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 5. Theoreme. The 5. Proposition. If vnto three right lines which touch the one the other, be erected a per∣pendicular line from the common point where those three lines touch: those three right lines are in one and the selfe same plaine superficies.

SVppose that vnto these three right lines BC, BD, and BE, touching the one the other in the poynt B, be erected perpendicularly from the poynt B, the line AB. Then I say, that those thre right lines BC, BD and BE, are in one & the selfe same plaine superficies. For if not, then if it be possible, let the lines BD &

BE be in the ground superficies, and let the line BC be e∣rected

[illustration]

vpward (now the lines AB and BC are in one and the same playne superficies (by the 2. of the eleuenth) for they touch the one the other in the point B). Extend the plaine superficies wherein the lines AB and BC are,* 1.1 and it shall make at the length a common section with the ground superficies, which common section shall be a right line (by the 3. of the eleuenth): let that common section be the line BF. Wherefore the three right lines AB, BC, and BF are in one and the selfe same su∣perficies, namely, in the superficies wherein the lines AB and BC are. And forasmuch as the right line AB is erected per∣pendicularly to either of these lines BD and BE, therefore the line AB is also (by the 4. of the eleuenth) erected perpendicu∣larly to the plaine superficies, wherein the lines BD and BE are. But the superficies wherein the lines BD and BE are is the ground superficies. Wherefore the line AB is erected per∣pendicularly to the ground plaine superficies. Wherefore (by the 2. definition of the eleuenth) the line AB maketh right angles with all the lines which are drawne vpon the ground super∣ficies and touch it. But the line BF which is in the ground superficies doth touch it. Wherfore the angle ABF is a right angle. And it is supposed that the angle ABC is a right angle. Wherefore the angle ABF is equall to the angle ABC, and they are in one and the selfe same plaine superficies which is impossible. Wherefore the right line BC is not in an higher superficies. Wherefore the right lines BC, BD, BE are in one and the selfe same plaine su∣perficies. If therefore vnto three right lines touching the one the one the other, be erected a perpendicular line from the common point where those three lines touch: those three right lines are in one and the selfe same plaine superficies: which was required to be demon∣strated.

This figure here set more playnely

[illustration]

declareth the demonstration of the for∣mer proposition, if ye erect perpendicu∣larly vnto the ground superficies, the s••••perficies wherein is drawne the line 〈◊〉〈◊〉 and so compare it with the sayd de••••••••stration.

Notes

• * 1.1

  Demonstrati∣on leading to an impossibi∣litie.