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the . Elementes, the Arte of Graduation, and some good vnderstding in _Musike_: and yet moreouer, with an other great Arte, hereafter following, though I, here, set this before, for some considerations me mouing. Sufficient (you see) is the stuffe, to make this rare and secrete Arte, of: and hard enough to frame to the Conclusion Syllogisticall. Yet both the
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manifolde and continuall trauailes of the most auncient and wise Philosophers, for the atteyning of this Arte: and by examples of effectes, to confirme the same: hath left vnto vs sufficient proufe and witnesse: and we, also, daily may perceaue, That mans body, and all other Elementall bodies, are altered, disposed, ordred, pleasured, and displeasured, by the Influentiall working of
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the _Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth _Aristotle_, in the first of his _Meteorologicall_ bookes, in the second Chapter: _Est autem necessari Mundus iste, supernis lationibus fer continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Caus prima putanda omnibus est, vnde motus principium existit._ That is: +_This =[Elementall]= World is of necessitie, almost, next
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adioyning, to the heauenly motions: That, from thence, all his vertue or force may be gouerned. For, that is to be thought the first Cause vnto all: from which, the beginning of motion, is._+ And againe, in the tenth Chapter. _Oportet igitur & horum principia sumamus, & causas omnium similiter. Principium igitur vt mouens, prcipuum[que] & omnium primum, Circulus ille
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est, in quo manifeste Solis latio, &c._ And so forth. His _Meteorologicall_ bookes, are full of argumentes, and effectuall demonstrations, of the vertue, operation, and power of the heauenly bodies, in and vpon the fower Elementes, and other bodies, of them (either perfectly, or vnperfectly) composed. And in his second booke, _De Generatione & Corruptione_, in the tenth Chapter. _Quocirca
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& prima latio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea nam[que] & continua est, & duobus motibus fit:_ In Englishe, thus. +_Wherefore the vppermost motion, is not the cause of Generation and Corruption, but the motion of the Zodiake: for, that, both, is continuall, and is caused of two mouinges._+ And in his second booke, and
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second Chapter of hys _Physikes_. _Homo nam[que] generat hominem, at[que] Sol._ +_For Man (sayth he) and the Sonne, are cause of mans generation._+ Authorities may be brought, very many: both of . . yea and . yeares Antiquitie: of great _Philosophers_, _Expert_, _Wise_, and godly men, for that Conclusion: which, daily and hourely, we men, may discerne and perceaue by
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sense and reason: All beastes do feele, and simply shew, by their actions and passions, outward and inward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally, the Elementes, and all thinges of the Elementes composed, do geue Testimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions, vertues, and naturall motions, depend of the Actiuitie of the heauenly motions and Influences.
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Whereby, beside the specificall order and forme, due to euery seede: and beside the Nature, propre to the Indiuiduall Matrix, of the thing produced: What shall be the heauenly Impression, the perfect and circumspecte Astrologien hath to Conclude._+ Not onely (by _Apotelesmes_) ]. but by Naturall and Mathematicall demonstration . Whereunto, what Sciences are requisite (without exception) I partly haue
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here warned: And in my _Propdeumes_ (besides other matter there disclosed) I haue Mathematically furnished vp the whole Method: To this our age, not so carefully handled by any, that euer I saw, or heard of. I was, [* Anno. and . in Louayn.] (for * . yeares ago) by certaine earnest disputations, of the Learned _Gerardus Mercator_, and _Antonius
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Gogaua_, (and other,) therto so prouoked: and (by my constant and inuincible zeale to the veritie) in obseruations of Heauenly Influencies (to the Minute of time,) than, so diligent: And chiefly by the Supernaturall influence, from the Starre of Iacob, so directed: That any Modest and Sober Student, carefully and diligently seking for the Truth, will both finde & cfesse,
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therin, to be the Veritie, of these my wordes: And also become a Reasonable Reformer, of three Sortes of people: about these Influentiall Operations, greatly erring from the truth. [Note.] Wherof, the one, is +Light Beleuers+, the other, +Light Despisers+, and the third +Light Practisers+. The first, & most cmon Sort, thinke the Heauen and Sterres, to be answerable to
