferred prematurely. Geography, chemistry, and political economy are all now sciences. The first has been recognized among the sciences from an early day, though it has advanced rapidly during the present century. The last two are comparatively new members, having held their place in the “Circle” scarcely a hundred years. True, many of the facts of chemistry, and the principles of political economy had been known for ages, but the knowledge men had of them lacked either system or certainty, or both. So, also, in respect to mineralogy, botany, and zoÖlogy, a store of known facts had been for ages accumulating, before they could rightly be called sciences. To reach that distinction the quality and orderly arrangement of the things known are as necessary as the quantity.

In the heading of this series of articles, “Circle” does not suggest the rim of a wheel, or a curved line all the points of which are equally distant from the center around which it is drawn, but rather a group of sciences, just as “social circle,” and “circle of friends” indicate the amicable relations of the persons without saying anything of their positions in the place of their meeting. It is a goodly group, this family of the sciences, and the members now so numerous and having such distinctive characteristics will be introduced, not as a body but severally, and in five classes: The Mathematical, Physical, Mental, Moral, and Social Sciences. They hold such intimate relations with each other, mutually giving and receiving aid, that we will not attempt to keep the members of classes from mixing occasionally in our account of them, as they often do in reality.

Mathematics is the science of quantities and numbers. Its principles are of the first importance, and are of service in all the departments of science. In several of its subdivisions, of which brief mention will be made, it uses known quantities for the determination of those unknown, reasoning from certain relations existing between them. The qualities it discusses are represented by diagrams, figures, or symbols, adopted for the purpose. It is customary to speak of pure and mixed, or abstract and applied mathematics; the former treating of laws, principles, and relations in the abstract, or without any special reference to anything as actual or existing. The latter discusses the principles, laws and relations in connection with existing phenomena. The operations with numbers and symbols in pure mathematics, dealing only with abstract quantities, do not necessarily imply the idea of matter. Those of the science as applied have much to do with material phenomena. The elements that enter into the calculations in both cases are axioms or self-evident truths, things that are known intuitively, or grasped by the reason soon as presented, only in applied mathematics, used more or less in all sciences, these same axiomatic, self-evident truths are employed in the discussion of natural objects, the laws, properties, and relations of which are learned mostly by experience and induction.

The sciences classed as pure mathematics are Arithmetic, Geometry, Algebra, Analytical Geometry and Calculus. Arithmetic is eminently the science of numbers, and treats of, or practically illustrates their nature and uses. It employs the nine Arabic digits or figures with the addition of the cipher, giving them various positions to express numerical values, and not the native qualities or functions of the things to which they are applied. The methods are the same, and the results obtained equally true, whatever may be the nature of the quantities about which inquiry is made. The elementary or fundamental idea in arithmetic is unity, expressed by the figure 1, from which, with the help of the other eight digits, and the individually valueless cipher, 0, expressions for all the other values, whole or fractional, are formed.

As arithmetical processes underlie, or enter into, the work of nearly all mathematical calculations, its great importance as a science is evident; though as often taught in our schools and used in business, it is simply a method of reckoning or computation.

Algebra is a kindred science, that, by the use of letters and symbols, enables us to solve more readily all difficult questions relating to numbers. It is, indeed, a kind of universal arithmetic. In the ordinary arithmetic the numbers or figures employed, taken separately, have always the same value, and the result, when, sometimes by a tedious process, obtained, is applicable only to the particular question proposed, but in solving the problem by algebra, since we employ letters to which any values may be attributed at pleasure, the result obtained is largely applicable to all questions of a particular class. Thus, having the sum and difference of two quantities given, we readily obtain an algebraic expression for the quantities themselves. By the new method the goal is reached speedily, and the cabalistic terms, that may, at his first attempts, perplex and discourage the young student, become his delight; and in many difficult processes greatly shorten the work, enabling him with ease to solve problems that to the common arithmetician are tedious, if not impossible.

Geometry, one of the oldest of sciences, measures extension, treats of order and proportion in space. Its working elements are not numbers or symbols, but points, and lines, either straight or curved, and surfaces, with volumes, or solids. The simpler problems, when successfully demonstrated, are used in solving those more complicated, making the progress easy.

Lines are made up of points, and have extension only in one direction. Surfaces have length and breadth, and are distinguished as triangles, quadrilaterals, polygons, etc., according to the number of lines that circumscribe them. Solids have length, breadth, and thickness. From a few elementary facts, much geometrical science has been deduced, by very simple, logical processes. It is intimately related to other sciences, and of much practical importance; but, if there were no other advantage derived, as a discipline of the reasoning faculty there can be nothing better. To pursue the study profitably there is little need of an instructor. Class recitations are helpful, but let any one intent on personal culture, and having only a little time for the work, get a good elementary treatise on plane and solid geometry, and study it. The exercise will become a delight, will give strength and grip to the faculties, and furnish protection against the mental dissipation caused by spending much time in the hasty, careless reading of what is fitly called light literature.

