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 let ae be cut into three parts, of which the first let it bee the 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 part ao: Then double this in ou: Lastly let ui be taken double to ou, and let the whole diagramme be made up with three parallels ie, uy, and os, The fourth parallell in the toppe, as afore-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 vj.
Let the right lines given, making an angle, be ae, and ai: and ae, the first, let it be continued equally to the same ai, and the same ai, let it be drawne out infinitly: Then the parallels ei, and ou, 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 iu, the third proportionall sought. For by the 28 e, as ae, is unto eo, so is ai, unto iu.
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 from 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 ae, the second ei, the third ao, and let the whole diagramme be made up according to the prescript of the consectary. Here by 28. e, as ae, is to ei so is ao, to ou. Thus farre Ramus.
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.
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. p vj.
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 vj.
The Diagramme here also, and demonstration is in all respects 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 ae. and ai, cut one another at a, and let two parallell lines uo, and ei, cut them; I say, as au, is to uo, so ae, is to ei. For from the end i, let is, be erected parallell to ae, and let uo, be drawne out untill it doe meete with it. Then from the end s, let sy, be made parallell to ai: and lastly, let ea, be drawne out, untill it doe meete with it. Here now ay, shall be equall to the right line is, that is, by the 26. e, to ue: and at length, by the 28. e, as ua, is to uo; so is ay, that is, ue, to os. Therefore, by composition or addition of proportions, as ua, is unto uo, so ua, and ue, shall be unto uo, and os, that is, ei, by the 27. e.
The same demonstration 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 ai, and us, doe cut one another vertically in o, let them be cut with the parallels au, and si: the precedent fabricke or figure being made up, it shall be by 28. e. as au, is unto ao, the segment next unto it: so ay, that is, is, shall be unto oi, 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 the continually meane proportionall single or double.
Therefore
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 ae, and ei. The meane proportional between these two is desired. For the finding of which, let it be granted that as ae, is to eo, so eo, is to ei: therefore let ae, be continued or drawne out unto i, so that ei, be equall to the other given. Then from e, the point of the continuation, let eo, an infinite perpendicular be erected. Now about this perpendicular, up and downe, this way and that way, let the squire ao, be moved, so that with his angle it may comprehend at eo, and with his shanks it may include the whole right line ai. I say that eo, the segment of the perpendicular, is the meane proportionall between ae, and ei, the two lines given. For let ea, be continued or drawne out into u, so that the continuation au, be equall unto eo: and unto a, the point of the continuation, let the angle uas, be made equall, and equicrurall to the angle oei, that is, let the shanke as, be made equall to the shanke ei. Wherefore knitting u, and s, together, the right lines us, and oi, shall be equall; and the angles eoi, aus, by the 7. e iij. And by the 21. e, the lines sa, and oe, are parallell: and the angle sao, is equall to the angle aoe. But the angles sae, and aoi, are right angles by the Fabricke and by the grant; and therefore they are equall, by the 14. e iij. Wherefore the other angles oae, and eoi, that is, sua, are equall. And therefore by the 21. e. us, and ao are parallell; and us, and eo, continued shall meete, as here in y: and by the 26. e. oy, and as are equall. Now, by the 32. e. as ue, is to ua, so is ey, to as. Therefore by subduction or subtraction of proportions, as ea, is to ua, so is eo, that is, ua, to oy, that is as.
And
34 If 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 be ae, and ei; and let there be two meane right lines, continually proportionall betweene them sought, to wit, that may be as ae, is unto one 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 ei. Let therefore ae, 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 ei, towards u, but ae, 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 au, 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 auoi, the moveable side be au; thus 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 ua, and oi, may touch the ends of the assigned a, and i. By the former consectary it shall be as ei, is to eo, so eo, shall be unto eu: and as eo, is to eu, so shall eu, be unto ea.
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.
Ptolomey hath two speciall examples of this Theorem: to those Theon addeth other foure.
