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 6. Probleme. The 11. Proposition. Vpon a right line geuen, to rayse vp from a poynt geuen in the same line a perpendicular line.

SVppose that the right line geuen be AB, & let the point in it geuen be C. It is required from the poynte C to rayse vp vnto the right line AB a perpendicular line. Take in the line AC a poynt at all aduentures,* 1.1 & let the same be D, and (by the 3. proposition) put vnto DC an equall line CE. And by the first proposition) vpon the line DE describe an equilater triangle FDE, & draw a line frō F to C. Then I say that vnto the right line geuen AB, and from the poynte in it geuen, namely, C is raysed vp a perpendicular line FC. For forasmuch

[illustration]

as DC is equal to CE,* 1.2 & the line CF is cōmon to them both: therfore these two DC and CF, are e∣qual to these two EC & CF, the one to the other: and (by the first proposition) the base DF is equal to the base EF: wherefore (by the 8. proposition) the angle DCF is equall to the angle ECF: and they be side angles. But whē a right line standing vpon a right line doth make the two side angles equall the one to the other, ether of those equall angles is (by the. 10. definition) a right angle: & the line standing vpon the right line is called a perpēdicular line. VVherfore the angle DCF, & thangle FCE are right angles. VVherfore vnto the right line geuē AB, & frō the poynt in it C, is raysed vp a perpendicular line CF: which was required to be done.

Although the poynte geuen should be set in one of the endes of the righte line geuen, it is easy so do it as it was before. For producing the line in length from the poynt by the second peticion, you may worke as you did before. But if one require to erect a right line perpendicularly from the poynt at the end of the

lyne, without producing the rightlyne, that also may well bee done after thys maner.

Suppose that the right line geuen be AB, & let

[illustration]

the point in it geuen be in one of the endes therof,* 1.3 namely, in A. And take in the line AB a point at all aduentures,* 1.4 and let the same be C. And from the said point raise vp (by the foresaid proposition) vn∣to AB a perpendiculer line, which let ••e OE. And (by the 3. proposition) from the line CE cut of the line CD equall to the line CA, And (by the 9. Pro∣position) deuide the angle ACD into two equall partes by the line CF. And from the point D raise vp vnto the line CE a perpēdiculer line, DF, which let concurre with the line CF in the point F. And drawe a right line from F to A. Then I say that the angle at the point A is a right angle. For, foras∣much as DC is equall to CA,* 1.5 and CF is common to them both, and they containe ••∣quall angles (for the angle at the point C is deuided into two equall partes) therefore (by the 4. Proposition) the line DF is equall to the line FA, and so the angle at the point A is equal to the angle at the point D. But the angle at the point D is a right an∣gle. Wherfore also the angle at the point A is a right angle. Wherefore from the point A vnto the line AB, is raised vp a perpendiculer line AF, without producing the line AB. Which was required to be done.

Appollonius teacheth to rayse vp vnto a line geuen, from a point in it geuen, a perpendiculer line, after this maner.* 1.6

Suppose that the right

[illustration]

line geuen be AB. And let the point in it geuē, be C. And in the line AC, take a point at all ad∣uētures, & let the same be D. And frō the lyne CB, cut of a line equall to the line CD, whiche let be CE•• and makyng the centre D, and the space DE, describe a circle. And againe ma∣king the centre C, & the space ED, describe an oth••r circle, and let the point of their intersection be F. And draw a right line from F to C. Then I say that the line FC is erected perpendiculerly vnto the line AB. For drawe these lin••s ED and FE: which shal by the definition of a circl•• be either of them equal to the line DE.* 1.7 and therfore (by the first common sentence) are equall the one to the other. But the lines DC and CE are by construction equall, and the line FC is common to them both. Wherfore the angles also at the point C are equal (by the 8. propositiō:) wherfore they are right angles. Wherfore the 〈◊〉〈◊〉 CE is erected perpendiculerly vnto the line AB from the point C•• which was required to be done.

By this way of erecting a perpendiculer line inuented by Appollonius,* 1.8 it is also manifest, that if a man will mechanically, without d••monstration, erect vnto

a line geuen from a point geuē in it a perpendi∣culer

[illustration]

line: he neede onely on either side of the pointe geuen, to cut of equall lines: and so ma∣king either of the endes of the said lines (either of th'endes I say, which haue not one point cō∣mon to them both) the centres, and the space both the lynes added together, or wider then both, or at the lest wider thē one of them, to de∣scribe those portions of the circles where they cut the one the other, and from the point of the intersection to the point geuen, to draw a lyne, which shall be perpendicular vnto the lyne ge∣uē: as in the figure here put it is manifest to see.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstration.
• * 1.3

  An other case in this proposition.
• * 1.4

  Construction.
• * 1.5

  Demonstration.
• * 1.6

  An other way to erect a perpē∣dicular line in∣uented by Ap∣polonius.

  Construction••.
• * 1.7

  Demonstration.
• * 1.8

  How to erect a perpendicular line mechani∣cally.