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any their doutes or desires: [.] which is not so: and, in dede, they, to much, ouer reache. The Second sorte thinke no Influentiall vertue (fr the heauenly bodies) to beare any Sway in Generation [.] and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and _Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so
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farre in distance, so manifold in their motions, so constant in their periodes. &c.) they assigne a sleight, simple office or two, and so allow vnto th (according to their capacities) as much vertue, and power Influentiall, as to the Signe of the _Sunne_, _Mone_, and seuen Sterres, hanged vp (for Signes) in London, for distinction of houses, & such
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grosse helpes, in our worldly affaires: And they vnderstand not (or will not vnderstand) of the other workinges, and vertues of the Heauenly _Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husband man: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, as the Porpentine doth: nor will allow these perfect, and incorruptible mighty bodies,
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so much vertuall Radiation, & Force, as they see in a litle peece of a _Magnes stone_: which, at great distance, sheweth his operation. And perchaunce they thinke, the Sea & Riuers (as the Thames) to be some quicke thing, and so to ebbe, and flow, run in and out, of them selues, at their owne fantasies. God helpe, God
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helpe. Surely, these men, come to short: and either are to dull: or willfully blind: or, perhaps, to malicious. The third man, is the common and vulgare _Astrologien_, or Practiser: who, being not duely, artificially, and perfectly [.] furnished: yet, either for vaine glory, or gayne: or like a simple dolt, & blinde Bayard, both in matter and maner, erreth:
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to the discredit of the _Wary_, and modest _Astrologien_: and to the robbing of those most noble corporall Creatures, of their Naturall Vertue: being most mighty: most beneficiall to all elementall Generation, Corruption and the appartenances: and most Harmonious in their Monarchie: For which thinges, being knowen, and modestly vsed: we might highly, and continually glorifie God, with the princely
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Prophet, saying. +_The Heauens declare the Glorie of God: who made the Heaus in his wisedome: who made the Sonne, for to haue dominion of the day: the Mone and Sterres to haue dominion of the nyght: whereby, Day to day vttereth talke: and night, to night declareth knowledge. Prayse him, all ye Sterres, and Light. Amen._+ In order, now
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foloweth, of +Statike+, somewhat to say, what we meane by that name: and what commodity, doth, on such Art, depend. +Statike, is an Arte Mathematicall, which demonstrateth the causes of heauynes, and lightnes of all thynges: and of motions and properties, to heauynes and lightnes, belonging.+ And for asmuch as, by the Bilanx, or Balance (as the chief sensible Instrument,)
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Experience of these demonstrations may be had: we call this Art, _Statike:_ that is, _the Experimentes of the Balance_. Oh, that men wist, what proffit, (all maner of wayes) by this Arte might grow, to the hable examiner, and diligent practiser. Thou onely, knowest all thinges precisely (O God) who hast made weight and Balance, thy Iudgement: who hast created
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all thinges in _Number, Waight, and Measure_: and hast wayed the mountaines and hils in a Balance: who hast peysed in thy hand, both Heauen and earth. We therfore warned by the Sacred word, to Consider thy Creatures: and by that consideration, to wynne a glyms (as it were,) or shaddow of perceiuerance, that thy wisedome, might, and goodnes is
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infinite, and vnspeakable, in thy Creatures declared: And being farder aduertised, by thy mercifull goodnes, that, three principall wayes, were, of the, vsed in Creation of all thy Creatures, namely, _Number_, _Waight_ and _Measure_, And for as much as, of _Number_ and _Measure_, the two Artes (auncient, famous, and to humaine vses most necessary,) are, all ready, sufficiently knowen and
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extant: This third key, we beseche thee (through thy accustomed goodnes,) that it may come to the nedefull and sufficient knowledge, of such thy Seruauntes, as in thy workemanship, would gladly finde, thy true occasions (purposely of the vsed) whereby we should glorifie thy name, and shew forth (to the weaklinges in faith) thy wondrous wisedome and Goodnes. Amen. Meruaile
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nothing at this pang (godly frend, you Gentle and zelous Student.) An other day, perchaunce, you will perceiue, what occasion moued me. Here, as now, I will giue you some ground, and withall some shew, of certaine commodities, by this Arte arising. And bycause this Arte is rare, my wordes and practises might be to darke: vnleast you had some
