The Way to Geometry, Book 15
The fifteenth Booke of Geometry, Of the Lines in a Circle.
AS yet we have had the Geometry of rectilineals: The Geometry of Curvilineals, of which the Circle is the chiefe, doth follow.
1. A Circle is a round plaine. … 15 dj.
As here thou seest. A Rectilineall plaine was at the 3 e vj, defined to be a plaine comprehended of right lines. And so also might a circle have beene defined to be a plaine comprehended of a periphery or bought-line, but this is better.
The meanes to describe a Circle, is the same, which was to make a Periphery: But with some difference: For there was considered no more but the motion, the point in the end of the ray describing the periphery: Here is considered the motion of the whole ray, making the whole plot conteined within the periphery.
A Circle of all plaines is the most ordinate figure, as was before taught at the 10 e iiij.
2 Cir…les are as the quadrates or squares made of their diameters 2 p. x ij.
For Circles are like plaines. And their homologall sides are their diameters, as was foretold at the 24 e iiij. And therefore by the 1 e vj, they are one to another, as the quadrates of their diameters are one to another, which indeed is the double reason of their homologall sides. As here the Circle a e i, is unto the Circle o u y, as 25, is unto 16, which are the quadrates of their Dieameters, 5 and 4.
Therefore
3. The Diameters are, as their peripheries Pappus, 5 l x j, and 26 th. 18.
As here thou seest in a e, and i o.
4. Circular Geometry is either in Lines, or in the segments of a Circle.
This partition of the subject matters howsoever is taken for the distinguishing and severing with some light a matter somewhat confused; And indeed concerning lines, the consideration of secants is here the foremost, and first of Inscripts.
5. If a right line be bounded by two points in the periphery, it shall fall within the Circle. 2 p iij.
As here a e, because the right within the same points is shorter, than the periphery is, by the 5 e ij.
From hence doth follow the Infinite section, of which we spake at the 6 e j.
This proposition teacheth how a Rightline is to be inscribed in a circle, to wit, by taking of two points in the periphery.
6. If from the end of the diameter, and with a ray of it equal to the right line given, a periphery be described, a right line drawne from the said end, unto the meeting of the peripheries, shall be inscribed into the circle, equall to the right line given. 1 p iiij.
As let the right line given be a: And from e, the end of the diameter e i: And with e o, a part of it equall to a, the line given, describe the circle e u: A right line e u, drawne from the end e, unto u, the meeting of the two peripheries, shall be inscribed in the circle given, by the 5 e, equall to the line given; because it is equall to e o, by the 10 e v, seeing it is a ray of the same Circle.
And this proposition teacheth, How a right line given is to be inscribed into a Circle, equall to a line given.
Moreover of all inscripts the diameter is the chiefe: For it sheweth the center, and also the reason or proportion of all other inscripts. Therefore the invention and making of the diameter of a Circle is first to be taught.
7. If an inscript do cut into two equall parts, another inscript perpendicularly, it is the diamiter of the Circle, and the middest of it is the center. 1 p iij.
As let the Inscript a e, cut the inscript i u, perpendicularly: dividing it into two equall parts in o. I say that the once inscript thus halfing the other, is the diameter of the Circle: And that the middest of it is the center thereof: As in the circle, let the Inscript i s, cut the inscript a e, and that perpendicularly dividing into two equall parts in o. I say that i u, thus dividing a e, is the Diameter of the Circle: And y, the middest of the said i u, is the Center of the same.
The cause is the same, which was of the 5 e x j. Because the inscript cut into halfes if for the side of the inscribed rectangle, and it doth subtend the periphery cut also into two parts; By the which both the Inscript and Periphery also were in like manner cut into two equall parts: Therefore the right line thus halfing in the diameter of the rectangle: But that the middle of the circle is the center, is m…nifest out of the 7 e v, and 29 e iiij.
Euclide, thought better of Impossibile, than he did of the cause: And thus he forceth it. For if y be not the Center, but s, the part must be equall to the whole: For the Triangle a o s, shall be equilater to the triangle e o s. For a o, oe, are equall by the grant: Item s a, and s e, are the rayes of the circle: And s o, is common to both the triangles. Therefore by the 1 e vij, the angles no each side at o are equall; And by the 13 e v, they are both right angles. Therefore s o e is a right angle; It is therefore equall by the grant, to the right angle y o e, that is, the part is equall to the whole, which is impossible. Wherefore y is not the Center. The same will fall out of any other points whatsoever …ut of y.
