Figure for demonstration 11.
Let the periphery of the sectionaoe, to be halfed or cut into two equall parts. Let the baseae, be cut into two equall parts by the pendiculario, which shall cut the periphery ino, I say, thatao, andoe, are bisegments. For draw two right linesao, andoe, and thou shalt have two trianglesaio, andeio, equilaters by the2 e vij. Therefore the basesao, andoe, areequall: And by the32. e xv. equall peripheries to the subtenses.
HereEuclidedoth by congruency comprehende two peripheries in one, and so doe we comprehend them.
12An angle in a section is an angle comprehended of two right lines joyntly bounded in the base and in the periphery joyntly bounded 7 d iij.
Or thus: An angle in the section, is an angle comprehended under two right lines, having the same tearmes with the bases, and the termes with the circumference:H. Asaoe, in the former example.
Figure for demonstration 13.
13The angles in the same section are equall. 21. p iij.
Let the section beeauo, And in it the angles ata, &u: These are equall, because, by the5 e, they are the halfes of the angleeyo, in the center: Or else they are equall, by the7 e, because they insist upon the same periphery.
Here it is certaine that angles in a section are indeed angles in a periphery, and doe differ onely in base.
14The angles in opposite sections are equall to two right angles. 22. p iij.
Figure for demonstration 14.
For here the opposite angles ata, andi, are equall to the three angles of the triangleeoi, which are equall to two right angles, by the13 e vj. For firsti, is equall to it selfe: Thena, by parts is equall to the two other. Foreai, is equall toeoi, andiao, tooei, by the13 e. Therefore the opposite angles are equall to two right angles.
The reason or rate of a section is thus: The similitude doth follow.
15If sections doe receive [or containe] equall angles, they are alike è 10. d iij.
Figure for demonstration 15.
As hereaei, andouy. The triangle here inscribed, seeing they are equiangles, by the grant; they shall also be alike, by the12 e vij.
16If like sections be upon an equall base, they are equall: and contrariwise. 23, 24. p iij.
Figure for demonstration 16.
In the first figure, let the base be the same. And if they shall be said to unequall sections; and one of them greater than another, the angle in thataoe, shall be lesse than the angleaie, in the lesser section, by the16 e vj. which notwithstanding, by the grant, is equall.
In the second figure, if one section be put upon another, it will agree with it: Otherwise against the first part, like sections upon the same base, should not be equall. But congruency is here sufficient.
By the former two propositions, and by the9 e xv. one may finde a section like unto another assigned, or else from a circle given to cut off one like unto it.
Angle of a section.
17Angle of a section is that which is comprehended of the bounds of a section.
As hereeai: Andeia.
18A section is either a semicircle: or that which is unequall to a semicircle.
A section is two fold, a semicircle, to wit, when it is cut by the diameter: or unequall to a semicircle, when it is cut by a line lesser than the diameter.
19A semicircle is the half section of a circle.
Or it is that which is made the diameter.
Therefore
Therefore
Therefore
Semicircle and sections.
20A semicircle is comprehended of a periphery and the diameter 18 d j.
Asaei, is a semicircle: The other sections, asoyu, andoeu, are unequall sections: that greater; this lesser.
21The angle in a semicircle is a right angle: The angle of a semicircle is lesser than a rectilineall right angle: But greater than any acute angle: The angle in a greater section is lesser than a right angle: Of a greater, it is a greater. In a lesser it is greater: Of a lesser, it is lesser, è 31. and 16. p iij.
Or thus: The angle in a semicircle is a right angle, the angle of a semicircle is lesse than a right rightlined angle, butgreater than any acute angle: The angle in the greater section is lesse than a right angle: the angle of the greater section is greater than a right angle: the angle in the lesser section is greater than a right angle, the angle of the lesser section, is lesser than a right angle:H.
Figure for demonstration 21.
There are seven parts of this Element: The first is thatThe angle in a semicircle is a right angle: as inaei: For if the rayoe, be drawne, the angleaei, shall be divided into two anglesaeo, andoei, equall to the angleseao, andeio, by the17 e vj. Therefore seeing that one angle is equall to the other two, it is a right angle, by the6 e viij.Aristotlesaith that the angle in a semicircle is a right angle, because it is the halfe of two right angles, which is all one in effect.
The second part,That the angle of a semicircle is lesser than a right angle; is manifest out of that, because it is the part of a right angle. For the angle ofthe semicircleaie, is part of the rectilineall right angleaiu.
Figure for demonstration 21, fourth to seventh.
The third part, That it is greater than any acute angle; is manifest out of the23. e xv. For otherwise a tangent were not on the same part one onely and no more.
The fourth part is thus made manifest: The angle ati, in the greater sectionaei, is lesser than a right angle; because it is in the same triangleaei, which ata, is a right angle. And if neither of the shankes be by the center, not withstanding an angle may be made equall to the assigned in the same section.
The fifth is thus: The angle of the greater sectioneai, is greater than a right angle: because it containeth a right-angle.
The sixth is thus, the angleaoe, in a lesser section, is greater than a right angle, by the14 e xvj. Because that which is in the opposite section, is lesser than a right angle.
The seventh is thus. The angleeao, is lesser than a right-angle: Because it is part of a right angle, to wit of the outter angle, ifia, be drawne out at length.
And thus much of the angles of a circle, of all which the most effectuall and of greater power and use is the angle in a semicircle, and therefore it is not without cause so often mentioned ofAristotle. This Geometry therefore ofAristotle, let us somewhat more fully open and declare. For from hence doe arise many things.
Therefore
Therefore
Therefore
22If two right lines jointly bounded with the diameter of a circle, be jointly bounded in the periphery, they doe make a right angle.
Or thus; If two right lines, having the same termes with the diameter, be joyned together in one point, of the circomference, they make a right angle.H.
This corollary is drawne out of the first part of the former Element, where it was said, that an angle in a semicircle is a right angle.
And
And
And
23If an infinite right line be cut of a periphery of an externall center, in a point assigned and contingent, and the diameter be drawne from the contingent point, a right line from the point assigned knitting it with the diameter, shall be perpendicular unto the infinite line given.
Figure for demonstration 23.
Let the infinite right line beae, from whose pointa, a perpendicular is to be raised.
The right lineae, let it be cut by the peripheryaei, (whose centero, is out of the assignedae,) and that in the pointa, and a contingent point, as ine: And frome, let thediamiter beeoi: The right lineai, froma, the point given, knitting it with the diameterioe, shall be perpendicular upon the infinite lineae; Because with the said infinite, it maketh an angle in a semicircle.
And
And
And
24If a right line from a point given, making an acute angle with an infinite line, be made the diameter of a periphery cutting the infinite, a right line from the point assigned knitting the segment, shall be perpendicular upon the infinite line.
As in the same example, having an externall point given, let a perpendicular unto the infinite right lineaebe sought: Let the right lineioe, be made the diameter of the peripherie; and withall let it make with the infinite right line given an acute angle ine, from whose bisection for the center, let a periphery cut the infinite, &c.
And
And
And
Figure for demonstration 25.
25If of two right lines, the greater be made the diameter of a circle, and the lesser jointly bounded with the greater and inscribed, be knit together, the power of the greater shall be more than the power of the lesser by the quadrate of that which knitteth them both together. ad 13 p. x.
As in this example; The power of the diameterae, is greater than the power ofei, by the quadrate ofai. For the triangleaei, shall be a rectangle; And by the9 e xij.ae, the greater shall be ofpower equall to the shankes. Out of an angle in a semicircle Euclide raiseth two notable fabrickes; to wit, the invention of a meane proportionall betweene two lines given: And the Reason or rate in opposite sections. Thegenesisor invention of the meane proportionall, of which we heard at the9 e viij. is thus:
26If a right line continued or continually made of two right lines given, be made the diameter of a circle, the perpendicular from the point of their continuation unto the periphery, shall be the meane proportionall betweene the two lines given. 13 p vj.
Figure for demonstration 26.
As for example, let the assigned right lines beae, andei, of the whichaei, is continued. And leteo, be perpendicular from the peripheryaoi, untoe, the point of continuation or joyning together of the lines given. Thiseo, say I, shall be the meane proportionall: Because drawing the right linesao, andio, you shall make a rectangled triangle, seeing thataoi, is an angle in a semicircle: And, by the9 e viij.oe, shall be proportionall betweeneae, andei.
So if the side of a quadrate of 10. foote content, were sought; let the sides 1. foote and 10. foote an oblong equall to that same quadrate, be continued; the meane proportionall shall be the side of the quadrate, that is, the power of it shall be 10. foote. The reason of the angles in opposite sections doth follow.
Figure for demonstration 27.
27The angles in opposite sections are equall in the alterne angles made of the secant and touch line. 32. p iij.
If the sections be equall or alike, then are they the sections of a semicircle, and the matter is plaine by the21 e. But if they be unequall or unlike the argument of demonstrationis indeed fetch'd from the angle in a semicircle, but by the equall or like angle of the tangent and end of the diameter.
As let the unequall sections beeio, andeao: the tangent let it beuey: And the angles in the opposite sections,eao, andeio. I say they are equall in the alterne angles of the secant and touch lineoey, andoeu. First that which is ata, is equall to the alterneoey: Because also three anglesoey,oea, andaeu, are equall to two right angles, by the14 e v. Unto which also are equall the three angles in the triangleaeo, by the13 e vj. From three equals take away the two right anglesaue, andaoe: (Foraoe, is a right angle, by the21 e; because it is in a semicircle:) Take away also the common angleaeo: And the remainderseao, andoey, alterne angles, shall be equall.
Figure for demonstration 27.
Secondarily, the angles ata, andi, are equall to two right angles, by the14, e: To these are equall bothoey, andoeu. Buteao, is equall to the alterneoey. Therefore that which is ati, is equall to, the other alterneoeu. Neither is it any matter, whether the angle ata, be at the diameter or not: For that is onely assumed for demonstrations sake: For wheresoever it is, it is equall, to wit, in the same section. And from hence is the making of a like section, by giving a right line to be subtended.
Therefore
Therefore
Therefore
28If at the end of a right line given a right lined angle be made equall to an angle given, and from thetoppe of the angle now made, a perpendicular unto the other side do meete with a perpendicular drawn from the middest of the line given, the meeting shall be the center of the circle described by the equalled angle, in whose opposite section the angle upon the line given shall be made equall to the assigned è 33 p iij.
Figure for demonstration 28.
This you may make triall of in the three kindes of angles, all wayes by the same argument: as here the angle given isa: The right line givenei: at the ende, the equalled angle,ieo: The perpendicular to the sideeo, let it beeu: But from the middest of the line given let it beyu. Hereu, shall be the center desired. And from hence one may make a section upon a right line given, which shall receive a rectilineall angle equall to an angle assigned.
And
And
And
29If the angle of the secant and touch line be equall to an assigned rectilineall angle, the angle in the opposite section shall likewise be equall to the same. 34. p iij.
As in this figure underneath. And from hence one may from a circle given cut off a section, in which there is anangle equall to the assigned. As let the angle given bea: And the circleeio. Thou must make at the pointe, of the secanteo, and the tangentyu, an angle equall to the assigned, by the11 e iij. such as here isoeu: Then the sectionoei, shall containe an angle equall to the assigned.
Figure for demonstration 29.
Hitherto we have spoken of the Geometry of Rectilineall plaines, and of a circle: Now followeth the Adscription of both: This was generally defined in the first book12 e. Now the periphery of a circle is the bound therof. Therefore a rectilineall is inscribed into a circle, when the periphery doth touch the angles of it 3 d iiij. It is circumscribed when it is touched of every side by the periphery; 4 d iij.
Figure for demonstration 1.
1.If rectilineall ascribed unto a circle be an equilater, it is equiangle.
Of the inscript it is manifest; And that of a Triangle by it selfe: Because if it be equilater, it is equiangle, by the19 e vj. But in a Triangulate the matter is to be prooved by demonstration. As here, if the inscriptsou, andsy, be equall, then doe they subtend equall peripheries, by the 32e xv. Then if you doe omit the periphery in the middest betweene them both, as hereuy, and shalt addeoiesthe remainder to each of them, the wholeoiesy, subtended to the angle atu: Anduoies, subtended to the angle aty, shall be equall. Therefore the angles in the periphery, insisting upon equall peripheries are equall.
Of the circumscript it is likewise true, if the circumscript be understood to be a circle. For the perpendiculars from the centera, unto the sides of the circumscript, by the9 e xij, shal make triangles on each side equilaters, & equiangls, by drawing the semidiameters unto the corners, as in the same exÄple.
2.It is equall to a triangle of equall base to the perimeter, but of heighth to the perpendicular from the center to the side.
As here is manifest, by the8 e vij. For there are in one triangle, three triangles of equall heighth.
Figure for demonstration 2.
The same will fall out in a Triangulate, as here in a quadrate: For here shal be made foure triangles of equall height.
Lastly every equilater rectilineall ascribed to a circle, shall be equall to a triangle, of base equall to the perimeter of the adscript. Because the perimeter conteineth the bases of the triangles, into the which the rectilineall is resolved.
3.Like rectilinealls inscribed into circles, are one to another as the quadrates of their diameters, 1 p. xij.
Figure for demonstration 3.
Because by the1 e vj, like plains have a doubled reasó of their homologall sides. But in rectilineals inscribed the diameters are the homologall sides, or they are proportionall to their homologall sides. As let the like rectangled triangles beaei, andouy; Here becauseaeandou, are the diameters, the matter appeareth to be plaine at the first sight. But in the Obliquangled triangles,sei, andruy, alike also, the diameters are proportionall to their homologall sides, to wit,eianduy. For by the grant, asseis toru: so iseitouy, And therefore, by the former,asthe diametereaanduo.
In like Triangulates, seeing by the4 e x, they may be resolved into like triangles, the same will fall out.
Therefore
Therefore
Therefore
4.If it be as the diameter of the circle is unto the side of rectilineall inscribed, so the diameter of the second circle be unto the side of the second rectilineall inscribed, and the severall triangles of the inscripts be alike and likely situate, the rectilinealls inscribed shall be alike and likely situate.
ThisEuclidedid thus assume at the 2 p xij, and indeed as it seemeth out of the 18 p vj. Both which are conteined in the23 e iiij. And therefore we also have assumed it.
Adscription of a Circle is with any triangle: But with a triangulate it is with that onely which is ordinate: And indeed adscription of a Circle is commonto all.
5.If two right lines doe cut into two equall parts two angles of an assigned rectilineall, the circle of the ray from their meeting perpendicular unto the side, shall be inscribed unto the assigned rectilineall. 4 and 8. p. iiij.
Figure for demonstration 5.
As in the Triangleaei, let the right linesao, andeu, halfe the anglesaande: And fromy, their meeting, let the perpendiculars unto the sides beyo,yu,ys; I say that the centery, with the rayyo, orya, orys, is the circle inscribed, by the17 e xv. Because the halfing lines with the perpendiculars shall make equilater triangles, by the2 e vij. And therefore the three perpendiculars, which are the bases of the equilaters, shall be equall.
The same argument shall serve in a Triangulate.
6.If two right lines do right anglewise cut into two equall parts two sides of an assigned rectilineall, the circle of the ray from their meeting unto the angle, shall be circumscribed unto the assigned rectilineall. 5 p iiij.
As in former figures. The demonstration is the same with the former. For the three rayes, by the2 e vij, are equall: And the meeting of them, by the17 e x, is the center.
And thus is the common adscription of a circle: The adscription of a rectilineall followeth, and first of a Triangle.
Figure for demonstration 7.
7.If two inscripts, from the touch point of a right line and a periphery, doe make two angles on each side equall to two angles of the triangle assigned be knit together, they shall inscribe a triangle into the circle given, equiangular to the triangle given è 2 p iiij.
Let the Triangleaeibe given: And the circle,o, into which a Triangle equiangular to the triangle given, is to be inscribed. Therefore let the right lineuys, touch the peripheryyrl: And from the touchy, let the inscriptsyr, andyl, make with the tangent two anglesuyr, andsyl, equall to the assigned anglesaei, andaie: And let them be knit together with the right linerl: They shall by the27 e xvj, make the angle of the alterne segments equall to the anglesuyr, andsyl. Therefore by the4 e vijseeing that two are equall, the other must needs be equall to the remainder.
The circumscription here is also speciall.
8If two angles in the center of a circle given, be equall at a common ray to the outter angles of a triangle given, right lines touching a periphery in the shankes of the angles, shall circumscribe a triangle about the circle given like to the triangle given. 3 p iiij.
Figure for demonstration 8.
Let there be a Triangle, and in it the outter anglesaei, andaou: The Circle let it beysr; And in the centerl, let the anglesylr, andslr; at the common sidelr, bee made equall to the said outter anglesaei, andaou. I say the angles of the circumscribed triangle, are equall to the angles of the triangle given. For the foure inner angles of the quadrangleylrm, are equall to the foure right angles, by the6 e x: And two of them, to wit, atyandr, are right angles, by the construction: For they are made by the secant and touch line, from the touch point by the center, by the20 e xv. Therefore the remainders atlandm, are equall to two right angles: To which twoaeiandaeoare equall. But the angle atl, is equall to the outter: Therefore the remainderm, is equall toaeo. The same shall be sayd of the anglesaoe, andaou. Therefore two being equall, the rest ataandi, shall be equall.
Therefore
Therefore
Therefore
9.If a triangle be a rectangle, an obtusangle, an acute angle, the center of the circumscribed triangle is in the side, out of the sides, and within the sides: And contrariwise. 5 e iiij.
Figure for demonstration 9.
As, thou seest in these three figures, underneath, the centera.
Such is the Adscription of a triangle: The adscription of an ordinate triangulate is now to be taught. And first the common adscription, and yet out of the former adscription, after this manner.
1.If right lines doe touch a periphery in the angles of the inscript ordinate triangulate, they shall onto a circle cirumscribe a triangulate homogeneall to the inscribed triangulate.
The examples shall be laid downe according as the species or severall kindes doe come in order. The speciall inscription therefore shall first be taught, and that by one side, which reiterated, as oft as need shall require, may fill up the whole periphery. For thatEuclidedid in the quindecangleone of the kindes, we will doe it in all the rest.
Figure for demonstration 2.
2.If the diameters doe cut one another right-angle-wise, a right line subtended or drawne against the right angle, shall be the side of the quadrate. è 6 p iiij.
As here. For the shankes of the angle are the raies whose diameters knit together shall make foure rectangled triangles, equall in shankes: And by the2 e vij, equall in bases. Therefore they they shall inscribe a quadrate.
Therefore
Therefore
Therefore
3.A quadrate inscribed is the halfe of that which is circumscribed.
Because the side of the circumscribed (which here is equall to the diameter of the circle) is of power double, to the side of the inscript, by the9 e xij.
And
And
And
4.It is greater than the halfe of the circumscribed Circle.
Because the circumscribed quadrate, which is his double, is greater than the whole circle.
For the inscribing or other multangled odde-sided figures we must needes use the helpe of a triangle, each of whose angles at the base is manifold to the other: In aQuinquanglefirst, that which is double unto the remainder, which is thus found.
Figure for demonstration 5.
5.If a right line be cut proportionally, the base of that triangle whose shankes shall be equall to the whole line cut, and the base to the greater segment of the same, shall have each of the angles at base double to theremainder: And the base shall be the side of the quinquangle inscribed with the triangle into a circle. 10, and 11. p iiij.
Here first thou shalt take for the fabricke or making of the Triangle, for the ray the right lineaeby the3 e xiiij, cut proportionally ino: A circle also shalt thou make upon the centera, with the rayae: And then shalt thou by the6 e xv, inscribe a right line equall to the greater segment: And shalt knit the same inscript with the whole line cut with another right line. This triangle shall be your desire. For by the17 e vj, the angles at the baseeiare equall, so that looke whatsoever is prooved of the one, is by and by also prooved of the other. Then letoibe drawne; And a Circle, by the8 e xvij, circumscribed about the triangleaoi. This circle the right lineei, shall touch, by the27 e xv. Because, by the grant, the right lineae, is cut proportionally, therefore the Oblong of the secant and outter segment, is equall to the quadrate of the greater segment, to which by the grant, the baseei, is equall. Here therefore the angleaieis the double of the angle ata: because it is equall to the anglesaio, andoai, which are equall betweene themselves. For by the27 e xvjit is equall to the angleoaiin the alterne segment. And the remainderaio, is equall to it selfe. Therefore also the angleaei, is equall to the same two angles, because it is equall to the angleaie. But the outter angleeoi, is equall to the same two, by the15 e vj. Therefore the anglesioeandoei(because they are equall to the same) they are equall betweene themselves. Wherefore by the17 e vj, the sidesoiandeiare equall. And there alsoaoandoi: And the anglesoai&oiaare equall by the17 e vj. Wherefore seeingthat to both the angleaieis equall, it shall be the double ofeitherof the equalls.
Figure for demonstration 5.
But the baseei, is the side of the equilater quinquangle. For if two right lines halfing both the angles of a triangle which is the double of the remainder, be knit together with a right line, both one to another, and with the angles, shall inscribe unto a circle an equilater triangle, whose one side shall be the base it selfe: As here seeing the angleseoa,eoi,uio,uia,iao, are equal in the periphery, the peripheries, by the7 e, xvj. subtending them are equall: And therefore, by the32 e, xv. the subtensesae,ei,io,ou,ua, are also equall. Now of those five, one isae. Therefore a right line proportionall cut, doth thus make the adscription of a quinquangle: And from thence againe is afforded a line proportionally cut.
6If two right lines doe subtend on each side two angles of an inscript quinquangle, they are cut proportionally, and the greater segments are the sides of the said inscript è 8, p xiij.
Figure for demonstration 6.
As here, Letai, andeu, subtending the angles on each sideaei, andeau: I say, That they are proportionally cut in the points: And the greater segmentssi, andsu, are equall toae, the side of the quinquangle. For here two triangles are equiangles: Firstaei, anduae, are equall by the grant, and by the2 e, vij. Therefore the anglesaie, andaes, are equall. Thenaei, andase, are equall: Because theangle ata, is common to both: Therefore the other is equall to the remainder, by the4 e, 7. Now, by the12. e, vij.asia, is untoae, that is, as by and by shall appeare, untois: so isea, untoas: Therefore, by the1 e, xiiij.ia, is cut proportionally ins. But the sideea, is equall tois: Because both of them is equall to the sideei, that by the grant, this by the17. e, vj. For the angles at the base,ise, andies, are equall, as being indeed the doubles of the same. Forise, by the16. e vj. is equall to the two inner, which are equall to the angle atu, by the17 e vj. and by the former conclusion. Therefore it is the double of the anglesaes: Whose double also is the angleuei, by the7 e. xvj. insisting indeede upon a double periphery.
And from hence the fabricke or construction of an ordinate quinquangle upon a right line given, is manifest.