triangles which have one angle of the one equal to one angle of the other, have to one another the ratio which is compounded of the ratios of their sides.
Then VI. 19 is an immediate consequence of this theorem. For lett ABC and DEF be simmilar triangles, so that AB is to DE as BC is to EF; and therefore, alternately, AB is to DE as BC is to EF. Then, by the theorem, the triangle ABC has to the triangle DEF the ratio which is compounded of the ratios of AB to BE and of BC to EF, that is, the ratio which is compounded of the ratios of BC to EF and of BC to EF. And, from the definitions of duplicate ratio and of compound ratio, it follows that the ratio compounded of the ratios of BC to EF and of BC to EF is the duplicate ratio of BC to EF.
VI. 25. It will be easy for the student to exhibit in detail the process of shewing that BC and CF are in one straight line, and also LE and EM; the process is exactly the same as that in I. 45, by which it is shewn that KH and HM are in one straight line, and also FG and GL.
It seems that VI. 25 is out of place, since it separates propositions so closely connected as VI. 24 and VI. 26. We may enunciate VI. 25 in familiar language thus:
to make a figure which shall have the form of one figure and the size of another.
VI. 26. This proposition is the converse of VI. 24; it might be extended to the case of two similar and similarly situated parallelograms which have a pair of angles vertically opposite.
We have omitted in the sixth Book Propositions 27, 28, 29, and the first solution which Euclid gives of Proposition 30, as they appear now to be never required, and have been condemned as useless by various modern commentators; see Austin, Walker, and Lardner, Some idea of the nature of these propositions may be obtained from the following statement of the problem proposed by Euclid in VI, 29. AB is a given straight line; it has to be produced through B to a point O, and a parallelogram described on AO subject to the following conditions; the parallelogram is to be equal to a given rectilineal figure, and the parallelogram on the base BO which can be cut of by a straight line through B is to be similar to a given parallelogram.
VI. 32. This proposition seems of no use. Moreover the enunciation is imperfect. For suppose ED to be produced