Therefore
8. If two r…ght lines doe perpendicularly halfe two inscripts, the meeting of these two bisecants shall be the Center of the circle è 25 p iij.
As here a e, and i o, let them cut into halfes the right lines u y, and y s. And let them meete, that they cut one another in r. I say r is the center of the circle a y o s e i u. For before, at the 6, and 7 e v, it was manifest that the Center was in the Diameter. And in the meeting of the diameters. [Therefore two manner of wayes is the Center found; First by the middle of the diameter: And then againe by the concourse, or meeting of the diameters, in the middest of the lines halfed or cut into two equall portions.] Here is no neede of the meeting of many diameters, one will serve well enough
And one may
9. Draw a periphery by three points, which doe not fall in a right line.
As here, by a e i, (First from a, to e, let a right line be drawne; And likewise from e to i. Then, by the 12 e v, let both these lines be cut into equall parts, by two infinite right lines: These halfing lines also shall meete: And in their meeting shall be the Center, by the 8 e. And therefore from that meeting unto any of the sayd points given is the ray of the periphery desired.)
10. If a diameter doe halfe an inscript, that is, n…t a diameter, it doth cut it perpendicularly: And contrariwise: 3 p iij.
As let the diameter a e, halfe the inscript i o, which is not a diameter: And let the raies of the circle bee u i, and u o. The cause in all is the same, which was of the 5 e x j.
11. If inscripts which are not diameters doe cut one another, the segments shall be unequall. 4 p iij.
This is a consectary drawne out of the 28 e iiij. For if the inscripts were halfed, they should be diameters, against the grant.
But rate hath beene hitherto in the parts of inscripts: Proportion in the same parts followeth.
12 If two inscripts doe cut one another, the rectangle of the segments of the one is equall to the rectangle of the segments of the other. 35 p iij.
If the inscripts thus cut be diameters, the proportion is manifest, as in the first figure. For the Rectangle of the segments, of the one is equall to the rectangle of the segments of the other, seeing they be both quadrates of equall sides. If they be not diameters let them otherwise as a e, and i o: I say the Oblong of a u, and u e, is equall to the Oblong of o u, and u i. For let the raies from the Center y, be y e, and y i. To the quadrate of each of these both the rectangles of the segments shall be equall. For by the 7 e, let the diameter y u, fall upon the point of the common section u; And let y s, and s r, be perpendiculars. Here by the 5 e x j. the inscripts are cut equally in the points r and s: And unequally in the point u: Therefore by the 7 e xiij, the oblong, of o u, and u i, with the quadrate s u, is equall to the quadrates s i: And adding y s, the same oblong, with the quadrates u s and s y, that is, by the 9 e x i j, with the quadrate y u, is equall to the quadrates i s and s y, that is, by the 9 e xij, to the quadrate i y, that is, by the 5 e xij, to y e, to the which by the same cause it is manifest the other oblong with the quadrate y u is equall. Let the quadrate y u, bee taken from each of them: And then the oblongs shall be equall to the same: And therefore betweene themselves.
And this is the comparison of the parts inscripts. The rate of whole inscripts doth follow, the which whole one diameter doth make:
13 Inscripts are equall distant from the center, unto which the perpendiculars from the center are equall 4 d iij.
As it appeareth in the next figure, of the lines a e and i o, unto which the perpendiculars u y, and u s, from the Center u, are equall.
14. If inscripts be equall, they be equally distant from the center: And contrariwise. 13 p iij.
The diameters in the same circle, by the 28 e iiij… are equall: And they are equally distant from the center, seeing they are by the center, or rather are no whit at all distant from it: Other inscripts are judged to be equall, greater, or lesser one than another, by the diameter, or by the diameters center.
Euclide doth demonstrate this proposition thus: Let first a e and i o be equall; I say they are equidistant from the center. For let u y, and u y, be perpendiculars: They shall cut the assigned a e, & i o, into halfes, by the 5 e xj: And y a and s i a…e equall, because they are the halfes of equals. Now let the raies of the circle be u a, aund u i: Their quadrates by the 9 e xij, are equall to the paire of quadrates of the shankes, which paires are therefore equall betweene themselves. Take from equalls the quadrates y a, and s i, there shall remaine y u and u s, equalls: and therefore the sides are equall, by the 4 e 12.
The converse likewise is manifest: For the perpendiculars given do halfe them: And the halfes as before are equall.
15 Of unequall inscripts the diameter is the greatest: And that which is next to the diameter, is greater than that which is farther off from it: That which is farthest off from it, is the least: And that which is next to the least, is lesser than that which is farther off: And those two onelyonly which are on each side of the diameter are equall è 15 e iij.