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light, holden before the matter: and that, best will be, in giuing you, out of _Archimedes_ demonstrations, a few principal Conclusions, as foloweth. +.+ +The Superficies of euery Liquor, by it selfe consistyng, and in quyet, is Sphricall: the centre whereof, is the same, which is the centre of the Earth.+ +.+ +If Solide Magnitudes, being of the same bignes,
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or qutitie, that any Liquor is, and hauyng also the same Waight: be let downe into the same Liquor, they will settle downeward, so, that no parte of them, shall be aboue the Superficies of the Liquor: and yet neuertheles, they will not sinke vtterly downe, or drowne.+ +.+ +If any Solide Magnitude beyng Lighter then a Liquor, be let
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downe into the same Liquor, it will settle downe, so farre into the same Liquor, that so great a quantitie of that Liquor, as is the parte of the Solid Magnitude, settled downe into the same Liquor: is in Waight, quall, to the waight of the whole Solid Magnitude.+ +.+ +Any Solide Magnitude, Lighter then a Liquor, forced downe into
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the same Liquor, will moue vpward, with so great a power, by how much, the Liquor hauyng quall quantitie to the whole Magnitude, is heauyer then the same Magnitude.+ +.+ +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into the same Liquor, will sinke downe vtterly: And wilbe in that Liquor, Lighter by so much, as is the
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waight or heauynes of the Liquor, hauing bygnes or quantitie, quall to the Solid Magnitude.+ +.+ [I. D. The Cutting of a Sphre according to any proportion assigned may by this proposition be done Mechanically by tempering Liquor to a certayne waight in respect of the waight of the Sphre therein Swymming.] +If any Solide Magnitude, Lighter then a Liquor,
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be let downe into the same Liquor, the waight of the same Magnitude, will be, to the Waight of the Liquor. (Which is quall in quantitie to the whole Magnitude,) in that proportion, that the parte, of the Magnitude settled downe, is to the whole Magnitude.+ By these verities, great Errors may be reformed, in Opinion of the Naturall Motion
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of thinges, Light and Heauy. Which errors, are in Naturall Philosophie (almost) of all m allowed: to much trusting to Authority: and false Suppositions. As, +Of any two bodyes, the heauyer, to moue downward faster then the lighter.+ [A common error, noted.] This error, is not first by me, Noted: but by one _Iohn Baptist de Benedictis_. The chief of
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his propositions, is this: which seemeth a Paradox. +If there be two bodyes of one forme, and of one kynde, quall in quantitie or vnquall, [A paradox.] they will moue by quall space, in quall tyme: So that both theyr mouynges be in ayre, or both in water: or in any one Middle.+ Hereupon, in the feate of +Gunnyng+, [N.
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T.] certaine good discourses (otherwise) may receiue great amendement, and furderance. [The wonderfull vse of these Propositions.] In the entended purpose, also, allowing somwhat to the imperfection of Nature: not aunswerable to the precisenes of demonstration. Moreouer, by the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth, the Fire, may be nerely, knowen, how light or heauy they
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are (Naturally) in their assigned partes: or in the whole. And then, to thinges Elementall, turning your practise: you may deale for the proportion of the Elementes, in the thinges Compounded. Then, to the proportions of the Humours in Man: their waightes: and the waight of his bones, and flesh. &c. Than, by waight, to haue consideration of the Force
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of man, any maner of way: in whole or in part. Then, may you, of Ships water drawing, diuersly, in the Sea and in fresh water, haue pleasant consideration: and of waying vp of any thing, sonken in Sea or in fresh water &c. And (to lift vp your head a loft:) by waight, you may, as precisely, as by
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any instrument els, measure the Diameters of _Sonne_ and _Mone. &c._ Frende, I pray you, way these thinges, with the iust Balance of Reason. And you will finde Meruailes vpon Meruailes: And esteme one Drop of Truth (yea in Naturall Philosophie) more worth, then whole Libraries of Opinions, vndemonstrated: or not aunswering to Natures Law, and your experience. Leauing these
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thinges, thus: I will giue you two or three, light practises, to great purpose: and so finish my Annotation _Staticall_. In Mathematicall matters, by the Mechaniciens ayde, we will behold, here, the Commodity of waight. [The practise Staticall, to know the proportion, betwene the Cube, and the Sphre.] Make a Cube, of any one Vniforme: and through like heauy stuffe:
