As here the foure particular right angles are equall to the whole, which are made of ae, one of his sides, and of ei, io, ou, uy, the segments of the other.
The Demonstration of this is from the rule of congruency: Because the whole agreeth to all his parts. But the same reason in numbers is more apparent by an induction of the parts: as foure times eight are 32. I breake or divide 8. into 5. and 3. Now foure times 5. are 20. And foure times 3. are 12. And 20. and 12. are 32. And 32. and 32. are equall. Therefore 20. and 12. are also equall to 32.
Lastly, every arithmeticall multiplication of the whole numbers doth make the same product, that the multiplication of the one of the whole numbers given, by the parts of the other shall make: yea, that the multiplication of the parts by the parts shall make. This proportion is cited by Ptolomey in the 9. Chapter of the 1 booke of his Almagest.
8 If foure right lines be proportionall, the rectangle of the two middle ones, is equall to the rectangle of the two extremes. 16. p vj.
It is a speciall consectary out of the [28 e x]. As here are foure right lines proportionall betweene themselves: And the rectangle of the extremes, or first and last let it be ay: Of the middle ones, let it be se.
9 The figurate of a rationall rectangle is called a rectinall plaine. 16. d vij.
A rationall figure was defined at the [12. e iiij]. of which
sort amongst all the rectilineals hitherto spoken of, we have not had one: The first is a Right angled parallelogramme; And yet not every one indifferently: But that onely whose base is rationall to the highest: And that reason of the base and heighth is expressable by a number, where also the Figurate is defined. A rectangle or irrational sides, such as were mentioned at the [9 e j]. is irrationall. Therefore a rectangled rationall of rationall sides, is here understood: And the figurate thereof, is called, by the generall name, A Plaine: Because of all the kindes of Plaines, this kinde onely is rationall.