This is the fifth example of irrationality and incommensurability. The first was of the diagony and side of a quadrate or square. The second was of a line proportionally cut and his segments: The third is of the diameter of a Circle and the side of an inscribed quinquangle. The fourth was of the diagony and side of an icosahedrum. The fifth now is of the diagony and side of a dodecahedrum.

If the side of a cube be cut proportionally, the greater segment shall be the side of a dodecahedrum

For that hath beene told you even now.

But from hence also doth arise the geodesy or māner of measuring of a dodecahedrum. For if the quadrate of the line subtending the angle of a quinquangle be trebled, the half of the treble shall be the side of the semidiagony of the dodecahedrum: Because by the

6 e xxiiij

, the diagony of the cube, that is of the dodecahedrum is of treble power to the side of the cube. But if the quadrate of the side of the decangle be taken out of the quadrate of the side of the quinquangle; The side of the remainder shall be the ray of the circle circumscribed about a quinquangle. Lastly if the quadrate of the ray, be taken of the quadrate of the half-diagony; the side of the remainder shall be the heighth of perpendicular. As if the side of the decangle be 7.3/5: The quadrate of that shall be 57.19/25: the treble of which is 173.7/25 whose side is about 13.107/131 for the side of the Dodecahedrum, therefore 6.119/131 the halfe shall be the semidiagony of the dodecahedrum. The ray of the Circle shall now thus be found. If the quadrate of the side of the decangle be taken out of the quadrate of the side of the sexangle; the side of the remainder, shall be the Ray of the Circle, by the

and

9 e xviij

. As here the side of the Quinquangle is 4.2/3. The side of the Decangle 2.2/5: And the quadrates therefore are 21.7/9, and 5.19/25. This subducted from that leaveth 16.4/225 whose side is 4.2/15 for the Ray of the Circle.

The semidiagony and ray of the circle thus found, the altitude remaineth. Take out therefore the quadrate of the ray of the circle, 16.4/225 out of the quadrate of the semidiagony 47.12458/17161, the side of the remainder 31.2714406/3861225 is for the altitude or heighth: whose 1/3 is 5/3. The quinquangled base is almost 38. Which multiplied by 5/3 doth make 63.1/3 for the solidity of one Pyramis; which multiplied by 12, doth make 760. for the soliditie of the whole dodecaedrum.

There are but five ordinate solid plaines

This appeareth plainely out of the nature of a solid angle, by the kindes of plaine figures. Of two plaine angles a solid angle cannot be comprehended. Of three angles of an ordinate triangle is the angle of a Tetrahedrum comprehended: Of foure, an Octahedrum: Of five, an Icosahedrum: Of sixe none can be comprehended: For sixe such like plaine angles, are equall to 12 thirds of one right angle, that is to foure right angles. But plaine angles making a solid angle, are lesser than foure right angles, by the

8 e xxij

. Of seven therefore, and of more it is, much lesse possible. Of three quadrate angles the angle of a cube is comprehended: Of 4. such angles none may be comprehended for the same cause. Of three angles of an ordinate quinquangle, is made the angle of a Dodecahedrum. Of 4. none may possibly be made; For every such angle: For every one of them severally doe countervaile one right angle and 1/5 of the same, Therefore they would be foure, and three fifths. Of more therefore much lesse may it be possible.

This demonstration doth indeed very accurately and manifestly appeare, Although there may be an innumerable sort of ordinate plaines, yet of the kindes of angles five onely ordinate bodies may be made; From whence the Tetrahedrum, Octahedrum, and Icosahedrum are made upon a triangular base: the Cube upon a quadrangular: And the Dodecahedrum, upon a quinquangular.

Geometry

the twenty sixth Booke; Of a

Spheare

An imbossed solid is that which is comprehended of an imbossed surface

And it is either a spheare or a Mingled forme

A speare is a round imbossement

It may also be defined to be that which is comprehended of a sphearical surface. A sphearicall body in Greeke is called

Sphæra

, in Latine

Globus

, a Globe.

Therefore

A Spheare is made by the conversion of a semicircle, the diameter standing still. 14 d xj.

As here thou seest.

The greatest circle of a spheare, is that which cutteth the spheare into two equall parts

Therefore

That circle which is neerest to the greatest, is greater than that which is farther off

And

Those which are equally distant from the greatest are equall

As in the example above written.

