If four lines (AB, CD, EF, GH) be proportional, and any pair of similar rectilineal figures (ABK, CDL) be similarly described on the first and second, and also any pair (EI, GJ) on the third and fourth, these figures are proportional. Conversely, if any rectilineal figure described on the first of four right lines: the similar and similarly described figure described on the second :: any rectilineal figure on the third : the similar and similarly described figure on the fourth, the four lines are proportional. Dem. 1.—ABK : CDL :: AB2 : CD2. [xx.]; and EI : GJ :: EF2 : GH2 [xx.]. But since AB : CD :: EF : GH, AB2 : CD2 :: EF2 : GH2 [V. xxii., Cor. 1]; therefore ABK : CDL :: EI : GJ. If ABK : CDL :: EI : GJ, AB : CD :: EF : GH. 3 Dem. 2.—ABK : CDL :: AB2 : CD2 [xx.], and EI : GJ :: EF2 : GH2 [xx.]; therefore AB2 : CD2 :: EF2 : GH2. Hence AB : CD :: EF : GH. The enunciation of this Proposition is wrongly stated in Simson’s Euclid, and in those that copy it. As given in those works, the four figures should be similar. PROP. XXIII.—Theorem. Equiangular parallelograms (AD, CG) are to each other as the rectangles contained by their sides about a pair of equal angles. Dem.—Let the two sides AB, BC about the equal angles ABD, CBG, be placed so as to form one right line; then it is evident, as in Prop. xiv., that GB, BD form one right line. Complete the parallelogram BF. Now, denoting the parallelograms AB, BF, CG by X, Y , Z, respectively, we have— X : Y :: AB : BC [i.], Y : Z :: BD : BG [i.]. Hence XY : Y Z :: AB.BD : BC.BG; or X : Z :: AB.BD : BC.BG. Observation.—Since AB.BD : BC.BG is compounded of the two ratios AB : BC and BD : BG [V. Def. of compound ratio], the enunciation is the same as if we said, “in the ratio compounded of the ratios of the sides,” which is Euclid’s; but it is more easily understood as we have put it. Exercises. 1. Triangles which have one angle of one equal or supplemental to one angle of the other, are to one another in the ratio of the rectangles of the sides about those angles. 2. Two quadrilaterals whose diagonals intersect at equal angles are to one another in the ratio of the rectangles of the diagonals. PROP. XXIV.—Theorem. In any parallelogram (AC), every two parallelograms (AF, FC) which are about a diagonal are similar to the whole and to one another. Dem.—Since the parallelograms AC, AF have a common angle, they are equiangular [I. xxxiv.], and all that is required to be proved is, that the sides about the equal angles are proportional. Now, since the lines EF, BC are parallel, the triangles AEF, ABC are equiangular; therefore [iv.] AE : EF :: AB : BC, and the other sides of the parallelograms are equal to AE, EF; AB, BC: hence the sides about the equal angles are proportional; therefore the parallelograms AF, AC are similar. In the same manner the parallelograms AF, FC are similar.