This proposition consisteth of five members: The first is, The diameter is the greatest iuscript: The second, That which is next to the diameter is greater than that which is farther off: The third, That which is farthest off from the diameter is the least: The fourth, That next to the least is lesser, than that farther off: The fifth, That two onelyonly on each side of the diameter are equall betweene themselves. All which are manifest out of that same argument of equalitie, that is the center the beginning of decreasing, and the end of increasing. For looke how much farther off you goe from the center, or how much nearer you come unto it, so much les…er or greater doe you make the inscript.
Let there be in a circle many inscripts, of which one, to wit, a e, let it be the diameter: I say, that it is of them all the greatest or longest. But let i o, be nearer to the diameter, (or as in the former Elements was said) nearer to the center, than u y. I say that i o, is longer than u y. Moreover, let u y, be the farthest off from the same diameter or center; I say the same u y, is the shortest of them all. Now to this shortest u y, let i o, be nearer than a e … I say therefore that i o, also is lesser than a e. Let at length i o, be not the diameter: I say that beyond the diameter a e, there may onelyonly a line be inscribed equall unto it, such as is s r. And those equal betweene themselves on each side of the diametry may only be given, not three, nor more. And after the same manner also, onelyonly one beyond the diameter, may possibly be equall to u y, to wit, that which is as farre off from the diameter as it is; and so in others.
But Euclides conclusion is by triangles of two sides greater than the other; and of the greater angle.
The first part is plaine thus: Because the diameter a e, is equall to i l, and l o, viz. to the raies: And to those which are greater than i o, the base by the 9. e v j &c.
The second part of the nearer, is manifest by the 5 e vij. because of the triangle i l o, equicrurall to the triangle u l y, is greater in angle: And therefore it is also greater in base.
The third and fourth are consectaries of the first.
The fifth part is manifest by the second: For if beside i o, and s r, there be supposed a third equall, the same also shall be unequall, because it shall be both nearer and farther off from the diameter.
16 Of right lines drawne from a point in the diameter which is not the center unto the periphery, that which passeth by the center is the greatest: And that which is nearer to the greatest, is greater than that which is farther off: The other part of the greatest is the lest. And that which is nearest to the least, is lesser than that which is farther off: And two on each side of the greater or least are only equall. 7 p iij.
The first part of a e, and a i, is manifest, as before, by the 9 e v j. The second of a i, and a o; Item of a o, and a u, is plaine by the 5 e vij.
The third, that a y, is lesser than a u, because s y, which is equall to s u, is lesser than the right lines s a, and a u, by the 9 e v j: And the common s a, being taken away, a y shall be left, lesser than a u.
The fourth part followeth of the third.
The fifth let it be thus: s r, making the angle a s r, equall to the angle a s u, the bases a u, and a r, shall be equall by the 2 e v ij. To these if the third be supposed to be equall, as a l, it would follow by the 1 e v ij. that the whole angle s a, should be equall to r s a, the particular angle, which is impossible. And out of this fifth part issueth this Consectary.
Therefore
17 If a point in a circle be the bound of three equall right lines determined in the periphery, it is the center of the circle. 9 p iij.
Let the point a, in a circle be the common bound of three right lines, ending in the periphery and equall betweene themselves, be a e, a i, a u. I say this point is the center of the Circle.
Otherwise from a point of the diameter which is not the center, not onelyonly two right lines on each side should be equall. For by any point whatsoever the diameter may be drawne. Such was before observed in a quinquangle; If three angles be equall, all are equall; so in a Circle: If three right lines falling from the same point unto the perephery be equall, all are equall.
18 Of right lines drawne from a point assigned without the periphery, unto the concavity or hollow of the same, that which is by the center is the greatest; And that next to the greatest, is greater than that which is farther off: But of those which fall upon the convexiti… of the circumference, the segment of the greatest is least… And that which is next unto the least is lesser than that is farther off: And two on each side of the greatest or least are onelyonly equall. 8 piij.
The demonstration of this is very like unto the above mentioned, of five parts. And thus much of the secants, the Tangents doe follow.
19 If a right line be perpendicular unto the end of the diameter, it doth touch the periphery: And contrariwise è 16 p iij.
As for example, Let the circle given a e, be perpendicular to the end of the diameter, or the end of the ray, in the end a, as suppose the ray be i a: I say, that e a, doth touch, not cut the periphery in the common bound a.
