The Way to Geometry, Book 5

The Way to Geometry · Book 5

p. 51The fifth Booke, of Ramus his Geometry, which is of Lines and Angles in a plaine Surface.

1. A lineate is either a Surface or a Body.

LIneatum, (or Lineamentum) a magnitude made of lines, as was defined at 1. e. iij. is here divided into two kindes: which is easily conceived out of the said definition there, in which a line is excluded, and a Surface & a body are comprehended. And from hence arose the division of the arte Metriall into Geometry, of a surface, and Stereometry, of a body, after which maner Plato in his vij. booke of his Common-wealth, and Aristotle in the 7. chapter of the first booke of his Posteriorums, doe di…tinguish betweene Geometry and Stereometry: And yet the name of Geometry is used to signifie the whole arte of measuring in generall.

2. A Surface is a lineate only broade. 5. dj.

As here a e i o. and u y s r. The definition of a Surface doth comprehend the distance or dimension of a line, to p. 52witt Length: But it addeth another distance, that is Breadth. Therefoce a Surface is defined by some, as Proclus saith, to be a magnitude of two dimensions. But two doe not so specially and so properly define it. Therefore a Surface is better defined, to bee a magnitude onelyonly long and broad. Such, saith Apollonius, are the shadowes upon the earth, which doe farre and wide cover the ground and champion fields, and doe not enter into the earth, nor have any manner of thicknesse at all.

Epiphania, the Greeke word, which importeth onelyonly the outter appearance of a thing, is here more significant, because of a Magnitude there is nothing visible or to bee seene, but the surface.

3. The bound of a surface is a line. 6. dj.

The matter in Plaines is manifest. For a three cornered surface is bounded with 3. lines: A foure cornered su…face, with foure li…es, and so forth: A Circle is bounded with one line. But in a Sphearicall surface the matter is not so plaine: For it being whole, seemeth not to be bounded with a line. Yet if the manner of making of a Sphearicall surface, by the conversiō or turning about of a semiperiphery, the beginning of it, as also the end, shalbe a line, to wit a semiperiphery: And as a point doth not only actu, or indeede bound and end a line: But is potentia, or in power, the middest of it: So also a line boundeth a Surface actu, and an innumerable company of lines may be taken or supposed to be throughout the whole surface. A Surface therefore is made by the motion of a line, as a Line was made by the motion of a point.

4. A surface is either Plaine or Bowed.

The difference of a Surface, doth answer to the difference of a Line… in straightnesse and obliquity or crookednesse.

Obliquum, oblique, there signified crooked; Not righ… or straight: Here, uneven or bowed, either upward or downeward. Sn.

p. 535. A plaine surface is a surface, which lyeth …qually betweene his bounds, out of the 7. dj.

As here thou seest in a e i o. That therefore a Right line doth looke two contrary waies, a Plaine surface doth looke all about every way, that a plaine surface should, of all surfaces within the same bounds, be the shortest: And that the middest thereof should hinder the sight of the extreames. …astly, it is equall to the dimension betweene the lines: It may also by one right line every way applyed be tryed, as Proclus at this place doth intimate.

Planum, a Plaine, is taken and used for a plaine surface: as before Rotundum, a Round, was used for a round figure.

Therefore,

6. From a point unto a point we may, in a plaine surface, draw a right line. 1 and 2. post. j.

Three things are from the former ground begg'd: The first is of a Right line. A right line and a periphery were in the ij. booke defined: But the fabricke or making of them both, is here said to bee properly in a plaine.

The fabricke or construction of a right line is the 1. petition. And justly is it required that it may bee done onelyonly upon a plaine: For in any other surface it were in vaine to aske it. For neither may wee possibly in a sphericall betweene two points draw a right line: Neither may wee possibly in a Conicall and Cylindraceall betweene any two points assigned draw a right line. For from the toppe p. 54unto the base that in these is only possible: And then is it the bounde of the plaine which cutteth the Cone and Cylinder. Therefore, as I said, of a right plaine it may onelyonly justly bee demanded: That from any point assigned, unto any point assigned, a right line may be drawne, as here from a unto e.

Now the Geometricall instrument for the drawing of a right plaine is called Amussis, & by Petolemey, in the 2. chapter of his first booke of his Musicke, Regula, a Rular, such as heere thou seest.

