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 their parts, when they may be so placed as to coincide throughout their whole

extent.

ELEMENTS OF
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 sometimes 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.

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

-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 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.

B

D

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

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

The angles formed by the

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 mutually equilateral, when their sides, taken in the same order, are equal, each to each: that is, following their perimeters in the same direction, the first

side of the one is equal to the first side of the other, the second side of the one, to the second side of the other, and so on.

22. Two polygons are mutually equiangular, when their angles, taken in the same order, are equal, each to each.

23. A DIAGONAL of a polygon is a straight line joining the vertices of two angles, not consecutive.

24. A BASE of a polygon is any one of its sides on which the polygon is supposed to stand.

25. Triangles may be classified with reference either to their sides, or their angles.

When classified with reference to their sides, there are two classes: scalene and isosceles.

1st. A SCALENE TRIANGLE is one which

has no two of its sides equal.

2d. An IsosCELES TRIANGLE is one which

has two of its sides equal.

When all of the sides are equal, the

triangle is EQUILATERAL.

When classified with reference to their angles, there are are two classes: right-angled and oblique-angled.

1st. A RIGHT-ANGLED TRIANGLE is one

that has one right angle.

The side opposite the right angle, is called the hypothe

nuse.

2d. An OBLIQUE-ANGLED TRIANGLE is

one whose angles are all oblique.