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INSTRUCTIONAL PLANE GEOMETRY.
DEFINITIONS, EXPLANATIONS, AND AXIOMS.
1. A Point, has position, but theoretically no length, breadth, or thickness, and probably rays of light, or the line formed between the eye and an object on which it looks, would best elucidate the nature of points.
In practice a point is marked by a fine dot.
2. A Line, has length without breadth or thickness theoretically, but is practically drawn by a pen or pencil, with as little breadth as will make the line clearly perceptible.
3. A Surface or Superficies, has two dimensions, length and breadth, but without thickness, or in practice noticeable thickness ; for instance a sheet of paper.
4. A Body or Solid, has three dimensions, length, breadth, and thickness ; for example a block of wood.
Thus surfaces are the extremities of solids, lines of surfaces, and points of lines.
5. Lines, are right (or straight), curved (or bent), and mixed, or broken.
A right line is the direct or shortest line between any
two points; a curved line continually changes its direction, in no part being straight, and such lines are indefinite in their variety
In Fig. C 1, the lines ac, gh, and bd, are right lines, the lines bc, and ik, are curved lines, as would be any other bending lines although not parts of a circle ; a mixed line would be one in part right or straight and in part curved or bent, ib, bc; and a broken line would be either a mixed line, or two right lines with different directions, kf, fa.
6. Lines are in their direction and relation to other lines, parallel, oblique, perpendicular, or tangental.
Parallel, when in the same plane and continued for any length in either direction, they would at their nearest points have the same perpendicular distance between them.
Thus in Fig. C 1, the lines ac and gh are parallel. The curved lines bc and il in that figure are also parallel, as explained Div. C Sect. 2 : but in geometrical operations, unless otherwise stated, the term “ Parallels " is understood to mean right-lined parallels.
Oblique, when changing their distance, and which would meet on the side of the least distance if continued ; thus in Fig. A 3, the lines CD and EF are oblique.
Perpendicular, when with one end joining another line, otherwise than as its continuation it does not incline more on one side than on the other :- Thus in Fig. A 3, the line a a' is perpendicular to the line EF.
If joining at the end and not inclining, it would form a continuation of the right line, as with line aE joining aF.
Tangental, touching another line without cutting it, when
both lines are continued: Thus in Fig. C 1, the line min, is tangental to the circle.
7. An Angle is the inclination or opening of two lines (unless otherwise stated, two right lines) having different directions and meeting in a point :—Thus in Fig. C 1, the lines kf and If, as also if and lf, and other linės meeting at f severally form an angle at point f.
Angles are according to the extent of the opening acute, right, obtuse, flat (or double right), and reflex or more than two and less than four right angles.
Before generally explaining the various angles it may be remarked that in Fig. C 1, the angles bfa, bfc, dfa, and dfc are respectively right angles, and that together they include all the angles in the circle, being four right angles, and that the entire angles on one side of the diameter ac equal two right angles.
The different kinds of Angles are
Acute, or less than a right angle, as kfl or any angle less than ifl (dotted arc 1 to 1).
Right, when the one side is perpendicular to the other, ifl (2 to 2).
Obtuse, when greater than a right angle and less than two right angles, sf, fl, or fc, the length of sides being immaterial, the opening giving the angle (3 to 3).
Flat, or double right, when including two right angles (4 to 4).
Reflex, when more than two right angles, and less than four, as marked by cft or the arc 5 to 5.
Oblique, when otherwise than a right angle, whether more or less.
Angles are also salient or re-entering in different forms of surfaces.
The salient angle projects, the re-entering angle recedes.
Thus in Fig. C 19, the angle cdf is a salient angle, and the angle cba a re-entering angle.
8. SURFACES are either plane or curved, plane where a right line may in every point coincide with the figure, as for instance a flat table with which a right or straight rule will in all positions when placed on it coincide; and curved where such right line will only in part coincide, as in the case of a globe or sphere.
Plane surfaces may be bounded either by right lines or curved lines ; by the former as to triangles, squares, &c., and by the latter as to circles, ellipses, &c., or the figure may be in part bounded by right lines and in part by curved lines.
9. Plane Figures bounded by right lines have names according to the sides or their angles, as they have as many sides as angles, the least number being 3.
10. Figures having 3 sides only and 3 angles are called Triangles, and there are 6 varieties, 3 in relation to the sides and 3 in relation to the angles, as shown in Fig. A l.
As regards the varieties in the sides.
Equilateral Triangles have the 3 sides and the 3 angles equal (abc).
Isosceles Triangles have 2 equal sides and the angles at base of those sides equal (adc), or the summit might be any point (say f) in the perpendicular (do), from the summit including the equal sides.