And from a point unto a point is this justly demanded to be done, not unto points; For neither doe all points fall in a right line: But many doe fall out to be in a crooked line. And in a Spheare, a Cone & Cylinder… a Ruler may be applyed, but it must be a sphearicall, Conicall, or Cylindraceall. But by the example of a right line doth Vitellio, 2 p j. demaund that betweene two lines a surface may be extended: And so may it seeme in the Elements, of many figures both plaine and solids, by Euclide to be demanded; That a figure may be described, at the 7. and 8. e ij. Item that a figure may be made vp, at the 8. 14. 16. 23.28. p.vj. which are of Plaines. Item at the 25. 31. 33. 34. 36. 49. p.xj. which are of Solids. Yet notwithstanding a plaine surface, and a plaine body doe measure their rectitude by a right line, so that jus postulandi, this right of begging to have a thing granted may seeme primarily to bee in a right plaine line.

Now the Continuation of a right line is nothing else, but the drawing out farther of a line now drawne, and that from a point unto a point, as we may continue the right line a e. unto i. wherefore the first and second Petitions of Eu…lde do agree in one.

p. 55And] 7. To set at a point assigned a Right line equall to another right line given: And from a greater, to cut off a part equall to a lesser. 2. and 3. pj.

As let the Right line given be a e. And to i. a po…nt assigned, grant that i o. equall to the same a e. may bee set. Item, in the second example, let a e. bee greater then i o. And let there he cut off from the same a e. by applying of a rular made equall to i o. the the lesser, portion a u. as here For if any man shall thinke that this ought only to be don in the minde, hee also, as it were, beares a ruler in his minde, that he may doe it by the helpe of the ruler. Neither is the fabricke in deede, or making of one right equal to another: And the cutting off from greater Right line, a portion equall to a lesser, any whit harder, then it was, having a point and a distance given, to describe a circle: Then having a Triangle, Parallelogramme, and semicircle given, to describe or make a Cone, Cylinder, and spheare, all which notwithstanding Euclide did account as principles.

Therefore,

8. One right line, or two cutting one another, are in the same plaine, out of the 1. and 2. p xj.

One Right line may bee the common section of two plaines: yet all or the whole in the same plaine is one: And all the whole is in the same other: And so the whole is the same plaine. Two Right lines cutting one another, may bee in two plaines cutting one of another; But then a plain… may be drawne by them: Therefore both p. 56of them shall be in the same plaine. And this plaine is geometrically to be conceived: Because the same plaine is not alwaies made the ground whereupon one oblique line, or two cutting one another are drawne, when a periphery is in a sphearicall: Neither may all peripheries cutting one another be possibly in one plaine.

And

9. With a right line given to describe a peripherie.

This fabricke or construction is taken out of the 3. Petition which is thus. Having a center and a distance given to describe, make, or draw a circle. But here the terme or end of a circle is onelyonly sought, which is better drawne out of the definition of a periphery, at the 10. e ij. And in a plaine onelyonly may that conversion or turning about of a right line bee made: Not in a sphearicall, not in a Conicall, not in a Cylindraceall, except it be in top, where notwithstanding a periphery may bee described. Therefore before (to witt at the said 10. e ij.) was taught the generall fabricke or making of a Periphery: Here we are informed how to discribe a Plaine periphery, as here.

Now as the Rular was the in…trument invented and …sed for the drawing of a right line: so also may the same Rular, used after another manner, be the instrument to describe or draw a periphery withall. And indeed such is that instrument used by the Coopers (and other like artists) for the rounding of their bottomes of their tubs, heads of barrells and otherlike vessells: But the Compasses, whether straight shanked or bow-legg'd, such as here thou seest, it skilleth not, are for al purposes and practises, in this case the best and readiest. And in deed the Compasses, of all geometricall p. 57instruments, are the most excellent, and by whose help famous Geometers have taught: That all the problems, of geometry may bee wrought and performed: And there is a booke extant, set out by Iohn Baptist, an Italian, teaching, How by one opening of the Compasses all the problems of Euclide may be resolved: And Ieronymus Cardanus, a famous Mathematician, in the 15. booke of his Subtilties, writeth, that there was by the helpe of the Compasses a demonstration of all things demonstrated by Euclide, found out by …im and one Ferrarius.

