Therefore
Therefore
Therefore
Figure for demonstration 7.
7If a right line given, cut proportionall, be continued at each end with the greater segment, and sixe peripheries at the distance of the line given shall meete, two on each side from the ends of the line given and the continued, two others from their meetings, right lines drawne from their meetings, & the ends of the assigned shall make an ordinate quinquangle upon the assigned.
The example is thus.
8If the diameter of a circle circumscribed about a quinquangle be rationall, it is irrationall unto the side of the inscribed quinquangle, è 11. p xiij.
So before the segments of a right line proportionally cut were irrationall.
The other triangulates hereafter multiplied from the ternary, quaternary, or quinary of the sides, may be inscribed into a circle by an inscript triangle, quadrate, or quinquangle. Therefore by a triangle there may be inscribed a triangulate of 6. 12, 24,48, angles: By a quadrate, a triangulate of 8. 16, 32, 64, angles. By a quinquangle, a triangulate of 10, 20, 40, 80. angles, &c.
9The ray of a circle is the side of the inscript sexangle. è 15 p iiij.
Figure for demonstration 8.
A sexangle is inscribed by an inscript equilaterall triangle, by halfing of the three angles of the said triangle: But it is done more speedily by the ray or semidiameter of the circle, sixe times continually inscribed. As in the circle given, let the diameter beae; And upon the centero, with the rayie, let the peripheryuio, be described: And from the pointsoandu, let the diameters beoy, andus; These knit both one with another, and also with the diameteraeshall inscribe an equilaterall sexangle into the circle given, whose side shal be equal to the ray of the same circle. Aseu, is equal toui, because they both equall to the sameie, by the29 e, iiij. There foreeiu, is an equilater triangle: And likewiseeio, is an equilater. The angles also in the center are ⅔ of one rightangle: And therefore they are equall. And by the14. e v, the anglesio, is ⅓. of two rightangles: And by the15. e v. the angles at the toppe are also equall. Wherefore sixe are equall: And therefore, by the7 e xvj. and32. e, xv, all the bases are equall, both betweene themselves, and as was even now made manifest, to the ray of the circle given. Therefore the sexangle inscript by the ray of a circle is anequilater; And by the1 e xvij. equiangled.
Therefore
Therefore
Therefore
10Three ordinate sexangles doe fill up a place.
Figure for demonstration 10.
As here. For they are sixe equilater triangles, if you shal resolve the sexangles into sixe triangls: Or els because the angle of an ordinate sexangle is as much as one right angle and ⅓. of a right angle.
Furthermore also no one figure amongst the plaines doth fill up a place. A Quinquangle doth not: For three angles a quinquangle may make only 3.3/5 angles which is too little. And foure would make 4.4/5 which is as much too great. The angles of a septangle would make onely two rightangles, and 6/7 of one: Three would make 3, and 9/7, that is in the whole 4.2/7, which is too much, &c. to him that by induction shall thus make triall, it will appeare, That a plaine place may be filled up by three sorts of ordinate plaines onely.
And
And
And
11If right lines from one angle of an inscript sexangle unto the third angle on each side be knit together, they shall inscribe an equilater triangle into the circle given.
Figure for demonstration 11.
As here; Because the sides shall be subtended to equall peripheries: Therefore by the32 e xv. they shall be equall betweene themselves: And againe, on the contrary, by such a like triangle, by halfing the angles, a sexangle is inscribed.
12The side of an inscribed equilater triangle hath atreble power, unto the ray of the circle 12. p xiij.
Figure for demonstration 12.
As here, withae, one side of the triangleaei, two third parts of the halfe periphery are imployed: For with one side one third of the wholeeu, is imployed: Thereforeeu, is the other third part, that is, the sixth part of the whole periphery. Therefore the inscripteu, is the ray of the circle, by the9 e. Now the power of the diameteraou, by the14 e xij. is foure times so much as is the power of the ray, that is, ofeu: And by21. e xvj, and9 e xij,ae, andeu, are of the same power; take awayeu, and the sideae, shall be of treble power unto the ray.
13If the side of a sexangle be cut proportionally, the greater segment shall be the side of the decangle.
