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THE PROPOSITIONS of Euclid are divided into two kinds, Problems and Theorems. Problems give something to be done, as the making of a Triangle. Theorems state something to be proved, as that the angles at the base of an Isosceles Triangle are equal to each other. But in Problems as well as in Theorems, argument is employed. In a Theorem the necessity of argument is apparent. In a Problem, after we have done what is required, we have to prove the accuracy of our work. Every Proposition, therefore, is to be considered as a process of reasoning. This process is carried on step by step. We start from known truths or suppositions admitted, through others, plainly flowing from or connected with them, till we arrive at the conclusion to which they unavoidably lead us. Mere assertion is allowed no place in this process. The truth of everything stated must be capable of, and have, its necessary proof. This fact the scholar must bear in mind. He must remember that to learn a Proposition is to get up an argument by which the statement contained in that Proposition is established. To say, or to write out, a Proposition is to reproduce that argument complete. When the scholar has grasped this idea of a Proposition, he will have made a great step, not only towards the study of Euclid successfully, but also attractively : 'a consummation devoutly to be wished.'
From this general view of a Proposition, in Euclid, we can now proceed to a closer one.
Every Proposition contains what is called, 1. The Enunciation. 2. The Construction. 3. The Demonstration, or Proof.
1. The Enunciation is the original statement of the Problem, or Theorem, and it is of two kinds, general and particular.
The General Enunciation is the statement asserted generally, and printed here, and usually, in Italics.
The Particular Enunciation is the repetition of the General, asserted with reference to the particular case in which we are about to consider it. This gives us also, in distinct terms, the statement to be proved, which is printed here in red type.
2. The Construction is the addition to the lines or figures, originally given, of such other lines, or figures, as are necessary to the argument required. These Construction lines and figures are also printed here, as far as Prop. XXVI. inclusive, in red.
3. The Demonstration, or Proof, follows on the principles, above explained, of starting from truths known, or suppositions admitted, through others flowing from, or connected with, them-such as Definitions, Axioms, Postulates, Hypotheses, and other and previous Propositions—till we reach the evident conclusion, and this is the statement required in the Particular Enunciation to be proved. This conclusion is printed here in red type, which is thus used to show that the argument, or process of reasoning, employed is only an intervening part of the entire Proposition, and that the statement, originally made, has been demonstrated, as required.
1. A Point is that which has position, but not magnitude.
A geometrical point cannot be represented without magnitude. Its position is denoted by a letter, as the point A.
A geometrical line cannot be represented without breadth. It is denoted by a letter, or letters, placed on the line, or at its extremities.
4. A STRAIGHT LINE is that which lies evenly between its extreme points.
This is sometimes called a right line.
5. A SUPERFICIES is that which has only length and breadth.
A geometrical superficies, or, as it is sometimes called, surface, cannot be represented without depth, or thickness. The shadow of
any object gives us the best idea of a superficies, or surface. A superficies is denoted by letters placed at its sides, or extremities.
THE EXTREMITIES OF A SUPERFICIES are lines.
A PLANE SUPERFICIES is that in which any two points being taken, the straight line between them lies wholly in that superficies.
A Plane Superficies is sometimes called 'A Plane.'
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 direction.
The term 'angle ' in this definition denotes the opening which exists between two lines meeting in a point, in a plane. These
two lines’ may be straight, or curved, either or both. The only restriction is that they must meet each other, not in the same direction, in a Plane. These Plane Angles are not introduced in Elementary Geometry, which refers entirely to the Plane Rectilineal Angle spoken of in Definition 9.
9. A PLANE RECTILINEAL ANGLE is the inclination of two straight lines to each other, which meet together, but are not in the same straight line.
This definition is a very important one, and it must be distinctly understood.
a. The plane rectilineal angle—angulus, a corner—is simply the opening between two straight lines meeting at, or starting from, the same point, as A.
b. The point where these lines meet is called the vertex, and the lines themselves the arms of the angle.
C. An angle is denoted by a letter placed at its vertex, or by this letter with two others placed on, or at the extremities of, the lines or arms of the angle.
Thus we have the angle A, or the angle BAC. The vertex letter is always the middle one of the three. Hence the angle BAC is the same as the angle CAB.
d. When several lines meet and form several angles, at the same vertex, we may consider one of such angles as a part of two or more ; and we may consider two or more of such angles combined as one angle.
Thus we may consider the angle BAC as a part of the angle BAD, or BAE; and we may consider the angles BAC and CĂT) and DAE combined as one angle BAE, &c.
e. The magnitude of an angle is explained under def. 15. f. When a straight line meets another straight line at a point