MISCELLANEOUS PAPERS. 225 



These are precisely the laws dominating the affinity of figures in a 

 plane, which itself is a special case of homology (when the center is 

 infinitely distant ), The propositions which we have established in 

 connection with homologous triangles may be specialized for triangles 

 related by affinity and their proof does not present the slightest diffi- 

 culty. In fact, affinity results from homology by considering a 

 triangular prism instead of a triangular pyramid, and it is clear that 

 our previous reasoning applies also to this case. 



I shall now mention two metrical specializationsof affinity. Assume 

 two triangles ABC and A'B'C related by affinity and assume that the 

 parallel lines AA', BB', CO' be perpendicular to the axis of affinity, 

 then the ratios between the distances of corresponding points from 

 the axis of affinity are equal and the proposition holds : 



The areas of the two triangles are to each other as the distances of 

 corresponding points from the axis of affinity. 



If this ratio is unity then their areas are equal and the triangles 

 are in axial symmetry. 



If AA', BB'. CC are parallel to the axis of affinity then the areas 

 of ABC and A'B'C are equal. 



This is a case which is never considered in plane geometry. 



I might extend this subject still further, but I hope the pre- 

 vious treatment will be sufficient to show in what a simjjle and effect- 

 ive manner, and without losing much time, important geometrical 

 IDropositions may be obtained from the study of elementary de- 

 script ve geometry. It is hardly necessary to point out that these 

 propositions are conversely the most efficient and rapid means to make 

 projections of plane figures. 



Suppose, for instance, that the horizontal projection of a plane fig- 

 ure A, B, C, ..., the vertical projection C" of C and the line of inter- 

 section u of the plane of ABC... wuth U be given. The vertical 

 projection A"B"C"... may be constructed by the previous principles 

 alone. Thus, to find B", connect B'C and prolong to the intersec- 

 tion with ui; join the latter point to C", and from B' draw a perpen- 

 dicular to the ground line. Where this perpendicular intersects the 

 last line, is the required vertical projection B" of B. By this method 

 three lines are sufficient to find a required point, while the ordinary 

 method by means of the traces of the plane requires four or five 

 ( two parallels and two perpendiculars in one case, and two connecting 

 lines and three perpendiculars in the other). The same principle 

 may be applied to find the true shape of a plane figure from its projec- 

 tions ; three lines give a point of the rabatted figure. This method is, 

 therefore, also in the line of Lemoin's Geom^trographie,* where fig- 



*M. E. Lamoia : Priacip3S de la Geometrographie, Archiv der Mathematik und Physik, 

 Yol. 1, 1901, p. 99. 



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