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8. Magnitudes which coincide with one another-that is, which exactly fill the same space—are equal to one another.
9. The whole is greater than its part.
10. Two straight lines cannot inclose a space.
11. All right angles are equal to one another.
12. If a straight line meets two straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles.
EXPLANATION OF TERMS. 1, A Proposition in geometry, as its name implies, is something proposed either to be done or demonstrated.
2. Propositions fall into two classes, problems and theorems..
3. A Problem proposes some geometrical construction to be done -2.g., the construction of a figure.
4. A Theorem proposes some geometrical property to be demonstrated.
5. A Postulate is a problem so simple that it is unnecessary to point out the method of doing it.
6. An Axiom is a theorem, the truth of which is self-evident.
7. A Corollary is an inference made immediately from the discussion of the proposition to which it is subjoined.
PROPOSITION 1.- Problem.
To describe an equilateral triangle upon a given finite straight line.
Let AB be the given straight line.
Construction. From the centre A, at the distance AB, describe the circle BCD (Postulate 3); from the centre B, at the distance BA, describe the circle ACE; and from C, one of the points in which the circles cut one another, draw the straight lines CA, CB to the points A, B (Post. 1).
Then ABC shall be an equilateral triangle.
1. AC is equal to AB (Definition 15);
2. BC is equal to BA. But it has been proved that AC is equal to AB; therefore AC, BC are each of them equal to AB; but things which are equal to the same thing are equal to one another (Axiom 1); therefore
3. AC is equal to BC. Wherefore AB, BC, CA are equal to one another; and therefore
4. The triangle ABC is equilateral ; and it is described upon the given straight line AB.
Which was required to be done.
Problem. From a given point, to draw a straight line equal to a given straight
EUCLID'S ELEMENTS OF GEOMETRY.
1. A POINT is that which has no parts, or which has no magnitude.
2. A line is length without breadth.
3. The extremities of a line are points.
4. A straight line is that which lies evenly between its extreme points.
5. A superficies is that which has only length and breadth.
6. The extremities of superficies are lines.
7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
8. A plane angle is the inclination of two lines to each other in a plane, which meet together, but are not in the same straight line.
9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
N.B. When several angles are at one point B, either of them is expressed by three letters, of which the letter that is at the vertex of the angle—that is, at the point in which the straight lines that contain the angle meet one another-is put between the other two letters, and one of these two is somewhere upon one of these straight lines, and the other upon the other line. Thus the angle which is contained by the straight lines AB, CB, is named the angle ABC, or CBA ; that which is contained by AB, DB, is named the angle ABD, or DBA; and that which is contained by DB, CB, is called the angle DBC, or CBD. But, if there be only one angle at a point, it may be expressed by the letter at that point; as the angle at E.
10. When a straight line standing on another straight line, makes the adjacent angles equal to each other, each of these angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.
An obtuse angle is that which is greater than a right angle.