Commentary on Proposition XII.8

Back to Proposition XII.8.

[Folio 0401 | p. 397]

It is at once proved that, the pyramids being similar, the parallelepipeds constructed as shown in the figure are also similar.

Consequently, as these latter are in the triplicate ratio of their corresponding sides [XI. 33], so are the pyramids which are their sixth parts respectively (being one third of the respective prisms on the same bases, i.e. of the halves of the respective parallelepipeds, XI. 28).

As the Porism is not used where Euclid might have been expected to use it (see note on XII. 12, p. 416), there is some reason to doubt its genuineness. P only has it in the margin, though in the first hand.