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Of Geometry, the thirteenth Booke, Of an Oblong
1 An Oblong is a rectangle of inequall sides, 31. d j.
Or thus: An Oblong is a rectangled parallelogramme, being not equilater: H. As here is ae, io.
This second kinde of rectangle is of Euclide in his elements properly named for a definitions sake onely.
The rate of Oblongs is very copious, out of a threefold section of a right line given, sometime rationall and expresable by a number: The first section is as you please, that is, into two segments, equall or unequall: From whence a five-fold rate ariseth.
2 An oblong made of an whole line given, and of one segment of the same, is equall to a rectangle made of both the segments, and the square of the said segment. 3. p ij.
It is a consectary out of the 7 e xj. For the rectangle of the segments, and the quadrate, are made of one side, and of the segments of the other.
As let the right line ae, be 6. And let it be cut into two parts ai, 2. and ie, 4. The rectangle 12. made of ae, 6. the whole, and of ai, 2. the one segment, shall be equall to iu, 8. the rectangle made of the same ai, 2. and of ie, 4. And also to ao, 4. the quadrate of the said segment ai, 2.
Now a rectangle is here therefore proposed, because it may be also a quadrate, to wit, if the line be cut into to equall parts.
Secondarily,
3 Oblongs made of the whole line given, and of the segments, are equall to the quadrate of the whole 2 p ij.
This is also a Consectary out of the 7. e xj.
As let the line ae, 6. be cut into ai, 2. io. 2. and oe, 2. The Oblongs as, 12. ir, 12. and oy, 12. made of the whole ae, and of those segments, are equall to ay, the quadrates of the whole.
Here the segments are more than two, and yet notwithstanding from the first the rest may be taken for one, seeing that the particular rectangle in like manner is equall to them. This proposition is used in the demonstration of the 9. e xviij.
Thirdly,
4 Two Oblongs made of the whole line given, and of the one segment, with the third quadrate of the other segment, are equall to the quadrates of the whole, and of the said segment. 7 p ij.
As for example, let the right line ae, 8. be cut into ai, 6. and ie, 2. The oblongs ao, and iu, of the whole, and 2. the segments, are 32. The quadrate of 6. the other segment 36. And the whole 68. Now the quadrate, of the whole ae. 8. is 64. And the quadrate of the said segment 2, is 4. And the summe of these is 68.
5. The base of an acute triangle is of lesse power than the shankes are, by a double oblong made of one of the shankes, and the one segment of the same, from the said angle, unto the perpendicular of the toppe. 13 p. ij.
As in the triangle aei, let the angle at i, be taken for an acute angle. Here by the 4. e, two obongs of ei, and oi, with the quadrate of eo, are equall to the quadrates of ei, and oi. Let the quadrate of ao, be added to both in common. Here the quadrate of ei, with the quadrates of io, and oa, that is the 9 e xij, with the quadrate of ia, is equall to two oblongs of ei, and oi, with two quadrates of eo & oa, that is by the 9 e xij, with the quadrate of ea. Therefore two oblongs with the quadrate of the base, are equall to the quadrates of the shankes: And the base is exceeded of the shankes by two oblongs.
And from hence is had the segment of the shanke toward the angle, and by that the perpendicular in a triangle.
Therefore
6. If the square of the base of an acute angle be taken out of the squares of the shankes, the quotient of the halfe of the remaine, divided by the shanke, shall be the segment of the dividing shanke from the said angle unto the perpendicular of the toppe.
As in the acute angled triangle aei, let the sides be 13, 20, 21. And let ae be the base of the acute angle. Now the quadrate or square of 13 the said base is 169: And the quadrate of 20, or ai, is 400: And of 21, or ei, is 441. The summe of which is 841. And 841, 169, are 672: Whose halfe is 336. And the quotient of 336, divided by 21, is 16, the segment of the dividing shanke ei, from the angle aei, unto ao, the perpendicular of the toppe. Now 21, 16, are 5. Therefore the other segment or portion of the said ei, is 5.
Now againe from 169, the quadrate of the base 13, take 25, the quadrate of 5, the said segment: And the remaine shall be 144, for the quadrate of the perpendicular ao, by the 9 e xij.
Here the perpendicular now found, and the sides cut, are the sides of the rectangle, whose halfe shall be the content of the Triangle: As here the Rectangle of 21 and 12 is 252; whose halfe 126, is the content of the triangle.
The second section followeth from whence ariseth the fourth rate or comparison.
7. If a right line be cut into two equall parts, and otherwise; the oblong of the unequall segments, with the quadrate of the segment betweene them, is equall to the quadrate of the bisegment. 5 p ij.
