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The twenty third Booke of Geometry, of a Prisma

1 A Pyramidate is a plaine solid comprehended of pyramides.

2. A pyramidate is a Prisma, or a mingled polyedrum.

3. A prisma is a pyramidate whose opposite plaines are equall, alike, and parallell, the rest parallelogramme. 13 d xj.

As here thou seest. The base of a pyramis was but one: Of a Prisma, they are two, and they opposite one against another, First equall; Then like: Next parallell. The other are parallelogramme.

4. The flattes of a prisma are two more than are the angles in the base.

And indeed as the augmentation of a Pyramis from a quaternary is infinite: so is it of a Prisma from a quinary: As if it be from a triangular, quadrangular, or quinquangular base; you shal have a Pentraedrum, Hexaedrum, Heptaedrum, and so in infinite.

5. The plaine of the base and heighth is the solidity of a right prisma.

6. A prisma is the triple of a pyramis of equall base and heighth. è 7 p. xij.

As in the example a prisma pentaedrum is cut into three equall pyramides. For the first consisting of the plaines aei, aeo, aoi, eio; is equall to the second consisting of the plaines aoi, aou, aiu, iou, by the 10 e vij. Because it is equall to it both in common base and heighth. Therefore the first and second are equall. And the same second is equall to it selfe, seeing the base is iou, and the toppe a. Then also it is equall to the third consisting of the plaines aiu, aiy, uiy, auy. Therefore three are equall.

If the base be triangular, the Prisma may be resolved into prisma's of triangular bases, and the theoreme shall be concluded as afore.

7. The plaine made of the base and the third part of the heighth is the solidity of a pyramis of equall base and heighth.

The heighth of a pyramis shall be found, if you shall take the square of the ray of the base out of the quadrate of the side: for the side of the remainder, by the 9 e xij, shall be the altitude or heighth, as in the example following.

Here the content of the triangle by the 18 e xij, is found to be 62.44/125 for the base of the pyramis. The altitude is 9.15/19: Because by the 12 e xviij, the side is of treble power to the ray. But if from 144, the quadrate of 12 the side, you take the subtriple i. 48, the remainder 96, by the 9 e xij, shall be the square of the heighth. And the side of the quadrate shall be 9.15/19. Now the third part of 9.15/19 is 3.5/19. And the plaine of 62.44/125 and 3.5/19, shall be 203.1103/2375 for the solidity of the pyramis.

So in the example following, Let 36, the quadrate of 6 the ray, be taken out of 292.9/1156 the quadrate of the side 17.3/34 the side 16.3/34 of 256.9/1156 the remainder shall be the height, whose third part is 5.37/102; the plaine of which by the base 72.1/4 shall be 387.11/24 for the solidity of the pyramis given.

If the pyramis be unperfit, first measure the whole, and then that part which is wanting: Lastly from the whole subtract that which was wanting, and the remaine shall be the solidity of the unperfect pyramis given: As here, let ao, the side of the whole be 16.5/12, eo the side of the particular be 8.1/16. Therefore the perpendicular of the whole ou, shall be 15.5/32: Whose third part is 5.5/96: Of which, and the base 93.3/11 the plaine shall be 471.134/1056 for the whole pyramis. But in the lesser pyramis, 9 the square of the ray 3, taken out of 65.1/256 the quadrate of the side 8.1/16 the remaine shall be 56.1/256; whose side is almost 7-1/2 for the heighth. The third part of which is 2-1/2. The base likewise is almost 22. The plaine of which two is 55, for the solidity of the lesser pyramis: And 471 – 55 is 416, for the imperfect pyramis.

After this manner you may measure an imperfect Prisma.

8. Homogeneall Prisma's of equall heighth are one to another as their bases are one to another, 29, 30, 31, 32 p xj.

The reason is, because they consist equally of like number of pyramides. Now it is required that they be homogeneall or of like kindes; Because a Pentaedrum with an Hexaedrum will not so agree.

This element is a consectary out of the 16 e iiij.

9. If they be reciprocall in base and heighth, they are equall.

This is a Consectary out the 18 e iiij.

