The Way to Geometry, Book 18
Of Geometry, the eighteenth Booke, Of the adscription of a Triangulate.
SVch 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 unto a circle circumscribe 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 that Euclide did in the quindecangle one of the kindes, we will doe it in all the rest.
2. If the diameters doe cut one another right-anglewise, 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 the 2 e v i j, equall in bases. Therefore they shall inscribe a quadrate.
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 the 9 e x i j.
An…
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 of 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 a Quinguangle first, that which is double unto the remainder, which is thus found.
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 the base double to the remainder: And the base shall be the side of the quinquangle inscribed with the triangle into a circle. 10, and 11. p i i i j.
Here first thou shalt take for the fabricke or making of the Triangle, for the ray the right line a e, by the 3 e x iiij, cut proportionally in o: A circle also shalt thou make upon the center a, with the ray a e: And then shalt thou by the 6 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 the 17 e v j, the angles at the base e i are equall, so that looke whatsoever is prooved of the one, is by and by also prooved of the other… Then let o i be drawne; And a Circle, by the 8 e xvij, circumscribed about the triangle a o i. This circle the right line e i, shall touch, by the 27 e xv. Because, by the grant, the right line a e, 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 base e i, is equall. Here therefore the angle a i … is the double of the angle at a: because it is equall to the angles a i o, and o a i, which are equall betweene themselves. For, by the 7 e xvj it is equall to the angle o a i, in the alterne segment. And the remainder a i o, is equall to it selfe. Therefore also the angle a e i, is equall to the same two angles, because it is equall to the angle a i e. But the outter angle e o i, is equall to the same two, by the 15 e vj. Therefore the angles i o e and … e i (because they are equall to the same) they are equall betweene themselves. Wherefore by the 17 e vj, the sides o i and e i are equall: And therefore also a o and o i: And the angles o a i, & o a i … are equall, by the 17 e vj. Wherefore seeing that to both the angle a i e is equall, it shall be the double of ther of the equalls.
But the base e i, is the side of the equilater quinquangle. For if two right lines halfi…g 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 angles e o a, e o i, u i o, u i a, i a o, are equal in the periphery, the peripheries, by the 7 e, x vj. subtending them are equall: And therefore, by the 32 e, xv. the subtenses a e, e i, i o, o u, u a, are also equall. Now of those five, one is a e. Therefore a right line proportionall cut, doth thus make the adscription of a quinquangle: And from thence againe is afforded a line proportionally cut.
6 If 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 x iij.
As here, Let a i, and e u, subtending the angles on each side a e i, and e a u: I say, That they are proportionally cut in the point s: And the greater segments s i, and s u, are equall to a e, the side of the quinquangle. For here two triangles are equiangles: First a e i, and u a e, are equall by the grant, and by the 2 e, v i j. Therefore the angles a i e, and a e s, are equall. Then a e i, and a s e, are equall: Because the angle at a, is common to both: Therefore the other is equall to the remainder, by the 4 e, 7. Now, by the 12. e, vij. as i a, is unto a e, that is, as by and by shall appeare, unto i s: so is e a, unto a s: Therefore, by the 1 e, xiiij. ia, is cut proportionally in s. But the side e a, is equall to i s: Because both of them is equall to the side e i, that by the grant, this by the 17. e, vj. For the angles at the base, i s e, and i e s, are equall, as being indeed the doubles of the same. For i s e, by the 16. e v j. is equall to the two inner, which are equall to the angle at u, by the 17 e vj. and by the former conclusion. Therefore it is the double of the angles a e s: Whose double also is the angle u e i, by the 7 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.
Therefore
7 If 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.
8 If 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. 12,24,46, angles: By a quadrate, a triangulate of 8. 16,32,64. angles. By a quinquangle, a triangulate of 10, 20. 40,80. angles &c.
9 The ray of a circle is the side of the inscript sexangle è 15 p iiij.
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 be a e: And upon the center o, with the ray i e, let the periphery u i o, be described: And from the points o and u, let the diameters be o y, and u s; These knit both one with another, and also with the diameter a e, shall inscribe an equilaterall sexangle into the circle given, whose side shal be equal to the ray of the same circle. As e u, is equal to u i, because they both equall to the same ie, by the 29 e, iiij. Therefore eiu, is an equilater triangle: And likewise e i o, is an equilater. The angles also in the center are ⅔. of one right angle: And therefore they are equall. And by the 14. e v, the angle s i o, is ⅓. of two rightangles: And by the 15. e v. the angles at the toppe are also equall. Wherefore sixe are equall: And therefore, by the 7 e xvj. and 32. 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 an equilater: And by the 1 e x vij. equiangled.
Therefore
10 Three ordinate sexangles doe fill up a place.
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 …/5 angles which is too little. And foure would make 4 …/5. which is as much too great. The angles of a septangle would make onelyonly 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 onelyonly.
