Educationally, however, we are forced to proceed as we do. Just as Dedekind's theory of numbers is a simple one for college students, so is the ancient theory of proportion; but as the former is not suited to pupils in the high school, so the latter must be relegated to the college classes. And in this we merely harmonize educational progress with world progress, for the numerical theory of proportion long preceded the theory of Eudoxus. The ancients made much of such terms as duplicate, triplicate, alternate, and inverse ratio, and also such as composition, separation, and conversion of ratio. These entered into such propositions as, "If four magnitudes are proportional, they will also be proportional alternately." In later works they appear in the form of "proportion by composition," "by division," and "by composition and division." None of these is to-day of much importance, since modern symbolism has greatly simplified the ancient expressions, and in particular the proposition concerning "composition and division" is no longer a basal theorem in geometry. Indeed, if our course of study were properly arranged, we might well relegate the whole theory of proportion to algebra, allowing this to precede the work in geometry. We shall now consider a few of the principal propositions of Book III. THEOREM. _If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally._ In addition to the usual proof it is instructive to consider in class the cases in which the parallel is drawn through the two sides produced, either below the base or above the vertex, and also in which the parallel is drawn through the vertex. THEOREM. _The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides._ The proposition relating to the bisector of an exterior angle may be considered as a part of this one, but it is usually treated separately in order that the proof shall appear less involved, although the two are discussed together at this time. The proposition relating to the exterior angle was recognized by Pappus of Alexandria. If _ABC_ is the given triangle, and _CP__{1}, _CP__{2} are respectively the internal and external bisectors, then _AB_ is divided harmonically by _P__{1} and _P__{2}. [therefore]_AP__{1} : _P__{1}_B_ = _AP__{2} : _P__{2}_B_. [therefore]_AP__{2} : _P__{2}_B_ = _AP__{2} - _P__{1}_P__{2} : _P__{1}_P__{2} - _P__{2}_B_, and this is the criterion for the harmonic progression still seen in many algebras. For, letting _AP__{2} = _a_, _P__{1}_P__{2} = _b_, _P__{2}_B_ = _c_, we have _a_/_c_ = (_a_ - _b_)/(_b_ - _c_), which is also derived from taking the reciprocals of _a_, _b_, _c_, and placing them in an arithmetical progression, thus: 1/_b_ - 1/_a_ = 1/_c_ - 1/_b_, whence (_a_ - _b_)/_ab_ = (_b_ - _c_)/_bc_,