The plaine of the diameter and sixth part of the sphearicall is the solidity of the spheare

As before there was an analogy betweene a Circle and a Sphericall: so now is there betweene a Cube and a spheare. A cubicall surface is comprehended of sixe quadrate or square and equall bases: And a spheare in like manner is comprehended of sixe equall sphearicall bases compassing the cubicall bases. A cube is made by the multiplication of the sixth part of the base, by the side: And a spheare likewise is made by multiplying the sixth part of the sphearicall by the diameter, as it were by the side: so the plaine of 616/6 and 14, the diameter is 1437.1/3 for the solidity of the spheare.

Therefore

As 21 is unto 11, so is the cube of the diameter unto the spheare

As here the Cube of 14 is 2744. For it was an easy matter for him that will compare the cube 2744, with the spheare, to finde that 2744 to be to 1437.1/3 in the least boundes of the same reason, as 21 is unto 11.

Thus much therefore of the Geodesy of the spheare: The geodesy of the Sectour and section of the spheare shall follow in the next place.

And

10.

The plaine of the ray, and of the sixth part of the sphearicall is the hemispheare

But it is more accurate and preciser cause to take the halfe of the spheare.

11.

Spheares have a trebled reason of their diameters

So before it was told you; That circles were one to another, as the squares of their diameters were one to another, because they were like plaines: And the diameters in circles were, as now they are in spheares, the homologall sides. Therefore seeing that spheres are figures alike, and of treble dimension, they have a trebled reason of their diameters.

12.

The five ordinate bodies are inscribed into the same spheare, by the conversion of a semicicle having for the diameter, in a tetrahedrum, a right line of value sesquialter unto the side of the said tetrahedrum; in the other foure ordinate bodies, the diagony of the same orginate

The Adscription of ordinate plaine bodies is unto a spheare, as before the Adscription plaine surfaces was into a circle; of a triangle, I meane, and ordinate triangulate, as Quadrangle, Quinquangle, Sexangle, Decangle, and Quindecangle. But indeed the Geometer hath both inscribed and circumscribed those plaine figures within a circle. But these five ordinate bodies, and over and above the Polyhedrum the Stereometer hath onely inscribed within the spheare. The Polyhedrum we have passed over, and we purpose onely to touch the other ordinate bodies.

Out of the reason of the axeltree of the sphearicall the sides of the tetraedrum, cube, octahedrum and dodecahedrum are found out

The axeltree in the three first bodies is rationall unto the side, as was manifested in the former. For it is of the sesquialter valew unto the side of the tetrahedrum; of treble, to the side of the cube: Of double, to the side of the Octahedrum. Therefore if the axis

, be cut by a double reason in

: And the perpendicular

, be knit to

, and

, shall be the side of the tetrahedrum; and

, of the cube, as was manifest by the

10 e viij

, and

25 iiij

: And the greater segment of the side of the cube proportionally cut, is by the

24 e, xxv

If the same axis be cut into two halfes, as in

: And the perpendicular

, be erected: And

, and

, be knit together, the same

, thus knitting them, shall be the side of the Octahedrum, as is manifest in like manner, by the said

10 e, viij

, and

25 e iiij

The side of the Icosahedrum is had by this meanes.

14.

If a right line equall to the axis of the sphearicall, and to it from the end of the perpendicular be knit unto the center, a right line drawne from the cutting of the periphery unto the said end shall be the side of the Icosahedrum

As here let the Axis

; be the diameter of the circle

aue

, and

, equall to the same axis, and perpendicular from the end, be knit unto the center, by the right line

: A right drawne from the section

, unto

, shall be the side of the Icosahedrum. From

, let the perpendicular

, be drawne: Here the two triangles

iao

, &

uyo

, are equiangles by the

13 e, vij

. Therfore by the

12 e, vij

. as

, is unto

: so is

, unto

. But

, is the double of the said

: Therefore

, is the double of the same

: Therefore by the

14 e, xij

, it is of quadruple power unto it: And therefore also

, and

, that is, by the

9 e xij

, that is againe by the

28 e, iiij

, is of quintuple power to

. But

, is lesser than

, that is, than

: Let therefore

, be cut off equall to it. Now as the halfe of

, is of quintuple valew to the halfe of

: so the double

, is of quintuple power to the double

. Therefore, by the

18 e xxv

. seeing that the diagony

, is of quintuple power to

; the said

, shall be the side of the sexangle inscribed into a circle, circumscribing the quinquangle of the Icosahedrum. But the perpendicular

, is equall to

; because each of them is the double of

. Wherefore

, is the side of the sexangle. But

, is the side of the Decangle: For it is equall to

: Because if from equall rayes

, and

, you take equall portions

, and

: There shall remaine equall,

, and

. And the Diagony of an Icosahedrum by the

16 e xxv

, is compounded of the side of the sexangle, continued at each end with the side of the decangle. Wherefore

, is the side of the decangle. Lastly,

, whose power is as much as the sides of the sexangle and decangle, by the

15. e, xviij

, shall be the side of an Icosahedrum.

