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

Corollary. And hereby it is manifest, that as the sector is to the sector, so is angle to angle by the 11. of the fifth.

Flussates here addeth fiue Propositions wherof one is a Probleme hauing three Corollaryes following of it, and the rest are Theoremes : which for that they are both witty, & also serue to great vse, as we shall afterward see, I thought not good to omitte, but haue here placed them: but onely that I haue not put them to fol∣lowe in order with the Propositions of Euclide as he hath done.

¶The first Proposition added by Flussates.

To describe two rectiline figures equall and like vnto a rectiline figure geuen and in like sort situate, which shall haue also a proportion geuen.

Suppose that the rectiline figure geuen be ABH.* 1.1 And let the proportion geuen be the proportion of the lines GC and CD. And (by the 10. of this booke) deuide the line AB like vnto the line GD in the poynt E (so that as the line GC is to the line CD, so let the line AE be to the line EB). And vpon the line AB describe a semicircle AFB. And from the poynt E erect (by the 11. of the first) vnto the line AB a perpendicular line EF cutting the circumference in the poynt F. And draw these lines AF and FB. And vpon either of these lines describe rectiline figures like vnto the rectiline figure AHB and in like sort situate (by the 18. of the sixt): which let be AKF, & FIB. Then I say, that the rectiline figures AKF, and FIB,

[illustration]

haue the proportion geuē (namely, the propor∣tion of the line GC to the line CD) and are e∣quall to the rectiline figure geuen ABH vnto which they are described like and in like sort si∣tuate. For forasmuch as AFB is a semicircle, therefore the angle AFB is a right angle (by the 31. of the third) and FE is a perpendicular line.* 1.2 Wherefore (by the 8. of this booke) the triangles AFE and FBE are like both to the whole triangle AFB and also the one to the other. Wherefore (by the 4. of this booke) as the line AF is to the line FB, so is the line AE to the line EF, and the line EF to the line EB, which are sides cōtayning equall angles. Wher∣fore (by the 22. of this booke) as the rectiline fi∣gure described of the line AF is to the rectiline figure described of the line FB, so is the recti∣line figure described of the line AE to the recti∣line figure described of the line EF, the sayd rectiline figures being like and in like sort

••ituate. But as the rectiline figure described of the line AE being the ••irst, is to the rectiline ••igure described of the line EF being the second, so is the line AE the first, 10. the line ••B the third (by the 2. Corollary of the 20. of thys booke). Wherfore the recti∣line figure described of the line AF is to the rectiline figure described of the line FB, as the line A•• is to the line EB. But the line AE is to, the line EB (by construction) as the line GC is to the line CD. Wherefore (by the 11. of the fift) as the line GC is to the line CD, so is the r••••tiline ••igure described of the line AF to the rectiline ••igure described 〈◊〉〈◊〉 the line ••B, the sayd rectiline figures being like and in like sort described. But the 〈…〉〈…〉 described o•• the lines AF and FB, are equall to the rectiline ••igure d••••••••••bed o•• the line AB, vnto which they are (by construction) described lyke and in like sort situate. Wherefore there are described two rectiline figures AKF and FIB equ••ll and like vnto the rectiline figure geuen ABH and in like sort situate, and they ha••e also the one to the other the proportion geuen, namely, the proportion of the line GC to the line CD: which was required to be done.

¶The first Corollary.

To resolue a rectiline figure geue into two like rectiline ••igures which shall hau•• also a proporti•• ge∣••ē. For i•• there be put three right lines in the proportiō geuē, and if the line AB be cu•• in the same proportion that the first line is to the third,* 1.3 the rectiline ••igures described of the line•• A•• and FB (which figures haue the same proportion that the lines AE and EB haue) shall be in double proportion to that which the lines AF and FB are (by the ••irs•• ••orollary o•• the 20. o•• this booke). Wherefore the right lines AF and FB are the o••e to the other in the same proportion that the first of the three lines put is to the 〈◊〉〈◊〉. ••or t••e 〈◊〉〈◊〉 line to the third, namely, the line AE to the line EB is in dou∣ble propo••tion that it is to the second, by the 10. de••inition of the fi••t.

¶ The second Corollary.

Hereby may we learne, how from a rectiline ••igure geuen to take away a part appointed, lea••ing, the rest of the rectiline ••igure like vnto the whole.* 1.4 For if frō the right line AB be cut of a part appoynted, namely, EB (by the 9. of this booke) as the line AE is to the line EB, so is the rectiline ••igure described of the line AF to the rectiline figure described of the line FB (the sayd ••igures being supposed to be like both the one to the other and also to the rectiline ••igure described of the line AB, and being also in like sort situate). Wherfore taking away ••rom the rectiline ••igure described of the line AB, the rectiline figure de∣scribed of the line FB, the residue, namely, the rectiline figure described of the line AF shall be both like vnto the whole rectiline ••igure geuen described of the line AB, and in like sort situate.

¶ The third Corollary.

To compose two like rectiline ••igures into one rectiline figure like and equall to the same figures. Let their sides of like proportiō be set so that they make a right angle,* 1.5 as the lines AF and FB are. And vpō the line subtending the said angle, namely, the line AB, describe a rectiline ••igure like vnto the rectiline figures geuen and in like sort situate (by the 18. of this booke) and the same shall be equall to the two rectiline figures geuen (by the 31. of this booke).

¶ The second Proposition.

If two right lines cut the one the other obtuseangled wise, and from the endes of the lines which ••ut the one the other be drawen perpendicular lines to either line: the lines which are betwene the endes and the perpendicular lines are cut reciprokally.

