ELEMENTS OF GEOMETRY. INTRODUCTION. DEFINITIONS OF TERMS. 1. QUANTITY is anything which can be increased, diminished, and measured. To measure a thing, is to find out how many times it contains some other thing of the same kind, taken as a standard. The assumed standard is called the unit of measure. 2. In GEOMETRY, there are four species of quantity, viz.: LINES, SURFACES, VOLUMES, and ANGLES. These are called, GEOMETRICAL MAGNITUdes. Since the unit of measure of a quantity is of the same kind as the quantity measured, there are four kinds of units of measure, viz.: Units of Length, Units of Surface, Units of Volume, and Units of Angular Measure. 3. GEOMETRY is that branch of Mathematics which treats of the properties, relations, and measurement of the Geometrical Magnitudes. 4. In Geometry, the quantities considered are generally represented by means of the straight line and curve. The operations to be performed upon the quantities and the rela tions between them, are indicated by signs, as in Analysis. The following are the principal signs employed: The Sign of Addition, +, called plus: Thus, A+ B, indicates that The Sign of Subtraction, B is to be added to A. called minus : Thus, AB, B, indicates that B is to be subtracted from A. The Sign of Multiplication, x: Thus, A× B, indicates that A is to be multiplied by B. Thus, A3, indicates that A is to be taken three times as a factor, or raised to the third power. The Radical Sign, √: 3 Thus, A, B, indicate that the square root of A, and the cube root of B, are to be taken. When a compound quantity is to be operated upon as a single quantity, its parts are connected by a vinculum or by a parenthesis: Thus, A+B x C, indicates that the sum of A and B is to be multiplied by C; and (A + B) ÷ C, indicates that the sum of A and B is to be divided by C. A number written before a quantity, shows how many times it is to be taken. Thus, 3(A+B), 3(A + B), indicates that the sum of A and I is to be taken three times. The Sign of Equality, = =: Thus, A B+ C, indicates that A is equal to the suum of B and C. The The expression, A B+ C, is called an equation. part on the left of the sign of equality, is called the first member; that on the right, the second member. The Sign of Inequality, <: Thus, √A</B, indicates that the square root of A is less than the cube root of B. The opening of the sigu is towards the greater quantity. The sign, ... is used as an abbreviation of the word hence, or consequently. The symbols, 1o, 2o, etc., mean, 1st, 2d, etc. 5. The general truths of Geometry are deduced by a course of logical reasoning, the premises being definitions and principles previously established. The course of reasoning employed in establishing any truth or principle, is called a demonstration. 6. A THEOREM is a truth requiring demonstration. 7. An AXIOM is a self-evident truth. 8. A PROBLEM is a question requiring a solution. 9. A POSTULATE is a self-evident Problem. Theorems, Axioms, Problems, and Postulates, are all called Propositions. 10. A LEMMA is an auxiliary proposition. 11, A COROLLARY is an obvious consequence of one or more propositions. 12. A SCHOLIUM is a remark made upon one or more propositions, with reference to their connection, their use, their extent, or their limitation. |