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

* 1.122 A Tetrahedron is a solide which is contained vnder fower triangles equall and equilater.

A forme of this solide ye may see in these two examples here set,

[illustration]

whereof one is as it is commonly described in a playne. Neither is it hard to conceaue. For (as we before taught in a Pyramis) if ye imagine the triangle BCD to lie vpon a ground plaine superficies, and the point A to be pulled vp together with the lines AB, AC, and AD, ye shall perceaue the forme of the Tetrahedron to be contayned vnder 4. triangles, which ye must imagine to be al fower equilater and equiangle, though they can not so be drawen in a plaine. And a Te∣trahedron thus described, is of more vse in these fiue bookes follow∣ing, then is the other, although the other appeare in forme to the eye more bodilike.* 1.2

Why this definition is here left out both of Campane and of Flussas,

[illustration]

I can not but maruell, considering that a Tetrahedron, is of all Philo∣sophers counted one of the fiue chiefe solides which are here de∣fined of Euclide, which are called cōmonly regular bodies, with∣out mencion of which, the entreatie of these should seeme much maimed: vnlesse they thought it sufficiently defined vnder the definition of a Pyramis,* 1.3 which plainly and generally taken, inclu∣deth in deede a Tetrahedron, although a Tetrahedron properly much differe••h from a Pyramis, as a thing speciall or a particular, from a more generall. For so taking it, euery Tetrahedron is a Py∣ramis, but not euery Pyramis is a Tetrahedron. By the generall definition of a Pyramis, the superficieces of the sides may be as many in number as ye list, as 3.4. 5.6. or moe, according to the forme of the base, whereon it is set, whereof before in the defi∣nition of a Pyramis were examples geuen. But in a Tetrahedron the superficieces erected can be but three in number according to the base therof, which is euer a triangle. Againe, by the generall definition of a Pyrami••, the superfi∣cieces erected may ascend as high as ye list, but in a Tetrahedron they must all be equall to the base. Wherefore a Pyramis may seeme to be more generall then a Tetrahedron, as before a Prisme seemed to be more generall then a Parallelipipedon, or a sided Columne: so that euery Parallelipipedon is a Prisme, but not euery Prisme is a Parallelipipedon. And euery axe in a Sphere is a diameter: but not euery diameter of a Sphere is the axe therof. So also noting well the definition of a Pyramis, euery Te∣trahedron may be called a Pyramis, but not euery Pyramis a Tetrahedron. And in dede Psellus in num∣bring of these fiue solides or bodies, calleth a Tetrahedron a Pyramis in manifest wordes.* 1.4 This I say might make Flussas & others (as I thinke it did) to omitte the definition of a Tetrahedron in this place, as sufficiently comprehended within the definition of a Pyramis geuen before. But why then did he not count that de••inition of a Pyramis faultie, for that it extendeth it selfe to large, and comprehendeth vnder it a Tetrahedron (which differeth from a Pyramis by that it is contayned of equall triangles) as he not so aduisedly did before the definition of a Prisme.

Notes

• * 1.1

  Twenty two diffinition.
• * 1.2

  A Tetrahe∣dron one of the fiue regu∣lar bodyes.
• * 1.3

  Di••••erence betwene a Tetrahedron and a Pira∣mis.
• * 1.4

  Psellus calleth a Tetrahedron a Piramis.