Figure for demonstration 13.
Pappus lib. 5. ca. 24.&Campanus ad 3 p xiiij.Let the rayao, or side of the sexangle be cut proportionally, by the3 e xiiij: And letae, be equall to the greater segment. I say thatae, is the side of the decangle. For if it be moreover continued with the whole ray untoi, the wholeaei, shallbythe4 e xiiij. be cut proportionally: and the greater segmentei, shal be the same ray. For the if the right lineiea, be cut proportionally, it shall be asia, is untoie, that is tooa, to wit, unto the ray: soao, shal be untoae. Therefore, by the15. e vij. the trianglesiao, andoae, are equiangles: And the angleaoe, is equall to the angleoia. But the angleuoe, is foure times as great as the angleaoe: for it is equall to the two inner ata, ande, by the15 e vj: which are equall between themselves, by the10 e v. and by the17 e vj. And therefore it is the double ofaeo, which is the double, for the same cause, ofaio, equall to the sameaoe. Thereforeuoe, is the quadruple of the saidaoe. Thereforeue, is the quadruple of the peripheryea. Therefore the wholeuea, is the quintuple of the sameea: And the whole periphery is decuple unto it. And the subtenseae, is the side of the decangle.
Therefore
Therefore
Therefore
14If a decangle and a sexangle be inscribed in the same circle, a right line continued and made of both sides, shall be cut proportionally, and the greater segment shall be the side of a sexangle; and if the greater segment of a right line cut proportionally be the side of an hexagon, the rest shall be the side of a decagon. 9. p xiij.
The comparison of the decangle and the sexangle with the quinangle followeth.
15If a decangle, a sexangle, and a pentangle be inscribed into the same circle the side of the pentangle shall in power countervaile the sides of the others. And if a right line inscribed do countervaile the sides of the sexangle and decangle, it is the side of the pentangle. 10. p xiiij.
Figure for demonstration 15.
Let the side of the inscribed quinquangle beae: of the sexangle,ei: Of the decangleao. I say, the sideae, doth in power countervaile the rest. For let there be two perpēdiculars: The firstio, the secondiu, cutting the sides of the quinquangle and decangle into halves: And the meeting of the second perpendicular with the side of the quinquangle let it bey. The syllogisme of the demonstration is this: The oblongs of the side of the quinquangle, and the segments of the same, are equall to the quadrates of the other sides. But the quadrate of the same whole side, is equall to the oblongs of the whole, and the segments, by the3 e, xiij. Therefore it is equall to the quadrates of the other sides.
Let the proportion of this syllogisme be demonstrated: For this part onely remaineth doubtfull. Therefore two triangles,aei, andyei, are equiangles, having one common angle ate: And also two equall onesaei, andeiy, the halfes, to wit, of the sameeis: Because that is, by the17 e, vj: one of the two equalls, unto the whicheis, the out angle, is equall, by the15 e. vj. And this doth insist upon a halfe periphery. For the halfe peripheryals, is equall to the halfe peripheryars: and alsoal, is equall toar. Therefore the remnantls, is equall to the remnantrs: And the wholerl, is the double of the samers: And thereforeer, is the double ofeo: Andrs, the double ofou. For the bisegments are manifest by the10 e, xv. and the11 e, xvj. Therefore the peripheryers, is the double of the peripheryeou: And therefore the angleeiu, is the halfe of the angleeis, by the7 e, xvj. Therefore two angles of two triangles are equall: Wherefore the remainder, by the4 e vij, is equall to the remainder. Wherefore by the12 e, vij, as the sideae, is toei: so isei, toey. Therefore by the8 e xij, the oblong of the extreames is equall to the quadrate of the meane.
Now letoy, be knit together with a straight: Here againe the two trianglesaoe, andaoy, are equiangles, having one common angle ata: Andaoy, andoea, therefore also equall: Because both are equall to the angle ata: That by the17 e, vj: This by the2 e, vij: Because the perpendicular halfing the side of the decangle, doth make two triangles, equicrurall, and equall by the right angle of their shankes: And therefore they are equiangles. Therefore asea, is toao: so isea, toay. Wherefore by the8 e, xij. the oblong of the two extremes is equall to the quadrate of the meane: And the proposition of the syllogisme, which was to be demonstrated. The converse from hence as manifestEuclidedoth use at the 16 p xiij.
