return to Vignettes of Ancient Mathematics
|1||Similar rectilinear figures are those that have the angles equal, one by one, and the sides about the equal equals proportional.|
|2.||and reciprocal figures are when in each of the figures leaders and followers are ratios.||This definition is somewhat opaque. It is dubiously a part of the text. Nonetheless, it had meaning to whoever wrote it.|
|3.||A straight line is said to be cut in extreme and mean ratio when it is as the whole to the larger section, so the larger to the smaller.||The construction has already taken place at II 11.|
|4.||An altitude of any figure is the perpendicular drawn from the vertex to the base.|
|5.||A ratio is said to be composed from ratios when the sizes of the ratios being multiplied times themselves make some ratio/size(?).||This definition is unlikely to a part of the Elements. Additionally, it suggests a tradition that treats ratios as numbers, Heron, Diophantus, etc. What is significant is that it is a Greek definition that involves the notion of the size of the ratio, an important concept in Medieval and Renaissance reductions of ratio to number.|
Prop. 1: Triangles and parallelograms which are under the same height are to one another as the bases.
Prop. 2: If some straight-line is drawn parallel to one of the sides of a triangle it will cut the sides of the triangle proportionally; and if the sides of a triangle are cut proportionally, the straight-line joining at the sections will be parallel to the remaining side of the triangle.
Prop. 19: Similar triangles are to one another in duplicate ratio of the corresponding sides.
Corollary: It is, in fact, obvious from this, that if three straight-lines are proportional, it is as the first to the third, so the form from the first to the second that's similar and similarly described up,
Prop. 20: Similar polygons are divided into similar triangles and into triangles equal in plêthos and icorresponding with the whole, and the polygon has a duplicate ratio to the polygon that the corresponding side has to the corresponding side.
Prop. 23: Equal-angled parallelograms have a ratio to one another that's composed from the sides.