Right Lines Cutting.
15.If two right lines doe cut one another, they doe make the angles at the top equall and all equall to foure right angles. 15. p j.
Anguli ad verticem, Angles at the top or head, are called Verticall angles which have their toppes meeting in the same point. The Demonstration is: Because the lines cutting one another, are either perpendiculars, and then allright angles are equall as heere: Or else they are oblique, and then also are the verticalls equall, as areaui, andoue: And againe,auo, andiue. Nowaui, andoue, are equall, because by the14. e.withauo, the common angle, they are equall to two right angles: And therefore they are equall betweene themselves. Whereforeauo, the said common angle beeing taken away, they are equall one to another.
And
And
And
16.If two right lines cut with one right line, doe make the inner angles on the same side greater then two right angles, those on the other side against them shall be lesser then two right angles.
Ashere, ifauy, anduyi, bee greater then two right angleseuy, anduyo, shall bee lesser then two right angles.
Figure for demonstration 16.
17.If from a point assigned of an infinite right line given, two equall parts be on each side cut off: and then from the points of those sections two equall circles doe meete, a right line drawne from their meeting unto the point assigned, shall bee perpendicular unto the line given. 11. p j.
Figure for demonstration 17.
As leta, be the point assigned of the infinite line given: and from that on each side, by the7. e.cut off equallportionsae, andai, Then let two equall peripheries from the pointse, andi, meete, as ino, I say that a right line drawne fromo, the point of the meeting of the peripheries. untoa.the point given, shalbe perpendicular upon the line given. For drawing the right linesoe, &oi, the two angleseao, andiao, on each side, equicrurall by the construction of equall segments on each side, andoa, the common side, are equall in base by the9. e. And therefore the angles themselves shall be equall, by the7. e iij. and therefore againe, seeing thatao, doth lie equall betweene the partsea, andia, it is by the13. e ij. perpendicular upon it.
18.If a part of an infinite right line, bee by a periphery for a point given without, cut off a right line from the said point, cutting in two the said part, shall bee perpendicular upon the line given. 12. p j.
Figure for demonstration 18.
Of an infinite right line given, let the part cut off by a periphery of an externall center beae: And then letio, cut the said part into two parts by the12. e. I say thatiois perpendicular unto the said infinite right line. For it standeth upright, and makethaoi, andeoi, equall angles, for the same cause, whereby the next former perpendicular was demonstrated.
19.If two right lines drawne at length in the same plaine doe never meete, they are parallells. è 35. d j.
Parallel lines.
Thus much of the Perpendicularity of plaine right lines:Parallelissmus, or their parallell equality doth follow.Eucliddid justly require these lines so drawne to be granted paralels: for then shall they be alwayes equally distant, as hereae.andio.
Therefore
20.If an infinite right line doe cut one of the infinite right parallell lines, it shall also cut the other.
As in the same exampleuy.cuttingae.it shall also cutio.Otherwise, if it should not cut it, it should be parallell unto it, by the18 e. And that against the grant.
21.If right lines cut with a right line be pararellells, they doe make the inner angles on the same side equall to two right angles: And also the alterne angles equall betweene themselves: And the outter, to the inner opposite to it: And contrariwise,29, 28, 27. p 1.
Perpendicular to Parallel lines.
The paralillesme, or parallell-equality of right lines cut with a right line, concludeth a threefold equality of angles: And the same is againe of each of them concluded. Therefore in this one element there are sixe things taught; all which are manifest if a perpendicular, doe fallupon two parallell lines. The first sort of angles are in their owne words plainely enough expressed. But the wordAlternum, alterne [oralternate,H.] here, asProclussaith, signifieth situation, which in Arithmeticke signified proportion, when the antecedent was compared to the consequent; notwithstanding the metaphor answereth fitly. For as an acute angle is unto his successively following obtuse; So on the other part is the acute unto his successively following obtuse: Therefore alternly, As the acute unto the acute: so is the obtuse, unto the obtuse. But the outter and inner are opposite, of the which the one is without the parallels; the other is within on the same part not successively; but upon the same right line the third from the outer.
The cause of this threefold propriety is from the perpendicular or plumb-line, which falling upon the parallells breedeth and discovereth all this variety: As here they are right angles which are the inner on the same part or side: Item, the alterne angles: Item the inner and the outter: And therefore they are equall, both, I meane, the two inner to two right angles: and the alterne angles between themselvs: And the outter to the inner opposite to it.