Let therefore the two right lines be ae, and ai: and from the ends of these other two reflected, be iu, and eo, cutting themselves in y; and the two former in u, and o. The reason of the particular right lines made shall be as the 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 Consequent 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 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 ia, unto ao, is made of the reason of iu, unto uy, multiplied by the reason of ye, unto eo. 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 ey, which shall meete with ae, continued or drawne out infinitely in n. Therefore, by the 32. e, as ia, is to ao: so is the parallell drawne to eo, the Consequent of the second of the makers. Therefore now the multiplied proportions are thus iu, uy, in, ey, by the 32. e: ye, eo, ey, eo. Therefore as the product of iu, by ye, is unto the product of uy, by eo: So in, is to eo, that is, ia, to ao.
So let the second of Ptolemy to be taught, which in our Table aforegoing is the fifth. I say therefore that the reason of io, unto oa; is made of the reason of iy, unto yu, and the reason of ue, unto ea. For now againe, from the beginning of the Antecedent of the Product i, let a line be drawne parallell unto ea, the Consequent of the second of the Makers, which shall meete with eo, drawne out at length, in n: therefore, by the 32. e. as io, is to ao; so is en, unto ea. Therefore now again the multiplied proportions are thus:
by the 32. e. Therefore, by multiplication of proportions, the reason of en, unto ea, that is, of io, unto oa, is made of the reason of iy, unto yu, by the reason of ue, 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 oe, unto ey, is made of the reason oa, unto ai, by the reason of iu, unto uy. For from the end o, to wit, from the beginning of the Antecedent of the product, let the right line no, be drawne parallell to uy. It shall be by the 32. e. as oe, is to ey: so the parallell no, shall be to uy: but the reason of no, unto uy, is made of the reason of oa, unto ai, and of iu, unto uy: for the multiplied proportions are,
by the 32. e.
Againe, I say, that the reason of ey, unto yo, is compounded of the reason of eu, unto ua, and of ai, unto io.
Theon here draweth a parallell from o, unto ui. By the generall fabricke it may be drawne out of e, unto ui.
It shall be therefore as ey, is unto yo, so en, shall be unto oi. Now the proportions multiplied are,
by the 32. e.
Therefore the reason of en, unto io , that is of ey, unto yo, 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 Althindus, concerning this matter, are in many mens hands. And Regiomontanus in his Algorithmus: and Maurolycus upon the 1 p iij. of Menelaus, 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.
Of Geometry, the sixt Booke, of a Triangle
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 the 24. 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 the 4. 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.
4. A rightilineall doth make all his angles equall to right angles; the inner ones generally to paires from two forward: the outter always to foure.
Or thus: A right lined plaine maketh his angles equall unto right angles: Namely the inward angles generally, are equall unto the even numbers from two forward, but the outward angles are equall but to 4. right angles. H.
The first kinde I meane of rectilineals, that is a triangle doth make all his inner angles equall to two right angles, that is, to a binary, the first even number of right angles: the second, that is a quadrangle, to the second even number, that is, to a quaternary or foure: The third, that is, a Pentangle, of quinqueangle to the third, that is a senary of right angles, or 6. and so farre forth as thou seest in this Arithmeticall progression of even numbers,
Notwithstanding the outter angles, every side continued and drawne out, are alwayes equall to a quaternary of right angles, that is to foure. The former part being granted (for that is not yet demonstrated) the latter is from thence concluded: For of the inner angles, that of the outter, is easily proved. For the three angles of a triangle are equall to two right angles. The foure of a quadrangle to foure: of a quinquangle, to sixe: of a sexe angle, to eight: Of septangle, to tenne, and so forth, form a binarie by even numbers: Whereupon, by the 14. e. V. a perpetuall quaternary of the outer angles is concluded.
5. A rectilineall is either a Triangle or a Triangulate.
As before of a line was made a lineate: so here in like manner of a triangle is made a triangulate.
6. A triangle is a rectilineall figure comprehended of three rightlines. 21. d j.
As here aei. A triangular figure is of Euclide defined from the three sides; whereupon also it might be called Trilaterum, that is three sided, of the cause: rather than Trianglum, three cornered, of the effect; especially seeing that three angles, and three sides are not reciprocall or to be converted. For a triangle may have foure sides, as is Acidoides, or Cuspidatum, the barbed forme, which Zonodorus called Cœlogonion, or Cavangulum, an hollow cornered figure. It may also have both five, and sixe sides, as here thou seest. The name therefore of Trilaterum would more fully and fitly expresse the thing named: But use hath received and entertained the name of a triangle for a trilater: And therefore let it be still retained, but in that same sense:
7. A triangle is the prime figure of rectilineals.
A triangle or threesided figure is the prime or most simple figure of all rectilineals. For amongst rectilineall figures there is none of two sides: For two right lines cannot inclose a figure. What is meant by a prime figure, was taught at the 11. e. iiij.
