PAGR. Limiting value of Circular Functions, Relations of Circular Functions, Particular values of Certain Functions, Formulas of Relation between Functions and Arcs,... Formula for the Sine of Half an Angle,. To find the Circumference of a Circle, 118 119 PAGB, Area of the Surface of a Prism, Area of the Surface of a Pyramid, Area of the Frustum of a Cone, Area of the Surface of a Sphere, 182 ELEMENTS OF G E O M E T-RY. 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 relations between them, are indicated by signs, as in Analysis. The following are the principal signs employed : · B, indicates that B is to be subtracted from A. The Sign of Multiplication, X: Thus, A XB, indicates that A is to be multiplied by B. The Sign of Division, :: A indicates that A is to be B' divided by B. · The Exponential Sign: Thus, A3, indicates that A is to be taken three times as a factor, or raised to the third power. The Radical Sign, V: Thus, VA, VB, 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 limes it is to be taken. Thus, 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 sium of B and C. The expression, A = B + C, is called an equation. Tho 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, VA < 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. an abbreviation of the word The sign, ... is used as hence, or consequently. The symbols, 1°, 2°, 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. %. 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. an obvious consequence of one or 11, A COROLLARY is more propositions. one or more 12. A SCHOLIUM is a remark made upon propositions, with reference to their connection, their use, their extent, or their limitation. |