Talus, the nephew of Daedalus by his sister, is said in the viij. booke of Ovids Metamorphosis, to have beene the inventour of this instrument: For there he thus writeth of him and this matter: —Et ex uno duo ferrea brachia nodo: Iunxit, ut aequali spatio distantibus ipsis: Altera pars staret, pars altera duce…et orbem.

Therfore

10. The rai…s of the same, or of an equall periphery, are equall.

The reason is, because the same right line is every where converted or turned about. But here by the Ray of the …eriphery, must bee understood the Ray the figure contained within the periphery.

p. 5811. If two equall perip…eries, from the ends of equall shankes of an assigned rectilineall angle, doe meete before it, a right line drawne from the meeting of them unto the toppe or point of the angle, shall cut it into two equall parts. 9. pj.

Hitherto we have spoken of plaine lines: Their affection followeth, and first in the Bisection or dividing of an Angle into two equall parts.

Let the right lined Angle to bee divided into two equall parts bee e a i. whose equall shankes let them be a e. and a i. (or if they be unequall, let them be made equall, by the 7 e.) Then two equall peripheries from the ends e and i. meet before the Angle in o. Lastly, draw a line from …. unto a. I say the angle given is divided into two equall parts. For by drawing the right lines o e. and o i. the angles o a e. and o a i. equicrurall, by the grant, and by their common side a o. are equall in base e o. and i o. by the 10 e (Because they are the raies of equall peripheries.) Therefore by the 7. e iij. the angles o a e. and o a i. are equall: And therefore the Angle e a i. is equally divided into two parts.

12. If two equall peripheries from the ends of a right line given, doe meete on each side of the same, a right line drawne from those meetings, shall divide the right line given into two equall parts. 10. pj.

Let the right line given bee a e. And let two equall peripheries from the ends a. and e. meete in i. and o. Then from those meetings let the right line i o. be drawne. I say, That a e. is divided into two equall parts, by the said line thus p. 59drawne. For by drawing the raies of the equall peripheries i a. and i e. the said i o. doth cut the angle a i e. into two equall parts, by the 11. e. Therefore the angles a i u. and u i e. being equall and equicrurall (seeing the shankes are the raies of equall peripheries, by the grant.) have equall bases a u. and u e. by the 7. e iij. Wherefore seeing the parts a u. and u e. are equall, a e. the assigned right line is divided into two equall portions.

13. If a right line doe stand perpendicular upon another right line, it maketh on each side right angles: And contrary wise.

A right line standeth upon a right line, which cutteth, and is not cut againe. And the Angles on each side, are they which the falling line maketh with that underneath it, as is manifest out of Proclus, at the 15. pj. of Euclide; As here a e. the line cut: and i o. the insisting line, let them be perpendicular; The angles on each side, to witt a i o. and e i o. shall bee right angles, by the 13. e iij.

The Rular, for the making of straight lines on a plaine, was the first Geometricall instrument: The Compasses, for the describing of a Circle, was the second: The Norma or Square for the true …recting of a right line in the same plaine upon another right line, and then of a surface and body, upon a surface or body, is the third. The figure therefore is thus.

Now Perpendiculū, an instrument with a line & a plumme… of leade appendant upon it, used of Architects, Carpenters, and Masons, is meerely physicall: because heavie things naturally p. 60by their weight are in straight lines carried perpendicularly downeward. This instrument is of two sorts: The first, which they call a Plumbe-rule, is for the trying of an erect perpendicular, as whether a columne, pillar, or any other kinde of building bee right, that is plumbe unto the plaine of the horizont & doth not leane or reele any way. The second is for the trying or examining of a plaine or floore, whether it doe lye parallell to the horizont or not. Therefore when the line from the right angle, doth fall upon the middest of the base, it shall shew that the l…ngth is equally poysed. The Latines call it Libra, or Libella, a ballance: of the Italians Livello, and vel Archipendolo, Achildulo: of the French, Nivelle, or Niueau: of us a Levill.

Therefore

14. If a right line do stand upon a right line, it maketh the angles on each side equall to two right angles: and contrariwise out of the 13. and 14. pj.