And
8. If an infinite right line doe cut the angle of a triangle, it doth also cut the base of the same: Vitell. 29. t j.
9. Any two sides of a triangle are greater than the other.
Thus much of the difinition of a triangle; the reason or rate in the sides and angles of a triangle doth follow. The reason of the sides is first.
Let the triangle be aei; I say, the side ai, is shorter, than the two sides ae, and ei, because by the 6. e ij, a right line is betweene the same bounds the shortest.
Therefore
10. If of three right lines given, any two of them be greater than the other, and peripheries described upon the ends of the one, at the distances of the other two, shall meete, the rayes from that meeting unto the said ends, shall make a triangle of the lines given.
Let it be desired that a triangle be made of these three lines, aei, given, any two of them being greater than the other: First let there be drawne an infinite right; From this let there be cut off continually three portions, to wit, ou, uy, and ys, equall to ae, and i, the three lines given. Then upon the ends y, and u, at the distances ou, and ys; let two peripheries meet in the point r. The rayes from that meeting unto the said ends, u, and y, shall make the triangle ury: for those rayes shall be equall to the right lines given, by the 10. e v.
And
11. If two equall peripheries, from the ends of a right line given, and at his distance, doe meete, lines drawne from the meeting, unto the said ends, shall make an equilater triangle upon the line given. 1 p. j.
As here upon ae, there is made the equilater triangle, aei; And in like manner may be framed the construction of an equicrurall triangle, by a common ray, unequall unto the line given; and of a scalen or various triangle, by three diverse raies; all which are set out here in this one figure. But these specialls are contained in the generall probleme: neither doe they declare or manifest unto us any new point of Geometry.
12. If a right line in a triangle be parallell to the base, it doth cut the shankes proportionally: And contrariwise. 2 p vj.
Such therefore was the reason or rate of the sides in one triangle; the proportion of the sides followeth.
As here in the triangle aei, let ou, be parallell to the base; and let a third parallel be understood to be in the toppe a; therefore, by the 28. e. v. the intersegments are proportionall.
The converse is forced out of the antecedent: because otherwise the whole should be lesse than the part. For if ou, be not parallell to the base ei, then yu, is: Here by the grant, and by the antecedent, seeing ao, oe, ay, ye, are proportionall: and the first ao, is lesser than ay, the third: oe, the second must be lesser than ye, the fourth, that is the whole then the part.
13. The three angles of a triangle, are equall to two right angles. 32. p j.
Hitherto therefore is declared the comparison in the sides of a triangle. Now is declared the reason or rate in the angles, which joyntly taken are equall to two right angles.
The truth of this proposition, saith Proclus, according to common notions, appeareth by two perpendiculars erected upon the ends of the base: for looke how much by the leaning of the inclination, is taken from two right angles at the base, so much is assumed or taken in at the toppe, and so by that requitall the equality of two right angles is made; as in the triangle aei, let, by the 24. e v, ou, be parallell against ie. Here three particular angles, iao, iae, eau, are equall to two right lines; by the 14. e v. But the inner angles are equall to the same three: For first, eai, is equall to it selfe: Then the other two are equall to their alterne angles, by the 24. e v.
Therefore
14. Any two angles of a triangle are lesse than two right angles.
For if three angles be equall to two right angles, then are two lesser than two right angles.
And
15. The one side of any triangle being continued or drawne out, the outter angle shall be equal to the two inner opposite angles.
This is the rate of the inner angles in one and the same triangle: The rate of the outter with the inner opposite angles doth followe. As in the triangle aei, let the side ei, be continued or drawne out unto o; the two angles on each side aio and aie, are by the 14 e v. equall to two right angles: and the three inner angles, are by the 13. e. equall also to two right angles; take away aie, the common angle, and the outter angle aio, shall be left equall to the other two inner and opposite angles.