If so be that the cutting line be oblique, that is, fall not upon them plumbe or perpendicularly, the same shall on the contrary befall the parallels. For by that same obliquation or slanting, the right lines remaining and the angles unaltered, in like manner both one of the inner, to wit,euy, is made obtuse, the other, to wit,uyo, is made acute: And the alterne angles are made acute and obtuse: As also the outter and inner opposite are likewise made acute and obtuse.
If any man shall notwithstanding say, That the inner angles are unequall to two right angles: By the same argument may he say (saithPtolomeinProclus) That on each side they be both greater than two right angles, and also lesser: As in the parallel right linesaeandio, cut withthe right lineuy, if thou shalt say thatauyandiyu, are greater then two right angles, the angles on the other side, by the16 e, shall be lesser then two right angles, which selfesame notwithstanding are also, by the gainesayers graunt, greater then two right angles, which is impossible.
Line crossing Parallel lines.
The same impossibility shall be concluded, if they shall be sayd, to be lesser than two right angles.
The second and third parts may be concluded out of the first. The second is thus: Twise two angles are equall to two right anglesoyu, andeuy, by the former part: Item,auy, andeuy, by the14 e. Therefore they are equall betweene themselves. Now from the equall, Take awayeuy, the common angle, And the remainders, the alterne angles, atu, andyshall be least equall.
The third is thus: The angleseuy, andoys, are equall to the sameuyi, by the second propriety, and by the15 e. Therefore they are equall betweene themselves.
The converse of the first is here also the more manifest by that light of the common perpendicular, And if any man shall thinke, That although the two inner angles be equall to two right angles, yet the right may meete, as if those equall angles were right angles, as here; it must needes be that two right lines divided by a common perpendicular, should both leane, the one this way, the other that way, or at least one of them, contrary to the13 e ij.
Converse cases.
If they be oblique angles, as here, the lines one slanting orobliquely crossing one another, the angles on one side will grow lesse, on the other side greater. Therefore they would not be equall to two right angles, against the graunt.
From hence the second and third parts may be concluded. The second is thus: The alterne angles atuandy, are equall to the foresayd inner angles, by the 14 e: Because both of them are equall to the two right angles: And so by the first part the second is concluded.
The third is therefore by the second demonstrated, because the outteroys, is equall to the verticall or opposite angle at the top, by the15 e. Therefore seeing the outter and inner opposite are equall, the alterne also are equall.
Wherefore asParallelismus, parallell-equality argueth a three-fold equality of angles: So the threefold equality of angles doth argue the same parallel-equality.
Therefore,
Therefore,
Therefore,
22.If right lines knit together with a right line, doe make the inner angles on the same side lesser than two right Angles, they being on that side drawne out at length, will meete.
Lines that will meet.
As hereae, andio, knit together witheo, doe make two anglesaeo, andioe, lesser than two right angles: They shall therefore, I say, meete if they be continued out that wayward. The assumption and complexion is out of the21 e, of right lines in the same plaine. If right lines cut with a right line be parallels, they doe make the inner angles on the same part equall to two right-angles. Therefore if they doe not make them equall, but lesser, they shall not be parallel, but shall meete.
And
And
And
23.A right line knitting together parallell right lines, is in the same plaine with them.7p xj.
Line knitting together parallell right lines.
As hereuy, knitting or joyning together the two parallelsae, andio, is in the same plaine with them as is manifest by the8 e.
And
And
And
Alternate angles.
24.If a right line from a point given doe with a right line given make an angle, the other shanke of the angle equalled and alterne to the angle made, shall be parallell unto the assigned right line.31p j.
As let the assigned right line beae: And the point given, let it bei. From which the right line, making with the assignedae, the angle,ioe, let it beio: To the which ati, let the alterne angleoiu,be made equall: The right lineui, which is the other shanke, is parallel to the assignedae.
An angle, I confesse, may bee made equall by the first propriety: And so indeed commonly the Architects and Carpenters doe make it, by erecting of a perpendicular. It may also againe in like manner be made by the outter angle: Any man may at his pleasure use which hee shall thinke good: But that here taught we take to be the best.
And
And
And
Parallel shanks.
25.The angles of shanks alternly parallell, are equall.Or Thus,The angles whose alternate feete are parallells, are equall. H.
This consectary is drawne out of the third property ofthe21 e. The thing manifest in the example following, by drawing out, or continuing the other shanke of the inner angle. ButLazarus Schonerusit seemeth doth thinke the adverbealterne, (alternelyoralternately) to be more then needeth: And therefore he delivereth it thus: The angles of parallel shankes are equall.