For two such angles doe occupy or fill the same place that two right angles doe: Therefore p. 61they are equall to them by the 11. e j. If the insisting line be perpendicular unto that underneath it, it then shall make 2. right angles, by the 13. e. If it bee not perpendicular, & do make two oblique angles, as here a i o. and o i e. are yet shall they occupy the same place that two right angles doe: And therefore they are equall to two right angles, by the same.

The converse is forced by an argument ab impossibli, or ab absurdo, from the absurdity which otherwise would follow of it: For the part must otherwise needes bee equall to the whole. Let therefore the insisting or …standing line which maketh two angles a e o. and a e u. on each side equall to two right angles, be a e. I say that o e. and e i. are but one right line. Otherwise let o e. bee continued unto u. by the 6.e. Now by the 14.e. or next former element, a e o. & a e u. are equall to two right angles; To which also o e a. & a e i. are equall by the grant: Let a e o. the common angle be taken away: then shall there be left a e u. equall to a e i. the part to the whole, which is absurd and impossible. Herehence is it certaine that the two right lines o e, and e i, are in deede but one continuall right line.

And

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. pj.

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 all p. 62right angles are equall as heere: Or else they are oblique, and then also are the verticalls equall, as are a u i, and o u e: And againe, a u o, and i u e. Now a u i, and o u e, are equall, because by the 14.e. with a u o, the common angle, they are equall to two right angles: And therefore they are equall betweene themselves. Wherefore a u o, the said common angle beeing taken away, they are equall one to another.

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.

A here, if a u y, and u y i, bee greater then two right angles e u y, and u y o, shall bee lesser then two right angles.

17. If from … …oint 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. pj.

As let a, be the point assigned of the infinite line given: and from that on each side, by the 7. e… cut off equall portions p. 63a e, and a i, Then let two equall peripheries from the points e, and i, meete, as in o, I say that a right line drawne from o, the point of the meeting of the peripheries, unto a. the point given, shalbe perpendicular upon the line given. For drawing the right lines o e, & o i, the two angles e a o, and i a o, on each side, equicrurall by the construction of equall segments on each side, and o a, the common side, are equall in base by the 9. e. And therefore the angles themselves shall be equall, by the 7. e iij. and therefore againe, seeing that a o, doth lie equall betweene the parts e a, and i a, it is by the 13. e ij. perpendicular upon it.

18. If a part of an infin…te right line, bee by a periphery from 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. pj.

Of an infinite right line given, let the part cut off by a periphery of an externall center be a e: And then let i o, cut the said part into two parts by the 12. e. I say that i o is perpendicular unto the said infinite right line. For it standeth upright, and maketh a o i, and e o i, equall angles, for the same cause, whereby the next former perpendicular was demonstrated.

19. If two right lines drawne at l…ngth in the same plaine doe never meete, they are parallell…. è 35. dj.

Thus much of the Perpendicularity of plaine right lines: Parallelissmus, or their parallell equality doth follow. Euclid p. 64did justly require these lines so drawne to be granted paralels: for then shall they be alwayes equally distant, as here a e. and i o.

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 example u y. cutting a e. it shall also cu… i o. Otherwise, if it should not cut it, it should be parallell unto it, by the 18 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.

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 fall p. 65upon two parallell lines. The first sort of angles are in their owne words plainely enough expressed. But the word Alternum, alterne [or alternate, H.] here, as Proclus saith, signifieth situation, which in Arithmeticke signifyed proportion, when the antecedent was compared to the consequent; notwithstanding the metaphor answereth fitly. For as an acute angle is unto his successively following ob…use; 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 outter.

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 obliqua…ion or slanting, the right lines remaining and the angles unaltered, in like manner both one of the inner, to wit, e u y, is made obtuse, the other, to wi…, u y o, 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 (saith Ptolome in Proclus) That on each side they be both greater than two right angles, and also lesser: As in the parallel right lines a e and i o, cut with p. 66the right line u y, if thou shalt say that a u y and i y u, are greater then two right angles, the angles on the other side, by the 16 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.

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 angles o y u, and e u y, by the former part: Item, a u y, and e u y, by the 14 e. Therefore they are equall betweene themselves. Now from the equall, Take away e u y, the common angle, And the remainders, the alterne angles, at u, and y shall be least equall.

