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

¶ The 16. Theoreme. The 18. Proposition. If a right line be erected perpēdicularly to a plaine superficies: all the superficieces extended by that right line, are erected perpendicularly to the selfe same plaine superficies.

SVppose that a right line AB be erected perpendicularly to a ground superficies. Thē I say, that all the superficieces passing by the line AB, are erected perpendicularly to the ground superficies. Extend a superficies by the line AB, and let the same be ED, & let the cōmon section of the plaine

[illustration]

superficies and of the ground superficies be the right line CE. And take in the line CE a point at all aduentures,* 1.1 and let the same be F: and (by the 11. of the first) from the point F drawe vnto the line CE a perpendicular line in the superficies DE, and let the same be FG. And forasmuch as the line AB is erected perpendicularly to the ground super∣ficies, therefore (by the 2. definition of the e∣leuenth) the line AB is erected perpendicu∣larly to all the right lines that are in the ground plaine superficies,* 1.2 and which touch it. Wher∣fore it is erected perpendicularly to the line CE. Wherefore the angle ABE is a right angle. And the angle GFB is also a right angle (by construction). Wherefore (by the ••8. of the first) the line AB is a parallel to the line FG. But the line AB is erected perpendicularly to the ground superficies: wherefore (by the 8. of the eleuenth) the line FG is also erected per∣pendicularly to the ground superficies. And forasmuch as (by the 3. definition of the eleuenth) a plaine superficies is then erected perpendicularly to a plaine superficies, when all the right lines drawen in one of the plaine superficieces vnto the common section of those two plaine su∣perficieces making therwith right angles, do also make right angles with the other plaine su∣perficies and it is proued that the line FG drawen in one of the plaine superficieces, namely, in DE, perpendicularly to the common section of the plaine superficieces, namely, to the line CE, is erected perpendicularly to the ground superficies: wherefore the plaine superficies DE is erected perpendicularly to the ground superficies. In like sort also may we proue, that all the plaine superficieces which passe by the line AB, are erected perpendicularly to the ground su∣perficies. If therefore a right line be erected perpendicularly to a plaine superficies all the su∣perficieces passing by the right line, are erected perpendicularly to the selfe same plaine super∣ficies: which was required to be demonstrated.

In this figure here set ye may erect per∣pēdicularly

[illustration]

at your pleasure the superficies wherin are drawen the lines DC, GF, AB, and HE, to the ground superficies wherin is drawen the line CFBE, and so plainly compare it with the demonstration before put.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.