ELEMENTS

OF

GEOMETRY.

INTRODUCTION,

1. QUANTITY is anything that can be increased, diminished, and measured.

The

To measure a thing is to find how many times it contains some other thing of the same kind, taken as a standard. assumed standard is called the unit of measure.

2. Since the unit of measure is of the same kind as the thing measured, there are as many kinds of units as there are species of quantity. In Geometry, there are four species of quantity, called GEOMETRICAL MAGNITUDES, ViZ., LINES, SURFACES, VOLUMES, and ANGLES: hence, 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 and relations of Geometrical Magnitudes.

4. In Geometry, the quantities considered are generally represented by pictorial symbols. The operations to be performed upon them, and the relations 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 B is to be added to A.

The Sign of Subtraction,

called minus :

Thus, AB, indicates that B is to is to be subtracted from A.

The Sign of Multiplication, x: Thus, Ax B, indicates that A by B.

The Sign of Division, ÷:

Thus, AB, or,

divided by B.

A

В

B'

The Exponential Sign:

is to be multiplied

indicates that A is to be

Thus, 43, indicates that A is to be taken three times as a factor, or raised to the third power.

The Radical Sign, √:

Thus, A, B, indicate that the square root of A, and the cube root of B, are to be taken.

or

When a compound quantity is to be operated upon as a single quantity, its parts are connected by a vinculum 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), indicates that the sum of A and B is to be taken three times.

The Sign of Equality, =:

Thus, ABC, indicates that A is equal to the sum of B and C.

The expression, A = B+ C, is called an equation. The 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 sign is towards the greater quantity.

The sign, .. is used as an abbreviation of the word hence, or consequently.

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 problem whose solution is selfevident.

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 more propositions.

or

12. A SCHOLIUM is a remark made upon one or more propositions, with reference to their connection, their use, their extent, or their limitation.

13. An HYPOTHESIS is a supposition made, either in the statement of a proposition, or in the course of a demonstration.

14. Two magnitudes are equal, when they are equal in

measure,

When they may be so placed as to coincide throughout their whole extent, they are equal in all their parts.

ELEMENTS OF GEOMETRY.

BOOK I.

ELEMENTARY

PRINCIPLES.

DEFINITIONS.

1. GEOMETRY is that branch of Mathematics which treats of the properties and relations of Geometrical Magnitudes.

2. A POINT is that which has position, but not magnitude.

3. A LINE is that which has length, but neither breadth nor thickness.

Lines are divided into two classes, straight and curved.

4. A. STRAIGHT LINE is one which does not change its direction at any point.

5. A CURVED LINE is one which changes its direction at every point.

The word line, alone, is used for straight line; and the word curve, alone, for curved line.

6. A line made up of straight lines, not lying in the sam direction, is called a broken line.

7. A SURFACE is that which has length and breadth, without thickness.