Figure for demonstration 16.
16.If a triangle and a quinquangle be inscribed into the same Circle at the same point, the right line inscribed betweene the bases of the both opposite to the saidpoint, shall be the side of the inscribed quindecangle. 16. p. iiij.
For the side of the equilaterall triangle doth subtend 1/3 of the whole pheriphery. And two sides of the ordinate quinquangle doe subtend 2/5 of the same. Now 2/5 - 1/3 is 1/15: Therefore the space betweene the triangle, and the quinquangle shall be the 1/15 of the whole periphery.
Therefore
Therefore
Therefore
17.If a quinquangle and a sexangle be inscribed into the same circle at the same point, the periphery intercepted beweene both their sides, shall be the thirtieth part of the whole periphery.
As here. Therefore the inscription of ordinate triangulates, of a Quadrate, Quinquangle, Sexangle, Decangle, Quindecangle is easie to bee performed by one side given or found, which reiterated as oft as need shall require, shal subtend the whole periphery.Jun. 4.A. C.MDCXXII in apostrophus formCampana pulsante pro. H. W.
Figure for demonstration 17.
Out of the Adscription of a Circle and a Rectilineall is drawne the Geodesy of ordinate Multangles, and first of the Circle it selfe. For the meeting of two right lines equally, dividing two angles is the center of the circumscribed Circle: From the center unto the angle is the ray: And then if the quadrate of halfe the side be taken out of the quadrate of the ray, the side of the remainder shall be the perpendicular, by the9 e xij. Therefore a speciall theoreme is here thus made:
Figure for demonstration 1 in a quinquangle.
1.A plaine made of the perpendicular from the center unto the side, and of halfe the perimeter, is the content of an ordinate multangle.
As here; The quadrate of 10, the ray is 100. The quadrate of 6, the halfe of the side 12, is 36: And 100. 36 is 64, the quadrate of the Perpendicular, whose side 8, is the Perpendicular it selfe. Now the whole periphery of the Quinquangle, is 60. The halfe thereof therefore is 30. And the product of 30, by 8, is 240, for the content of the sayd quinquangle.
The Demonstration here also is of the certaine antecedent cause thereof. For of five triangles in a quinquangle, the plaine of the perpendicular, and of halfe the base is one of them, as in the former hath beene taught: Therefore fivesuch doe make the whole quinquangle. But that multiplication, is a multiplication of the Perpendicular by the Perimeter or bout-line.
Figure for demonstration 1 in a sexangle.
In an ordinate Sexangle also the ray, by the9 e xviij, is knowne by the side of the sexangle. As here, the quadrate of 6, the ray is 36. The quadrate of 3, the halfe of the side, is 9: And 36 - 9. are 27, for the quadrate of the Perpendicular, whose side 5.2/11 is the perpendicular it selfe. Now the whole perimeter, as you see, is 36. Therefore the halfe is 18. And the product of 18 by 5.2/11 is 93.3/11 for the content of the sexangle given.
Lastly in all ordinate Multangles this theoreme shall satisfie thee.
2The periphery is the triple of the diameter and almost one seaventh part of it.
Figure for demonstration 2.
Or the Periphery conteineth the diameter three times and almost one seventh of the same diameter. That it is triple of it, sixe raies, (that is three diameters) about which the periphery, the9 e xviij, is circumscribed doth plainely shew: And therefore the continent is the greater: But the excesse is not altogether so much as one seventh part. For there doth want an unity of one seventh: And yet is the same excesse farre greater than one eighth part. Therefore because the difference was neerer to one seventh, than it was to one eighth, therefore one seventh was taken, as neerest unto the truth, for the truth it selfe.
Therefore
Therefore
Therefore
3.The plaine of the ray, and of halfe the periphery is the content of the circle.
Figure for demonstration 3.
