Chapter II: Part 2

Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting, is the workemanship of our minde: which, of distinct and discrete Vnits, maketh a Number: by vniformitie, resulting of a certaine multitude of Vnits. And so, euery number, may haue his least part, giuen: namely, an Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be giuẽ: by cause, infinitly, diuision therof, may be conceiued. All Magnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine, or Solid, of no Sense, can be perceiued, nor exactly by hãd (any way) represented: nor of Nature produced: But, as (by degrees) Number did come to our perceiuerance: So, by visible formes, we are holpen to imagine, what our Line Mathematicall, is. What our Point, is. So precise, are our Magnitudes, that one Line is no broader then an other: for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we can haue Smaller, then either Arte or Nature can produce any: and Greater also, then all the world can comprehend. Our least Magnitudes, can be diuided into so many partes, as the greatest. As, a Line of an inch long, (with vs) may be diuided into as many partes, as may the diameter of the whole world, from East to West: or any way extended: What priuiledges, aboue all manual Arte, and Natures might, haue our two Sciences Mathematicall? to exhibite, and to deale with thinges of such power, liberty, simplicity, puritie, and perfection? And in them, so certainly, so orderly, so precisely to procede: as, excellent is that workemã Mechanicall Iudged, who nerest can approche to the representing of workes, Mathematically demonstrated?

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And our two Sciences, remaining pure, and absolute, in their proper termes, and in their owne Matter: to haue, and allowe, onely such Demonstrations, as are plaine, certaine, vniuersall, and of an æternall veritye?

[Geometrie.]

This Science of _Magnitude_, his properties, conditions, and appertenances: commonly, now is, and from the beginnyng, hath of all Philosophers, ben called _Geometrie_. But, veryly, with a name to base and scant, for a Science of such dignitie and amplenes. And, perchaunce, that name, by cõmon and secret consent, of all wisemen, hitherto hath ben suffred to remayne: that it might carry with it a perpetuall memorye, of the first and notablest benefite, by that Science, to common people shewed: Which was, when Boundes and meres of land and ground were lost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of _Nilus_, the greatest and longest riuer in the world) or, that ground bequeathed, were to be assigned: or, ground sold, were to be layd out: or (when disorder preuailed) that Commõs were distributed into seueralties. For, where, vpon these & such like occasiõs, Some by ignorãce, some by negligẽce, Some by fraude, and some by violence, did wrongfully limite, measure, encroach, or challenge (by pretence of iust content, and measure) those landes and groundes: great losse, disquietnes, murder, and warre did (full oft) ensue: Till, by Gods mercy, and mans Industrie, The perfect Science of Lines, Plaines, and Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. The people then, by this art pleasured, and greatly relieued, in their landes iust measuring: & other Philosophers, writing Rules for land measuring: betwene them both, thus, confirmed the name of _Geometria_, that is, (according to the very etimologie of the word) Land measuring. Wherin, the people knew no farder, of Magnitudes vse, but in Plaines: and the Philosophers, of thẽ, had no feet hearers, or Scholers: farder to disclose vnto, then of flat, plaine _Geometrie_. And though, these Philosophers, knew of farder vse, and best vnderstode the etymologye of the worde, yet this name _Geometria_, was of them applyed generally to all sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and _Pythagoras_: When they would precisely declare their owne doctrine. Then, was

[* Plato. 7. de Rep.]

* _Geometria_, with them, _Studium quod circa planum versatur_. But, well you may perceiue by _Euclides Elementes_, that more ample is our Science, then to measure Plaines: and nothyng lesse therin is tought (of purpose) then how to measure Land. An other name, therfore, must nedes be had, for our Mathematicall Science of Magnitudes: which regardeth neither clod, nor turff: neither hill, nor dale: neither earth nor heauen: but is absolute _Megethologia_: not creping on ground, and dasseling the eye, with pole perche, rod or lyne: but “liftyng the hart aboue the heauens, by inuisible lines, and