10. If a Prisma be cut by a plaine parallell to his opposite flattes, the segments are as the bases are. 25 p. xj.

The segments are homogeneall because the prismas. Therefore seeing they are of equall heighth (by the heighth I meane of plaine dividing them) they shall be as their bases are: And here the bases are to be taken opposite to the heighth.

11. A Prisma is either a Pentaedrum, or Compounded of pentaedra's.

Here the resolution sheweth the composition.

12 If of two pentaedra's, the one of a triangular base, the other of a parallelogramme base, double unto the triangular, be of equall heighth, they are equall 40. p xj.

The cause is manifest and briefe: Because they be the halfes of the same prisma: As here thou maist perceive in a prisma cut into two halfes by the diagoni's of the opposite sides.

Euclide doth demonstrate it thus: Let the Pentaedra's aeiou, and ysrlm, be of equall heighth: the first of a triangular base eio: The second of a parallelogramme base sl, double unto the triangular. Now let both of them be double and made up, so that first be aeioun. The second ysrlvf. Now againe, by the grant, the base sl, is the double of the base eio,: whose double is the base eo, by the 12 e x. Therefore the bases sl, and eo, are equall: And therefore seeing the prisma's, by the grant, here are of equall heighth, as the bases by the conclusion are equall, the prisma's are equall; And therefore also their halfes aeiou, and ysmlr, are equall.

The measuring of a pentaedrall prisma was even now generally taught: The matter in speciall may be conceived in these two examples following.

The plaine of 18. the perimeter of the triangular base, and 12, the heighth is 216. This added to the triangular base, 15.18/31. or 15.3/5, almost twise taken, that is, 31.1/5, doth make 247.1/5, for the summe of the whole surface. But the plaine of the same base 15.3/5, and the heighth 12. is 187.1/5, for the whole solidity.

So in the pentaedrum, the second prisma, which is called Cuneus, (a wedge) of the sharpnesse, and which also more properly of cutting is called a prisma, the whole surface is 150, and the solidity 90.

13 A prisma compounded of pentaedra's, is either an Hexaedrum or Polyedrum: And the Hexaedrum is either a Parallelepipedum or a Trapezium.

14 A parallelepipedum is that whose opposite plaines are parallelogrammes ê 24. p xj.

Therefore a Parallelepipedum in solids, answereth to a Parallelogramme in plaines. For here the opposite Hedræ or flattes are parallell: There the opposite sides are parallell.

15 It is cut into two halfes with a plaine by the diagonies of the opposite sides. 28 p xj. It answereth to the 34. p j.

Let the Prisma be of sixe bases ai, yo, ye, ui, si, au. The diagonies doe cut into halfes, by the 10. e x. the opposite bases: And the other opposite bases or the two prisma's cut, are equall by the 3 e. Wherefore two prisma's are comprehended of bases, equall both in multitude and magnitude: therfore they are equall.

16 If it be halfed by two plaines halfing the opposite sides, the common bisection and diagony doe halfe one another 39. p xj.

Because here the diameters (such as is that bisection) are halfed betweene themselves [or doe halfe one another.] Let the parallelepipedum aeiouy, be cut in to y the halfs by two plains, fro srlm, uivf, halfing the opposite sides: Here the common section ts, and the diagony ao, doe cut one another.

17 If three lines be proportionall, the parallelepipedum of meane shall be equall to the equiangled parallelepipedum of all them. è 36. p xj.

It is a consectary out of the 8 e.

18 Eight rectangled parallelepiped's doe fill a solid place.

19 The Figurate of a rectangled parallelepipedum is called a solid, made of three numbers 17. d vij.

As if thou shalt multiply 1, 2, 3. continually, thou shalt make the solid 6. Item if thou shalt in like manner multiply 2, 3, 4. thou shalt make the solid 24. And the sides of that solid 6 solid shall be 1, 2, 3. Of 24, they shall be 2, 3, 4.

20 If two solids be alike, they have their sides proportionalls, and two meane proportionalls 21 d vij, 19. 21. p viij.

It is a consectary out of the 5 e xxij. But the meane proportionalls are made of the sides of the like solids, to wit, of the second, third, and fourth: Item of the third, fourth, and fifth, as here thou seest.

Of Geometry the twentie fourth Book. Of a Cube

1 A Rightangled parallelepipedum is either a Cube, or an Oblong.

2 A Cube is a right angled parallelepipedum of equall flattes, 25. d. xj.

As here thou seest in these two figures.

3 The sides of a cube are 12. the plaine angles 24. the solid 8.

4 If sixe equall quadrates be joyned with solid angles, they shall comprehend a cube.

As here in these two examples.

5 If from the angles of a quadrate, perpendiculars equall to the sides be tied together aloft, they shall comprehend a Cube. è 15 p xj.

It is a consectary following upon the former consectary: For then shall sixe equall quadrates be knit together:

6 The diagony of a Cube is of treble power unto the side.

For the Diagony of a quadrate is of double power to the side, by the 12 e, xij. And the Diagony of a Cube is of as much power as the side the diagony of the quadrate, by the same e. Therefore it is of treble power to the side.

7 If of foure right lines continually, proportionally the first be the halfe of the fourth, the cube of the first shall be the halfe of the Cube of the second è 33 p xj.

It is a consectary out of the 25 e, iiij. From hence Hippocrates first found how to answer Apollo's Probleme.

8 The solid plaine of a cube is called a Cube, to wit, a solid of equall sides. 19, d vij.

9 It is made of a number multiplied into his owne quadrate.

So is a Cube made by multiplying a number by it selfe, and the product againe by the first. Such are these nine first cubes made of the nine first Arithmeticall figures.

This is the generall invention of a Cube, both Geometricall and Arithmeticall.

10 If a right line be cut into two segments, the Cube of the whole shall be equall to the Cubes of the segments, and a double solid thrice comprehended of the quadrate of his owne segment and the other segment.

As for example, the side 12, let it be cut into two segments 10 and 2. The cube of 12. the whole, which is 1728, shall be equall to two cubes 1000, and 8 made of the segments 10. and 2. And a double solid; of which the first 600. is thrise comprehended of 100. the quadrate of his segment 10. and of 2. the other segment: The second 120. is thrice comprehended of 4, the quadrate of his owne segment, and of 10. the other segment. Now 1000 + 600 + 120. + 8, is equall to 1728: And therefore a right. &c.

But the genesis of the whole cube will make all this whole matter more apparant, to wit, how the extreme and meane solids are made. Let therefore a cube be made of three equall sides, 12, 12, and 12: And first of all let the second side be multiplied by the first, after this manner: And not adding the severall figures of the same degree, as was taught in multiplication, but multiply againe every one of them by the other side; and lastly, add the figures of the same degrees severally, thus:

11. The side of the first severall cube is the other side of the second solide: And the quadrate of the same side is the other side of the first solide, whose other side is the side of the second cube; and the quadrate of the same other side is the other side of the second solid.

In that equation therefore of foure solids with one solid, thou shalt consider a peculiar making and composition: First that the last cube be made of the last segment 2: Then that the second solid of 4, the quadrate of his owne segment, and of 10, the other segment be thrise comprehended: Lastly that the first solid of 100, the square of his owne segment 10 and the other segment 2, be also thrice comprehended: Lastly, that the Cube 1000, be made of the greater segment 10. Out of this making &c.

And thus much of the Cube: Of other sorts of parallelepipedes, as of the Oblong, the Rhombe, the Rhomboides, and of the Trapezium, and many flatted pentaedra's there is no peculiar stereometry. The measuring of a Prisma hath in the former beene generally declared, and is now onely farther be made more plaine by speciall examples; as here:

The plaine of the perimeter of the base 20, and the altitude 5 is 100. This added to 25 and 25, both the bases that is to 50, maketh 150, for the whole surface. Now the plaine of 25 the base, and the heighth 5 is 125, for the whole solidity.

So in the Oblong, the plaine of the base's perimeter 20, and the heighth 11, is 220, which added to the bases 24 and 24, that is 48, maketh 268, for the whole surface. But the plaine of the base 24, and the height 11, is 264, for the solidity.

The same also Geodesie or manner of measuring is used in the measuring of rectangled walls or gates and doores, which have either any window, or any hollow or voyde space cut out of them, if those voyde places be taken out of them; as here thou seest in the next following example. The thickenesse is 3 foote; the breadth 12, the heighth 11. Therefore the whole solidity is 396. Now the Gate way is of thickenesse 3 foote, of breadth 4: of heighth 6. And therefore the whole solidity of the Gate is 72 foote. But 396 – 72 are 314. Therefore the solidity of the rest of the wall remaining is 324.

In the second example, the length is 10. The breadth 8, the heighth 7. Therefore the whole body if it were found, were 560 foote. But there is an hollow in it, whose length is 6, breadth 5, heighth 7. Therefore the cavity or hollow place is 168. Now 560 – 168 is 392, for the solidity of the rest of the sound body.

Thus are such kinde of walls whether of mudde, bricke, or stone, of most large houses to bee measured. The same manner of Geodesy is also to be used in the measuring of a Rhombe, Rhomboides, Trapezium or mensall, and any kinde of multangled body. The base is first to be measured, as in the former: Then out of that and the heighth the solidity shall be manifested: As in the Rhombe the base is 24, the heighth 4. Therefore the solidity is 96.

In the Rhomboides, the base is 64.35/129: The heigh 16. Therefore the solidity is 1028.44/129.

The same is the geodesy of a trapezium, as in these examples: The surface of the first is 198: The solidity 192.1/2.

The surface of the second is 158.3/49: The solidity is 91.29/49.

The same shall be also the geodesy of a many flatted Prisma: As here thou seest in an Octoedrum of a sexangular base: The surface shall bee 762.6/11: The solidity 1492.4/11.

And from hence also may the capacity or content of vessels or measures, made after any manner of plaine solid bee esteemed and judged of as here thou seest. For here the plaine of the sexangular base is 41.1/7; (For the ray, by the 9 e xviij, is the side:) and the heighth 5, shall be 205.5/7. Therefore if a cubicall foote doe conteine 4 quarters, as we commonly call them, then shall the vessell conteine 822.6/7 quartes, that is almost 823 quartes.

Of Geometry the twenty fifth Booke; Of mingled ordinate Polyedra's

1. A mingled ordinate polyedrum is a pyramidate, compounded of pyramides with their toppes meeting in the center, and their bases onely outwardly appearing.

Seeing therefore a Mingled ordinate pyramidate is thus made or compounded of pyramides the geodesy of it shall be had from the Geodesy of the pyramides compounding it: And one Base multiplyed by the number of all the bases shall make the surface of the body. And one Pyramis by the number of all the pyramides; shall make the solidity.

2 The heighth of the compounding pyramis is found by the ray of the circle circumscribed about the base, and by the semidiagony of the polyedrum.

The base of the pyramis appeareth to the eye: The heighth lieth hidde within, but it is discovered by a right angle triangle, whose base is the semidiagony or halfe diagony, the shankes the ray of the circle, and the perpendicular of the heighth. Therefore subtracting the quadrate of the ray, from the quadrate of the halfe diagony the side of the remainder, by the 9 e xij. shall be the heighth. But the ray of the circle shall have a speciall invention, according to the kindes of the base, first of a triangular, and then next of a quinquangular.

3 A mingled ordinate polyedrum hath either a triangular, or a quinquangular base.

The division of a Polyhedron ariseth from the bases upon which it standeth.

4 If a quadrate of a triangular base be divided into three parts, the side of the third part shall be the ray of the circle circumscribed about the base.

As is manifest by the 12 e. xviij. And this is the invention or way to finde out the circular ray for an octoedrum, and an icosoedrum.

5 A mingled ordinate polyedrum of a triangular base, is either an Octoedrum, or an Icosoedrum.

This division also ariseth from the bases of the figures.

6 An octoedrum is a mingled ordinate polyedrum, which is comprehended of eight triangles. 27 d xj.

As here thou seest, in this Monogrammum and solidum, that is lines and solid octahedrum.

7 The sides of an octoedrum are 12. the plaine angles 24, and the solid 6.

8 Nine octoedra's doe fill a solid place.

For foure angles of a Tetraedrum are equall to three angles of the Octoedrum: And therefore 12. are equall to nine. Therefore nine angles of an octaedrum doe countervaile eight solid right angles.

9 If eight triangles, equilaters and equall be joyned together by their edges; they shall comprehend an octaedreum.

This construction is easie, as it is manifest in the example following: Where thou seest as it were two equilater and equall triangles of a double pentaedrum to cut one another.

10 If a right line of each side perpendicular to the center of a quadrate and equall to the halfe diagony be tied together with the angles, it shall comprehend an octaedrum, 14. d xiij.

For the perpendicular yu, and su, with the semidiagoni's, ua, uo, ui, ue, shall be made equall by the 2 e vij, the eight sides ya, ye, yo, yi, se, si, sa, so; And also eight triangles.

Therefore

11 The Diagony of an octaedrum is of double power to the side.

As is manifest by the 9 e xij.

And

12 If the quadrate of the side of an octaedrum, be doubled, the side of the double shall be the diagony.

As in the figure following, the side is 6. The quadrate is 36. the double is 72. whose side 8.8/17, is the diagony.

And from hence doth arise the geodesy of the octaedrum. For the semidiagony is 4.4/17. whose quadrate is 17.171/289. And the quadrate of 6, the side of the equilater triangle, being of treble power to the ray, by the 12 e, xviij. is 36. And the side of 12. the third part 3.3/7 is the ray of the circle. Wherefore 8.8/17. that is 5.21/289. is the quadrate of the perpendicular, whose side 2.1/5 is the height of the same perpendicular: whose third part againe 11/25. multiplied by 15.18/31. the triangular base doe make 11.66/155 for one of the eight pyramides: Therefore the same 11.66/155 multiplied by eight, shall make 91.63/155 for the whole octoedrum.

13 An Icosaedrum is an ordinate polyedrum comprehended of 20 triangles 29 d xj.

14 The sides of an Icosaedrum are 30. plaine angles 60. the solid 12.

15 If twentie ordinate and equall triangles be joyned with solid angles, they shall comprehend an Icosaedrum.

This fabricke is ready end easie, as is to be seene in this example following.

16. If ordinate figures, to wit, a double quinquangle, and one decangle be so inscribed into the same circle, that the side of both the quinquangle doe subtend two sides of the decangle, sixe right lines perpendicular to the circle and equall to his ray, five from the angles of one of the quinquangles, knit together both betweene themselves, and with the angles of the other quinquangle; the sixth from the center on each side continued with the side of the decangle, and knit therewith the five perpendiculars, here with the angles of the second quinquangle, they shall comprehend an icosaedrum. è 15 p xiij.

For there shall be made 20 triangles, both equilaters and equall. Let there be therefore two ordinate quinquangles, the first aeiou; The second ysrlm; each of whose sides let them subtend two sides of a decangle; to wit, utym, let it subtend ya, and am. Then let there be five perpendiculars from the angles of the second quinquangle yj, sy, rv, lf, mt. And let them be knit first one with another, by the lines nj, jv, vf, ft, tn. Secondarily, with the angles of the first quinquangle, by the lines ne, ej, ji, iv, of, fu, ut, ta, an. The sixth perpendicular from the center d, let it be bg, the ray dc, continued at each end with the side of the decangle, cg, and db, tied together about with the perpendiculars, as by the lines ng, tg: Beneath with the angles of the first quinquangle, as by the lines be, bi, and in other places in like manner, and let all the plaines be made up. This say I, is an Icosaedrum; And is comprehended of 20. triangles, both equilaters and equall. First, the tenne middle triangles, leaving out the perpendiculars, that they are equilaters and equall, one shall demonstrate, as nat. For mt and yu, because they are perpendiculars, they are also, by the 6 e xxj. parallells: And by the grant, equall. Therefore by the 27 e, v, nt, is equall to ym, the side of the quinquangle. Item na, by the 6 e xij. is of as great power, as both the shankes ny, and ya, that is, by the construction, as the sides of the sexangle and decangle: And, by the converse of the 15. e xviij. it is the side of the quinquangle. The same shall fall out of ot. Wherefore nat, is an equilater triangle. The same shall fall out of the other nine middle triangles, nae, nej, eji, jiv, ivo, vof, fou, fut, uta, tan.

