The following are the principal signs employed:

The Sign of Addition, +,
+, called plus:

Thus, AB, indicates that B is to be added to A.

Thus, A3, indicates that A is to be taken three times as a factor, or raised to the third power.

The Radical Sign, √:

Thus, √Ā, 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:

indi

Thus, A+B x C, indicates that the sum of A and B is to be multiplied by C; and (A + B) ÷ C, cates 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, A

of B and C.

B+ C, indicates that A is equal to the sum

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, √Ā< VB, 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.

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 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.

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

14. Magnitudes are equal to each other, when each contains the same unit an equal number of times.

15. Magnitudes are equal in all respects, when they may be so placed as to coincide throughout their whole extent; they are equal in all their parts when each part of one is equal to the corresponding part of the other, when taken either in the same or in the reverse order.

ELEMENTS OF GEOMETRY.

BOOK I.

ELEMENTARY PRINCIPLES.

DEFINITIONS.

1. GEOMETRY is that branch of Mathematics which treats of the properties, relations, and measurements of the 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.

When the sense is obvious, to avoid repetition, the word line, alone, is commonly used for straight line; and the word curve, alone, for curved line.

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

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

Surfaces are divided into two classes, plane and curved surfaces.

8. A PLANE is a surface, such, that if any two of its points be joined by a straight line, that line will lie wholly in the surface.

9. A CURVED SURFACE is a surface which is neither a plane nor composed of planes.

10. A PLANE ANGLE is the amount of divergence of two straight lines lying in the same plane.

-B

Thus, the amount of divergence of the lines AB and AC, is an angle. The lines AB and AC are called sides, and their common point A, is called the vertex. An angle is designated by naming its sides, or sometimes by simply naming its vertex; thus, the above is called the angle BAC, or simply, the angle A.

11. When one straight line meets another, the two angles which they form are called adjacent angles. Thus, the angles ABD and DBC are adjacent.

A

12. A RIGHT ANGLE is formed by one straight line meeting another so as to make the adjacent angles equal. The first line is then said to be perpendicular to the second.

13. An OBLIQUE ANGLE is formed by

one straight line meeting another so as

to make the adjacent angles unequal.

Oblique angles are subdivided into two classes, acute

angles, and obtuse angles.

14. An ACUTE ANGLE is less than a

right angle.