FIRST BOOK OF MATHEMATICS.

I. ELEMENTARY DEFINITIONS.

Point-Line-Plane-Circle-Angle-Parallel Lines.

It

1. Each page or side of this leaf is A SURFACE. has a certain shape or form. It has a certain size or magnitude, depending on its length and breadth. It may be quite flat, or more or less bent-that is, curved.

2. The edges or boundaries of the leaf are LINES. Each line has length, but not breadth, or next to none. In geometry, we speak of lines as if they had no breadth, though we cannot draw any line entirely without breadth. But we may reason about them as if they had no breadth. Also, the edges of the leaf may be straight or curved.

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3. But the book has more than length and breadth. It has also thickness or depth. In geometry, a thing that has these three properties is called a SOLID. solid, in a geometrical sense, need not have body or substance all through. An empty room, a vessel from which the air has been taken out, or a portion of space considered by itself, are solids, speaking geometrically.

4. The boundary of a solid is a surface, or surfaces.

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5. These three, length, breadth, and depth, are called dimensions.

6. The forms and dimensions of solids, surfaces, and lines are the objects of geometry.

7. The ends of the edges of the leaf and the corners of the book are POINTS. A point has neither length nor breadth—no magnitude of any kind. But it is in a certain place; that is, it has position; and it is useful in geometry to refer to points, to enable us to indicate exactly particular places or positions.

Hence flow the following definitions :

8. A point is that which has position, but not magnitude.

A point is named by a letter placed close to it.

9. A line is that which has length without breadth. A line is named by letters placed at its ends.

The ends of a line are points, and the intersection (crossing) of one line with another is also a point.

10. A STRAIGHT LINE is a line which lies evenly between its ends; that is, which points all in one direction.

The adjoining is a straight line, called AB, as indicating the direction from A to B, or BA, denoting the opposite direction, from B to A.

A.

FIG. I

B

II. A straight line is the shortest way between two points. The distance between two points means the straight line between them. Only one straight line can be drawn between two points.

12. The word rectilineal, means

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straight-lined."

ELEMENTARY DEFINITIONS.

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13. A CURVED LINE, or CURVE, is a line which is continually changing its

direction.

CD is a curved line.

14. A SURFACE is that

which has only length and breadth.

FIG. 2

The edge or boundary of a surface is a line or lines. 15. A PLANE SURFACE, or PLANE, is a surface in which, any two points being taken, the straight line between them lies wholly in that surface; that is, every point of such straight line touches the surface.

A plane is a perfectly flat surface, as the top of a table, the board of a book, the wall or floor of a room. The surface of any liquid of moderate extent may be considered a plane, but not that of a drop, which is a curved or round surface, nor that of a large body of water, which partakes of the roundness of the earth's surface.

Every point in the straight edge of a well-made flat ruler touches a true plane, at whatever part of the plane it is laid along it.

16. There are some curved surfaces on which straight lines may be drawn, coinciding with them at every point, as a cylinder or a cone. But this cannot be done between two points anywhere, only at particular parts.

17. A SOLID is that which has the three dimensions, length, breadth, and depth.

18. A PLANE FIGURE is a portion of a plane surface enclosed within a line or lines. If contained by several lines, they are called its sides.

19. THE PERIMETER of a plane figure is the whole length of the line or lines which contain it.

20. THE AREA of a plane figure is the quantity or extent of surface which it contains.

The Circle.

21. A CIRCLE is a plane figure, bounded by one

FIG. 3

C

D

curved line, every point of which is at the same distance from a point within it called THE CENTRE.

The bounding line, or perimeter, is called the circumference of the circle.

22. The adjoining figure, B EABD, is a circle; the point C is the centre, which is equidistant from every point in the circumference; that is, the straight lines CE, CA, CB, CD, and all

such lines, are equal in length.

23. A RADIUS (plural, radii) is any straight line from the centre of a circle to the circumference; as, the lines just named.

24. All radii of the same circle are equal in length. 25. AN ARC of a circle is any part of the circumference; as, the curved lines AE, ED, DB, AB.

26. A CHORD of a circle is a straight line joining any two points of the circumference.

The straight lines AB, AD, are chords.

27. A chord is called the chord of either of the arcs whose ends it joins. AB is the chord of the large arc from A by E and D to B. It is also the chord of the small arc forming the remainder of the circumference.

28. A DIAMETER of a circle is a chord passing through its centre. AD is a diameter.

The diameter of a circle is manifestly double of its radius. The radii, CA, CD, are equal, and the diameter AD is double of either of them.