In like manner also shall it be proved of the five upper triangles, by drawing the right lines dy and cn which as afore (because they knit together equall parallells, to wit, dc, and yn) they shall be equall. But dy, is the side of a sexangle: Therefore cn, shall be also the side of a sexangle: And cg, is the side of a decangle: Therefore an, whose power is equall to both theirs by the 9 e xij. shall by the converse of the 15 e xviij, be the side of a quinquangle: And in like manner gt, shall be concluded to be the side of a quinquangle. Wherefore ngt, is an equilater: And the foure other shall likewise be equilaters.

The other five triangles beneath shall after the like manner be concluded to be equilaters. Therefore one shall be for all, to wit, ibe, by drawing the raies di, and de. For ib, whose power, as afore, is as much as the sides of the sexangle, and decangle, shall be the side of the quinquangle: And in like sort be, being of equall power with de, and do, the sides of the sexangle and decangle, shall be the side of the quinquangle. Wherefore the triangle ebi, is an equilater: And the foure other in like manner may be shewed to be equilaters. Therefore all the side of the twenty triangles, seeing they are equall, they shall be equilater triangles: And by the 8 e, vij. equall.

17 The diagony of an icosaedrū is irrational unto the side.

This is the fourth example of irrationality, or incommensurability. The first was of the Diagony and side of a square or quadrate. The second was of the segments of a line proportionally cut. The third of the Diameter of a circle and the side of a quinquangle.

18 The power of the diagony of an icosaedrum is five times as much as the ray of the circle.

For by the 13 e, xviij. the line continually made of the side of the sexangle and decangle is cut proportionally, and the greater segment is the side of the sexangle: As here. Let the perpendicular ae, be cut into two equall parts in i. Then eo, that is the lesser segment continued with the halfe of the greater, that is, with ie. it shall by the 6 e xiiij, be of power five times so great as is the power of the same halfe. Therefore seeing that io, the halfe of the diagony is of power fivefold to the halfe: the whole diagony shall be of power fivefold to the whole cut.

And from hence also shall be the geodesy of the Icosaedrum. For the finding out of the heighth of the pyramis, there is the semidiagony of the side of the decangle and the halfe ray of the circle: But the side of the decangle is a right line subtending the halfe periphery of the side of the quinquangle, or else the greater segment of the ray proportionally cut. For so it may be taken Geometrically, and reckoned for his measure. Therefore if the quadrate of the side of the decangle, be taken out of the quadrate of the side of the quinquangle, there shall by the 15 e xviij, remaine the quadrate of the sexangle, that is of the ray. The side of the decangle (because the side of the quinquangle here is 6) shall be 3.3/35 to wit a right line subtending the halfe periphery. Now the halfe ray shall thus be had. The quadrates of the quinquangle and decangle are 36, and 9.639/1225. And this being subducted fro that, the remaine 26.386/1225 by the 15 e xviij, shall be the quadrate or square of the sexangle: And the side of it, 5, and almost 5/7 shall be the ray: The halfe ray therefore shall be 2.6/7. To the side of the decangle 3.3/35 adde 2.6/7: the whole shal be 5.33/35 for the semi-diagony of the Icosaedrum. The ray of the circle circumscribed about the triangle, is by the 12 e xviij, the same which was before 3.3/7 to wit of the quadrate 12. Therefore if the quadrate of the circular ray, be taken out of the quadrate of the halfe diagony, there shall remaine the quadrate of the heighth and perpendicular: the quadrate of the halfe-diagony is 35.389/1225: the quadrate of the circular ray is 12. This taken out of that beneath 23.639/1225: whose side is almost 5, for the perpendicular and heighth proposed: From whence now the Pyramis is esteemed. The case of a triangular pyramis is 15.18/31. The Plaine of this base and the third part of the heighth is 25.30/31 for the solidity of one Pyramis. This multiplyed by 20 maketh 519.11/31 for the summe or whole solidity of the Icosaedum. And this is the geodesy or manner of measuring of an Icosaedrum.

