The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

A Corrollary added by Campane and Flussas.

Hereby it is manifest that equall pyramids hauing to their bases Poligonon figures, haue their bases reciprokall with their altitudes. And Pyramids whose bases being poligonon figures are recipro∣kall with their altitudes, are equall the one to the other.

Suppose that vpon the poligonon figures A

[illustration]

and B, be set equall pyramids. Then I say that their bases A and B are reciprokall with their al∣titudes. Describe by the 25. of the sixth, triangles equall to the bases A and B. Which let be C and D. Vpon which let there be set pyramids equall in altitude with the pyramids A and B. Wherfore the pyramids C and D, being set vpō bases equall with the bases of the pyramids A and B, and ha∣uing also their altitudes equall with the altitudes of the sayd pyramids A and B, shall be equall by the 6. of this booke. Wherefore by the first part of this proposition, the bases of the pyramids, C to D are reciprokall with the altitudes of D to C. But in what proportion are the bases C to D, in the same are the bases A to B, forasmuch as they are equall. And in what proportion are the altitudes of D to C, in the same are the altitudes of B to A, which altitudes are likewise equall. Wherefore by the 11. of the fifth, in what proportion the bases A to B are, in the same reciprokally are the altitudes of the pyramids B to A. In like sort by the second part of this proposition may be proued the conuerse of this corollary. The same thing followeth also in a Prisme, and in a sided columne, as hath before at large bene declared in the corollary of the 40. propo∣sition of the 11. booke. For those solides are in proportiō the one to the other, as the pyramids or paral∣lelipipedons, for they are either partes of equemultiplices or equemultiplices to partes.