The third is thus; The angles e u y, and o y s, are equall to the same u y i, by the second propriety, and by the 15 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 the 13 eij.

If they be oblique angles, as here, the lines one slanting or p. 67liquely 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 at u and y, 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 outter o y s, is equall to the verticall or opposite angle at the top, by the 15 e. Therefore seeing the outter and inner opposite are equall, the alterne also are equall.

Wherefore as Parallelismus, parallell-equality argueth a three-fold equality of angels: So the threefold equality of angles doth argue the same parallel-equality.

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.

As here a e, and i o, knit together with e o, doe make two angles a e o, and i o e, 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 the 21 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

p. 6823. A right line knitting together parallell right lines, is in the same plaine with them. 7 p xj.

As here u y, knitting or joyning together the two parallels a e, and i o, is in the same plaine with them as is manifest by the 8 e.

And

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. 31 pj.

As let the assigned right line be a e: And the point given, let it be i. From which the right line, making with the assigned a e, the angle, i o e, let it be i o: To the which at i, let the alterne angle o i u, be made equall: The right line u i, which is the other shanke, is parallell, to the assigned a e.

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

25. The angles of shanks alternly parallell, are equall. Or Thus, The angles whose altenate feete are parallells, are equall. H.

This consectary in drawne out of the third property of p. 69the 21 e. The thing is manifest in the example following, by drawing out, or continuing the other shanke of the inner angle. But Lazarus Schonerus it seemeth doth thinke the adverbe alternè, (alternely or alternately) to be more then needeth: And therefore he delivereth it thus: The angles of parallel shankes are equall.

And

26 If parallels doe bound parallels, the opposite lines are equall è 34 p.j. Or thus: If parallels doe inclose parallels, the opposite parallels are equall. H.

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, onelyonly the corners will be changed.

And

27. If right lines doe joyntly bound on the same side equall and parallell lines, they are also equall and parallell.

This element might have beene concluded out of the next precedent: But it may also be learned out of those p. 70which went before. As let a e, and i o, equall parallels be bounded joyntly of a i, and e o: and let e i be drawn. Here because the right line e i, falleth upon the parallels a e, and i o, the alterne angles a e i and e i o, are equall, by the 21 e. And they are equall in shankes a e, and i o, by the grants, and e i, is the common shanke: Therefore they are also equall in base a i, and e o, by the 7 e iij. This is the first: Then by 21 e, the alterne angles e i a, and i e o, are equall betweene themselves: And those are made by a i, and e o, cut by the right line e i: 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 the 2 p vj and 17 p x j.

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 portions intercepted p. 71betweene the two parallels shall be equall: for common perpendiculars doe make parallell equality, a… 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.

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.

Schoner hath altered this Consectary, and delivereth it p. 72thus: If a right making an angle with a right line given, and … it 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 v j.

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 many patts, 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 be a e. And the right line making an angle with it, let it be the infinite right line a i. Let a o, one portion thereof be cut off. And then by the 7 e, let o i, another part thereof be taken equall to it. And lastly, by the 24 e, draw parallels from the points i, and o, unto e, the end of the line given, and to u, a contingent point therein. Now the third parallell is understood by the point a, neither is it necessary that it should be expressed. Therefore the line a e, by the 28, is cut into two equall portions: And as a o, is to oi: So is a u, to u e. But a o, and o i, are halfe parts. Therefore a u, and u e, are also halfe parts.

And here also is the 12 e comprehended, although not in the same kinde of argument, yet in effect the same. But that argument was indeed shorter, although this be more generall.

Now … be cut into three parts… … which the first let it bee p. 73the halfe of the second: And the second, the halfe of the third: And the conter minall or right line making an angle with the sayd assigned line, let it be cut one part a o: Then double this in o u: Lastly let u i be taken double to o u, and let the whole diagramme be made up with three parallels i e, u y … and os, The fourth parallell in the toppe, as a fore-sayd, shall be understood. Therefore that section which was made in the conterminall line, by the 28 e, shall be in the assigned line: Because the segments or portions intercepted are betweene the parallels.

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 v j.