And
And
And
26.If parallels doe bound parallels, the opposite lines are equall è34p. j.Or thus:If parallels doe inclose parallels, the opposite parallels are equall. H.
Parallels bounding parallels.
Otherwise they should not be parallell. This is understood by the perpendiculars, knitting them together, which by the definition are equall betweene two parallells: And if of perpendiculars they bee made oblique, they shall notwithstanding remaine equall, onely the corners will be changed.
And
And
And
27.If right lines doe joyntly bound on the same side equall and parallell lines, they are also equall and parallell.
Not parallel bounds.
Parallel bounds.
This element might have beene concluded out of the next precedent: But it may also be learned out of thosewhich went before. As letae, andio, equall parallels be bounded joyntly ofai, andeo: and leteibe drawn. Here because the right lineeifalleth upon the parallelsae, andio, the alterne anglesaeiandeio, are equall, by the21 e. And they are equall in shankesae, andio, by the grants, andei, is the common shanke: Therefore they are also equall in baseai, andeo, by the7 e iij. This is the first: Then by21 e, the alterne angleseia, andieo, are equall betweene themselves: And those are made byaiandeo, cut by the right lineei: Therefore they are parallell; which was the second.
On the same part or side it is sayd, least any man might understand right lines knit together by opposite bounds as here.
28.If right lines be cut joyntly by many parallell right lines, the segments betweene those lines shall bee proportionall one to another, out of the2p vj and17p xj.
First case: perpendiculars.
Thus much of the Perpendicle, and parallell equality of plaine right lines: Their Proportion is the last thing to be considered of them.
The truth of this element dependeth upon the nature of the parallells: And that throughout all kindes of equality and inequality, both greater and lesser. For if the lines thus cut be perpendiculars, the portionsintercepted betweene the two parallels shall be equall: for common perpendiculars doe make parallell equality, as before hath beene taught, and here thou seest.
If the lines cut be not parallels, but doe leane one toward another, the portions cut or intercepted betweene them will not be equall, yet shall they be proportionall one to another. And looke how much greater the line thus cut is: so much greater shall the intersegments or portions intercepted be. And contrariwise, Looke how much lesse: so much lesser shall they be.
Cases of non-perpendiculars.
The third parallell in the toppe is not expressed, yet must it be understood.
This element is very fruitfull: For from hence doe arise and issue, First the manner of cutting a line according to any rate or proportion assigned: And then the invention or way to finde out both the third and fourth proportionalls.
29.If a right line making an angle with another right line, be cut according to any reason [or proportion] assigned, parallels drawne from the ends of the segments, unto the end of the sayd right line given and unto some contingent point in the same, shall cut the line given according to the reason given.
Schonerhath altered this Consectary, and delivereth itthus:If a rightlinemaking an angle with a right line given, and knit unto it with a base, be cut according to any rate assigned, a parallell to the base from the ends of the segments, shall cut the line given according to the rate assigned.9 and 10 p vj.
Division into two parts.
Punctum contingens, A contingent point, that is falling or lighting in some place at al adventurs, not given or assigned.
This is a marvelous generall consectary, serving indifferently for any manner of section of a right line, whether it be to be cut into two parts, or three parts, or into as manyparts, as you shall thinke good, or generally after what manner of way soever thou shalt command or desire a line to be cut or divided.
Let the assigned Right line to be cut into two equall parts beae. And the right line making an angle with it, let it be the infinite right lineai.Letao, one portion thereof be cut off. And then by the7 e, letoi, another part thereof be taken equall to it. And lastly, by the24 e, draw parallels from the pointsi, ando, untoe, the end of the line given, and tou; a contingent point therein. Now the third parallell is understood by the pointa, neither is it necessary that it should be expressed. Therefore the lineae, by the28, is cut into two equall portions: And asao, is tooi: So isau, toue. Butao, andoi, are halfe parts. Thereforeau, andue, are also halfe parts.
And here also is the12 ecomprehended, although not in the same kinde of argument, yet in effect the same. But that argument was indeed shorter, although this be more generall.
Division into three parts.
Now letaebe cut into three parts, of which the first let it beethe halfe of the second: And the second, the halfe of the third: And the conterminall or right line making an angle with the sayd assigned line, let it be cut one partao: Then double this inou: Lastly letuibe taken double toou, and let the whole diagramme be made up with three parallelsie,uy, andos, The fourth parallell in the toppe, as afore-sayd, shall be understood. Therefore that section which was made in the conterminall line, by the28 e, shall be in the assigned line: Because the segments or portions intercepted are betweene the parallels.