For here 7, the ray, of halfe the diameter 14, Multiplying 22, the halfe of the periphery 44, maketh the oblong 154, for the content of the circle. In the diameter two opposite sides, and likewise in the perimeter the two other opposite sides of the rectangle are conteined. Therefore the halfes of those two are taken, of the which the rectangle is comprehended.
And
And
And
4.As 14 is unto 11, so is the quadrate of the diameter unto the Circle.
For here 3 bounds of the proportion are given inpotentia: The fourth is found by the multiplication of the third by the second, and by the Division of the product by the first: As here the Quadrate of the diameter 14, is 196. The product of 196 by 11 is 2156. Lastly 2156 divided by 14, the first bound, giveth in the Quotient 154, for the content of the circle sought. This ariseth by an analysis out of the quadrate and Circle measured. For the reason of 196, unto a 154; is the reason of 14 unto 11, as will appeare by the reduction of the bounds.
This is the second manner of squaring of a circle taught byEuclideasHerotelleth us, but otherwise layd downe, namely after this manner.If from the quadrate of the diameter you shall take away 3/14 parts of the same, the remainder shall be the content of the Circle.As if 196, the quadrate be divided by 14, the quotient likewise shall be 14. Now thrise 14, are 42: And 196 - 42, are 154, the quadrate equall to the circle.
Out of that same reason or rate of the pheriphery anddiameter ariseth the manner of measuring of the Parts of a circle, as of a Semicircle, a Sector, a Section, both greater and lesser.
And
And
And
5.The plaine of the ray and one quarter of the periphery, is the content of the semicircle.
Figure for demonstration 5.
As here thou seest: For the product of 7, the halfe of the diameter, multiplyed by 11, the quarter of the periphery, doth make 77, for the content of the semicircle.
This may also be done by taking of the halfe of the circle now measured.
And
And
And
6.The plaine made of the ray and halfe the base, is the content of the Sector.
Figure for demonstration 6.
Here are three sectours,aethe base of 12 foote: Andeiin like manner of 12 foote. The other or remainderiaof 7f. and 3/7 of one foote. The diameter is 10 foote. Multiply therefore 5, halfe of the diameter, by 6 halfe of the base, and the product 30, shall be the content of the first sector. The same shall also be for the second sectour. Againe multiply the same ray or semidiameters 5, by 3.5/7, the halfe of 7.3/7, the product of 18.4/7 shall be the content of the third sector. Lastly, 30 + 30 + 18.4/7 are 78.4/7, the content of the whole circle.
And
7.If a triangle, made of two raies and the base of the greater section, be added unto the two sectors in it, the whole shall be the content of the greater section: If the same be taken from his owne sector, the remainder shall be the content of the lesser.
In the former figure the greater section isaei: The lesser isai. The base of them both is as you see, 6. The perpendicular from the toppe of the triangle, or his heighth is 4. Therefore the content of the triangle is 12. Wherefore 30 + 30 + 12, that is 72, is the content of the greater sectionaei. And the lesser sectour, as in the former was taught, is 18.4/7. Therefore 18.4/7 - 12, that is, 6.4/7, is the content ofai, the lesser section.
And
And
And
8.A circle of unequall isoperimetrall plaines is the greatest.
Figures for demonstration 8.
The reason is because it is the most ordinate, andcomprehended of most bounds; see the7, and15 e iiij. As the Circlea, of 24 perimeter, is greater then any rectilineall figure, of equall perimeter to it, as the Quadratee, or the Trianglei.
Figures for demonstration 8.
1.A bossed surface is a surface which lyeth unequally betweene his bounds.
It is contrary unto a Plaine surface, as wee heard at the4 e v.
Sphericall surface.
2.A bossed surface is either a sphericall, or varium.
3.A sphericall surface is a bossed surface equally distant from the center of the space inclosed.
Therefore
Therefore
Therefore
4.It is made by the turning about of an halfe circumference the diameter standeth still. è 14 d xj.
Figure for demonstration 4.
As here if thou shalt conceive the space betweene the periphery and the diameter to be empty.
5.The greatest periphery in a sphericall surface is that which cutteth it into two equall parts.
Those things which were before spoken of a circle, the same almost are hither to bee referred. The greatest periphery of a sphericall doth answere unto the Diameter of a Circle.