This was to have beene made a postulatum out of the definition of a perpendicle: Because if this should leane never so little, it should cut the periphery, and should not be perpendicular: Notwithstanding Euclide doth force it thus: Otherwise let the right line a e, be perpendicular to the diameter a i. And a right line from o, with the center i, let it fall within the circle at o, and let o i, joyned together. Here in the triangle a o i, two angles, contrary to the 13. e v j. should be right angles at a, by the grant: And at o, by the 17 e vj.
The demonstration of the converse is like unto the former. For if the tangent, or touch-line a e, be not perpendicular to the diameter i o u, let o e, from the center o, be drawne perpendicular: Then shall the angle o e i, be right angle: And o i e an acutangle: And therefore by the 22 e vj, o i, that is o y, shall be greater then o y e, that is the part, then the whole.
Therefore
20 If a right line doe passe by the center and touch-point, it is perpendicular to the tangent or touch-line. 18 p iij.
And
Or thus, as Schoner amendeth it: If a right line be the diameter by the touch point, it is perpendicular to the tangent.
21 If a right line be perpendicular unto the tangent, it doth passe by the center and touch-point. 19. piij.
Or thus: if it be perpendicular to the tangent, it is a diameter by the touch point: Schoner.
For a right line either from the center unto the touch-point; or from the touch point unto the center is radius or semidiameter.
And
22 The touch-point is that, into which the perpendicular from the center doth fall upon the touch line.
23 A tangent on the same side is onelyonly one.
Or touch line is but one upon one, and the same side: H. Or. A tangent is but one onelyonly in that point of the periphery Schoner.
It is a consectary drawne out of the xiij. e ij. Because a tangent is a very perpendicular.
Euclide propoundeth this more specially thus; that no other right line may possibly fall betweene the periphery and the tangent.
And
24 A touch-angle is lesser than any rectilineall a…ute angle, è 16 p ij.
Angulus contractus, A touch angle is an angle of a straight touch-line and a periphery. It is commonly called Angulus contingentiae: Of Proclus it is named Cornicularis, an hornelike corner… because it is made of a right line and periphery like unto a horne. It is lesse therefore than any acute or sharpe right-lined angle: Because if it were not lesser, a right line might fall between the periphery and the perpendicular.
And
25 All touch-angles in equall peripheries are equall.
But in unequall peripheries, the cornicular angle of a lesser periphery, is greater than the Cornicular of a greater.
26 If from a ray out of the center of a periphery given, a periphery be described unto a point assigned without, and from the meeting of the assigned and the ray, a perpendicular falling upon the said ray unto the now described periphery, be tied by a right line with the said center, a right line drawne from the point given unto the meeting of the periphery given, and the knitting line shall touch the assigned periphery 17 p iij.
As with the ray a e, from the center a, of the periphery assigned, unto the point assigned e, let the periphery e o, be described: And let i o, be perpendicular to the ray unto the described periphery. This knit by a right line unto the center a, let e u, be drawne. I say, that e u, doth touch the periphery i u, assigned: Because it shall be perpendicular unto the end of the diameter. For the triangles e a u, and o a i, by the 2 e v i j, seeing they are equicrurall; And equall in shankes of the common angle; they are equall in the angles at the base. But the angle a i o, is a right angle: Therefore the angle e u a, shall be a right angle. And therefore the right line e u, by the 13 e ij, is perpendicular to a o.
Thus much of the Secants and Tangents severally: It followeth of both kindes joyntly together.
27 If of two right lines, from an assigned point without, the first doe cut a periphery unto the concave, the other do touch the same; the oblong of the secant, and of the outter segment of the secant, is equall to the quadrate of the tangent: and if such a like oblong be equall to the quadrate of the other, that same other doth touch the periphery: 36 , and 37 .p iij.
If the secant or cutting line do passe by the center, the matter is more easie and as here, Let a, cut; And a i, touch: The outter segment is a o, and the center u, Now u i, shall be perpendicular to the tangent a i, by the 20. e: Then by 8 e xiij, the oblong of e a, and a o, with the quadrate of a u, that is, of i u, is equall to the quadrate of a u, that is, by the 9 e x i j. to the quadrates of a i, and i u. Take i u, the common quadrate: The Rectangle shall be equall to the quadrate of the tangent.
