Part 5

+‡Of Cosmographie‡+, I appointed briefly in this place, to geue you some intelligence. +Cosmographie, is the whole and perfect description of the heauenly, and also elementall parte of the world, and their homologall application, and mutuall collation necessarie.+ This Art, requireth _Astronomie_, _Geographie_, _Hydrographie_ and _Musike_. Therfore, it is no small Arte, nor so simple, as in common practise, it is (slightly) considered. This matcheth Heauen, and the Earth, in one frame, and aptly applieth parts Correspõdent: So, as, the Heauenly Globe, may (in practise) be duely described vpon the Geographicall, and Hydrographicall Globe. And there, for vs to consider an _Æquonoctiall Circle_, _an Ecliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes, Latitudes, Declinations, and Verticalitie: also Climes, and Parallels: and by an _Horizon_ annexed, and reuolution of the earthly Globe (as the Heauen, is, by the _Primouant_, caried about in 24. æquall Houres) to learne the Risinges and Settinges of Sterres (of _Virgill_ in his _Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sphære_, to Perdicca King of the Macedonians: of _Diocles_, to King _Antigonus_, and of other famous _Philosophers_ prescribed) a thing necessary, for due manuring of the earth, for _Nauigation_, for the Alteration of mans body: being, whole, Sicke, wounded, or brused. By the Reuolution, also, or mouing of the Globe Cosmographicall, the Rising and Setting of the Sonne: the Lengthes, of dayes and nightes: the Houres and times (both night and day) are knowne: with very many other pleasant and necessary vses: Wherof, some are knowne: but better remaine, for such to know and vse:

[☞]

who of a sparke of true fire, can make a wonderfull bonfire, by applying of due matter, duely.

+‡Of Astrologie‡+, here I make an Arte, seuerall from _Astronomie_: not by new deuise, but by good reason and authoritie: for, +Astrologie, is an Arte Mathematicall, which reasonably demonstrateth the operations and effectes, of the naturall beames, of light, and secrete influence: of the Sterres and Planets: in euery element and elementall body: at all times, in any Horizon assigned.+ This Arte is furnished with many other great Artes and experiences: As with perfecte _Perspectiue_, _Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4. Elementes, the Arte of Graduation, and some good vnderstãding 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 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 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 necessariò Mundus iste, supernis lationibus ferè continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causà prima putanda omnibus est, vnde motus principium existit._ That is: +_This =[Elementall]= World is of necessitie, almost, next 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, præcipuum[que] & omnium primum, Circulus ille 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 & 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 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 1000. 2000. yea and 3000. yeares Antiquitie: of great _Philosophers_, _Expert_, _Wise_, and godly men, for that Conclusion: which, daily and hourely, we men, may discerne and perceaue by 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. 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_) τὸ ὁτὶ]. but by Naturall and Mathematicall demonstration τὸ διότι. Whereunto, what Sciences are requisite (without exception) I partly haue here warned: And in my _Propædeumes_ (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. 1548 and 1549. in Louayn.]

(for * 21. yeares ago) by certaine earnest disputations, of the Learned _Gerardus Mercator_, and _Antonius 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 & cõfesse, 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 cõmon Sort, thinke the Heauen and Sterres, to be answerable to any their doutes or desires:

[1.]

which is not so: and, in dede, they, to much, ouer reache. The Second sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare any Sway in Generation

[2.]

and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and _Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so 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 thẽ (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 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, 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 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

[3.]

furnished: yet, either for vaine glory, or gayne: or like a simple dolt, & blinde Bayard, both in matter and maner, erreth: 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 Prophet, saying. +_The Heauens declare the Glorie of God: who made the Heauẽs 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 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,) 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 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 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 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 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 light, holden before the matter: and that, best will be, in giuing you, out of _Archimedes_ demonstrations, a few principal Conclusions, as foloweth.

+1.+

+The Superficies of euery Liquor, by it selfe consistyng, and in quyet, is Sphæricall: the centre whereof, is the same, which is the centre of the Earth.+

+2.+

+If Solide Magnitudes, being of the same bignes, or quãtitie, 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.+

+3.+

+If any Solide Magnitude beyng Lighter then a Liquor, be let 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, æquall, to the waight of the whole Solid Magnitude.+

+4.+

+Any Solide Magnitude, Lighter then a Liquor, forced downe into the same Liquor, will moue vpward, with so great a power, by how much, the Liquor hauyng æquall quantitie to the whole Magnitude, is heauyer then the same Magnitude.+

+5.+

+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 waight or heauynes of the Liquor, hauing bygnes or quantitie, æquall to the Solid Magnitude.+

+6.+

[I. D. The Cutting of a Sphære 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 Sphære therein Swymming.]

+If any Solide Magnitude, Lighter then a Liquor, be let downe into the same Liquor, the waight of the same Magnitude, will be, to the Waight of the Liquor. (Which is æquall 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 of thinges, Light and Heauy. Which errors, are in Naturall Philosophie (almost) of all mẽ 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 his propositions, is this: which seemeth a Paradox.

+If there be two bodyes of one forme, and of one kynde, æquall in quantitie or vnæquall,

[A paradox.]

they will moue by æquall space, in æquall 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. 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 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 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 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 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 Sphære.]

Make a Cube, of any one Vniforme: and through like heauy stuffe: of the same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall 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 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, 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 grayne.

[I. D. * For, so, haue you .256. partes of a Graine.]

Your Cube, Sphære, apt Balance, and conuenient waightes, being ready: fall to worke.❉. First, way your Cube. Note the Number of the waight. Way, after that, your Sphære. Note likewise, the Nũber of the waight. If you now find the waight of your Cube, to be to the waight of the Sphære, as 21. is to 11: 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 Sphære. Then, chaunge, your Sphære 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, bycause it is well demonstrated, that a Cylinder, whose heith, and Diameter of his base, is æquall to the Diameter of the Sphære, is Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of the waight of the Sphære, adde halfe so much, as it is: and so haue you the number of the waight 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 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: of him, by great trauaile of minde, sought and found. Wherfore, to any Circle giuen, you can giue a Square æquall:

[* =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 æquall:

[* =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 Sphære, & the Cube, what their proportion is: as now, you came from the Sphære to the Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough one hole drawẽ, 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 Circumference and Diameter.]

That we may 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 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 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 Circumference 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 æquall.=]

+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. æquall 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.=]