And
And
And
30.If two right lines given, making an angle, be continued, the first equally to the second, the second infinitly, parallels drawne from the ends of the first continuation, unto the beginning of the second, and some contingent point in the same, shall intercept betweene them the third proportionall. 11. p vj.
Third proportional.
Let the right lines given, making an angle, beae, andai: andae, the first, let it be continued equally to the sameai, and the sameai, let it be drawne out infinitly: Then the parallelsei, andou, drawne from the ends of the first continuation, untoi, the beginning of the second: andu, a contingent point in the second, doe cut offiu, the third proportionall sought. For by the28 e, asae, is untoeo, so isai, untoiu.
And
31.If of three right lines given, the first and the third making an angle be continued, the first equally to the second, and the third infinitly; parallels drawnefrom the ends of the first continuation, unto the beginning of the second, and some contingent point, the same shall intercept betweene them the fourth proportionall. 12. p vj.
Let the lines given be these: The firstae, the secondei, the thirdao, and let the whole diagramme be made up according to the prescript of the consectary. Here by28. e, asae, is toeiso isao, toou. Thus farreRamus.
Lazarus Schonerus, who, about some 25. yeares since, did revise and augment this worke of our Authour, hath not onely altered the forme of these two next precedent consectaries: but he hath also changed their order, and that which is here the second, is in his edition the third: and the third here, is in him the second. And to the former declaration of them, hee addeth these words: From hence, having three lines given, is the invention of the fourth proportionall; and out of that, having two lines given, ariseth the invention of the third proportionall.
2Having three right lines given, if the first and the third making an angle, and knit together with a base, be continued, the first equally to the second; the third infinitly; a parallel from the end of the second, unto the continuation of the third, shall intercept the fourth proportionall. 12. p vj.
The Diagramme, and demonstration is the same with our31. eor 3 c ofRamus.
3If two right lines given making an angle, and knit together with a base, be continued, the first equally to the second, the second infinitly; a parallell to the base from the end of the first continuation unto the second, shall intercept the third proportionall. 11. p vj.
The Diagramme here also, and demonstration is in allrespects the same with our30 e, or 2 c ofRamus.
Thus farreRamus: And here by the judgement of the learnedFinkius, two elements ofPtolomeyare to be adjoyned.
Parallels proportional to segments.
32If two right lines cutting one another, be againe cut with many parallels, the parallels are proportionall unto their next segments.
It is a consectary out of the28 e. For let the right linesae.andai, cut one another ata, and let two parallell linesuo, andei, cut them; I say, asau, is touo, soae, is toei. For from the endi, letis, be erected parallell toae, and letuo, be drawne out untill it doe meete with it. Then from the ends, letsy, be made parallell toai: and lastly, letea, be drawne out, untill it doe meete with it. Here noway, shall be equall to the right lineis, that is, by the26. e, toue: and at length, by the28. e, asua, is touo; so isay, that is,ue, toos. Therefore, by composition or addition ofproportions, asua, is untouo, soua, andue, shall be untouo, andos, that is,ei, by the27. e.
The samedemonstrationshall serve, if the lines do crosse one another, or doe vertically cut one another, as in the same diagramme appeareth. For if the assignedai, andus, doe cut one another vertically ino, let them be cut with the parallelsau, andsi: the precedent fabricke or figure being made up, it shall be by28. e.asau, is untoao, the segment next unto it: soay, that is,is, shall be untooi, his next segment.
The28. eteacheth how to finde out the third and fourth proportionall: This affordeth us a meanes how to find outthe continually meane proportionall single or double.
Therefore
Therefore
Therefore
Squire.
33.If two right lines given be continued into one, a perpendicular from the point of continuation unto the angle of the squire, including the continued line with the continuation, is the meane proportionall betweene the two right lines given.
A squire (Norma,Gnomon, orCanon) is an instrument consisting of two shankes, including a right angle. Of this we heard before at the13. e. By the meanes of this a meane proportionall unto two lines given is easily found: whereupon it may also be called aMesolabium, orMesographus simplex, or single meane finder.
