INTRODUCTION. ART. 1. Anything which can be multiplied, divided, or measured, is called QUANTITY. Thus, lines, weight, time, number, &c., are quantities. OBS. 1. A line is a quantity, because it can be measured in feet and inches; weight can be measured in pounds and ounces; time, in hours and minutes; numbers can be multiplied, divided, &c. 2. Color, and the operations of the mind, as love, hatred, desire, choice, &c., cannot be multiplied, divided, or measured, and therefore cannot properly be called quantities. 2. MATHEMATICS is the science of Quantity. 3. The fundamental branches of Mathematics are, Arithmetic, Algebra, and Geometry. 4. Arithmetic is the science of Numbers. 5. Algebra is a general method of solving problems, and of investigating the relations of quantities by means cf letters and signs. Овs. Fluxions, or the Differential and Integral Calculus, may be considered as belonging to the higher branches of Algebra. 6. Geometry is that branch of Mathematics which treats of Magnitude. 7. The term magnitude signifies that which is extended, or which has one or more of the three dimensions, length, breadth, and thickness, Thus, lines, surfaces, and solids are magnitudes. QUEST.-1. What is Quantity? Give some examples of quantity. O. Why is a line a quantity? Weight? Time? Numbers? Are color and the operations of the mind quantities? Why not? 2. What is Mathematics? 3. What are the fundamental branches of mathematics? 4. What is Arithmetic? 5. Algebra ? 6. Geometry ? 7. What is meant by magnitude? OBS. 1. A line is a magnitude, because it can be extended in length; a surface, because it has length and breadth; a solid, because it has length, breadth, and thickness. 2. Motion, though a quantity, is not, strictly speaking, a magnitude; for it has neither length, breadth, nor thickness. 3. The term magnitude is sometimes, though inaccurately, used as syncnymous with quantity. 8. Trigonometry and Conic Sections are branches of Mathematcs, in which the principles of Geometry are applied to triangles, and the sections of a cone. 9. Mathematics are either pure or mixed. In pure mathematics, quantities are considered, independently of any substances actually existing. 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 heavenly bodies. OBS. The science of 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. 10. A definition is an explanation of what is meant by a word, or phrase. OBS. It is essential to a complete definition, that it perfectly distinguishes the thing defined, from everything else. 11. A proposition is something proposed to be proved, or required to be done, and is either a Theorem, or a Problem. 12. A theorem is something to be proved. 13. A problem is something to be done, as a question to be solved. QUEST.-Obs. Why is a line a magnitude? A surface? A solid? Is motion a magnitude? Why not? 9. Of how many kinds are mathematics? In pure mathematics how are quantities considered? How in mixed mathematics? Obs. For what is the science of pure mathematics distinguished? 10. What is a definition? Obs. What is essential to a complete definition? 11. What is a proposition? 12. A theorem? 13. A problem? OBS. 1. In the statement of every proposition, whether theorem or problem, certain things must be given, or assumed to be true. These things are called the data of the proposition. 2. The operation by which the answer of a problem is found, is called a solution. 3. When the given problem is so easy, as to be obvious to every one without explanation, it is called a postulate. 14. One proposition is contrary, or contradictory to another, when what is affirmed in the one, is denied in the other. OBS. 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. 15. One proposition is the converse of another, when the order is inverted; so that, what is given or supposed in the first, becomes the conclusion in the last; and what is given in 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 are 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. OBS. 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 necessarily a circle. 16. The process of reasoning by which a proposition is shown to be true, is called a demonstration. OBS. A demonstration is either direct or indirect. A direct demonstration commences with certain principles or data which are admitted, or have been proved to be true; and from these, a series of other truths are deduced, each depending on the preceding, till we arrive at the truth which was required to be established. An indirect demonstration is the mode of establishing the truth of a proposition by proving that the supposition of its contrary, involves an absurdity. QUEST.-Obs. What is meant by the data of a proposition? By the solution of problem? What is a postulate? 14. When is one proposition contrary to another Obs. Can a proposition and its contrary both be true? 15. When is one proposition the converse of another? Obs. Can a proposition and its converse both be true? 16. What is a demonstration? Obs. Of how many kinds are demonstrations? What is a direct demonstration? An indirect demonstration? |