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.