Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/308

284 placed is very imperfectly described; their bases are to be in the same straight line and contiguous, their vertices are to be on the same side of the base, and each of the two angles which have a common vertex is to be equal to the remote angle of the other triangle.

By superposition we might deduce VI. 4 immediately from VI. 2.

VI. 5. The hypothesis in VI. 5 involves more than is directly asserted; the enunciation should be, "if the sides of two triangles, taken in order, about each of their angles;" that is, some restriction equivalent to the words taken in order should be introduced. It is quite possible that there should be two triangles ABC, DEF, such that AB is to BC as DE is to EF, and BC to CA as DF is to ED, and therefore, by V. 23, AB to AC as DF is to EF; in this case the sides of the triangles about each of their angles are proportionals, but not in the same order, and the triangles are not necessarily equiangular to one another. For a numerical illustration we may suppose the sides of one triangle to be 3, 4 and 5 feet respectively, and those of another to be 12, 15 and 20 feet respectively. Walker.

Each of the two propositions VI. 4 and VI. 5 is the converse of the other. They shew that if two triangles have either of the two properties involved in the definition of similar figures they will have the other also. This is a special property of triangles. In other figures either of the properties may exist alone. For example, any rectangle and a square have their angles equal, but not their sides proportional; while a square and any rhombus have their sides proportional, but not their angles equal.

VI. 7. In VI. 7 the enunciation is imperfect; it should be, "if two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals, so that the sides subtending the equal angles are homologous; then if each ....." The imperfection is of the same nature as that which is pointed out in the note on VI. 5. Walker.

The proposition might be conveniently broken up and the essential part of it presented thus: if two triangles have two sides of the one proportional to two sides of the other, and the angles opposite to one pair of homologous sides equal, the angles which are opposite to the other pair of homologous sides shall either be equal, or be together equal to two right angles.

For, the angles included by the proportional sides must be