VI. Def. 4. The fourth definition is strictly only applicable to a triangle, because no other figure has a point which can be exclusively called its vertex. The altitude of a parallelogram is the perpendicular drawn to the base from any point in the op-
VI. 2. The enunciation of this important proposition is open to objection, for the manner in which the sides may be cut is not sufficiently limited. Suppose, for example, that AD is double of DB, and CE double of EA; the sides are then cut proportionally, for each side is divided into two parts, one of which is double of the other; but DE is not parallel to BC. It should therefore be stated in the enunciation that the segments terminated at the vertex of the triangle are to he homologous terms in the ratios, that is, are to he the antecedents or the consequents of the ratios.
It will be observed that there are three figures corresponding
to three cases which may exist; for the straight line drawn pa-
rallel to one side may cut the other sides, or may cut the other sides when they are produced through the extremities of the base, or may cut the other sides when they are produced through the vertex. In all these cases the triangles which are shewn to be equal have their vertices at the extremities of the base of the given triangle, and have for their common base the straight line which is, either by hypothesis or by demonstration, parallel to the base of the triangle. The triangle with which these two triangles are compared has the same base as they have, and has its vertex coinciding with the vertex of the given triangle.
VI. A. This proposition was supplied by Simson.
VI. 4. We have preferred to adopt the term " triangles
which are equiangular to one another," instead of "equiangular
triangles," when the words are used in the sense they bear in
this proposition, Euclid himself does not use the term equian-
gular triangle in the sense in which the modern editors use it in the Corollary to I. 5, so that he is not prevented from using the term in the sense it bears in the enunciation of VI. 4. and else-
where; but modern editors, having already employed the term in one sense ought to keep to that sense. In the demonstrations, where Euclid uses such language as "the triangle ABC is equi-
angular to the triangle DEF," the modern editors sometimes adopt it, and sometimes change it to "the triangles ABC and DEF are equiangular."
In VI. 4 the manner in which the two triangles are to be