Mathematics in General
Art. 1. Mathematics is the science of quantity.
Any thing which can be multiplied, divided, or measured, is called quantity. Thus, a line is a quantity, because it can be doubled, trebled, or halved; and can be measured, by applying to it another line, as a foot, a yard, or an ell. Weight is a quantity, which can be measured, in pounds, ounces, and grains. Time is a species of quantity, whose measure can be expressed, in hours, minutes, and seconds. But color is not a quantity. It cannot be said, with propriety, that one color is twice as great, or half as great, as another. The operations of the mind, such as thought, choice, desire, hatred, &c. are not quantities. They are incapable of mensuration.
2. Those parts of the Mathematics, on which all the others are founded, are Arithmetic, Algebra, and Geometry.
3. Arithmetic is the science of numbers. Its aid is required to complete and apply the calculations, in almost every other department of the mathematics.
4. Algebra is a method of computing by letters and other symbols. Fluxions, or the Differential and Integral Calculus, may be considered as belonging to the higher branches of algebra.
5. Geometry is that part of the mathematics, which treats of magnitude. By magnitude, in the appropriate sense of the term, is meant that species of quantity, which is extended; that is, which has one or more of the three dimensions, length, breadth, and thickness. Thus a line is a magnitude, because it is extended, in length. A surface is a magnitude, having length and breadth. A solid is a magnitude, having length, breadth, and thickness. But motion, though a quantity, is not, strictly speaking, a magnitude. It has neither length, breadth, nor thickness.
6. Trigonometry and Conic Sections are branches of the mathematics, in which the principles of geometry are applied to triangles, and the sections of a cone.
7. Mathematics are either pure or mixed. In pure mathematics, quantities are considered, independently of any substances actually existing. But, in mixed mathematics, the relations of quantities are investigated, in connection with some of the properties of matter, or with reference to the common transactions of business. Thus, in Surveying, mathematical principles are applied to the measuring of land; in Optics, to the properties of light; and in Astronomy, to the motions 6f the heavenly bodies.
8. The science of the pure mathematics has long been distinguished, for the clearness and distinctness of its principles ; and the irresistible conviction, which they carry to the mind of every one who is once made acquainted with them. This is to be ascribed, partly to the nature of the subjects, and partly to the exactness of the definitions, the axioms, and the demonstrations.
9. The foundation of all mathematical knowledge must be laid in definitions. A definition is an explanation of what is meant, by any word or phrase. Thus, an equilateral triangle is defined, by saying, that it is a figure bounded by three equal sides.
It is essential to a complete definition, that it perfectly distinguish the thing defined, from every thing else. On many subjects it is difficult to give such precision to language, that it shall convey, to every hearer or reader, exactly the same ideas. But, in the mathematics, the principal terms may be so defined, as not to leave room for the least difference of apprehension, respecting their meaning. All must be agreed, as to the nature of a circle, a square, and a triangle, when they have once learned the definitions of these figures.
Under the head of definitions, may be included explanations of the characters which are used to denote the relations of quantities. Thus the character √ explained or defined, by saying that it signifies the same as the words square root.
10. The next step, after becoming acquainted with the meaning of mathematical terms, is to bring them together, in the form of propositions. Some of the relations of quantities require no process of reasoning, to render them evident. To be understood, they need only to be proposed. That a square is a different figure from a circle; that the whole of a thing is greater than one of its parts; and that two straight lines cannot enclose a space, are propositions so manifestly true, that no reasoning-upon them could make them more certain. They are, therefore, called self-evident truths, or axioms.
11. There are, however, comparatively few mathematical truths which are self-evident. Most require to be proved by a chain of reasoning. Propositions of this nature are denominated theorems; and the process, by which they are shown to be true, is called demonstration. This is a mode of arguing, in which, every inference is immediately derived, either from definitions, or from principles which have been previously demonstrated. In this way, complete certainty is made to accompany every step, in a long course of reasoning.
