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Explanation of Terms.
The Science of Geometry is developed in a series of Propositions, which are divided into Theorems and Problems.
A Theorem is the statement of a truth to be demon. strated from truths previously admitted or proved.
A Problem proposes a geometrical construction to be effected.
A Corollary is an inference easily deducible from a proposition.
In a Proposition we have (1) the General Enunciation, which is the statement of the Problem to be effected, or of the Theorem to be established; (2) the Particular Enunciation, in which the general enunciation is referred to a particular figure used as an illustration of the general case; (3) there is, when required, the Construction of the lines and circles necessary for the solution of the Problem, or of aid in the demonstration of the Theorem ; (4) the Demonstration or Proof that the Problem has been effected, or that the Theorem is true.
Definitions. 1. A point is that which has position, but not magnitude.
2. A line is length without breadth.
The extremities of a line are points, and the intersection of one line with another is a point.
3. A straight line is such that any part of it, however placed, will coincide with any other part, provided only that two points of the one part are placed on the other part.
Hence two straight lines cannot enclose a space ; neither can two straight lines have a common segment, that is, coincide in part without coinciding altogether.
Note.-Euclid defines a straight line as," that which lies evenly between its two extreme points.” Here he awears to refer to what one expects to find in looking along a straight edge, that “the end points darken the middle points."
4. A superficies, or surface, is that wbich has only length and breadth.
The extremities of a superficies are lines, and the intersection of one superficies with another is a line.
5. A plane superficies, or plane, is that in which any two points being taken, the straight line between them lies wholly in that superficies.
This is the test of planeness which we usually employ to find if there are inequalities in a surface--the top of a table, for instance--such as the eye is not able to detect, for we place a straight edge on it in different positions, and if in much sinaitian +!:
6. An angle is formed by two straight lines that meet together; the lines are said to contain the angle, or to make an angle with each other.
The amount of opening between the two lines, or the amount of turning required to bring either line to coincidence with the other line, is a measure of the magnitude of the angle.
Note.—Euclid defines a plane rectilineal angle as "the inclination of two straight lines to one another, which meet together, but are not in the same straight line." Here the synonym for angle, “the inclination of two straight lines," has this fault, that when the lines are much inclined, i.e., when the inclination is great, the angle is small, and vice versa. Also the phrase "are not in the same straight line" would not merely unnecessarily, but mischievously, restrict us to angles less than two right angles.
The point where the two lines meet is called the vertex of the angle; and the lines, which may be of any length, are its arms. When there is only one angle at a point it is named by a single letter placed at the
A vertex, as - N.
In other cases three letters name an angle, the middle
B one being the one at the vertex, and the others may be anywhere on the arms, as – ABC, Z CBD, _ ABD. Sometimes a number or a small letter is used : it is placed between the arms of the angle, as - 3, L a.
When two angles are formed at a point, having one arm common, they are called adjacent angles. The name is most commonly used C
B for those angles which have the other arms in the same straight line.
- ABC and - ABD are adjacent angles.
Vertically opposite angles are formed by two lines intersecting; L1 and _ 3, 2 and 44 are pairs of vertically opposite angles.
EXERCISE.-In this figure name
D - ABC, - A, etc., by the numbers; name 25, 46, etc., by the letters; name two adjacent angles ; name also two vertically opposite angles.
H 7. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and each of the lines is perpendicular to the other, or at right angles to the other.
8. An obtuse angle (A) is that which is greater than a right angle.
9. An acute angle (B) is that which is less than a right angle.
10. A figure is that which is enclosed by one or more boundaries.
Note.-The word figure is very commonly used for any diagram, as that of an angle, which is one not "enclosed."
11. A circle is a plane figure bounded by one line called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.
12. This point is called the centre of the circle.
A radius is a straight line from the centre to the circumference; and, from the definition of a circle, radii of the same circle are all equal to one another.
13. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
14. A semicircle is the figure enclosed by a diameter and the part of the circumference cut off by a diameter.
15. Rectilineal figures are bounded by straight lines.
16. A triangle, or trilateral figure, is a plane figure enclosed by three straight lines.
17. An equilateral triangle is one which has three equal sides.
18. An isosceles triangle is one which has two equal sides.
19. A scalene triangle is one which has its sides unequal.
20. A right-angled triangle is one which has a right angle.
21. An obtuse-angled triangle is one which has an obtuse angle.
22. An acute-angled triangle is one which has three acute angles.
Note.-There are three kinds of triangles named from considering their sides, and three kinds named from their angles, but not six different kinds altogether.