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immortall beames meteth with the reflexions, of the light incomprehensible: and so procureth Ioye, and perfection vnspeakable.” Of which true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_ seemed to haue good taste, and iudgement: and (by the name of _Geometrie_) so noted it: and warned his Scholers therof: as, in hys seuenth _Dialog_, of the Common wealth, may euidently be sene. Where (in Latin) thus it is: right well translated: _Profecto, nobis hoc non negabunt, Quicun[que] vel paululum quid Geometriæ gustârunt, quin hæc Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa versantur._ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuer haue, (but euen very litle) tasted of Geometrie, will not denye vnto vs, this: but that this Science, is of an other condicion, quite contrary to that, which they that are exercised in it, do speake of it._+ And there it followeth, of our _Geometrie_, _Quòd quæritur cognoscendi illius gratia, quod semper est, non & eius quod oritur quando[que] & interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur (ô Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum preparabit cogitationem, vt ad supera conuertamus: quæ, nunc, contra quàm decet, ad inferiora deijcimus. &c. Quàm maximè igitur præcipiendum est, vt qui præclarissimam hanc habitãt Civitatem, nullo modo, Geometriam spernant. Nam & quæ præter ipsius propositum, quodam modo esse videntur, haud exigua sunt. &c._ It must nedes be confessed (saith _Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, which is euer: and not of that, which, in tyme, both is bred and is brought to an ende. &c. Geometrie is the knowledge of that which is euerlastyng. It will lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by that meanes, it will prepare the Thought, to the Philosophicall loue of wisdome: that we may turne or conuert, toward heauenly thinges =[both mynde and thought]= which now, otherwise then becommeth vs, we cast down on base or inferior things. &c. Chiefly, therfore, Commaundement must be giuen, that such as do inhabit this most honorable Citie, by no meanes, despise Geometrie. For euen those thinges =[done by it]= which, in manner, seame to be, beside the purpose of Geometrie: are of no small importance. &c._+ And besides the manifold vses of _Geometrie_, in matters appertainyng to warre, he addeth more, of second vnpurposed frute, and commoditye, arrising by _Geometrie_: saying: _Scimus quin etiam, ad Disciplinas omnes facilius per discendas, interesse omnino, attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam, secundo loco discendam Iuuenibus statuamus._ That is. +_But, also, we know, that for the more easy learnyng of all Artes, it importeth much, whether one haue any knowledge in Geometrie, or no. &c. Let vs therfore make an ordinance or decree, that this Science, of young men shall be learned in the second place._+ This was _Diuine Plato_ his Iudgement, both of the purposed, chief, and perfect vse of _Geometrie_: and of his second, dependyng, deriuatiue commodities. And for vs, Christen men, a thousand thousand mo occasions are, to haue nede of the helpe of *

[I. D.
* Herein, I would gladly shake of, the earthly name,
of Geometrie.]

_Megethologicall_ Contemplations: wherby, to trayne our Imaginations and Myndes, by litle and litle, to forsake and abandon, the grosse and corruptible Obiectes, of our vtward senses: and to apprehend, by sure doctrine demonstratiue, Things Mathematicall. And by them, readily to be holpen and conducted to conceiue, discourse, and conclude of things Intellectual, Spirituall, æternall, and such as concerne our Blisse euerlasting: which, otherwise (without Speciall priuiledge of Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally) is hable to reach vnto, or to Compasse. And, veryly, by my small Talent (from aboue) I am hable to proue and testifie, that the litterall Text, and order of our diuine Law, Oracles, and Mysteries, require more skill in Numbers, and Magnitudes: then (commonly) the expositors haue vttered: but rather onely (at the most) so warned: & shewed their own want therin. (To name any, is nedeles: and to note the places, is, here, no place: But if I be duely asked, my answere is ready.) And without the litterall, Grammaticall, Mathematicall or Naturall verities of such places, by good and certaine Arte, perceiued, no Spirituall sense (propre to those places, by Absolute _Theologie_) will thereon depend.

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“No man, therfore, can doute, but toward the atteyning of knowledge incomparable, and Heauenly Wisedome: Mathematicall Speculations, both of Numbers and Magnitudes: are meanes, aydes, and guides: ready, certaine, and necessary.” From henceforth, in this my Preface, will I frame my talke, to _Plato_ his fugitiue Scholers: or, rather, to such, who well can, (and also wil,) vse their vtward senses, to the glory of God, the benefite of their Countrey, and their owne secret contentation, or honest preferment, on this earthly Scaffold. To them, I will orderly recite, describe & declare a great Number of Artes, from our two Mathematicall fountaines, deriued into the fieldes of _Nature_. Wherby, such Sedes, and Rotes, as lye depe hyd in the groũd of _Nature_, are refreshed, quickened, and prouoked to grow, shote vp, floure, and giue frute, infinite, and incredible. And these Artes, shalbe such, as vpon Magnitudes properties do depende, more, then vpon Number. And by good reason we may call them Artes, and Artes Mathematicall Deriuatiue: for (at this tyme) I Define

[An Arte.]

+An Arte, to be a Methodicall cõplete Doctrine, hauing abundancy of sufficient, and peculier matter to deale with, by the allowance of the Metaphisicall Philosopher: the knowledge whereof, to humaine state is necessarye.+ And that I account,

[Art Mathematicall Deriuatiue.]

+An Art Mathematicall deriuatiue, which by Mathematicall demonstratiue Method, in Nũbers, or Magnitudes, ordreth and confirmeth his doctrine, as much & as perfectly, as the matter subiect will admit.+ And for that, I entend to vse the name and propertie of a

[A Mechanitien.]

