The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara Part 5

[* =To any Square geuen, to geue a Circle, equall.=]

* If you describe a Circle, which shall be in that proportion, to your Circle inscribed, as the Square is to the same Circle: This, you may do, by my Annotations, vpon the second proposition of the twelfth boke of _Euclide_, in my third Probleme there. Your diligence may come to a proportion, of the Square to the Circle inscribed, nerer the truth, then is the proportion of 14. to 11. And consider, that you may begyn at the Circle and Square, and so come to conclude of the Sphaere, & the Cube, what their proportion is: as now, you came from the Sphaere to the Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough one hole draw?, as the maner is) if you make a Square figure & way it: and then, describing theron, the Circle inscribed: & cut of, & file away, precisely (to the Circle) the ouerplus of the Square: you shall then, waying your Circle, see, whether the waight of the Square, be to your Circle, as 14. to 11. As I haue Noted, in the beginning of _Euclides_ twelfth boke. &c. after this resort to my last proposition, vpon the last of the twelfth. And there, helpe your selfe, to the end.

And, here, Note this, by the way.

[Note Squaring of the Circle without knowledge of the proportion betwene Circ.u.mference and Diameter.]

That we may Square the Circle, without hauing knowledge of the proportion, of the Circ.u.mference to the Diameter: as you haue here perceiued. And otherwayes also, I can demonstrate it. So that, many haue c.u.mberd them selues superfluously, by trauailing in that point first, which was not of necessitie, first: and also very intricate. And easily, you may, (and that diuersly) come to the knowledge of the Circ.u.mference: the Circles Quant.i.tie, being first knowen. Which thing, I leaue to your consideration: making hast to despatch an other Magistrall Probleme: and to bring it, nerer to your knowledge, and readier dealing with, then the world (before this day,) had it for you, that I can tell of. And that is, _A Mechanicall Dubblyng of the Cube: &c._ Which may, thus, be done:

[To Dubble the Cube redily: by Art Mechanicall: depending vppon Demonstration Mathematicall.]

+Make of Copper plates, or Tyn plates, a foursquare vpright Pyramis, or a Cone: perfectly fashioned in the holow, within. Wherin, let great diligence be vsed, to approche (as nere as may be) to the Mathematicall perfection of those figures. At their bases, let them be all open: euery where, els, most close, and iust to. From the vertex, to the Circ.u.mference of the base of the Cone: & to the sides of the base of the Pyramis:+

[=I. D.= =The 4. sides of this Pyramis must be 4. Isosceles Triangles alike and aequall.=]

+Let 4. straight lines be drawen, in the inside of the Cone and Pyramis: makyng at their fall, on the perimeters of the bases, equall angles on both sides them selues, with the sayd perimeters. These 4. lines (in the Pyramis: and as many, in the Cone) diuide: one, in 12. aequall partes: and an other, in 24. an other, in 60, and an other, in 100. (reckenyng vp from the vertex.) Or vse other numbers of diuision, as experience shall teach you.+

[=I. D.= =* In all workinges with this Pyramis or Cone, Let their Situations be in all Pointes and Conditions, alike, or all one: while you are about one Worke. Els you will erre.=]

+Then, * set your Cone or Pyramis, with the vertex downward, perpendicularly, in respect of the Base. (Though it be otherwayes, it hindreth nothyng.) So let th? most stedily be stayed.+ Now, if there be a Cube, which you wold haue Dubbled. Make you a prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so proporti in respect of your Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the waight of them, in water, 3. or 4. times: at the least: what stuff so euer they be made of. Let not your Solid angle, at the vertex, be to sharpe: but that the water may come with ease, to the very vertex, of your hollow Cone or Pyramis. Put one of your Solid Cubes in a Balance apt: take the waight therof exactly in water. Powre that water, (without losse) into the hollow Pyramis or Cone, quietly. Marke in your lines, what numbers the water Cutteth: Take the waight of the same Cube againe: in the same kinde of water, which you had before:

[=I. D.= =* Consider well whan you must put your waters togyther: and whan, you must empty your first water, out of your Pyramis or Cone. Els you will erre.=]

put that* also, into the Pyramis or Cone, where you did put the first.