Therefore
Figure for demonstration 6.
6.That periphery that is neerer to the greatest, is greater than that which is farther off: And on eachside those two which are equally distant from the greatest, are equall.
The very like unto those which are taught at the15,16,17,18. e. xv. may here againe be repeated: As here.
7The plaine made of the greatest periphery and his diameter is the sphericall.
Figure for demonstration 7.
So the plaine made of the diameter 14. and of 44. the greatest periphery, which is 616. is the sphericall surface. So before the content of a circle was measured by a rectangle both of the halfe diameter, and periphery. But here, by the whole periphery and whole diameter, there is made a rectangle for the measure of the sphericall, foure times so great as was that other: Because by the1 e vj. like plaines (such as here are conceived to be made of both halfe the diameter, and halfe the periphery, and both of the whole diameter and whole periphery) are in a doubled reason of their homologall sides.
Therefore
8A plaine of the greatest circle and 4, is the sphericall.
This consectarium is manifest out of the former element.
And
And
And
9As 7 is to 22. so is the quadrate of the diameter unto the sphericall.
For 7, and 22, are the two least bounds in the reason of the diameter unto the periphery: But in a circle, as 14, is to 11, so is the quadrate of the diameter unto the circle. The analogie doth answer fitly: Because here thou multipliest by the double, and dividest by the halfe: There contrariwise thou multipliest by the halfe, and dividest by the double. Therefore there one single circle is made, here the quadruple of that. This is, therefore the analogy of a circle and sphericall; from whence ariseth the hemispherical, the greater and the lesser section.
And
And
And
10The plaine of the greatest periphery and the ray, is the hemisphericall.
As here, the greatest periphery is 44. the ray 7. The product therefore of 44. by 7. that is, 308. is the hemisphericall.
Figure for demonstration 11.
11If looke what the part be of the ray perpendicular from the center unto the base of the greater section, so much the hemisphericall be increased, the whole shall be the greater section of the sphericall: But if it be so much decreased, the remainder shall be the lesser.
As in the example, the part of the third ray, that is, of 3/7, is from the center: such like part of the hemispherical 308, is 132. (For the 7, part of 308. is 44. And three times 44. is 132.) Therefore 132. added to 308. do make 440. for the greater section of the sphericall. And 132. taken from 308. doe leave 176. for the lesser section of the same.
12The varium is a bossed surface, whose base is aperiphery, the side a right line from the bound of the toppe, unto the bound of the base.
13A varium is a conicall or a cylinderlike forme.
14A conicall surface is that which from the periphery beneath doth equally waxe lesse and lesse unto the very toppe.
Therefore
Therefore
Therefore
Conicall surface.
15.It is made by turning about of the side about the periphery beneath.
16The plaine of the side and halfe the base is the conicall surface.
As in the example next aforegoing, the side is 13. The halfe periphery is 15.5/7: And the product of 15.5/7 by 13. is 204.2/7. for the conicall surface. To which if you shall adde the circle underneath, you shall have the whole surface.
Cylinderlike forme.
17A cylinderlike forme is that which from the periphery underneath unto the the upper one, equall and parallell unto it, is equally raised.
Therefore
18It is made by the turning of the side about two equall and parallell peripheries.
19The plaine of his side and heighth is the cylinderlike surface.
As here the periphery is 22. as is gathered by the Diameter, which is 7. The heighth is 12. The base therefore is 38.1/2. And 38.1/2 by 12. are 462. for the cylinderlike surface. To which if you shall adde both the bases on each side, to wit, 38.1/2. twise, or 77. once, the whole surface shall be 539.
Body or solid.
1A body or solid is a lineate broad and high 1 d xj.
For length onely is proper to a line: Length and breadth, to a surface: Length breath, and heighth joyntly, belong unto a body: This threefold perfection of a magnitude, is proper to a body: Whereby wee doe understand that are in a body, not onely lines of length, and surfaces of breadth, (for so a body should consist of lines and surfaces.) But we do conceive a solidity in length, breadth and heighth. For every part of a body is also a body. And therefore a solid we doe understand the body it selfe. As in the bodyaeio, the length isae; the breadth,ai, And the heighth,ao.
