The Elements of Euclid for the Use of Schools and Colleges/Introductory Remarks
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.