Let the two right lines given, beae, andei. The meane proportional between these two is desired. For the finding of which, let it be granted that asae, is toeo, soeo, is toei: therefore letae, be continued or drawne out untoi, so thatei, be equall to the other given. Then frome, the point of the continuation, leteo, an infinite perpendicular be erected. Now about this perpendicular, up and downe, this way and that way, let the squireao, be moved, so that with his angle it may comprehend ateo, and with his shanks it may include the whole right lineai. I say thateo, the segment of the perpendicular, is the meane proportionall betweenae, andei, the two lines given. For letea, be continued or drawne out intou, so that the continuationau, be equall untoeo: and untoa, the point of the continuation, let the angleuas, be made equall, and equicrurall to the angleoei, that is, let the shankeas, be made equall to the shankeei. Wherefore knittingu, ands, together, the right linesus, andoi, shall be equall; and the angleseoi,aus, by the7. e iij. And by the21. e, the linessa, andoe, are parallell: and the anglesao, is equall to the angleaoe. But the anglessae, andaoi, are right angles by the Fabricke and by the grant; and therefore they are equall, by the14. e iij. Wherefore the other anglesoae, andeoi, that is,sua, are equall. And therefore by the21. e.us, andaoare parallell; andus, andeo, continued shall meete, as here iny: and by the26. e.oy, andasare equall. Now, by the32. e.asue, is toua, so isey, toas. Therefore by subduction or subtraction of proportions, asea, is toua, so iseo, that is,ua, tooy, that isas.
Figure for demonstration 33.
And
And
And
Plato's Mesographus in use.
34If two assigned right lines joyned together by their ends rightanglewise, be continued vertically; a square falling with one of his shankes, and another to it parallell and moveable upon the ends of the assigned, with the angles upon the continued lines, shall cut betweene them from the continued two meanes continually proportionall to the assigned.
The former consectary was of a single mesolabium; this is of a double, whose use in making of solids, to this or that bignesse desired is notable.
Let the two lines assigned beae, andei; and let there be two meane right lines, continually proportionall betweene them sought, to wit, that may be asae, is untoone of the lines found; so the same may be unto the second line found. And as that is unto this, so this may be untoei. Let thereforeae, andei, be joyned rightanglewise by their ends ate; and let them be infinite continued, but vertically, that is, from that their meeting from the lines ward, fromei, towardsu, butae, towardso. Now for the rest, the construction; it wasPlato's Mesographus; to wit, a squire with the opposits parallell. One of his sidesau, moueable, or to be done up and downe, by an hollow riglet in the side adjoyning. Therefore thou shalt make thee a Mesographus, if unto the squire thou doe adde one moveable side, but so that how so ever it be moved, it be still parallell unto the opposite side [which is nothing else, but as it were a double squire, if this squire be applied unto it; and indeed what is done by this instrument, may also be done by two squires, as hereafter shall be shewed.] And so long and oft must the moveable side be moved up and downe, untill with the opposite side it containe or touch the ends of the assigned, but the angles must fall precisely upon the continued lines: The right lines from the point of the continuation, unto the corners of the squire, are the two meane proportionals sought.
As if of the Mesographusauoi, the moveable side beau;thus thou shalt move up and downe, untill the anglesu, ando, doe hit just upon the infinite lines; and joyntly at the same instantua, andoi, may touch the ends of the assigneda, andi. By the former consectary it shall be asei, is toeo, soeo, shall be untoeu: and aseo, is toeu, so shalleu, be untoea.
Plato's Mesographus.
And thus wee have the composition and use, both of the single and double Mesolabium.
35.If of foure right lines, two doe make an angle, the other reflected or turned backe upon themselves, from the ends of these, doe cut the former; the reason of the one unto his owne segment, or of the segments betweene themselves, is made of the reason of the so joyntly bounded, that the first of the makers be joyntly bounded with the beginning of the antecedent made; the second of this consequent joyntly bounded with the end; doe end in the end of the consequent made.
Ptolomeyhath two speciall examples of thisTheorem: to thoseTheonaddeth other foure.
Figure for demonstration 35.
Let therefore the two right lines beae, andai: and from the ends of these other two reflected, beiu, andeo, cutting themselves iny; and the two former inu, ando. The reason of the particular right lines made shall be asthe draught following doth manifest. In which the antecedents of the makers are in the upper place: the consequents are set under neathe their owne antecedents.
The businesse is the same in the two other, whether you doe crosse the bounds or invert them.
Here for demonstrations sake we crave no more, but that from the beginning of an antecedent made a parallell be drawne to the second consequent of the makers, unto one of the assigned infinitely continued: then the multiplied proportions shall be,
The Antecedent, the Consequent; the Antecedent, theConsequent of the second of the makers; every way the reason or rate is of Equallity.