_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke it good, (for distinction sake) to giue you also a brief description, what I meane therby. +A Mechanicien, or a Mechanicall workman is he, whose skill is, without knowledge of Mathematicall demonstration, perfectly to worke and finishe any sensible worke, by the Mathematicien principall or deriuatiue, demonstrated or demonstrable.+ Full well I know, that he which inuenteth, or maketh these demonstrations, is generally called _A speculatiue Mechanicien_: which differreth nothyng from a _Mechanicall Mathematicien_. So, in respect of diuerse actions, one man may haue the name of sundry artes: as, some tyme, of a Logicien, some tymes (in the same matter otherwise handled) of a Rethoricien. Of these trifles, I make, (as now, in respect of my Preface,) small account: to fyle thẽ for the fine handlyng of subtile curious disputers. In other places, they may commaunde me, to giue good reason: and yet, here, I will not be vnreasonable.

[+1.+]

First, then, from the puritie, absolutenes, and Immaterialitie of Principall _Geometrie_, is that kinde of _Geometrie_ deriued, which vulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible magnitudes, their iust quãtities and contentes.+

[Geometrie vulgar.]

This, teacheth to measure, either at hand: and the practiser, to be by the thing Measured: and so, by due applying of Cumpase, Rule, Squire, Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to the Length, Plaine, or Solide measured,

[1.]

* to be certified, either of the length, perimetry, or distance lineall: and this is called, _Mecometrie_. Or

[2.]

* to be certified of the content of any plaine Superficies: whether it be in ground Surueyed, Borde, or Glasse measured, or such like thing: which measuring, is named _Embadometrie_.

[3.]

* Or els to vnderstand the Soliditie, and content of any bodily thing: as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels, small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale vessells, &c, the Measuring, commonly, hath a peculier name: and is called _Gaging_. And the generall name of these Solide measures, is _Stereometrie_.

[+2.+]

Or els, this _vulgar Geometrie_, hath consideration to teach the practiser, how to measure things, with good distance betwene him and the thing measured: and to vnderstand thereby, either

[1.]

* how Farre, a thing seene (on land or water) is from the measurer: and this may be called _Apomecometrie_:

[2.]

Or, how High or depe, aboue or vnder the leuel of the measurers stãding, any thing is, which is sene on land or water, called _Hypsometrie_.

[3.]

* Or, it informeth the measurer, how Broad any thing is, which is in the measurers vew: so it be on Land or Water, situated: and may be called _Platometrie_. Though I vse here to condition, the thing measured, to be on Land, or Water Situated:

[Note.]

yet, know for certaine, that the sundry heigthe of Cloudes, blasing Starres, and of the Mone, may (by these meanes) haue their distances from the earth: and, of the blasing Starres and Mone, the Soliditie (aswell as distances) to be measured: But because, neither these things are vulgarly taught: nor of a common practiser so ready to be executed: I, rather, let such measures be reckened incident to some of our other Artes, dealing with thinges on high, more purposely, then this vulgar Land measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c._

Of these Feates (farther applied) is Sprong the Feate of _Geodesie_, or Land Measuring: more cunningly to measure & Suruey Land, Woods, and Waters, a farre of. More cunningly, I say: But God knoweth (hitherto) in these Realmes of England and Ireland (whether through ignorance or fraude, I can not tell, in euery particular)

[Note.]

how great wrong and iniurie hath (in my time) bene committed by vntrue measuring and surueying of Land or Woods, any way. And, this I am sure: that the Value of the difference, betwene the truth and such Surueyes, would haue bene hable to haue foũd (for euer) in eche of our two Vniuersities, an excellent Mathematicall Reader: to eche, allowing (yearly) a hundred Markes of lawfull money of this realme: which, in dede, would seme requisit, here, to be had (though by other wayes prouided for) as well, as, the famous Vniuersitie of Paris, hath two Mathematicall Readers: and eche, two hundreth French Crownes yearly, of the French Kinges magnificent liberalitie onely. Now, againe, to our purpose returning: Moreouer, of the former knowledge Geometricall, are growen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and _Stratarithmetrie_.