Marke now againe, in what number or place of the lines, the water Cutteth them. Two wayes you may conclude your purpose: it is to wete, either by numbers or lines. By numbers: as, if you diuide the side of your Fundamentall Cube into so many aequall partes, as it is capable of, conueniently, with your ease, and precisenes of the diuision. For, as the number of your first and lesse line (in your hollow Pyramis or Cone,) is to the second or greater (both being counted from the vertex) so shall the number of the side of your Fundamentall Cube, be to the nuber belonging to the Radicall side, of the Cube, dubble to your Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it selfe, whether it be dubble or no, to the Cubik number of your Fundamentall Cube. By lines, thus: As your lesse and first line, (in your hollow Pyramis or Cone,) is to the second or greater, so let the Radical side of your Fundam?tall Cube, be to a fourth proportionall line, by the 12. proposition, of the sixth boke of _Euclide_. Which fourth line, shall be the Rote Cubik, or Radicall side of the Cube, dubble to your Fundamentall Cube: which is the thing we desired.

[? G.o.d be thanked for this Inuention, & the fruite ensuing.]

For this, may I (with ioy) say, ??????, ??????, ??????: thanking the holy and glorious Trinity: hauing greater cause therto, then

[* Vitruuius. Lib. 9. Cap. 3.]

* _Archimedes_ had (for finding the fraude vsed in the Kinges Crowne, of Gold): as all men may easily Iudge: by the diuersitie of the frute following of the one, and the other. Where I spake before, of a hollow Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it with water, precisely full, and poure that water into your Pyramis or Cone. And here note the lines cutting in your Pyramis or Cone. Againe, fill your coffen, like as you did before. Put that Water, also, to the first. Marke the second cutting of your lines. Now, as you proceded before, so must you here procede.

[* Note.]

* And if the Cube, which you should Double, be neuer so great: you haue, thus, the proportion (in small) betwene your two litle Cubes: And then, the side, of that great Cube (to be doubled) being the third, will haue the fourth, found, to it proportionall: by the 12. of the sixth of Euclide.

[Note, as concerning the Sphaericall Superficies of the Water.]

Note, that all this while, I forget not my first Proposition Staticall, here rehea.r.s.ed: that, the Superficies of the water, is Sphaericall.

Wherein, vse your discretion: to the first line, adding a small heare breadth, more: and to the second, halfe a heare breadth more, to his length. For, you will easily perceaue, that the difference can be no greater, in any Pyramis or Cone, of you to be handled. Which you shall thus trye. _For finding the swelling of the water aboue leuell._

"Square the Semidiameter, from the Centre of the earth, to your first Waters Superficies. Square then, halfe the Subtendent of that watry Superficies (which Subtendent must haue the equall partes of his measure, all one, with those of the Semidiameter of the earth to your watry Superficies): Subtracte this square, from the first: Of the residue, take the Rote Square. That Rote, Subtracte from your first Semidiameter of the earth to your watry Superficies: that, which remaineth, is the heith of the water, in the middle, aboue the leuell."

Which, you will finde, to be a thing insensible. And though it were greatly sensible, *

[* Note.]

yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides twelfth booke, noted: you may reduce all, to a true Leuell. But, farther diligence, of you is to be vsed, against accidentall causes of the waters swelling: as by hauing (somwhat) with a moyst Sponge, before, made moyst your hollow Pyramis or Cone, will preuent an accidentall cause of Swelling, &c. Experience will teach you abundantly: with great ease, pleasure, and cmoditie.

Thus, may you Double the Cube Mechanically, Treble it, and so forth, in any proportion.

[Note this Abridgement of Dubbling the Cube. &c.]

Now will I Abridge your paine, cost, and Care herein. Without all preparing of your Fundamentall Cubes: you may (alike) worke this Conclusion. For, that, was rather a kinde of Experimentall demstration, then the shortest way: and all, vpon one Mathematicall Demonstration depending. "Take water (as much as conueniently will serue your turne: as I warned before of your Fundamentall Cubes bignes) Way it precisely.

Put that water, into your Pyramis or Cone. Of the same kinde of water, then take againe, the same waight you had before: put that likewise into the Pyramis or Cone. For, in eche time, your marking of the lines, how the Water doth cut them, shall geue you the proportion betwen the Radicall sides, of any two Cubes, wherof the one is Double to the other: working as before I haue taught you:

[* ? Note.]

