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

* 1.13 A plaine superficies is then vpright or erected perpendicularly to a plaine superficies, when all the right lines drawen in one of the plaine su∣perficieces vnto the common section of those two plaine superficieces, ma∣king therwith right angles, do also make right angles to the other plaine superficies. Inclination or leaning of a right line, to a plaine superficies, is an acute angle, contained vnder a right line falling from a point aboue to the plaine superficies, and vnder an other right line, from the lower end of the sayd line (let downe) drawen in the same plaine superficies, by a certaine point assigned, where a right line from the first point aboue, to the same plaine superficies falling perpendicularly, toucheth.

In this third definition are included two definitions: the first is of a plaine superficies erected per∣pendicularly vpon a plaine superficies.* 1.2 The second is of the inclination or leaning of a right line vnto a superficies: of the first take this example. Suppose ye haue two super••icieces ABCD and CDEF. Of which let the superficies CDEF be a ground plaine superficies, and let the superficies ABCD be e∣rected vnto it, and let the line CD be a common terme or in∣tersection

[illustration]

to them both, that is, let it be the end or bound of either of them,* 1.3 & be drawen in either of them: in which line note at pleasure certaine pointes, as the point G, H. From which pointes vnto the line CD, draw perpendicular lines in the super••icies ABCD, which let be GL and HK, which fal∣ling vpon the superficies CDEF, if they cause right angles with it, that is, with lines drawen in it from the same pointes G and H, as if the angle LGM or the angle LGN contayned vnder the line ••G drawen in the superficies erected, and vnder the GM or GN drawen in the ground superficies CDEF ly∣ing flat, be a right angle, then by this definition, the superficies ABCD is vpright or erected vpon the superficies CDEF. It is also commonly called a superficies perpendicular vpon or vnto a superficies.

* 1.4For the second part of this definition, which is of the inclination of a right line vnto a plaine su∣perficies, take this example. Let ABCD be a ground plaine superficies, vpon which from a point being a loft, namely, the point E, suppose a right line to fall, which let be the line EG, touching the plaine superficies ABCD at the poynt G. Againe, from the point E, being the toppe or higher limite and end of the inclining line EG, let a perpendicular line fall vnto the plaine superficies ABCD, which let be the line EF, and let F be the point where EF toucheth the plaine superficies ABCD. Then from the point of the fall of the line inclining vpon the superficies vnto

[illustration]

the point of the falling of the perpendicular line vpon the same super∣ficies, that is, from the point G to the point F, draw a right line GF. Now by this definition, the acute angle EGF is the inclination of the line EG vnto the superficies ABCD. Because it is contayned of the inclining line, and of the right line drawen in the superficies, from the point of the fall of the line inclining to the point of the fall of the per∣pendicular line: which angle must of necessitie be an acute angle. For the angle EFG is by construction a right angle, and three angles in a triangle are equ••〈…〉〈…〉 ••ight angles. Wherefore the other two angles, namely, the angles EGF, and GEF, are equ••〈…〉〈…〉 right angle. Wherfore either of them is lesse then a right angle. Wherfore the angle EGF is an 〈…〉〈…〉gle.

Notes

• * 1.1

  Third diffini∣tion.
• * 1.2

  Two dif••ini∣tions included in this di••••i∣nition.
• * 1.3

  Declaratiō of the first part.
• * 1.4

  Declaration of the second part.