Of the five ordinate bodies inscribed into the same spheare, the tetrahedrum in respect of the greatnesse of his side is first, the Octahedrum, the second; the Cube, the third; the Icosahedrum, the fourth; and the Dodecahedrum, the fifth

As it will plainely appeare, if all of them be gathered into one, thus. For

, the side of the Tetrahedrum, subtendeth a greater periphery than

, the side of the Octahedrum; And

, a greater than

, the side of the Cube; because it subtendeth but the halfe: And

, greater than

, the side of the Icosahedrum: And

, greater than

, the side of Dodecahedrum.

The latter,

Euclide

doth demonstrate with a greater circumstance. Therefore out of the former figures and demonstrations, let here be repeated, The sections of the axis first into a double reason in

: And the side of the sexangle

: And the side of the Decangle

, inscribed into the same circle, circumscribing the quinquangle of an icosahedrum: And the perpendiculars

, and

Here the two triangles

aie

, and

ies

, are by the

8 e, viij

. alike; And as

, is unto

: So is

, unto

: And by

25 e, iiij

, as

, is to

: so is the quadrate of

, to the quadrate of

: And inversly or backward, as

, is to

: so is the quadrate of

, to the quadrate of

. But

, is the triple of

. Therefore the quadrate of

, is the triple of

. But the quadrate of

, by the grant, and

14 e xij

, the quadruple of the quadrate of

. Therefore also it is greater than the quadrate of

: And the right line

, is greater than

, and

, therefore is much greater. But

, is by the grant compounded of the sides of the sexangle and decangle

, and

. Therefore by the 1 c.

5 e, 18.

it is cut proportionally: And the greater segment is the side of the sexangle, to wit,

: And the greater segment of

, proportionally also cut, is

. Therefore the said

, is greeter than

: And even now it was shewed

, was equall to

. Therefore

, is greater than

: But

, the side of the Icosahedrum, by

22. e vj

. is greater than

. Therefore the side of the Icosahedrum is much greater, then the side of the dodecahedrum.

Geometry

the twenty seventh Book; Of the Cone and Cylinder

A mingled solid is that which is comprehended of a variable surface and of a base

For here the base is to be added to the variable surface.

If variable solids have their axes proportionall to their bases, they are alike. 24. d xj

It is a Consectary out of the

19 e, iiij

. For here the axes and diameters are, as it were, the shankes of equall angles, to wit, of right angles in the base, and perpendicular axis.

A mingled body is a Cone or a Cylinder

The cause of this division of a varied or mingled body, is to be conceived from the division of surfaces.

A Cone is that which is comprehended of a conicall and a base

Here the base is a circle.

Therefore

It is made by the turning about of a right angled triangle, the one shanke standing still

As it appeareth out of the definition of a variable body.

And

A Cone is rightangled, if the shanke standing still be equall to that turned about: It is Obtusangeld, if it be lesse: and acutangled, if it be greater. ê 18 d xj

Here a threefold difference of the heighth of a Cone is professed, out of the threefold difference of the angles, whereby the toppe of the halfed cone is distinguished: Notwithstanding this consideration belongeth rather to the Optickes, than to Geometry. For a Cone a farre off seeme like triangle. Therefore according to the difference of the heighth, it appeareth with a right angled, or obtusangled or acutangled toppe: As here the least Cone is obtusangled: the middle one rightangled: and the highest acutangled. But the cause of this threefold difference in the angles from of the difference of the shankes, is out of the consectaries of the threefold triangle of a right line cutting the base into two equall parts, as appeareth at the end of the

viij

Booke.

And

A Cone is the first of all variable

For a Cone is so the first in variable solids, as a triangle is in rectilineall plaines: As a Pyramis is in solid plaines: For neither may it indeed be divided into any other variable solids more simple.

And

Cones of equall heighth are as their bases are 11. p xij

As here you see.

