The Fourteenth Booke, of P. Ramus Geometry: Of a Right Line Proportionally Cut: And of Other Quadrangles, and Multangels
Thus farre of the threefold section, from whence we have the five rationall rates of equality: There followeth of the third section another section, into two segments proportionall to the whole. The section it selfe is first to be defined.
1. A right line is cut according to a meane and extreame rate, when as the whole shall be to the greater segment; so the greater shall be unto the lesser. 3 d vj.
This line is cut so, that the whole line it selfe, with the two segments, doth make the three bounds of the proportion: And the whole it selfe is first bound: The greater segment is the middle bound: The lesser the third bound.
2. If a right line cut proportionally be rationall unto the measure given, the segments are unto the same, and betweene themselves irrationall è 6 p xiij.
Euclide calleth each of these segments Ἀποτομὴ that is, Residuum, a Residuall or remaine: And surely these cannot otherwise be expressed, then by the name Residuum; As if a line of 7 foote should thus be given or put downe: The greater segment shall be called a line of 7 foote, from whence the lesser is substracted: Neither may the lesser otherwise be expressed, but by saying, It is the part residuall or remnant of the line of 7 foote, from which the greater segment was subtracted or taken.
A Triangle, and all Triangulates, that is figures made of triangles, except a Rightangled-parallelogramme, are in Geometry held to be irrationalls. This is therefore the definition of a proportionall section: The section it selfe followeth, which is by the rate of an oblong with a quadrate.
3. If a quadrate be made of a right line given, the difference of the right line from the middest of the conterminall side of the said quadrate made, above the same halfe, shall be the greater segment of the line given proportionally cut: 11 p ij.
Or thus: If a square be made of a right line given, the difference of a right line drawne from the angle of the square made unto the middest of the next side, above the halfe of the side, shall be the greater segment of the line given, being proportionally cut: H.
Let the right line gived be ae. The quadrate of the same let it be aeio: And from the angle e, unto u, the middest of the conterminal side, let the right line eu, be drawne; Then compare or lay it to the halfe ua; The difference of it above the said halfe shall be ay, This ay, say 1, is the greater segment of ae, the line given, proportionally cut.
For of ya, let the quadrate aysr, be made: And let sr, be continued unto l. Now by the 8 e, xiij. the oblong of oy, and ay, with the quadrate of ua, is equall to the quadrate of uy, that is by the construction of ue: And therefore, by the 9 e xij. it is equall to the quadrates ea, and au: Take away from each side the common oblong al, and the quadrate yr, shall be equall to the oblong ri. Therefore the three right lines, ea, ar, and re, by the 8 e xij. are continuall proportionall. And the right line ae, is cut proportionally.
Therefore
4. If a right line cut proportionally, be continued with the greater segment, the whole shall be cut proportionally, and the greater segment shall be the line given. 5 p xiij.
As in the same example, the right line oy, is continued with the greater segment, and the oblong of the whole and the lesser segment is equall to the quadrate of the greater. And thus one may by infinitely proportionally cutting increase a right line; and againe decrease it. The lesser segment of a right line proportionally cut, is the greater segment, of the greater proportionally cut. And from hence a decreasing may be made infinitely.
5. The greater segment continued to the halfe of the whole, is of power quintuple unto the said halfe, that is, five times so great as it is: and if the power of a right line be quintuple to his segment, the remainder made the double of the former is cut proportionally, and the greater segment, is the same remainder. 1. and 2. p xiij.
This is the fabricke or manner of making a proportionall section. A threefold rate followeth: The first is of the greater segment.
Let therefore the right line ae, be cut proportionally in i: And let the greater segment be ia: and let the line cut be continued unto io, so that oa, be the halfe of the line cut. I say, the quadrate of io, is in power five times so great, as ys, the power of the quadrate of ao. Let therefore of ao, be made the quadrate iosr: We doe see the quadrate ua, to be once contained in the quadrate si. Let us now teach that it is moreover foure times comprehended in lmn, the gnomon remaining: Let therefore the quadrate aeiu, be made of the line given: And let ri, be continued unto f. Here the quadrate ae, is (14. e xij.) foure times so much as is that au, made of the halfe: and it is also equall to the gnomon lmn: For the part iu, is equall to ry; first by the grant, seeing that ai, is the greater segment, from whence ry, is made the quadrate, because the other Diagonall is also a quadrate: Secondarily the complements sy, and yi, by the 19. e x, are equall: And to them is equall af. For by the 23. e x. and by the grant, it is the double of the complement yi. Therefore it is equall to them both. Wherefore the gnomon lmn, is equall to the quadruple quadrate of the said little quadrate: And the greater segment continued to the halfe of the right line given is of power five fold to the power of ao.
The converse is apparent in the same example: For seeing that io, is of power five times so much as is ao; the gnomon lmn, shall be foure times so much as is ua: Whose quadruple also, by the 14. e xij, is av. Therefore it is equall to the gnomon. Now aj, is equall to ae: Therefore it is the double also of ao, that is of ay: And therefore by the 24. e x. it is the double of at: And therefore it is equall to the complements iy, and ys: Therefore the other diagonall yr, is equall to the other rectangle iv. Wherefore, by the 8 e xij. as ev, that is, ae, is to yt, that is ai: so is ai, unto ie; Wherefore by the 1 e, ae, is proportionall cut: And the greater segment is ai, the same remaine.