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of the same Stuffe, make a Sphre or Globe, precisely, of a Diameter quall to the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne, Lead, Siluer. &c. (being, as I sayd, of like nature, condition, and like waight throughout.) And you may, by Say Balance, haue prepared a great number of the smallest waightes: which, by
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those Balance can be discerned or tryed: and so, haue proceded to make you a perfect Pyle, company & Number of waightes: to the waight of six, eight, or twelue pound waight: most diligently tryed, all. And of euery one, the Content knowen, in your least waight, that is wayable. [They that can not haue these waightes of precisenes: may,
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by Sand, Vniforme, and well dusted, make them a number of waightes, somewhat nere precisenes: by halfing euer the Sand: they shall, at length, come to a least common waight. Therein, I leaue the farder matter, to their discretion, whom nede shall pinche.] The _Venetians_ consideration of waight, may seme precise enough: by eight descentes progressionall, * halfing, from a
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grayne. [I. D. * For, so, haue you .. partes of a Graine.] Your Cube, Sphre, apt Balance, and conuenient waightes, being ready: fall to worke.. First, way your Cube. Note the Number of the waight. Way, after that, your Sphre. Note likewise, the Nber of the waight. If you now find the waight of your Cube, to be to
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the waight of the Sphre, as . is to : Then you see, how the Mechanicien and _Experimenter_, without Geometrie and Demonstration, are (as nerely in effect) tought the proportion of the Cube to the Sphere: as I haue demonstrated it, in the end of the twelfth boke of _Euclide_. Often, try with the same Cube and Sphre. Then, chaunge,
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your Sphre and Cube, to an other matter: or to an other bignes: till you haue made a perfect vniuersall Experience of it. Possible it is, that you shall wynne to nerer termes, in the proportion. When you haue found this one certaine Drop of Naturall veritie, procede on, to Inferre, and duely to make assay, of matter depending. As,
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bycause it is well demonstrated, that a Cylinder, whose heith, and Diameter of his base, is quall to the Diameter of the Sphre, is Sesquialter to the same Sphre (that is, as . to :) To the number of the waight of the Sphre, adde halfe so much, as it is: and so haue you the number of the waight
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of that Cylinder. Which is also Comprehended of our former Cube: So, that the base of that Cylinder, is a Circle described in the Square, which is the base of our Cube. But the Cube and the Cylinder, being both of one heith, haue their Bases in the same proportion, in the which, they are, one to an other, in
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their Massines or Soliditie. But, before, we haue two numbers, expressing their Massines, Solidities, and Quantities, by waight: wherfore, [* =The proportion of the Square to the Circle inscribed.=] we haue * the proportion of the Square, to the Circle, inscribed in the same Square. And so are we fallen into the knowledge sensible, and Experimentall of _Archimedes_ great Secret:
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of him, by great trauaile of minde, sought and found. Wherfore, to any Circle giuen, you can giue a Square quall: [* =The Squaring of the Circle, Mechanically.=] * as I haue taught, in my Annotation, vpon the first proposition of the twelfth boke, And likewise, to any Square giuen, you may giue a Circle quall: [* =To any Square
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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
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the Square to the Circle inscribed, nerer the truth, then is the proportion of . to . And consider, that you may begyn at the Circle and Square, and so come to conclude of the Sphre, & the Cube, what their proportion is: as now, you came from the Sphre to the Circle. For, of Siluer, or Gold, or Latton
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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
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. to . 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 Circumference and Diameter.] That we may
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Square the Circle, without hauing knowledge of the proportion, of the Circumference to the Diameter: as you haue here perceiued. And otherwayes also, I can demonstrate it. So that, many haue cumberd 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
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the knowledge of the Circumference: the Circles Quantitie, 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
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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.