As for example, Let the right line ae 8, be cut into two equall portions, ai 4, and ie 4. And otherwise that is into two unequall portions, ao 7, and oe 1: The oblong of 7 and 1, with 9, the quadrate of 3, the intersegment, (or portion cut betweene them) that is 16; shall bee equall to the quadrate of ie 4, which is also 16. Which is also manifest by making up the diagramme as here thou seest. For as the parallelogramme as is by the 26 e x, equall to the parallelogramme iu; And therefore by the 19 e x, it is equall to oy. For ou, is common to both the equall complements, Therefore if so be added in common to both; the ar, shall be equall to the gnomon mni: Now the quadrate of the segment betweene them is sl. Wherefore ar, the oblong of the unequall segments, with s the quadrate of the intersegment, is equall to iy the quadrate of the said bisegment.
The third section doth follow, from whence the fifth reason ariseth.
8. If a right line be cut into equall parts; and continued; the oblong made of the continued and the continuation, with the quadrate of the bisegment or halfe, is equall to the quadrate of the line compounded of the bisegment and continuation. 6 p ij.
As for example, let the line ae 6, be cut into two equall portions, ai 3, and ie 3: And let it be continued unto eo 2: The oblong 16, made of 8 the continued line, and of 2, the continuation; with 9 the quadrate of 3, the halfe, (that is 25.) shall be equall to 25, the quadrate of 3, the halfe and 2, the continuation, that is 5. This as the former, may geometrically, with the helpe of numbers be expressed. For by the 26 e x, as is equall to iy: And by the 19 e x, it is equall to yr, the complement. To these equalls adde so. Now the oblong au, shall be equall to the gnomon nju. Lastly, to the equalls adde the quadrate of the bisegment or halfe. The Oblong of the continued line and of the continuation, with the quadrate of the bisegment, shall be equall to the quadrate of the line compounded of the bisegment and continuation. These were the rates of an oblong with a rectangle.
From hence ariseth the Mesographus or Mesolabus of Heron the mechanicke; so named of the invention of two lines continually proportionall betweene two lines given. Whereupon arose the Deliacke probleme, which troubled Apollo himselfe. Now the Mesographus of Hero is an infinite right line, which is stayed with a scrue-pinne, which is to be moved up and downe in riglet. And it is as Pappus saith, in the beginning of his III booke, for architects most fit, and more ready than the Plato's mesographus. The mechanicall handling of this mesographus, is described by Eutocius at the 1 Theoreme of the II booke of the spheare; But it is somewhat more plainely and easily thus layd downe by us.
9. If the Mesographus, touching the angle opposite to the angle made of the two lines given, doe cut the said two lines given, comprehending a right angled parallelogramme, and infinitely continued, equally distant from the center, the intersegments shall be the meanes continually proportionally, betweene and two lines given.
Or thus: If a Mesographus, touching the angle opposite to the angle made of the lines given, doe cut the equall distance from the center, the two right lines given, conteining a right angled parallelogramme, and continued out infinitely, the segments shall be meane in continuall proportion with the line given: H.
As let the two right-lines given be ae, and ai: And let them comprehend the rectangled parallelogramme ao: And let the said right lines given be continued infinitely, ae toward u; and ai toward y. Now let the Mesographus uy, touch o, the angle opposite to a: And let it cut the sayd continued lines equally distant from the Center.
(The center is found by the 8 e iiij, to wit, by the meeting of the diagonies: For the equidistance from the center the Mesographus is to be moved up or downe, untill by the Compasses, it be found.)
Now suppose the points of equidistancy thus found to be u, and y. I say, That the portions of the continued lines thus are the meane proportionalls sought: And as ae is to iy: so is iy to eu, so is eu, to ai.
First let from s, the center, sr be perpendicular to the side ae: It shall therefore cut the said ae, into two parts, by the 5 e xj: And therefore againe, by the 7 e, the oblong made of au, and ue, with the quadrate of re, is equall to the quadrate of ru: And taking to them in common rs, the oblong with two quadrates er, and rs, that is, by the 9 e xij, with the quadrate se is equall to the quadrates ru and rs, that is by the 9 e xij, to the quadrate su. The like is to be said of the oblong of ay, and yi, by drawing the perpendicular sl, as afore. For this oblong with the quadrates li, and sl, that is, by the 9 e xij, with the quadrate is, is equall to the quadrates yl, and ls, that is, by the 9 e 12, to ys. Therefore the oblongs equall to equalls, are equall betweene themselves: And taking from each side of equall rayes, by the 11 e x, equall quadrates se and si, there shall remaine equalls. Wherefore by the 27 e x, the sides of equall rectangles are reciprocall: And as au is to ay: so by the 13 e vij, oi, that is, by the 8 e x, ea, to iy: And so therefore by the concluded, yi is to ue; And so by the 13 e vij, is ue to eo, that is, by the 8 e x, unto ai. Therefore as ea is to yi: so is yi to ue; and so is ue, to ai. Wherefore eu, iy, the intersegments or portions cut, are the two meane proportionals betweene the two lines given.
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.
