Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/344

320 The figure will vary slightly according to the situation of the four given points, but corresponding to an assigned situation there will be only one point such as is required. For suppose there could be such a point P, besides the point O which is determined by the construction given above; and that the points are in the order A,C,D, B, O, P. Join PE, and let it meet CF, produced at G; join BG. Then the rectangle PA, PB is, by hypothesis, equal to the rectangle PC, PD; and therefore PA is to PC as PD is to PB. But PA is to PC as PE is to PG ( VI. 2); therefore PD is to PB as PE is to PG (V. 11); therefore BG is parallel to DE.

But, by the construction, BF is parallel to ED; therefore BG and BF are themselves parallel (I, 30); which is absurd. Therefore P is not such a point as is required.

ON GEOMETRICAL ANALYSIS.

35. The substantives analysis and synthesis, and the corresponding adjectives analytical and synthetical, are of frequent occurrence in mathematics. In general analysis means decomposition, or the separating a whole into its parts, and synthesis means composition, or making a whole out of its parts. In Geometry however these words are used in a more special sense. In synthesis we begin with results already established, and end with some new result; thus, by the aid of theorems already demonstrated, and problems already solved, we demonstrate some new theorem, or solve some new problem. In analysis we begin with assuming the truth of some theorem or the solution of some problem, and we deduce from the assumption consequences which we can compare with results already established, and thus test the validity of our assumption.

36. The propositions in Euclid's Elements are all exhibited synthetically; the student is only employed in examining the soundness of the reasoning by which each successive addition is made to the collection of geometrical truths already obtained; and there is no hint given as to the manner in which the propositions were originally discovered. Some of the constructions and demonstrations appear rather artificial, and we are thus naturally induced to enquire whether any 'rules can be discovered by which we may be guided easily and naturally to the investigation of new propositions.