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At their bases, let them be all open: euery where, els, most close, and iust to. From the vertex, to the Circumference of the base of the Cone: & to the sides of the base of the Pyramis:+ [=I. D.= =The . sides of this Pyramis must be . Isosceles Triangles alike and quall.=] +Let . straight lines be drawen,
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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 . lines (in the Pyramis: and as many, in the Cone) diuide: one, in . quall partes: and an other, in . an other, in , and an other, in
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. (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
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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,
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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, . or . 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
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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
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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
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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 quall partes, as it is capable of, conueniently, with your ease, and precisenes of the diuision. For, as the
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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 nber belonging to the Radicall side, of the Cube, dubble to your Fundamentall Cube: Which being multiplied Cubik wise, will sone
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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 Fundamtall Cube, be to a fourth proportionall line, by the . proposition, of the sixth
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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. [ God 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
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[* Vitruuius. Lib. . Cap. .] * _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
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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.] *
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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 . of the sixth of Euclide. [Note, as concerning the Sphricall Superficies
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of the Water.] Note, that all this while, I forget not my first Proposition Staticall, here rehearsed: that, the Superficies of the water, is Sphricall. 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
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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
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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
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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
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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,
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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
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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
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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,
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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 assigned you, in right lines: Cut
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one of your perpendiculars (or a line equall to it, or lesse then it) likewise: by the . 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,
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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
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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 assigned:
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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
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Like Pyramids: or two Like Cones, in the proportions assigned: and like Pyramids or Cones, are in proportion, one to the other, in the proportion of their Homologall 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,
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shall be to the first, as the greater Pyramis or Cone, is to the lesse: by the . of the eleuenth of Euclide. If Pyramis to Pyramis, or Cone to Cone, be double, [I. D. = * Hereby, helpe your self to become a prcise practiser. And so consider, how, nothing at all, you are hindred (sensibly) by the Conuexitie
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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,
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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
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were one to an other: But our made Cube, is described of the second in proportion, of the fower proportionall lines: [= * By the . 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
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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.=]
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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 euidt, that the water Solide Parallelipipedons, are one to the other, as their heithes are, seing they haue one base. Wherfore the Pyramids
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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 assigned. But the Cubes made of lines, after the proporti of the Pyramidal or Conik _homologall_ lines, are one to the other, as the Pyramides or Cones are, one to the other (as we before
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did proue) therfore, the Cubes made, shalbe one to the other, as the lines assigned, are one to the other: Which was to be demonstrated. Note. [* _Note this Corollary._] * This, my Demonstrati is more generall, then onely in Square Pyramis or Cone: Consider well. Thus, haue I, both Mathematically and Mechanically, ben very long in wordes: yet (I
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trust) nothing tedious to them, who, to these thinges, are well affected. And verily I am forced (auoiding prolixitie) to omit sundry such things, easie to be practised: which to the Mathematicien, would be a great Threasure: and to the Mechanicien, no small gaine. [* The great Commodities following of these new Inuentions.] * Now may you, +Betwene two lines
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giuen, finde two middle proportionals, in Continuall proportion: by the hollow Parallelipipedon, and the hollow Pyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre right lines may be found, proportionall in any proportion assigned, of which, shal be produced a Parallelipipedon, quall to the Parallelipipedon giuen. Hereof, I noted somwhat, vpon the . proposition, of the . boke of
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_Euclide_. Now, all those thinges, which _Vitruuius_ in his Architecture, specified hable to be done, by dubbling of the Cube: Or, by finding of two middle proportionall lines, betwene two lines giuen, may easely be performed. Now, that Probleme, which I noted vnto you, in the end of my Addition, vpon the . of the . boke of _Euclide_, is
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proued possible. Now, may any regular body, be Transformed into an other, &c. Now, any regular body: any Sphere, yea any Mixt Solid: and (that more is) Irregular Solides, may be made (in any proporti assigned) like vnto the body, first giuen. Thus, of a _Manneken_, (as the _Dutch_ Painters terme it) in the same _Symmetrie_, may a Giant be
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made: and that, with any gesture, by the Manneken vsed: and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make one, of the same Mould (in any assigned proportion) bigger or lesser. [* ] Now, may you, of any * Gunne, or little peece of ordinace, make an other, with the same _Symmetrie_ (in all pointes)
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as great, and as little, as you will. Marke that: and thinke on it. Infinitely, +may you apply this, so long sought for, and now so easily concluded: and withall, so willingly and frankly communicated to such, as faithfully deale with vertuous studies.+ [Such is the Fruite of the Mathematicall Sciences and Artes.] Thus, can the Mathematicall minde, deale Speculatiuely
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in his own Arte: and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he can, by order, Descend, to frame Naturall thinges, to wonderfull vses: and when he list, retire home into his owne Centre: and there, prepare more Meanes, to Ascend or Descend by: and, all, to the glory of God, and our honest delectation in
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earth. Although, the Printer, hath looked for this Prface, a day or two, yet could I not bring my pen from the paper, before I had giuen you comfortable warning, and brief instructions, of some of the Commodities, by _Statike_, hable to be reaped: In the rest, I will therfore, be as brief, as it is possible: and with all,
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describing them, somwhat accordingly. And that, you shall perceiue, by this, which in order commeth next. For, wheras, it is so ample and wonderfull, that, an whole yeare long, one might finde fruitfull matter therin, to speake of: and also in practise, is a Threasure endeles: yet will I glanse ouer it, with wordes very few. This do I call
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