Let the right lines given, making an angle, be a e, and a i: and a e, the first, let it be continued equally to the same a i, and the same a i, let it be drawne out infinitly: Then the parallels e i, and o u, drawne from the ends of the first continuation, unto i, the beginning of the second: and u, a contingent point in the second, doe cut off i u, the third proportionall sought. For by the 28 e, as a e, is unto e o, so is a i, unto i u.

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 drawne p. 74from 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 first a e, the second e i, the third a o, and let the whole diagramme be made up according to the prescript of the consectary. Here by 28. e, as a e, is to e i so is a o, to o u. Thus farre Ramus.

Lazarus Schonerus, who, about some 25. yeares since, did revise and augment this worke of our Authour, hath not onelyonly 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.

2 Having 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. pvj.

The Diagramme, and demonstration is the same with our 31. e or 3 c of Ramus.

3 If 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 v j.

The Diagramme here also, and demonstration is in all p. 75respects the same with our 30 e, or 2 c of Ramus.

Thus farre Ramus: And here by the judgement of the learned Finkius, two elements of Ptolomey are to be adjoyned.

32 If 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 the 28 e. For let the right lines a e. and a i, cut one another at a, and let two parallell lines u o, and e i, cut them; I say, as a u, is to u o, so a e, is to e i. For from the end i, let i s, be erected parallell to a e, and let u o, be drawne out untill it doe meete with it. Then from the end s, let s y, be made parallell to a i: and lastly, let e a, be drawne out, untill it doe meete with it. Here now a y, shall be equall to the right line i s, that is, by the 26. e, to u e: and at length, by the 28. e, as u a, is to u o; so is a y, that is, u e, to o s. Therefore, by composition or addition of ptoportions, as u a, is unto u o, so u a, and u e, shall be unto u o, and o s, that is, e i, by the 27. e.

The same demonstation shall serve, if the lines do crosse one another, or doe vertically cut one another, as in the same diagramme appeareth. For if the assigned a i, and u s, doe cut one another vertically in o, let them be cut with the parallels a u, and s i: the precedent fabricke or figure being made up, it shall be by 28. e. as a u, is unto a o, the segment next unto it: so a y, that is, i s, shall be unto o i, his next segment.

The 28. e teacheth how to finde out the third and fourth proportionall: This affordeth us a meanes how to find out p. 76the continually meane proportionall single or double.

Thefore

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, or Canon) is an instrument consisting of two shankes, including a right angle. Of this we heard before at the 13.e. By the meanes of this a meane proportionall unto two lines given is easily found: whereupon it may also be called a Mesolabium, or Mesographus simplex, or single meane finder.

Let the two right lines given, be a e, and e i. The meane proportional between the…e two is desired. For the finding of which, let it be granted that as a e, is to e o, so e o, is to e i: therefore let a e, be continued or drawne out unto i, so that e i, be equall to the other given. Then from e, the point of the continuation, let e o, an infinite perpendicular be erected. Now about this perpendicular, up and downe, this way and that way, let the squire a o, be moved, so that with his angle it may comprehend at e o, and with his shanks it may include the whole right line a i. I say that e o, the segment of the perpendicular, is the meane proportionall between p. 77a e, and e i, the two lines given. For let e a, be continued or drawne out into u, so that the continuation a u, be equall unto e o: and unto a, the point of the continuation, let the angle u a s, be made equall, and equicrurall to the angle o e i, that is, let the shanke a s, be made equall to the shanke e i. Wherefore knitting u, and s, together, the right lines u s, and o i, shall be equall; and the angles e o i, a u s, by the 7. e i i j. And by the 21. e, the lines s a, and o e, are parallell: and the angle s a o, is equall to the angle a o e. But the angles s a e, and a o i, are right angles by the Fabricke and by the grant; and therefore they are equall, by the 14. e iij. Wherefore the other angles o a e, and e o i, that is, s u a, are equall. And therefore by the 21. e. u s, and a o are parallell; and u s, and e o, continued shall meete, as here in y: and by the 26. e. o y, and a s are equall. Now, by the 32. e. as u e, is to u a, so is e y, to a s. Therefore by subduction or subtraction of proportions, as e a, is to u a, so is e o, that is, u a, to o y, that is a s.

