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Definitions Brief comments
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 plĂȘthos 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.