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ELEMENTS OF GEOMETRY.
WHEN a block of stone is hewn from the rock, we call it a Solid Body. The stone-cutter shapes it, and brings it into that which we call regularity of form; and then it becomes a Solid Figure.
Now suppose the figure to be such that the block has six flat sides, each the exact counterpart of the others; so that, to one who stands facing a corner of the block, the three sides which are visible present the appearance represented in this diagram.
Each side of the figure is called a Surface; and when smoothed and polished, it is called a Plane Surface.
The sharp and well-defined edges, in which each pair of sides meets, are called Lines.
The place, at which any three of the edges meet, is called a Point.
A Magnitude is anything which is made up of parts in any way like itself.
Thus, a line is a magnitude; because we may regard it as made up of parts which are themselves lines.
The properties Length, Breadth (or Width), and Thickness (or Depth or Height) of a body are called its Dimensions.
We make the following distinction between Solids, Surfaces, Lines, and Points :
A Solid has three dimensions, Length, Breadth, Thickness.
I. A Point is that which has no parts.
This is equivalent to saying that a Point has no magnitude, since we define it as that which cannot be divided into smaller parts.
II. A LINE is length without breadth.
We cannot conceive a visible line without breadth ; but we can reason about lines as if they had no breadth, and this is what Euclid requires us to do.
III. The EXTREMITIES of finite Lines are points.
A point marks position, as for instance, the place where a line begins or ends, or meets or crosses another line.
IV. A STRAIGHT LINE is one which lies in the same direction from point to point throughout its length.
V. A SURFACE is that which has length and breadth only. VI. The EXTREMITIES of a SURFACE are lines.
VII. A PLANE SURFACE is one in which, if any two points be taken, the straight line between them lies wholly in that surface.
Thus the ends of an uncut cedar-pencil are plane surfaces ; but the rest of the surface of the pencil is not a plane surface, since two points may be taken in it such that the straight line joining them will not lie on the surface of the pencil.
In our introductory remarks we gave examples of a Surface, a Line, and a Point, as we know them through the evidence of the senses.
The Surfaces, Lines, and Points of Geometry may be regarded as mental pictures of the surfaces, lines, and points which we know from experience.
It is, however, to be observed that Geometry requires us to conceive the possibility of the existence
of a Surface apart from a Solid body,
of a Point apart from a Line. VIII. When two straight lines meet one another, the inclination of the lines to one another is called an ANGLE.
When two straight lines have one point common to both, they are said to form an angle (or angles) at that point. The point is called the vertex of the angle (or angles), and the lines are called the arms of the angle (or angles).
Thus, if the lines OA, OB are terminated at the same point 0, they form an angle, which is called the angle at 0, or the angle AOB, or the angle BOA,—the letter which marks the vertex being put between those that mark the arms.
Again, if the line CO meets the line DE at a point in the line DE, so that 0 is a point common to both lines, CO is said to make with DE the angles COD, COE; and these (as having one arm, CO, common to both) are called adjacent angles.
Lastly, if the lines FG, HK cut each other in the point 0, the lines make with each other four angles FOH, HOG, GOK, KOF; and of these GOH, FOK are called vertically opposite angles, as also are FOH and GOK.
When three or more straight lines as 0A, OB, OC, OD have a point 0 common to all, the angle formed by one of them, OD,
with OA may be regarded as being made up of the angles AOB, BOC, COD; that is, we may speak of the angle AOD as a whole, of which the parts are the angles AOB, BOC, and COD.
Hence we may regard an angle as a Magnitude, inasmuch as any angle may be regarded as being made up of parts which are themselves angles.
The size of an angle depends in no way on the length of the arms by which it is bounded.
We shall explain hereafter the restriction on the magnitude of angles enforced by Euclid's definition, and the important results that follow an extension of the definition.
IX. When a straight line (as AB) meeting another straight line (as CD) makes the adjacent angles (ABC and ABD) equal to one another, each of the angles is called a RIGHT ANGLE; and each line is said to be a PERPENDICULAR to the other.
X. An OBTUSE ANGLE is one which is greater than a right angle.
XI. An ACUTE ANGLE is one which is less than a right angle.
XII. A FIGURE is that which is enclosed by one or more bound