12. Demonstration is either direct or indirect. The former is the common, obvious mode of conducting a demonstrative argument. But in some instances, it is necessary to resort to indirect demonstration; which is a method of establishing a proposition, by proving that to suppose it not true, would lead to an absurdity. This is frequently called reductio ad absurdum. Thus, in certain cases in geometry! two lines may be proved to be equal, by showing that to suppose them unequal, would involve an absurdity.
13. Besides the principal theorems in the mathematics, there are also Lemmas and Corollaries. A Lemma is a proposition which is demonstrated, for the purpose of using it, in the demonstration of some other proposition. This preparatory step is taken to prevent the proof of the principal theorem from becoming complicated and tedious.
14. A Corollary is an inference from a preceding proposition. A Scholium is a remark of any kind, suggested by something which has gone before, though not, like a corollary, immediately depending on it.
15. The immediate object of inquiry, in the mathematics, is, frequently, not the demonstration of a general truth, but a method of performing some operation, such as reducing a vulgar fraction to a decimal, extracting the cube root, or inscribing a circle in a square. This is called solving a problem. A theorem is something to be proved. A problem is something to be done.
16. When that which is required to be done, is so easy, as to be obvious to every one, without an explanation, it is called a postulate. Of this nature is the drawing of a straight line, from one point to another.
17. A quantity is said to be given, when it is either supposed to be already known, or is made a condition, in the statement of any theorem or problem. In the rule of proportion in arithmetic, for instance, three terms must be given to enable us to find a fourth. These three terms are the data, upon which the calculation is founded. If we are required to find the number of acres, in a circular island ten miles in circumference, the circular figure, and the length of the circumference are the data. They are said to be given by supposition, that is, by the conditions of the problem. A quantity is also said to be given, when it may be directly and easily inferred, from something else which is given. Thus, if two numbers are given, their sum is given; because it is obtained, by merely adding the numbers together.
In Geometry, a quantity may be given, either in position, or magnitude, or both. A line is given in position, when its situation and direction are known. It is given in magnitude, when its length is known. A circle is given in position, when the place of its centre is known. It is given in magnitude, when the length of its diameter is known.
18. One proposition is contrary, or contradictory to another, when, what is affirmed, in the one, is denied, in the other. A proposition and its contrary, can never both be true. It cannot be true, that two given lines are equal, and that they are not equal, at the same time.
19. One proposition is the converse of another, when the order is inverted; so that, what is given or supposed in die first, becomes the conclusion in the last; and what is given tn the last, is the conclusion, in the first. Thus, it can be proved, first, that if the sides of a triangle are equal, the angles are equal; and secondly, that if the angles axe equal, the sides are equal. Here, in the first proposition, the equality of the sides is given; and the equality of the angles inferred: in the second, the equality of the angles is given, and the equality of the sides inferred. In many instances, a proposition and its converse are both true; as in the preceding example. But this is not always the case. A circle is a figure bounded by a curve; but a figure bounded by a curve is not of course a circle.
20. The practical applications of the mathematics, in the common concerns of business, in the useful arts, and in the various branches of physical science are almost innumerable. Mathematical principles are necessary in Mercantile transactions, for keeping, arranging, and settling accounts, adjusting the prices of commodities, and calculating the profits of trade: in Navigation, for directing the course of a ship on the ocean, adapting the position of her sails to the direction of the wind, finding her latitude and longitude, and determining the bearings and distances of objects on shove: in Surveying, for measuring, dividing, and laying out grounds, taking the elevation, of hills, and fixing the boundaries of fields, estates, and public territories : in Civil Engineering, for constructing ridges, aqueducts, locks, &c.: in Mechanics, for understanding the laws of motion, the composition of forces, the equilibrium of the mechanical powers, and the structure of machines : in Architecture, for calculating the comparative strength of timbers, the pressure which each will be required Nto sustain, the forms of arches, the proportions of columns, &c.: in Fortification, for adjusting the position, lines, and angles, of the several parts of the works : in Gunnery, for regulating the elevation of the cannon, the force of the powder, and the velocity and range of the shot: in Optics, for tracing the direction of the rays of light, understanding the formation of images, the laws of vision, the separation of colors, the nature of the rainbow, and the construction of microscopes and telescopes: in Astronomy, for computing the distances, magnitudes, and revolutions of the heavenly bodies; and the influence of the law of gravitation, in raising the tides, disturbing the motions of the moon, causing the return of the comets, and retaining the planets in their orbits: in Geography, for determining the figure and dimensions of the earth, the extent of oceans, islands, continents, and countries; the latitude and longitude of places, the courses of rivers, the height of mountains, and the boundaries of kingdoms: in History, for fixing the chronology of remarkable events, and estimating the strength of armies, the wealth of nations, the value of their revenues, and the amount of their population: and, in the concerns of Government, for apportioning taxes, arranging schemes of finance, and regulating national expenses. The mathematics have also important applications to Chemistry, Mineralogy, Music, Painting, Sculpture, and indeed to a great proportion of the whole circle of arts and sciences.
