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.