19. A mingled ordinate polyedrum of a quinquangular base is that which is comprehended of 12 quinquangles, and it is called a Dodecaedrum.

20. The sides of a Dodecaedrum are 30, the plaine angles 60. the solid 20.

21. If 12 ordinate equall quinquangles be joyned with solid angles, they shall comprehend a Dodecaedrum.

As here thou seest.

22. If the sides of a cube be with right lines cut into two equall parts, and three bisegments of the bisecants in the abbuting plaines, neither meeting one the other, nor parallell one unto another, two of one, the third of that next unto the remainder, be so proportionally cut that the lesser segments doe bound the bisecant: three lines without the cube perpendicular unto the sayd plaines from the points of the proportionall sections, equall to the greater segment knit together, two of the same bisecant, betweene themselves and with the next angles of cube; the third with the same angles, they shall comprehend a dodecaedrum. 17 p xiij.

Let there be two plaines for a cube for all, that one quinquangle for twelve may be described, and they abutting one upon another, aeio, and euyi, having their sides halfed by the bisecantes, sr, lm, rn, jv: And the three bisegments or portions of the bisegments lm, and rn, neither concurring or meeting, nor parallell one to another; two of the said lm, to wit, fl, and fm: The third next unto the remainder, that is lr. And let each bisegment be cut proportionally in the points d, c, g; so that the lesser segments doe bound the bisecant, to wit, dl, cm, and gr. Lastly let there be three perpendiculars from the points db, cg, to the said d, cp, gz: And the two first knit one to another, by bp: And againe with the angles of the cube, by be, and pi: The third knit with the same angles, by ze, and zi: And let all the plaines be made up. I say first, that the five sides bp, pi, iz, ze, and eb are equall; Because, every one of them severally are the doubles of the same greater segment. For in drawing the right lines de and eg, ig, it shall be plaine of two of them; And after the same manner of the rest. First therefore cd, and bp, are equall by the 6 e xxj, and by the 27 e v. Therefore bp, is the double of the greater segment. Then the whole fl, cut proportionally, and the lesser segment dl, they are by the 7 e xiiij, of treble power to the greater fd, that is, by the fabricke db. Therefore le wich is equall to lf, the line cut, and ld, are of treble power to the same db: But by the 9 e xij, de is of as much power as le, and ld too. Therefore de is of treble power to db. Therefore both ed, and db, are of quadruple power to db. But be, by the 9 e xij, is of as much power as ed, and db. And therefore be, is of quadruple value to db: And by the 14 e xij, it is the double of the said db. Therefore the two sides eb, and bp, are equall: And by the same argument pi, iz, and ze, are equall. Therefore the quinquangle is equilater.

I say also that it is a Plaine quinquangle: For it may be said to be an oblique quinquangle; and to be seated in two plaines. Let therefore fh be parallell to db, and cp: and be equall unto them. And let hz, be drawne: This hz shall be cut one line, by the 14 e vij. For as the whole tr, that is rf, is unto the greater segment that is to fh: so fh, that is zg, is unto gr. And two paire of shankes fh, gr, fc, gz, by the 6 e xxj, are alternely or crosse-wise parallell. Therefore their bases are continuall.

Hitherto it hath beene prooved that the quinquangle made is an equilater and plaine: It remaineth that it bee prooved to be Equiangled. Let therefore the right lines ep, and ec, be drawne: I say that the angles, pbe, and ezi, are equall: Because they have by the construction, the bases of equall shankes equall, being to wit in value the quadruple of le. For the right line lf, cut proportionally, and increased with the greater segment df, that is fc, is cut also proportionally, by the 4 e xiiij, and by the 7 e xiiij, the whole line proportionally cut, and the lesser segment, that is cp, are of treble value to the greater fl, that is of the sayd le. Therefore el, and lc, that is ec, and cp, that is ep, is of quadruple power to el: And therefore by the 14 e xij, it is the double of it: And ei, it selfe in like manner, by the fabricke or construction, is the double of the same. Therefore the bases are equall. And after the same manner, by drawing the right lines id, and ib, the third angle bpi, shall be concluded to be equal to the angle ezi. Therefore by the 13 e xiiij, five angles are equall.