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THE subject of Plane Geometry is here presented to the student arranged in six books, and each book is subdivided into propositions. The propositions are of two kinds, problems and theorems. In a problem something is required to be done; in a theorem some new principle is asserted to be true.
A proposition consists of various parts. We have first the general enunciation of the problem or theorem; as for example, To describe an equilateral triangle on a given finite straight line, or Any two angles of a triangle are together less than two right angles. After the general enunciation follows the discussion of the proposition. First, the enunciation is repeated and applied to the particular figure which is to be considered; as for example, Let AB be the given straight line: it is required to describe an equilateral triangle on AB. The construction then usually follows, which states the necessary straight lines and circles which must be drawn in order to constitute the solution of the problem, or to furnish assistance in the demonstration of the theorem. Lastly, we have the demonstration itself, which shews that the problem has been solved, or that the theorem is true.
Sometimes, however, no construction is required; and sometimes the construction and demonstration are combined,
The demonstration is a process of reasoning in which we draw inferences from results already obtained. These results consist partly of truths established in former propositions, or admitted as obvious in commencing the subject, and partly of truths which follow from the construction that has been made, or which are given in the supposition of the proposition itself. The word hypothesis is used in the same sense as supposition.
To assist the student in following the steps of the reasoning, references are given to the results already obtained which are required in the demonstration. Thus I. 5 indicates that we appeal to the result established in the fifth proposition of the First Book; Constr. is sometimes used as an abbreviation of Construction, and Hyp. as an abbreviation of Hypothesis.
It is usual to place the letters Q.E.F. at the end of the discussion of a problem, and the letters Q.E.D. at the end of the discussion of a theorem. Q.E.F. is an abbreviation for quod erat faciendum, that is, which was to be done; and Q.E.D. is an abbreviation for quod erat demonstrandum, that is, which was to be proved.
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 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.
8. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.
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.
Note. When several angles are at one point B, any one of them is expressed by three letters, of which the letter which is at the vertex of the angle, that is, at the point at which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two letters is somewhere on one of those straight lines, and the other letter on the other straight line. Thus, the angle which is contained by the
straight lines AB, CB is named the angle ABC, or CBA; the angle which is contained by the straight lines AB, DB is named the angle ABD, or DBA; and the angle which is contained by the straight lines DB, CB is named the angle DBC, or CBD; but if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at E.
10. 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 the straight line which stands on the other is called a perpendicular to it.
11. An obtuse angle is that which is greater than a right angle.
An acute angle is that which is less than a right angle.