The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara Part 4

(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 & cfesse, 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 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 (fr 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 a.s.signe 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 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. G.o.d helpe, G.o.d 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 n.o.ble 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 G.o.d, with the princely Prophet, saying. +_The Heauens declare the Glorie of G.o.d: who made the Heau?s 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 G.o.d) 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 princ.i.p.all 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 workemans.h.i.+p, 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 (G.o.dly 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 princ.i.p.al Conclusions, as foloweth.

+1.+

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

+2.+

+If Solide Magnitudes, being of the same bignes, or qut.i.tie, 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 quant.i.tie of that Liquor, as is the parte of the Solid Magnitude, settled downe into the same Liquor: is in Waight, aequall, 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 aequall quant.i.tie 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 quant.i.tie, aequall to the Solid Magnitude.+

+6.+

[I. D.

The Cutting of a Sphaere according to any proportion a.s.signed may by this proposition be done Mechanically by tempering Liquor to a certayne waight in respect of the waight of the Sphaere 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 aequall in quant.i.tie 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 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 his propositions, is this: which seemeth a Paradox.

+If there be two bodyes of one forme, and of one kynde, aequall in quant.i.tie or vnaequall,

[A paradox.]

they will moue by aequall s.p.a.ce, in aequall 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 a.s.signed 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 s.h.i.+ps 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 Sphaere.]

Make a Cube, of any one Vniforme: and through like heauy stuffe: of the same Stuffe, make a Sphaere or Globe, precisely, of a Diameter aequall 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, Sphaere, apt Balance, and conuenient waightes, being ready: fall to worke.?. First, way your Cube. Note the Number of the waight.

Way, after that, your Sphaere. Note likewise, the Nuber of the waight. If you now find the waight of your Cube, to be to the waight of the Sphaere, 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 Sphaere. Then, chaunge, your Sphaere 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 a.s.say, of matter depending. As, bycause it is well demonstrated, that a Cylinder, whose heith, and Diameter of his base, is aequall to the Diameter of the Sphaere, is Sesquialter to the same Sphaere (that is, as 3. to 2:) To the number of the waight of the Sphaere, 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 Ma.s.sines or Soliditie. But, before, we haue two numbers, expressing their Ma.s.sines, Solidities, and Quant.i.ties, 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 aequall:

[* =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 aequall: