The Sixteenth Booke of Geometry, of the Segments of a Circle
1. A Segment of a Circle is that which is comprehended outterly of a periphery, and innerly of a right line.
The Geometry of Segments is common also to the spheare: But now this same generall is hard to be declared and taught: And the segment may be comprehended within of an oblique line either single or manifold. But here we follow those things that are usuall and commonly received. First therefore the generall definition is set formost, for the more easie distinguishing of the species and severall kindes.
2. A segment of a Circle is either a sectour, or a section.
Segmentum a segment, and Sectio a section, and Sector a sectour, are almost the same in common acceptation, but they shall be distinguished by their definitions.
3. A Sectour is a segment innerly comprehended of two right lines, making an angle in the center; which is called an angle in the center: As the periphery is, the base of the sectour, 9 d iij.
As aei is a sectour. Here a sectour is defined, and his right lined angle, is absolutely called The greater Sectour which notwithstanding may be cut into two sectours by drawing of a semidiameter, as after shall be seene in the measuring of a section.
4. An angle in the Periphery is an angle comprehended of two right lines inscribed, and jointly bounded or meeting in the periphery. 8 d iij.
This might have beene called The Sectour in the Periphery, to wit, comprehended innerly of two right lines joyntly bounded in the periphery; as here aei.
5. The angle in the center, is double to the angle of the periphery standing upon the same base, 20 p iij.
The variety or the example in Euclide is threefold, and yet the demonstration is but one and the same: As here eai, the angle in the center, shall be prooved to be double to eoi, the angle in the periphery, the right line ou cutting it into two triangles on each side equicrurall; And, by the 17 e vj, at the base equiangles: Whose doubles severally are the angles, eau, of eoa: And iau, of ioa, For seeing it is equall to the two inner equall betweene themselves by the 15 e vj; it shall be the double of one of them. Therefore the whole eai, is the double of the whole eoi.
The second example is thus of the angle in the center aei: And in the periphery aoi. Here the shankes eo, and ei, by the 28 e iiij, are equall: And by the 17 e vj, the angles at o and i are equall: To both which the angle in the center is equall, by the 15 e vj. Therefore it is double of the one.
The third example is of the angle in the center, aei, And in the periphery aoi, Let the diameter be oeu. Here the whole angle ieu, by the 15 e vj, is equall to the two inner angles eoi, and eio, which are equall one to another, by the 17 e vj: And therefore it is double of the one. Item the particular angle aeu, is equall by the 15 e vj, to the angles eoa, and eao, equall also one to another, by the 17 e vj. Therefore the remainder aei, is the double of the other aoi, in the periphery.
Therefore
6. If the angle in the periphery be equall to the angle in the center, it is double to it in base. And contrariwise.
This followeth out of the former element: For the angle in the center is double to the angle in the periphery standing upon the same base: Wherefore if the angle in the periphery be to be made equall to the angle in the center, his base is to be doubled, and thence shall follow the equality of them both: S.
7. The angles in the center or periphery of equall circles, are as the Peripheries are upon which they doe insist: And contrariwise. è 33 p vj, and 26, 27 p iij.
Here is a double proportion with the periphery underneath, of the angles in the center: And of angles in the periphery. But it shall suffice to declare it in the angles in the center.
First therefore let the Angles in the center aei, and ouy be equall: The bases ai, and oy, shall be equall, by the 11 e vij: And the peripheries, ai, and oy, by the 32 e xv, shall likewise be equall. Therefore if the angles be unequall, the peripheries likewise shall be unequall.
The same shall also be true of the Angles in the Periphery. The Converse in like manner is true: From whence followeth this consectary:
Therefore
8. As the sectour is unto the sectour, so is the angle unto the angle: And Contrariwise.
And thus much of the Sectour.
9. A section is a segment of a circle within cōprehended of one right line, which is termed the base of the section.
As here, aei, and ouy, and srl, are sections.
10. A section is made up by finding of the center.
The Invention of the center was manifest at the 7 e xv: And so here thou seest a way to make up a Circle, by the 8 e xv.
11 The periphery of a section is divided into two equall parts by a perpendicular dividing the base into two equall parts. 20. p iij.
Let the periphery of the section aoe, to be halfed or cut into two equall parts. Let the base ae, be cut into two equall parts by the pendicular io, which shall cut the periphery in o, I say, that ao, and oe, are bisegments. For draw two right lines ao, and oe, and thou shalt have two triangles aio, and eio, equilaters by the 2 e vij. Therefore the bases ao, and oe, are equall: And by the 32. e xv. equall peripheries to the subtenses.
Here Euclide doth by congruency comprehende two peripheries in one, and so doe we comprehend them.
12 An 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. As aoe, in the former example.
13 The angles in the same section are equall. 21. p iij.
Let the section be eauo, And in it the angles at a, & u: These are equall, because, by the 5 e, they are the halfes of the angle eyo, in the center: Or else they are equall, by the 7 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.
14 The angles in opposite sections are equall to two right angles. 22. p iij.
For here the opposite angles at a, and i, are equall to the three angles of the triangle eoi, which are equall to two right angles, by the 13 e vj. For first i, is equall to it selfe: Then a, by parts is equall to the two other. For eai, is equall to eoi, and iao, to oei, by the 13 e. Therefore the opposite angles are equall to two right angles.
