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 33. Theoreme. The 38. Proposition. If a plaine superficies be erected perpendicularly to a plaine superficies, and from a point taken in one of the plaine superficieces be drawen to the other plaine superficies, a perpendicular line: that perpendicular line shall fall vpon the common section of those plaine superficieces.

SVppose that the plaine superficies CD be erected perpēdicularly to the plaine superfi∣cies AB, and let their common section be the line DA: and in the superficies CD take a point at all aduentures, and let the same be E. Then I say, that a perpendicu∣lar line drawen from the point E to the

[illustration]

plaine superficies AB, shall fall vpon the right line DA. For if not, then let it fall without the line DA, as the line EF doth, and let it fall vpon the plaine su∣perficies AB in the point F.* 1.1 And (by the 12. of the first) from the point F draw vnto the line DA, being in the superficies AB a perpendicular line F∣G, which line also is erected perpendicu∣larly to the plaine superficies CD: by the third diffinitiō: by reason we presuppose CD and AB to be perpendicularly erec∣ted ech to other. Draw a right line from the point E to the point G. And foras∣much as the line FG is erected perpendi∣cularly to the plaine superficies CD, and the line EG toucheth it being in the superficies CD. Wherefore the angle FGE is (by the 2. definition of the eleuenth) a right angle. But the line EF is also erected perpēdicularly to the superficies AB: wherefore the angle EFG is a right angle. Now therefore two angles of the triangle EFG, are equall to two right angles: which (by the 17. of the first) is impossible. Wherfore a perpendicular line drawen frō the point E to the s••••erficies AB, falleth not with∣out the line DA. Wherefore it falleth vpon the line DA: which was required to be proued.

¶ Note.

Campane maketh this as a Corollary, following vpon the 13: and very well, with small ayde of other Propositions, he proueth it•• whose demonstratiō there, Flussas hath in this place, and none other: though he sayth that Campane of such a Propositiō, as of Euclides, maketh no mention.

In this figure ye may more fully see the former Proposi∣tion

[illustration]

and demonstration if ye erecte perpendicularly vnto the ground plaine superficies AB the superficies CD, and imagine a line to be extended from the point E to the point F, instede whereof ye may extend if ye will a thred.

Notes

• * 1.1

  Demonstra∣tion leading to an impossibi∣litie.