The other propriety of the quintuple doth follow.
6 The lesser segment continued to the halfe of the greater, is of power quintuple to the same halfe è 3 p xiij.
As here, the right line ae, let it be cut proportionally in i: And the lesser ie, let it be continued even unto o, the halfe of the greater ai. I say, that the power of oe, shall be five times as much as is the power of io. Let a quadrate therefore be made of ae: And let the figure be made up (as you see:) And let the quadrate of the halfe be noted with su: And the gnomon rlm. Here the first quadrate oy, is five times as great, as the second su. For it doe containe it once: And the gnomon rlm, remaining containeth it foure times. For it is equall to the Oblong in; because os, the complement is equall to sy, by the 19 e x; And therefore also it is equall to in; seeing the whole complement as, is equall to the whole complement sn: And av, is equall to os, by the construction, and 23. e x: And adding to both the common oblong iy, the whole gnomon is equal to the whole oblong. But the oblong in, is equall to the quadrate ai, by the grant, & 8 e xij. which by the 14. e xij. is foure times as great, as the quadrate su. Wherefore the lesser segment ie, continued to io, the halfe of the greater segment, is of power five times as much as is the halfe of the same.
The rate of the triple followeth.
7 The whole line and the lesser segment are in power treble unto the greater. è 4 p xiij.
Let the right line ae be proportionally cut in i, and let the figure be made up: The oblong ay, and io, with the quadrate su, by the 4 e xiij, are equall to the quadrates of ae, and ie, whose power is treble to that of ai. For they doe once containe the quadrate su; And each of the oblongs is equall to the same quadrate su, by the grant, and 8 e xij. Therefore they doe containe it thrise.
8 An obliquangled parallelogramme is either a Rhombus, or a Rhomboides.
9 A Rhombus is an obliquangled equilater parallelogramme 32 d j.
Whereupon it is apparant that a Rhombus is a square having the angles as it were pressed, or thrust nearer together, by which name, both the Byrt or Turbot, a Fish; and a Wheele or Reele, which Spinners doe use; and the quarrels in glasse windowes, because they are cut commonly of this forme, are by the Greekes and Latines so called.
It is otherwise of some called a Diamond.
10 A Rhomboides is an obliquangled parallelogramme not equilater 33. d j.
And a Rhomboides is so opposed to an oblong, as a Rhombus is to a quadrate.
So also looke how much the straightening or pressing together is greater, so much is the inequality of the obtuse and acute angles the greater. As here.
And the Rhomboides is so called as one would say Rhombuslike, although beside the inequality of the angles it hath nothing like to a Rhombus. An example of measuring of a Rhomboides is thus.
11 A Trapezium is a quadrangle not parallelogramme. 34. d j.
Of the quadrangles the Trapezium remaineth for the last place: Euclide intreateth this fabricke to be granted him, that a Trapezium may be called as it were a little table: And surely Geometry can yeeld no reason of that name.
The examples both of the figure and of the measure of the same let these be.
Therefore triangulate quadrangles are of this sort.
12 A multangle is a figure that is comprehended of more than foure right lines. 23. d j.
By this generall name, all other sorts of right lined figures hereafter following, are by Euclide comprehended, as are the quinquangle, sexangle, septangle, and such like inumerable taking their names of the number of their angles.
In every kinde of multangle, there is one ordinate, as we have in the former signified, of which in this place we will say nothing, but this one thing of the quinquangle. The rest shall be reserved untill we come to Adscription.
13 Multangled triangulates doe take their measure also from their triangles.
As here, this quinquangle is measured by his three triangles. The first triangle, whose sides are 9. 10. and 17. by the 18. e xij. is 36. The second, whose sides are 6, 17, and 17. by the same e, is 50.20/101. The third, whose sides are 17, 15. and 8. by the same, is 60. And the summe of 36. 50.20/101. and 60. is 146.20/101, for the whole content of the Quinquangle given.
14 If an equilater quinquangle have three sides equall, it is equiangled. 7 p 13.
This of some, from the Greeke is called Pentagon; of others a Pentangle, by a name partly Greeke partly Latine.
As in the Quinquangle aeiou, the three angles at a, e, and i, are equall: Therefore the other two are equall: And they are equall unto these. For let eu, ai, ia, be knit together with right lines. Here the triangles aei, and eau. by the grant, and by the 2 and 1 e vij. are equilaters and equiangles: And the Bases ai, and eu, are equall: And the Angles, eai, and aue, are equall: Item aeu, and eia. Therefore ay, and ye, are equall, by the 17 e vj. Item the remainder uy, is equall to the remainder yi, when from equals equals be subtracted. Moreover by the grant, and by the 17 e vj, oui, and oiu, are equall. Wherefore three are equall; And therefore the whole angle is equall at u, to the whole angle at i. And therefore it is equall to those which are equall to it.
I say moreover that the angle at o, is likewise equall, if ao, and oe, be knit together with a right line, as here: For three in like manner do come to be equall.
But if the three angles non deinceps not successively following be equall, as aio, the businesse will yet be more easie, as here: For the angles eua, and eoi, are equall by the grant: And the inner also eou, and euo. Therefore the wholes of two are equall. Of the other at e, the same will fall out, if iu, be knit together with a right line iu, as here: For the wholes of two shall be equall.