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

¶An other demonstration after Campane of the 3. proposition.

Suppose that there be a Pyramis ABCD hauing to his base the triangle BCD, and let his toppe be the solide angle A: from which let there be drawne three subtended lines AB, AC, and AD to the three angles of the base, and deuide all the sides of the base into two equall partes in the three poyntes E, F, G: deuide also the three subtēded lines AB, AC, and AD in two equall partes in the three points H, K, L. And draw in the base these two lines EF and EG:

[illustration]

So shall the base of the pyramis be deuided into three su∣perficieces: whereof two are the two triangles BEF, and EGD, which are like both the one to the other, and also to the whole base, by the 2 part of the secōd of the sixth, & by the definitiō of like super••iciec••s, & they are equal the one to the other, by the 8. of the first: the third superficies is a quadrangled parallelogramme, namely, EFGC: which is double to the triangle EGD, by the 40. and 41. of the first. Now then agayne from the poynt H draw vnto the points E and F these two subtendent lines HE and HF: draw also a subtended line from the poynt K to the poynt G. And draw these lines HK, KL, and LH. Wherefore the whole pyramis ABCD is deuided into two pyramids, which are HBEF, and AHKL, and into two prismes of which the one is EHFGKC, and is set vpon the quadrangled base CFGE: the other is EGDHKL, and hath to his base the triangle EGD. Now as touching the two pyramids HBEF and AHKL, that they are equall the one to the other, and also that they are like both the one to the o∣ther and also to the whole, it is manifest by the definition of equall and like bodyes, and by the 10. of the eleuenth, and by 2. part of the second of the sixth. And that the two Prismes are equall it is manifest by the last of the eleuenth. And now that both the prismes taken together are greater then the halfe of the whole pyramis, hereby it is manifest, for that either of them may be deuided into two pyramids, of which the one is a triangular pyramis equall to one of the two pyramids into which together with the two prismes is deuided the whole pyramis, and the other is a quadrangled pyramis double to the other pyramis. Wherefore it is playne that the two prismes taken together are three quarters of the whole

pyramis deuided. But if ye are desirous to know the proportiō betwen them, reade the ••, of this booke. But now here to this purpose it shall be sufficient to know, that the two prismes taken together do ex∣ceede in quantity the two partial pyramids taken together, into which together with the two prismes the whole pyramis was deuided.