The Antecedent; the Consequent of the first of the makers; the Parallell; the Antecedent of the second of the makers, by the32. e. Therefore by multiplication of proportions, the reason of the Parallell, unto the Consequent of the second of the makers, that is, by the fabricke or construction, and the32. e.the reason of the Antecedent of the Product, unto the Consequent, is made of the reason, &c. after the manner above written.
Figure for several demonstrations in 35.
For examples sake, let the first speciall example be demonstrated. I say therefore, that the reason ofia, untoao, is made of the reason ofiu, untouy, multiplied by the reason ofye, untoeo. For from the beginning of the Antecedent of the product, to wit, from the pointi, let a line be drawne parallell to the right lineey, which shall meete withae, continued or drawne out infinitely inn. Therefore, by the32. e, asia, is toao: so is the parallell drawne toeo, the Consequent of the second of the makers. Therefore now the multiplied proportions are thusiu,uy,in,ey, by the 32. e:ye,eo,ey,eo. Therefore as the product ofiu, byye, is unto the product ofuy, byeo: Soin, is toeo, that is,ia, toao.
So let the second ofPtolemyto be taught, which in ourTable aforegoing is the fifth. I say therefore that the reason ofio, untooa; is made of the reason ofiy, untoyu, and the reason ofue, untoea. For now againe, from the beginning of the Antecedent of the Producti, let a line be drawne parallell untoea, the Consequent of the second of the Makers, which shall meete witheo, drawne out at length, inn: therefore, by the32. e.asio, is toao; so isen, untoea. Therefore now again the multiplied proportions are thus:
by the32. e. Therefore, by multiplication of proportions, the reason ofen, untoea, that is, ofio, untooa, is made of the reason ofiy, untoyu, by the reason ofue, untoea.
It shall not be amisse to teach the same in the examples ofTheon. Let us take therefore the reason of the Reflex, unto the Segment; And of the segments betweene themselves; to wit, the 4. and 6. examples of our foresaid draught: I say therefore, that the reason ofoe, untoey, is made of the reasonoa, untoai, by the reason ofiu, untouy. For from the endo, to wit, from the beginning of the Antecedent of the product, let the right lineno, be drawne parallell touy. It shall be by the32. e.asoe, is toey: so the parallellno, shall be touy: but the reason ofno, untouy, is made of the reason ofoa, untoai, and ofiu, untouy: for the multiplied proportions are,
by the32. e.
Againe, I say, that the reason ofey, untoyo, is compounded of the reason ofeu, untoua, and ofai, untoio.
Theonhere draweth a parallell fromo, untoui. By the generall fabricke it may be drawne out ofe, untoui.
It shall be therefore asey, is untoyo, soen, shall be untooi. Now the proportions multiplied are,
by the32. e.
Therefore the reason ofen, untoio, that is ofey, untoyo, shall be made of the foresaid reasons.
Of the segments of divers right lines, theArabianshave much under the name ofThe rule of sixe quantities.And theTheoremesofAlthindus, concerning this matter, are in many mens hands. AndRegiomontanusin hisAlgorithmus: andMaurolycusupon the 1 p iij. ofMenelaus, doe make mention of them; but they containe nothing, which may not, by any man skillfull in Arithmeticke, be performed by the multiplication of proportions. For all those wayes of theirs are no more but speciall examples of that kinde of multiplication.
1.Like plaines have a double reason of their homologall sides, and one proportionall meane, out of 20 p vj. and xj. and 18. p viij.
Or thus; Like plaines have the proportion of their correspondent proportionall sides doubled, & one meane proportionall: Hitherto wee have spoken of plaine lines and their affections: Plaine figures and their kindes doe follow in the next place. And first, there is premised a common corollary drawne out of the24. e. iiij. because in plaines there are but two dimensions.
2.A plaine surface is either rectilineall or obliquelineall,[or rightlined, or crookedlined. H.]
Straightnesse, and crookednesse, was the difference of lines at the4. e. ij. From thence is it here repeated and attributed to a surface, which is geometrically made of lines. That made of right lines, is rectileniall: that which is made of crooked lines, is Obliquilineall.
3.A rectilineall surface, is that which is comprehended of right lines.
A plaine rightlined surface is that which is on all sides inclosed and comprehended with right lines. And yet they are not alwayes right betweene themselves, but such lines as doe lie equally betweene their owne bounds, and without comparison are all and every one of them right lines.