2The bound of a solid is a surface 2 d xj.
The bound of a line is a point: and yet neither is a point a line, or any part of a line. The bound of a surface is a line: And yet a line is not a surface, or any part of a surface. So now the bound of a body is a surface: And yet a surface is not a body, or any part of a body. A magnitude is one thing;a bound of a magnitude is another thing, as appeared at the5 e j.
As they were called plaine lines, which are conceived to be in a plaine, so those are named solid both lines and surfaces which are considered in a solid; And their perpendicle and parallelisme are hither to be recalled from simple lines.
Figure for demonstration 3.
3If a right line be unto right lines cut in a plaine underneath, perpendicular in the common intersection, it is perpendicular to the plaine beneath: And if it be perpendicular, it is unto right lines, cut in the same plaine, perpendicular in the common intersection è 3 d and 4 p xj.
Perpendicularity was in the former attributed to lines considered in a surface. Therefore from thence is repeated this consectary of the perpendicle of a line with the surface it selfe.
If thou shalt conceive the right lines,ae,io,uy, to cut one another in the plaine beneath, in the common intersections: And the liners, falling from above, to be to every one of them perpendicular in the common points, thou hast an example of this consectary.
4If three right lines cutting one another, be unto the same right line perpendicular in the common section, they are in the same plaine 5. p xj.
For by the perpendicle and common section is understood an equall state on all parts, and therefore the same plaine: as in the former example,as,ys,os, suppose them to be tosr, the same loftie line, perpendicular, they shall be in the same nearer plaineaiueoy.
Figure for demonstration 5.
5If two right lines be perpendicular to the under-plaine, they are parallells: And if the one twoparallells be perpendicular to the under plaine, the other is also perpendicular to the same. 6. 8 p xj.
The cause is out of the first law or rule parallells. For if two right lines be perpendicular to the same under plaine, being joyned together by a right line, they shall make their inner corners equall to two right angles: And therefore they shall be parallells, by the21. e v. And if in two parallells knit together with a right line, one of the inner angles, be a right angle: the other also shall be a right angle. Because they are divided by a common perpendicular; As in the example. If the angles ata, ande, be right angles,ai, andeo, are parallells, and contrariwise, ifai, andeobe parallells, and the angle ata, be a right angle, the angle ate, also shall be a right angle.
6If right lines in diverse plaines be unto the same right line parallel, they are also parallell betweene themselves. 9 p xj.
Figure for demonstration 6.
As hereae, anduy, right lines in diverse plaines suppose them to be parallell toio: I say, they are parallell one to another. For from the pointi, letia, andiu, beerected at right angles toioto cut theparallells, by the17. e v. Therefore, by the3 e,oi, seeing that it is perpedicular toia, andiu, two lines cutting one another, it is perpendicular to the plaine beneath. Therefore by the the6 e,yu, andea, are perpendicular to the same plaine: And therefore, by the same, they are parallell.
Figure for demonstration 7.
7If two right lines be perpendiculars, the first from a point above, unto a right line underneath, the secondfrom the common section in the plaine underneath, a third, from the sayd point perpendicular to the second, shall be perpendicular to the plaine beneath. è 11 p xj.
It is a consectary out of the3 e. As for example, if from a lofty pointa,ae, be by the18 e v, perpendicular toe, a point of the right lineiounderneath: And fromethe common section, by the17 e v, there beeu, another perpendicular: Lastlyay, a lofty right line, be by the18 e v, perpendicular untoeu, at the pointy,ayshall be perpendicular unto the plaine underneath. For thataeis perpendicular toio, the sameaedeclineth neither to the right hand, nor to the left, by the13 e ij. And in that againeayis perpendicular toeu, it leaneth neither forward nor backeward. Therefore it lyeth equally or indifferently, betweene the foure quarters of the world.
If the right lineio, doe with equall angles agree tor, the third element.
8.If a right line from a point assigned of a plaine underneath, be parallell to a right line perpendicular to the same plaine, it shall also be perpendicular to the plaine underneath. ex 12 p xj.