21. It is true, that, in many of the branches which have been mentioned, the ordinary business is frequently transacted, and the mechanical operations performed, by persons who have not been regularly instructed in a course of Machines are framed, lands are surveyed, and ships are steered, by men who have never thoroughly investigated the principles, which lie at the foundation of their respective arts. The reason of this is, that the methods of proceeding, in their several occupations, have been pointed out to them, by the genius and labor of others. The mechanic often works by rules, which men of science have prpvided for his use, and of which he knows nothing more, than the practical application. The mariner calculates his longitude by tables, for which he is indebted to mathematicians and astronomers of no ordinary attainments. In this manner, even the abstruse parts of the mathematics are made to contribute their aid to the common arts of life.
22. But an additional and more important advantage, to persons of liberal education, is to be found, in the enlargement and improvement of the reasoning powers. The mind, like the body, acquires strength by exertion. The art of reasoning, like other arts, is learned by practice. It is perfected, only by long continued exercise. Mathematical studies are peculiarly fitted for this discipline of the mind. They are calculated to form it to habits of fixed attention; of sagacity, in detecting sophistry; of caution, in the admission of proof; of dexterity, in the arrangement of arguments; and of skill, in making all the parts of a long continued process tend to a result, in which the truth is clearly and firmly established. When a habit of close and accurate thinking is thus acquired, it may be applied to any subject, on whiph a man of letters or orbusiness may be called to employ his talents. "The youth," says Plato, "who are furnished with mathematical knowledge, are prompt and quick, at all other sciences."
It is not pretended, that an attention to other objects of inquiry is rendered unnecessary, by the study of the It is nx)t their office, to lay before us historical facts; to teach the principles of morals; to store the fancy with brilliant images; or to enable us to speak and write "with rhetorical vigor and elegance. The beneficial effects which they produce on the mind, are to be seen, principally, in the regulation and increased energy of the reasoning powers. These they are calculated to call into frequent and vigorous exercise. At the same time, mathematical studies may be so conducted, as not often to require excessive exertion and fatigue. Beginning with the more simple subjects, and ascending gradually to those which are more complicated, the mind a;ouires strength as it advances;*and by a succession of steps, nsing regularly one above another, is enabled to surmount »!ie obstacles which lie in its way. In a course of mathematics, the parts succeed each other in such a connected series, that the preceding propositions are preparatory to those which follow. The student who has made himself master of the former, is qualified for a successful investigation of the latter. But he who has passed over any of the ground superficially, will find that the obstructions to his future progress are yet to be removed. In mathematics as in war, it should be made a principle, not to advance, while any thing is left unconquered behind. It is important that the student should be deeply impressed with a conviction of the necessity of this. Neither is it sufficient that he understands the nature of one proposition or method of operation, before proceeding to another. He ought also to make himself familiar with every step, by careful attention to the examples. He must not expect to become thoroughly versed in the science, by merely reading the main principles, rules, and observations. It is practice only, which can put these completely m his possession. The method of studying here recommended, is not only that which promises success, but that which will be found, in the end, to be the most expeditious, and by far most pleasant. While a superficial attention occasions perplexity and consequent aversion; a thorough investigation is rewarded with a high degree of gratification. The peculiar entertainment which mathematical studies are calculated to furnish to the mind, is reserved for those who make themselves masters of the subjects to which their attention is called.