And

They which are reciprocall in base and heighth are equall, 15 p xij.

These are consectaries drawne out of the

and

18 e. iiij

. As here you see.

A Cylinder is that which is comprehended of a cylindricall surface and the opposite bases

For here two circles, parallell one to another are the bases of a Cylinder.

Therefore

It is made by the turning about of a right angled parallelogramme, the one side standing still. 21. d xj

As is apparant out the same definition of a varium.

12.

A plaine made of the base and heighth is the solidity of a Cylinder

The geodesy here is fetch'd from the prisma: As if the base of the cylinder be 38.1/2: Of it and the heighth 12, the solidity of the cylinder is 462.

This manner of measuring doth answeare, I say, to the manner of measuring of a prisma, and in all respects to the geodesy of a right angled parallelogramme.

If the cylinder in the opposite bases be oblique, then if what thou cuttest off from one base thou doest adde unto the other, thou shalt have the measure of the whole; as here thou seest in these cylinders,

and

From hence the capacity or content of cylinder-like vessell or measure is esteemed and judged of. For the hollow or empty place is to be measured as if it were a solid body.

As here the diameter of the inner Circle is 6 foote: The periphery is 18.6/7: Therefore the plot or content of the circle is 28.2/7. Of which, and the heighth 10, the plaine is 282.6/7 for the capacity of the vessell. Thus therefore shalt thou judge, as afore, how much liquor or any thing else conteined, a cubicall foote may hold.

13.

A Cylinder is the triple of a cone equall to it in base and heighth. 10 p xij

The demonstration of this proposition hath much troubled the interpreters. The reason of a Cylinder unto a Cone, may more easily be assumed from the reason of a Prisme unto a Pyramis: For a Cylinder doth as much resemble a Prisme, as the Cone doth a Pyramis: Yea and within the same sides may a Prisme and a Cylinder, a Pyramis and a Cone be conteined: And if a Prisme and a Pyramis have a very multangled base, the Prisme and Cylinder, as also the Pyramis and Cone, do seeme to be the same figure. Lastly within the same sides, as the Cones and Cylinders, so the Prisma and Pyramides, from their axeltrees and diameters may have the similitude of their bases. And with as great reason may the Geometer demand to have it granted him, That the Cylinder is the treble of a Cone: As it was demanded and granted him, That Cylinders and Cones are alike, whose axletees are proportionall to the diameters of their bases.

Therefore

14.

A plaine made of the base and thid part of the height, is the solidity of the cone of equall base & height;

The heighth is thus had. If the square of the ray of the base, be taken out of the square of the side, the side of the remainder shall bee the heighth, as is manifest by the

9 e xij

. Here therefore the square of the ray 5, is 25. The square of 13, the side is 169. And 169 – 25, are 144; whose side is 12 for the heighth: The third part of which is 4. Now the circular base is 78.4/7: And the plaine of these is 314.2/7 for the solidity of the Cone.

But the analogie of a conicall unto a Cylinder like surface doth not answeare, that the Conicall should be the subtriple of the Cylindricall, as the Cone is the subtriple of the Cylinder.

Of two cones of one common base is made

Archimede's Rhombus

, as here, whose geodæsy shall be cut of two cones.

And

15.

Cylinder of equall heighth are as their bases are. 11 p xij

Sackes in which they carry corne, are for the most part of a cylinderlike forme. If an husbandman therefore shall lend unto his neighbour a sacke full or corne, and the base of the sacke be 4 foote over. And the neighbour afterward for that one sacke, shall pay him 4 sacke fulls, every sacke being as long as that was, yet but one foote over in the diameter, he may be thought peradventure to have repayed that which he borrowed in equall measure, to wit in heighth and base. But it shall be indeed farre otherwise: For there is a great difference betweene the quadrate of the foure severall diameters, 1. 1. 1. 1. that is 4: and 16, the quadrate of 4, the diameter of that sacke by which it was lent. For Circles are one unto another as the quadrates of their diameters are one to another, by the

2 e xv

. Therefore he payd him but one fourth part of that which he borrowed of him.

And

Cylinders reciprocall in base and heighth are equall. 15 p xij

Both these affections are in common attributed to the equally manifold of first figures.

And

17.

If a cylinder be cut with a plaine surface parallell to his opposite bases, the segments are, as their axes are 13 p xij

As here thou seest. For the axes are the altitudes or heights. It is likwise a consectary following upon that generall theoreme of first figure, but

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The Way To Geometry

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