Suppose that there be two right lines AB and GD cutting the one the other in the point E, and making an obtuse angle in the section E. And from the endes of the lines, namely, A and G, let there be drawen to either line perpendicular lines, namely, from the point A to the line GD, which let be AD, and from the point G to the right line AB: which let be GB. Then I say, that the

[illustration]

right lines AB and GD do, betwene the end A and the perpendicular B, and the end G and the perpendicular D, cut the one the other reciprokally in the point E: so that as the line AE is to the line ED, so is the line GE to the line EB.* 1.6 For foras∣much as the angles ADE and GBE, are right angles, therfore they are equall. But the angles AED and GEB are also e∣quall (by the 15. of the first). Wherefore the angles remayning, namely, EAD, & EGB, are equall (by the Corollary of the 32. of the first). Wherefore the triangles AED and GEH, are equiangle. Wherfore the sides about the equall angles shall be proportionall (by the 4. of the sixt). Wherfore as the line AE is to the line ED, so is the line GE to the line EB. If therefore two right lines cut the one the other obtuseangled wife. &c: which was required to be proued.

¶ The third Proposition.

If two right lines make an acute angle, and from their endes be drawen to ech line perpendicular lines cutting them: the two right lines geuen shall be reciprokally pro∣portionall as the segmentes which are about the angle.

Suppose that there be two right lines AB and GB, making an acute angle ABG. And from the poyntes A and G let there be drawen vnto the lines AB and GB per∣pendicular lines AC and GE, cutting the lines AB and GB in the poyntes E and ••. Then I say, that the lines, namely AB to GB, are reciprokally proportionall, as the seg∣mentes, namely, CB to EB which are about the acute

[illustration]

angle B.* 1.7 For forasmuch as th•• right angles ACB and GER are equall, and the angle•• ABG is common to the triangles ABC, and GBE•• therefore the angles re∣mayning BAC and EGB are equall (by the Corolla∣ry of the 32. of the first). Wherfore the triangles ABC and GBE are equiangle. Wherefore the side•• about the equall angles are proportionall (by the 4. of the sixe)•• so that, as the line AB is to the line FC, so is the line GB to the line BE. Wherefore alternately as the line AB is to the line GB so is the line CB to the line BE. If therefore two right lines mak•• a•• ••c••te angle•• &c•• which was required to be proued.

•• The fourth Proposition••

If in a circle be drawen two right lines cutting the one the other, the sections of the one to the sections of the other shall be reciprokally proportionall.

In the circle, AGB let these two right lines 〈…〉〈…〉 one the other in the poynt E.* 1.8 Th•••• I say, that reciprokally 〈◊〉〈◊〉 ••h•• line AE is to the line ED, so is the line GE to the line EB. For forasmuch as (by the 35. of the third) the rectangle fi∣gure

contayned vnder the lines AE and

[illustration]

EB is equall to the rectangle figure contay∣ned vnder the lines GE and ED, but in e∣quall rectangle parallelogrammes the sides about the equall angles are reciprokall (by the 14. of the sixt). Therefore the line AE is to the line ED reciprokally as the line GE is to the line EB (by the second defi∣nition of the sixt). If therefore in a circle be drawen two right lines. &c: which was required to be proued.

¶The fift Proposition.

If from a poynt geuen be drawen in a plaine super••icies two right lines to the concaue circumference of a circle: they shall be reci∣prokally proportionall with their partes takē without the circle. And moreouer a right line drawen from the sayd poynt & touching the circle, shall be a meane proportionall betwene the whole line and the vtter segment.

Suppose that there be a circle ABD, and without it

[illustration]

take a certayne poynt, namely, G. And from the point G drawe vnto the concaue circumference two right lines GB and GD, cutting the circle in the poyntes C and E. And let the line GA touch the circle in the point A. Thē I say, that the lines, namely, GB to GD are reciprokally as their parts taken without the circle, namely, as GC to GE.* 1.9 For forasmuch as (by the Corollary of the 36. of the third) the rectangle figure contayned vnder the lines GB and GE is equall to the rectangle figure contayned vn∣der the lines GD and GC, therefore (by the 14. of the sixt) reciprokally as the line GB is to the line GD, so is the line GC to the line GE, for they are sides contayning equall angles. I say moreouer, that betwene the lines GB and GE, or betwene the lines GD and GC the touch line GA is a meane proportionall.* 1.10 For forasmuch as the re∣ctangle figure comprehended vnder the lines GB and GE is equall to the square made of the line AG (by the 36. of the third) it followeth that the touch line GA is a meane propor∣tionall betwene the extremes GB and GE (by the second part of the 17. of the sixt) for that by that Proposition the lines GB, GA, and GE are proportionall. And by the same reason may it be proued that the line GA is a meane proportionall betwene the lines GD and GC, and so of all others. If therefore from a poynt geuen•• &c•• which was required to be demonstrated.

Notes

• * 1.1

  Construction of the Pro∣bleme.
• * 1.2

  Demonstar∣tion of the same.
• * 1.3

  The first Co∣rollary.
• * 1.4

  The second Corollary.
• * 1.5

  The third Corollary.
• * 1.6

  Demonstra∣tion of this proposition.
• * 1.7

  Demonstra∣tion of this propositions
• * 1.8

  Demonstrati∣on of this pro∣position.
• * 1.9

  Demonstrati∣on of the first part of this proposition.
• * 1.10

  Demonstra∣tion of the se∣cond part.