Euclid's Elements of Geometry: Books I-IV, Vi and XI (Classic Reprint) - Tapa blanda

Excerpt from Euclid's Elements of Geometry: Books I-IV, Vi and XI

In the following School Edition of Euclid's Elements of Geometry the subject is treated in Euclid's order and manner, but with no special regard for the exact words of Simson's translation, which appears to have been scrupulously followed by many English editors.

Further explanation has been given whenever this appeared to be necessary or desirable, and we have not hesitated occasionally to give proofs different from those of Euclid. In Book I. we have, for example, discarded altogether Euclid's incomplete proof of Prop. 24, which would, we imagine, be now accepted as satisfactory by few examiners; and we have made Prop. 22 logically complete by shewing that, under the given conditions, the two circles will necessarily intersect. It may be interesting to remark that, with the additional axiom which must be explicitly or implicitly assumed (and which is indeed implied though not expressed in I. 1), I. 22 may be taken immediately after I. 3. No alternative proofs of I. 5 and I. 6 have been given, because the experience both of teachers and examiners appears to shew that the average beginner finds Euclid's proofs easier to understand - at any rate easier to reproduce - than the alternatives which have been suggested.

The changes we have made in Book II. are more considerable than in Book I. We have substituted for Euclid's proofs of Props. 9 and 10 the proof in which the equality to be established is shewn directly from the diagram. The advantage of this for educational purposes need not be dwelt upon; but as this proof is somewhat long, alternative proofs of these propositions are given, as also of II. 8, which do not require the construction of the different squares and rectangles. These alternative proofs being logically sound and strictly geometrical, may be given in examinations, except when a proof by means of a diagram is definitely asked for. Proofs of II. 12 and II. 13 are also given in which the equality that has to be established is proved at once from the diagram.

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