If the secant doe not passe by the center, as in this figure, the center u, found by the 7 e, i u, shall be by the 20 e perpendicular unto the tangent a i … then draw u a, and uo, and the perpendicular halving o e, by the 10 e. Here by the 8 e xiij, the oblong of a e, and a o, with the quadrate o y, is equal to the quadrate a y: Therefore y u, the common quadrate added, the same oblong, with the quadrates o y, and y u, that is by the 9 e xij. with the quadrate o u, is equall to the quadrates a y, and u y, that is, by the 9 e xij, to a u, that is, againe, to a i, and i u. Lastly, let u r, and i u, two equall quadrates be taken from each, and there wil remaine the oblong equall to the quadrate of the tangent.
The converse is likewise demonstrated in this figure. Let the Rectangle of a e, and a y, be equall to the quadrate of a i. I say, that a i doth touch the circle. For let, by the 26 e, a o the tangent be drawne: Item let a u, u i, and u o bee drawne. Here the oblong of e a, and a y, is equall to the quadrate of a o, by the 27 e: And to the quadrate of a i, by the grant. Therefore a i, and a o, are equall. Then is u o, by the 20 e, perpendicular to the tangent. Here the triangles a u o, and a u i, are equilaters: And by the … e vij, equiangles. But the angle at o is a right angle: Therefore also a right angle and equall to it is that at i, by the … 3 e iij, wherefore a i is perpendicular to the end of the diameter: And, by the 19 e, it toucheth the periphery.
Therefore
28. All tangents falling from the same point are equall.
Or, Touch lines drawne from one and the same point are equall: H.
Because their quadrates are equall to the same oblong.
And
29. The oblongs made of any secant from the same point, and of the outter segment of the secant are equall betweene themselves. Camp. 36 p iij.
The reason is because to the same thing.
And
30. To two right lines given one may so continue or joyne the third, that the oblong of the continued and the continuation may be equall to the quadrate remaining. Vitellio 127 p j.
As in the first figure, if the first of the lines given be e o, the second i a, the third o a.
Now are we come to Circular G…ometry, that is to the Geometry of Circles or Peripheries cut and touching one another: And of Right lines and Peripheries.
31. If peripheries doe either cut or touch one another, they are eccentrickes: And they doe cut one another in two points onelyonly, and these by the touch point doe continue their diameters, 5. 6. 10,11, 12 p iij.
All these might well have beene asked: But they have also their demonstrations, ex impossibili, not very dissicult.
The first part is manifest, because the part should be equall to the whole, if the Center were the same to both, as a. For two raies are equall to the common raie a o: And therefore a e and a i, that is, the part and the whole, are equall one to another.
The second part is demonstrated as the first: For otherwise the part must be equall to the whole, as here a e and a i, the raies of the lesser periphery; And a e, and a o, the raies of the greater are equall. Wherefore a i, should be equall to a o, the Part to the whole.
If the Peripheries be outwardly contiguall, the matter is more easie, and by the judgement of Euclide it deserved not a demonstration, as here.
The third part is apparent out of the first: Otherwise those which cut one another should be concentrickes. For, by the 7 e, the center being found: And by the 9 e, three right lines being drawne from the center unto three points of the sections, the three raies must be equall, as here.
The fourth part is demonstrated after the same manner: Because otherwise the Part must be greater then the whole. For let the right line a e i o, be drawne by the centers a and e: And let the particular raies be e u, and a u. Here two sides u e, and e a, of the triangle u e a, by the 9 e vj, are greater than u a: And therefore also then a o; Take away a e, the remainder u e, shall be greater than e o. But e i is equall to e u. Wherefore e i is greater than e o, the part, than the whole.
The same will fall out, if the touch be without, as here: For, by the 9 e vj, e a and i a, are greater than i e. But e o and i u, are equall to e a, and i a. Wherefore e o, and i u, are greater than i e, the parts than the whole.
Of right lines and Peripheries joyntly the rate is but one.
32. If inscripts be equall, they doe cut equall peripheries: And contrariwise, 28,29 p iij.
Or thus: If the inscripts of the same circle or of equall circles be equall, they doe cut equall peripheries: And contrariwise B.
Or thus; If lines inscribed into equall circles or to the same be equall, they cut equall peripheries: And contrariwise, if they doe cut equall peripheries, they shall themselves be equall: Schoner …
The matter is apparent by congruency or application: a… here in this example. For let the circles agree, and then shall equall inscripts and peripheries agree.
Except with the learned Rodulphus Snellius, you doe understand aswell two equall peripheries to be given, as two equall right lines, you shall not conclude two equall sections, and therefore we have justly inserted of the same, or of equall Circles; which we doe now see was in like manner by Lazarus Schonerus.