There is not any, either rate or proportion of the diagonall propounded, onely similitude is attributed to it, as in the same figure, the Diagonall auys, is like unto the whole parallelogramme aeio. For first it is equianglar to it. For the angle at a, is common to them both: And that is equall to that which is at y, (by the [10. e x]:) And therefore also it is equall to that at i by the [10. e x]. Then the angles auy, and asy, are equall, by the [21. e v]. to the opposite inner angles at e, and o. Therefore it is equiangular unto it.
Againe, it is proportionall to it in the shankes of the
equall angles. For the triangles auy, and aei, are alike, by the [12 e vij], because uy is parallell to the base. Therefore as au is uy; so is ai to ei: Then as uy is to ya; so is ei to ia. Againe by the [21 e v], because sy is parallell to the base io, as ay is to ys: so is ai, to io: Therefore equiordinately, as uy is to ys: so is ei to io: Item as sy is to ya, so is io to ia: And as ya is to as: so is ia to ao. Therefore equiordinately, as ys is to sa: so is io to oa. Lastly as sa is unto ay; so is oa unto ai: And as ay is to au; so is ai unto ae. Therefore equiordinately, as sa is to au: so is ao, to ae. Wherefore the Diagonall su is proportionall in the shankes of equall angles to the parallelogramme oe.
The demonstration shall be the same of the Diagonall rl. The like situation is manifest, by the [21 e iiij]. And from hence also is manifest, That the diagonall of a Quadrate, is a Quadrate: Of an Oblong, an Oblong: Of a Rhombe, a Rhombe: Of a Rhomboides, a Rhomboides: because it is like unto the whole, and a like situate.
Now the Diagonalls seeing they are like unto the whole and a like situate, they shall also be like betweene themselves and alike situate one to another, by the [21] and [22 e iiij].
Therefore
18. If the particular parallelogramme have one and the same angle with the whole, be like and alike situate unto it, it is the Diagonall. 26 p vj.
This might have beene drawn, as a consectary, out of the former: But it may also as it is by Euclide be forced, by an argument ab impossibili. For otherwise the whole should be equall to the part, which is impossible.
As for example, Let the particular parallelogramme auys, be