The reason or rate of a section is thus: The similitude doth follow.
15 If sections doe receive [or containe] equall angles, they are alike è 10. d iij.
As here aei, and ouy. The triangle here inscribed, seeing they are equiangles, by the grant; they shall also be alike, by the 12 e vij.
16 If like sections be upon an equall base, they are equall: and contrariwise. 23, 24. p iij.
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 that aoe, shall be lesse than the angle aie, in the lesser section, by the 16 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 the 9 e xv. one may finde a section like unto another assigned, or else from a circle given to cut off one like unto it.
17 Angle of a section is that which is comprehended of the bounds of a section.
As here eai: And eia.
18 A 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.
19 A semicircle is the half section of a circle.
Or it is that which is made the diameter.
Therefore
20 A semicircle is comprehended of a periphery and the diameter 18 d j.
As aei, is a semicircle: The other sections, as oyu, and oeu, are unequall sections: that greater; this lesser.
21 The 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, but greater 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.
There are seven parts of this Element: The first is that The angle in a semicircle is a right angle: as in aei: For if the ray oe, be drawne, the angle aei, shall be divided into two angles aeo, and oei, equall to the angles eao, and eio, by the 17 e vj. Therefore seeing that one angle is equall to the other two, it is a right angle, by the 6 e viij. Aristotle saith 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 of the semicircle aie, is part of the rectilineall right angle aiu.
The third part, That it is greater than any acute angle; is manifest out of the 23. 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 at i, in the greater section aei, is lesser than a right angle; because it is in the same triangle aei, which at a, 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 section eai, is greater than a right angle: because it containeth a right-angle.
The sixth is thus, the angle aoe, in a lesser section, is greater than a right angle, by the 14 e xvj. Because that which is in the opposite section, is lesser than a right angle.
The seventh is thus. The angle eao, is lesser than a right-angle: Because it is part of a right angle, to wit of the outter angle, if ia, 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 of Aristotle. This Geometry therefore of Aristotle, let us somewhat more fully open and declare. For from hence doe arise many things.
Therefore
22 If 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
23 If 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.
Let the infinite right line be ae, from whose point a, a perpendicular is to be raised.
The right line ae, let it be cut by the periphery aei, (whose center o, is out of the assigned ae,) and that in the point a, and a contingent point, as in e: And from e, let the diamiter be eoi: The right line ai, from a, the point given, knitting it with the diameter ioe, shall be perpendicular upon the infinite line ae; Because with the said infinite, it maketh an angle in a semicircle.
And
24 If 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 line ae be sought: Let the right line ioe, be made the diameter of the peripherie; and withall let it make with the infinite right line given an acute angle in e, from whose bisection for the center, let a periphery cut the infinite, &c.
And
25 If 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 diameter ae, is greater than the power of ei, by the quadrate of ai. For the triangle aei, shall be a rectangle; And by the 9 e xij. ae, the greater shall be of power 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. The genesis or invention of the meane proportionall, of which we heard at the 9 e viij. is thus:
26 If 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.
As for example, let the assigned right lines be ae, and ei, of the which aei, is continued. And let eo, be perpendicular from the periphery aoi, unto e, the point of continuation or joyning together of the lines given. This eo, say I, shall be the meane proportionall: Because drawing the right lines ao, and io, you shall make a rectangled triangle, seeing that aoi, is an angle in a semicircle: And, by the 9 e viij. oe, shall be proportionall betweene ae, and ei.
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.
27 The 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 the 21 e. But if they be unequall or unlike the argument of demonstration is 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 be eio, and eao: the tangent let it be uey: And the angles in the opposite sections, eao, and eio. I say they are equall in the alterne angles of the secant and touch line oey, and oeu. First that which is at a, is equall to the alterne oey: Because also three angles oey, oea, and aeu, are equall to two right angles, by the 14 e v. Unto which also are equall the three angles in the triangle aeo, by the 13 e vj. From three equals take away the two right angles aue, and aoe: (For aoe, is a right angle, by the 21 e; because it is in a semicircle:) Take away also the common angle aeo: And the remainders eao, and oey, alterne angles, shall be equall.
Secondarily, the angles at a, and i, are equall to two right angles, by the 14, e: To these are equall both oey, and oeu. But eao, is equall to the alterne oey. Therefore that which is at i, is equall to, the other alterne oeu. Neither is it any matter, whether the angle at a, 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
28 If at the end of a right line given a right lined angle be made equall to an angle given, and from the toppe 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.
This you may make triall of in the three kindes of angles, all wayes by the same argument: as here the angle given is a: The right line given ei: at the end e, the equalled angle, ieo: The perpendicular to the side eo, let it be eu: But from the middest of the line given let it be yu. Here u, 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
29 If 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 an angle equall to the assigned. As let the angle given be a: And the circle eio. Thou must make at the point e, of the secant eo, and the tangent yu, an angle equall to the assigned, by the 11 e iij. such as here is oeu: Then the section oei, shall containe an angle equall to the assigned.