And
11 If 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.
As here: Because the sides shall be subtended to equall peripheries: Therefore by the 32 e x v. they shall be equall betweene themselves: And againe, on the contrary, by such a like triangle, by halfing the angles, a sexangle is inscribed.
12 The side of an inscribed equilater triangle hath a treble power, unto the ray of the circle 12. p xiij.
As here, with a e, one side of the triangle a e i, two third parts of the halfe periphery are imployed: For with one side one third of the whole e u, is imployed: Therefore e u, is the other third part, that is, the sixth part of the whole periphery. Therefore the inscript e u, is the ray of the circle, by the 9 e. Now the power of the diameter a o u, by the 14 e xij. is foure times so much as is the power of the ray, that is, of e u: And by 21. e xvj, and 9 e xij, a e, and e u, are of the same power; take away e u, and the side a e, shall be of treble power unto the ray.
13 If the side of a sexangle be cut proportionally, the greater segment shall be the side of the decangle.
Pappus lib. 5. ca. 24. & Campanus ad 3 p xiiij… Let the ray a o, or side of the sexangle be cut proportionally, by the 3 e xiiij: And let a e, be equall to the greater segment. I say that a e, is the side of the decangle. For if it be moreover continued with the whole ray unto i, the whole a e i, shall be by the 4 e xiiij. be cut proportionally: and the greater segment e i, shal be the same ray. For if the right line iea, be cut proportionally, it shall be as i a, is unto i e, that is to o a, to wit, unto the ray: so a o, shal be unto a e. Therefore. by the 15. e vij. the triangles iao, and o a e, are equiangles: And the angle a o e, is equall to the angle o i a. But the angle u o e, is foure times as great as the angle a o e: for it is equall to the two inner at a, and e, by the 15 e v j: which are equall betweene themselves, by the 9 e v. and by the 17 e vj. And therefore it is the double of a e o, which is the double, for the same cause, of a i o, equall to the same a o e. Therefore u o e, is the quadruple of the said a o e. Therefore a e, is the quadruple of the periphery e a. Therefore the whole u e a, is the quintuple of the same e a: And the whole periphery is decuple unto it. And the subtense a e, is the side of the decangle.
Therefore
14 If 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 sexangle with the quinangle followeth.
15 If 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.
Let the side of the inscribed quinquangle be a e: of the sexangle, e i: Of the decangle a o. I say, the side a e, doth in power countervaile the rest. For let there be two perpēdiculars: The first i o, the second i u, 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 be y. 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 the 3 e, xiij. Theresore it is equall to the quadrates of the other sides.
Let the proportion of this syllogisme be demonstrated: For this part onelyonly remaineth doubtfull. Therefore two triangles a e i, and y e i, are equiangles, having one common angle at e: And also two equall ones a e i, and e i y, the halfes, to wit, of the same e i s: Because that is, by the 17 e, vj: one of the two equalls, unto the which e i s, the out angle, is equall, by the 15 e. vj. And this doth insist upon a halfe periphery. For the halfe periphery a l s, is equall to the halfe periphery a r s: and also a l, is equall to a r. Therefore the remnant l s, is equall to the remnant r s: And the whole r l, is the double of the same r s: And therefore e r, is the double of e o: And r s, the double of o u. For the bisegments are manifest by the 10 e, xv. and the 11 e, xvj. Therefore the periphery e r s, is the double of the periphery e o u: And therefore the angle e i u, is the halfe of the angle e i s, by the 7 e, xvj. Therefore two angles of two triangles are equall: Wherefore the remainder, by the 4 e vij, is equall to the remainder. Wherefore by the 12 e, vij, as the side a e, is to e i: so is e i, to e y. Therefore by the 8 e xij, the oblong of the extreames is equall to the quadrate of the meane.
Now let o y, be knit together with a straight: Here againe the two triangles a o e, and a o y, are equiangles, having one common angle at a: And a o y, and o e a, therefore also equall: Because both are equall to the angle at a: That by the 17 e, vj: This by the 2 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 as e a, is to a o: so is e a, to a y. Wherefore by the 8 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 manifest Euclide doth use at the 16 p xiij.
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 sayd point, shall be the side of the inscribed quindecangle. 16. p. iiij.
For the side of the equilaterall triangle doth subtend ⅓ of the whole pheriphery. And two sides of the ordinate quinquangle doe subtend …/… of the same. Now ⅓—⅔ is 1/…: Therefore the space betweene the triangle, and the quinquangle shall be the 1/… of the whole periphery.
Therefore
17. If a quinquangle and a sexangle be inscribed into the same circle at the same point, the periphery intercepted betweene 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. Iun. 4. A. C. MDCXXII Campana pulsante pro. H. W.