Therefore
16. The said outter angle is greater than either of the inner opposite angles. 16. p j.
This is a consectary following necessarily upon the next former consectary.
17. If a triangle be equicrurall, the angles at the base are equall: and contrariwise, 5. and 6. p. j.
The antecedent is apparent by the 7. e iij. The converse is apparent by an impossibilitie, which otherwise must needs follow. For if any one shanke be greater than the other, as ae: Then by the 7. e v. let oe, be cut off equall to it: and let oi, be drawne: then by 7. e iij. the base oi, must be equal to the base ae; but the base oi, is lesser than ae. For by the 9. e, ia; and ao, (to which ae, is equall, seeing that oe, is supposed to be equall to the same ai: and ao, is common to both) are greater than the said oi; therefore the same, oi, must be equall to the same ae, and lesser than the same, which is impossible. This was first found out by Thales Milesius.
Therefore
18. If the equall shankes of a triangle be continued or drawne out, the angles under the base shall be equall betweene themselves.
For the angles aei, and ieo: Item aie, and eiu, are equall to two right angles, by the 14. e v. Therefore they are equall betweene themselves: wherefore if you shall take away the inner angles, equall betweene themselves, you shall leave the outter equall one to another.
And
19. If a triangle be an equilater, it is also an equiangle: And contrariwise.
It is a consectary out of the condition of an equicrurall triangle of two, both shankes and angles, as in the example aei, shall be demonstrated.
And
20. The angle of an equilater triangle doth countervaile two third parts of a right angle. Regio. 23. p j.
For seeing that 3. angles are equall to 2. 1. must needs be equall to 2/3.
And
21. Sixe equilater triangles doe fill a place.
As here. For 2/3. of a right angle sixe lines added together doe make 12/3. that is foure right angles; but foure right angles doe fill a place by the 27. e. iiij.
22. The greatest side of a triangle subtendeth the greatest angle; and the greatest angle is subtended of the greatest side. 19. and 18. p j.
Subtendere, to draw or straine out something under another; and in this place it signifieth nothing else but to make a line or such like, the base of an angle, arch, or such like. And subtendi, is to become or made the base of an angle, arch, of a circle, or such like: As here, let ai, be a greater side than ae, I say the angle at e, shall be greater than that at i. For let there be cut off from ai, a portion equall to ae,; and let that be io: then the angle aei, equicrurall to the angle oie, shall be greater in base, by the grant. Therefore the angle shall be greater, by the 9 e iij.
The converse is manifest by the same figure: As let the angle aei, be greater than the angle aie. Therefore by the same, 9 e iij. it is greater in base. For what is there spoken of angles in generall, are here assumed specially of the angles in a triangle.
23. If a right line in a triangle, doe cut the angle in two equall parts, it shall cut the base according to the reason of the shankes; and contrariwise. 3. p vj.
The mingled proportion of the sides and angles doth now remaine to be handled in the last place.
Let the triangle be aei; and let the angle eai, be cut into two equall parts, by the right line ao: I say, as ea, is unto ai, so eo, is unto oi. For at the angle i, let the parallell iu, by the 24. e v. be erected against ao; and continue or draw out ea, infinitly; and it shall by the 20. e v. cut the same iu, in some place or other. Let it therefore cut it in u. Here, by the 28. e v. as ea, is to au, so is eo, to oi. But au, is equall to ai, by the 17. e. For the angle uia, is equall to the alterne angle oai, by the 21. e v. And by the grant it is equall to oae, his equall: And by the 21. e v. it is equall to the inner angle aui; and by that which is concluded it is equall to uia, his equall. Therefore by the 17. e, au, and ai, are equall. Therefore as ea, is unto ai, so is eo, unto oi.
The Converse likewise is demonstrated in the same figure. For as ea, is to ai; so is eo, to oi: And so is ea, to au, by the 12 e: therefore ai, and au, are equall, Item the angles eao, and oai, are equall to the angles at u, and i, by the 21. e v. which are equall betweene themselves by the 17. e.
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