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

measure.

When they may be so placed as to coincide through. out 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 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 lines lying in the same plane.

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 ver

A

-B

tex. 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 the Aangles ABD and DBC are adjacent.

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.

D

B

Oblique angles are subdivided into two classes, acute angles, and obtuse angles.

14. An ACUTE ANGLE is less than a

right angle

15. An OBTUSE ANGLE is greater than a right angle.

16. Two straight lines are parallel, when they lie in the same plane and cannot meet, how far soever, either way, both

may be produced. They then have the same direction. 17. A PLANE FIGURE is a portion of a plane bounded by lines, either straight or curved.

18. A POLYGON is a plane figure bounded by straight lines.

The bounding lines are called sides of the polygon. The broken line, made up of all the sides of the polygon, is called the perimeter of the polygon. The angles formed by the sides, are called angles of the polygon.

19. Polygons are classified according to the number of their sides or angles.

A Polygon of three sides is called a triangle; one of four sides, a quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of seven sides, a heptagon; one of eight sides, an octagon; one of ten sides, a decagon; one of twelve sides, a dodecagon, &c.

20. An EQUILATERAL POLYGON, is one whose sides are all equal.

An EQUIANGULAR POLYGON, is one whose angles are all equal.

A REGULAR POLYGON, is one which is both equilateral and equiangular.

21. Two polygons are equilateral, or mutually equilateral, when their sides, taken in the same order, are equal, each to each that is, following their perimeters in the :

same