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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 of letters and signs.
Obs. 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. 01. 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 fundadiental 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 le 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 Mathemat'cs, 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 distinguishe's 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 inixed 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 a 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. Or how many kinds are demonstrations ? What is a direct demonstration ? An indirect demonstration ?
This is commonly called reductio ad absurdum. The former is the more common method of conducting a demonstrative argument, and is the most satisfactory to the mind.
17. A Lemma is a subsidiary truth or proposition, demonstrated for the purpose of using it in the demonstration of a theorem, or the solution of a problem.
18. A Corollary is an inference or principle deduced from a preceding proposition.
19. A Scholium is a remark made upon a preceding prope osition, pointing out its connection, use, restriction, or extension.
20. An Hypothesis is a supposition, made either in the statement of a proposition, or in the course of a demonstration.
21. An Axiom is a self-evident proposition ; that is, a proposition whose truth is so evident at sight, that no process of reasoning can make it plainer. The following axioms are among the most common :
1. Quantities which are equal to the same quantity, are equal to each other.
2. If the same or equal quantities are added to equals, the sums will be equal.
3. If the same or equal quantities are subtracted from equals, the remainders will be equal.
4. If the same or equal quantities are added to unequals, the sums will be unequal.
5. If the same or equal quantities are subtracted from unequals, the remainders will be unequal.
6. If equal quantities are multiplied by the same or equal quantities, the products will be equal.
7. If equal quantities are divided by the same or equal quantities, the quotients will be equal.
8. If the same quantity is both added to and subtracted from another, the value of the latter will not be altered.
QUEST.-17. What is a lemma? 18. What is a corollary? 19. Whas is a scholium ? 20. What is an hypothesis ? 21. What is an axiom} Name some of the most common axioms.
9. If a quantity is both multiplied and divided by the same or an equal quantity, its value will not be altered.
10. The whole of a quantity is greater than a part. 11. The whole of a quantity is equal to the sum of all its parts.
22. Addition is represented by the sign (+), which is called plus. " It consists of two lines, one horizontal, the other perpendicular, forming a cross, and shows that the numbers between which it is placed, are to be added together. Thus, the expression 6+8, signifies that 6 is to be added to 8. It is read, “ 6 plus 8,”
« 6 added to 8.” OBS.—The term plus is a Latin word, originally signifying “more,” hence " added to."
23. Subtraction is represented by a short horizontal line (-), which is called minus. When placed between two numbers, it shows that the number after it is to be subtracted from the one before it. Thus, the expression 9—4, signifies that 4 is to be subtracted from 9; and is read, “9 minus 4,” or “9 less 4.” OBS.—The term minus is a Latin word, signifying less.
24. Multiplication is usually denoted by two oblique lines crossing each other (X), called the sign of multiplication. It shows that the numbers between which it is placed, are to be multiplied together. Thus, the expression (9X6), signifies that 9 and 6 are to be multiplied together, and is read, “ 9 multiplied by 6,” or simply, “9 into 6." Sometimes multiplication is denoted by a point (.) placed between the two numbers or quantities. Thus, 9.6 denotes the same as 9x6.
OBs. It is better to denote the multiplication of figures by a cross than by a point; for the latter is liable to be confounded with the decimal point.
24. a. When two or more numbers are to be subjected to the same operation, they must be connected by a line () placed
QUEST.-22. What is the sign of addition called? Of what does it consist? What does it show? Obs. What is the meaning of the term plus ? 23. How is subtraction represented ? What is the sign of subtraction called? What does it show? Obs. What does the term minus signify? 24. How is multiplication usually denoted? What does the sign of mul. tiplication show? In what other way is multiplication sometimes denoted ?