* sauing that for you Fundamentall Cube his Radicall side: here, you may take a right line, at pleasure."

Yet farther proceding with our droppe of Naturall truth:

[To giue Cubes one to the other in any proportion, Rationall or Irrationall.]

+you may (now) geue Cubes, one to the other, in any proporti geu?: Rationall or Irrationall+: on this maner. Make a hollow Parallelipipedon of Copper or Tinne: with one Base wting, or open: as in our Cubike Coffen. Fr the bottome of that Parallelipipedon, raise vp, many perpendiculars, in euery of his fower sides. Now if any proportion be a.s.signed you, in right lines: Cut one of your perpendiculars (or a line equall to it, or lesse then it) likewise: by the 10. of the sixth of Euclide. And those two partes, set in two sundry lines of those perpendiculars (or you may set them both, in one line) making their beginninges, to be, at the base: and so their lengthes to extend vpward.

Now, set your hollow Parallelipipedon, vpright, perpendicularly, steadie. Poure in water, handsomly, to the heith of your shorter line.

Poure that water, into the hollow Pyramis or Cone. Marke the place of the rising. Settle your hollow Parallelipipedon againe. Poure water into it: vnto the heith of the second line, exactly.

[* Emptying the first.]

Poure that water * duely into the hollow Pyramis or Cone: Marke now againe, where the water cutteth the same line which you marked before.

For, there, as the first marked line, is to the second: So shall the two Radicall sides be, one to the other, of any two Cubes: which, in their Soliditie, shall haue the same proportion, which was at the first a.s.signed: were it Rationall or Irrationall.

Thus, in sundry waies you may furnishe your selfe with such straunge and profitable matter: which, long hath bene wished for. And though it be Naturally done and Mechanically: yet hath it a good Demonstration Mathematicall.

[=The demonstrations of this Dubbling of the Cube, and of the rest.=]

Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones, in the proportions a.s.signed: and like Pyramids or Cones, are in proportion, one to the other, in the proportion of their h.o.m.ologall sides (or lines) tripled. Wherefore, if to the first, and second lines, found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in continuall proportion: that fourth line, shall be to the first, as the greater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of Euclide. If Pyramis to Pyramis, or Cone to Cone, be double,

[I. D.

= * Hereby, helpe your self to become a praecise practiser.

And so consider, how, nothing at all, you are hindred (sensibly) by the Conuexitie of the water.=]

then shall * Line to Line, be also double, &c. But, as our first line, is to the second, so is the Radicall side of our Fundamentall Cube, to the Radicall side of the Cube to be made, or to be doubled: and therefore, to those twaine also, a third and a fourth line, in continuall proportion, ioyned: will geue the fourth line in that proportion to the first, as our fourth Pyramidall, or Conike line, was to his first: but that was double, or treble, &c. as the Pyramids or Cones were, one to an other (as we haue proued) therfore, this fourth, shalbe also double or treble to the first, as the Pyramids or Cones were one to an other: But our made Cube, is described of the second in proportion, of the fower proportionall lines:

[= * By the 33. of the eleuenth booke of Euclide.=]

therfore * as the fourth line, is to the first, so is that Cube, to the first Cube: and we haue proued the fourth line, to be to the first, as the Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is to the Cube, as Pyramis is to Pyramis, or Cone is to Cone.

[I. D.

= * And your diligence in practise, can so (in waight of water) performe it: Therefore, now, you are able to geue good reason of your whole doing.=]

But we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double or treble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was to be demonstrated. And of the Parallelipiped, it is euid?t, that the water Solide Parallelipipedons, are one to the other, as their heithes are, seing they haue one base. Wherfore the Pyramids or Cones, made of those water Parallelipipedons, are one to the other, as the lines are (one to the other) betwene which, our proportion was a.s.signed. But the Cubes made of lines, after the proporti of the Pyramidal or Conik _h.o.m.ologall_ lines, are one to the other, as the Pyramides or Cones are, one to the other (as we before did proue) therfore, the Cubes made, shalbe one to the other, as the lines a.s.signed, are one to the other: Which was to be demonstrated. Note.