And

34 If two assigned right lines joyned together by their ends right anglewise, 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 be a e, and e i; and let there be two meane right lines, continually proportionall betweene them sought, to wit, that may be as a e, is unto p. 78one of the lines found; so the same may be unto the second line found. And as that is unto this, so this may be unto e i. Let therefore a e, and ei, be joyned rightanglewise by their ends at e; and let them be infinite continued, but vertically, that is, from that their meeting from the lines ward, from e i, towards u, but a e, towards o. Now for the rest, the construction; it was Plato's Mesographus; to wit, a squire with the opposits parallell. One of his sides a u, 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 Mesographus a u o i, the moveable side be a u; p. 78thus thou shalt move up and downe, untill the angles u, and o, doe hit just upon the infinite lines; and joyntly at the same instant u a, and o i, may touch the ends of the assigned a, and i. By the former consectary it shall be as e i, is to e o, so e o, shall be unto e u: and as e o, is to e u, so shall e u, be unto … a.

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 …egment, 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.

Ptolomey hath two speciall examples of this Theorem: to those Theon addeth other foure.

Let therefore the two right lines be … e, and a i: and from the ends of these other two reflected, be i u, and e o, cutting themselves in y; and the two former in u, and o. The reason of the particular right lines made shall be as p. 80the 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, the p. 81Consequent 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 Parallel; the Antecedent of the second of the makers, by the 32. 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 the 32. e. the reason of the Antecedent of the Product, unto the Consequent, is made of the reason, &c. after the manner above written.

For examples sake, let the first speciall example be demonstrated. I say therefore, that the reason of i a, unto a o, is made of the reason of i u, unto u y, multiplied by the reason of y e, unto e o. For from the beginning of the Antecedent of the product, to wit, from the point i, let a line be drawne parallell to the right line e y, which shall me…te with a e … continued or drawne out infinitely in n. Therefore, by the 32. e, as i a, is to a o: so is the parallell drawne to e o, the Consequent of the second of the makers. Therefore now the multiplied proportions are thus i u, u y, i u, e y, by the 32. e: y e, e o, e y, e o. Therefore as the product of i u, by y e, is unto the product of u y, by e o: So i n, is to e o, that is, i a, to a o.

So let the second of Ptolomey to be taught, which in our p. 82Table afore going is the fifth. I say therefore that the reason of i o, unto o a; is made of the reason of i y, unto y u, and the reason of u e, unto e a. For now againe, from the beginning of the Antecedent of the Product i, let a line be drawne parallell unto e a, the Consequent of the second of the Makers, which shall meete with e o, drawne out at length in n: therefore, by the 32. e. as i o, is to a o; so is i …, unto e a. Therefore now again the multiplied proportions are thus: by the 32. e. Therefore, by multiplication of proportions, the reason of i u, unto e a, that is, of i o, unto … a, is made of the reason of i y, unto y u, by the reason of u e, unto ea.

It shall not be amisse to teach the same in the examples of Theon. 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 of o e, unto e y, is made of the reason o a, unto a i, by the reason of i u, unto u y. For from the end o, to wit, from the beginning of the Antecedent of the product, let the right line n o, be drawne parallell to u y. It shall be by the 32. e. as o e, is to e y: so the parallell n o, shall be to u y: but the reason of n o, unto … y, is made of the reason of o a, unto a i, and of i u, unto … y: for the multiplied proportions are, by the 32. e.

Againe, I say, that the reason of e y, unto y o, is compounded of the reason of e u, unto u a, and of a i, unto i ….

Theon here draweth a parallell from o, unto u i. By the generall fabricke it may be drawne out of e, unto o i.

It shall be therefore as e y, is unto y o, so e n, shall be unto o i. Now the proportions multiplied are, by the 32. e.

Therefore the reason of e n, unto i o, that is of e y, unto p. 83y o, shall be made of the foresaid reasons.

Of the segments of divers right lines… the Arabians have much under the name of The rule of sixe quantities. And the Theoremes of Althin…us, concerning this matter, are in many mens hands. And Regiomontanus in his Algorithmus: and Maurolycus upon the 1 piij. of Menelaus, doe make mention of them; but they containe nothing, which may not; by any man skilfull 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.