Analytical geometry is that branch which examines, discusses and develops the properties of geometrical magnitudes by the use of algebraic symbols. The questions or problems are solved, not, as in plane geometry, by diagrams or figures drawn to show certain relations of magnitudes, but by making algebraic symbols represent them, and thus solving the problems. Analysis is much used in simple algebraic processes, but more in analytical geometry, and in differential and integral calculus, which has been called the transcendental analysis. It is useful as a higher branch of the science, and without it the best achievements of the greatest mathematicians would scarcely have been possible. These last named branches are generally best pursued in our higher academies and colleges. A college course would be sadly deficient without them, but only for exceptional cases would it be advisable to put them in a course of study to be pursued privately.

If this brief mention of the higher mathematics kindles desire for further knowledge, and you hesitate to grapple with them alone, by all means go to college, and after a proper introduction, wherein the chief embarrassment is felt, even calculus will be found an agreeable acquaintance.

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.

Navigation shows how vessels are directed in their course upon the great waters. In proportion as the “paths of the seas” have become open, safe and free for all, they are found paths of knowledge and civilization. The science, small at its beginning, has grown to its present advanced state by slow degrees, helped by contributions from the most opposite sources. Practical but uneducated seamen have doubtless done much, as their ingenuity is often, in emergencies, taxed to supply means of safety and success that are wanting. More has been contributed by scholars, secluded philosophic men whose lives are spent “in communion with the skies,” in observing the motions of the heavenly bodies and studying the laws by which they are regulated. But perhaps the most valuable service has been rendered by another class who combine an experience of the sea with much knowledge of astronomical science, men acquainted with the needs of seamen and qualified to meet them. The introduction of the mariner’s compass early in the fifteenth century was an epoch in the history of navigation, as it made seamen in a measure independent of the sun and stars. This was an incalculable advantage, as soon became apparent to those who adopted the compass as their guide. Of the many improvements and helps in the science of navigation we can only name, as conspicuous, the invention of Mercator’s chart[3] in 1569, Davis’s quadrant[4] about 1600, and Hadley’s quadrant a century later. The character of the instruments and a glance at the Nautical Almanac will show how largely both mathematics and astronomy enter into the science of navigation. Nor is it quite safe to take passage with a shipmaster who has but limited knowledge of either. He should at least thoroughly understand his instruments and be a ready, accurate computer.

Geometry grew out of the practice of surveying, and now embodies many of the laws and principles of the science. There are several distinct systems of surveying, classed according to the purposes contemplated. It is astronomically employed in determining the figure of the earth by the actual measurement of arcs. A fair knowledge of mathematics and trigonometry is required in what are known as coast surveys. Land surveying is of the plainest kind, and employed in finding the contents of areas, or in dividing large tracts into lots of smaller dimensions. The chief difficulty is in getting the exact bearing of the lines and the measure of the angles when the plot is an irregular polygon.

Topographical surveying, beside the measurement of lines and angles, takes note of variations of level, that the draft may properly represent superficial inequalities. Maritime surveying is an important branch, fixing the positions of shoals, rocks and shore-lines. Mine surveying determines the location of works in the mine and decides whether the excavations conform, as required, to lines on the surface. The compass and chain are the surveyor’s most common instruments, but others are used according to the nature of the surveys to be made. Incompetency or carelessness in surveys often occasions serious trouble and loss.

Fortifications for the defense of cities and the protection of soldiers are as ancient as the existence of armies. The former, built in time of peace, of such form and materials as military science and experience suggest, are called “permanent fortifications;” and the temporary works constructed as the exigencies of a campaign require are “field fortifications.” The art and science have been practiced and studied in all ages, and there is now an immense literature on the subject.

As methods of defense must be adjusted to those of attack the earlier permanent fortifications, in the progress of society and after the introduction of artillery, became nearly worthless. High stone walls are a protection while they stand, but, however strong, they can be battered down by heavy siege guns that have less effect when directed against earth works, which seem less formidable. A place thoroughly fortified is seldom taken by a sudden assault. The United States have fortified less than most of the great European nations, but are by no means defenseless. Previous to 1860 there had been expended on our forts more than $30,000,000; and all the exposed positions have been greatly strengthened within the last twenty-five years.

End of Required Reading for February.