“+‡Geographie‡+ teacheth wayes, by which, in sũdry formes, (as _Sphærike_, _Plaine_ or other), the Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, & such other things, vpõ the outface of the earthly Globe (either in the whole, or in some principall mẽber and portion therof cõtayned) may be described and designed, in cõmensurations Analogicall to Nature and veritie: and most aptly to our vew, may be represented.” Of this Arte how great pleasure, and how manifolde commodities do come vnto vs, daily and hourely: of most men, is perceaued. While, some, to beautifie their Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other some, for thinges past, as battels fought, earthquakes, heauenly fyringes, & such occurentes, in histories mentioned: therby liuely, as it were, to vewe the place, the region adioyning, the distance from vs: and such other circumstances. Some other, presently to vewe the large dominion of the Turke: the wide Empire of the Moschouite: and the litle morsell of ground, where Christendome (by profession) is certainly knowen. Litle, I say, in respecte of the rest. &c. Some, either for their owne iorneyes directing into farre landes: or to vnderstand of other mens trauailes. To conclude, some, for one purpose: and some, for an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, & Geographicall Globes. Of whose vse, to speake sufficiently, would require a booke peculier.

+‡Chorographie‡+ seemeth to be an vnderling, and a twig, of _Geographie_: and yet neuerthelesse, is in practise manifolde, and in vse very ample. “This teacheth Analogically to describe a small portion or circuite of ground, with the contentes: not regarding what commensuration it hath to the whole, or any parcell, without it, contained. But in the territory or parcell of ground which it taketh in hand to make description of, it leaueth out (or vndescribed) no notable, or odde thing, aboue the ground visible. Yea and sometimes, of thinges vnder ground, geueth some peculier marke: or warning: as of Mettall mines, Cole pittes, Stone quarries. &c.” Thus, a Dukedome, a Shiere, a Lordship, or lesse, may be described distinctly. But marueilous pleasant, and profitable it is, in the exhibiting to our eye, and commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true Symmetry: not approching to any of them: and out of Gunne shot. &c. Hereby, the _Architect_ may furnishe him selfe, with store of what patterns he liketh: to his great instruction: euen in those thinges which outwardly are proportioned: either simply in them selues: or respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also, terme this particular description of places, _Topographie_.

“+‡Hydrographie‡+, deliuereth to our knowledge, on Globe or in Plaine, the perfect Analogicall description of the Ocean Sea coastes, through the whole world: or in the chiefe and principall partes thereof:” with the Iles and chiefe particular places of daungers, conteyned within the boundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes, Rockes, Races, Countertides, Whorlepooles. &c. This, dealeth with the Element of the water chiefly: as _Geographie_ did principally take the Element of the Earthes description (with his appertenances) to taske. And besides thys, _Hydrographie_, requireth a particular Register of certaine Landmarkes (where markes may be had) from the sea, well hable to be skried, in what point of the Seacumpase they appeare, and what apparent forme, Situation, and bignes they haue, in respecte of any daungerous place in the sea, or nere vnto it, assigned: And in all Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes, come and go, the _Hydrographer_ ought to recorde. The Soundinges likewise: and the Chanels wayes: their number, and depthes ordinarily, at ebbe and flud, ought the _Hydrographer_, by obseruation and diligence of _Measuring_, to haue certainly knowen. And many other pointes, are belonging to perfecte _Hydrographie_, and for to make a _Rutter_, by: of which, I nede not here speake: as of the describing, in any place, vpon Globe or Plaine, the 32. pointes of the Compase, truely: (wherof, scarsly foure, in England, haue right knowledge: bycause, the lines therof, are no straight lines, nor Circles.) Of making due proiection of a Sphere in plaine. Of the Variacion of the Compas, from true Northe: And such like matters (of great importance, all) I leaue to speake of, in this place: bycause, I may seame (al ready) to haue enlarged the boundes, and duety of an Hydographer, much more, then any man (to this day) hath noted, or prescribed. Yet am I well hable to proue, all these thinges, to appertaine, and also to be proper to the Hydrographer. The chief vse and ende of this Art, is the Art of Nauigation: but it hath other diuerse vses: euen by them to be enioyed, that neuer lacke sight of land.

+‡Stratarithmetrie‡+, is the Skill, (appertainyng to the warre,) by which a man can set in figure, analogicall to any _Geometricall_ figure appointed, any certaine number or summe of men: of such a figure capable: (by reason of the vsuall spaces betwene Souldiers allowed: and for that, of men, can be made no Fractions. Yet, neuertheles, he can order the giuen summe of men, for the greatest such figure, that of them, cã be ordred) and certifie, of the ouerplus: (if any be) and of the next certaine summe, which, with the ouerplus, will admit a figure exactly proportionall to the figure assigned. By which Skill, also, of any army or company of men: (the figure & sides of whose orderly standing, or array, is knowen) he is able to expresse the iust number of men, within that figure conteined: or (orderly) able to be conteined.

[* Note.]

* And this figure, and sides therof, he is hable to know: either beyng by, and at hand: or a farre of. Thus farre, stretcheth the description and property of _Stratarithmetrie_: sufficient for this tyme and place.

[The difference betwene Stratarithmetrie and Tacticie.]

“It differreth from the Feate _Tacticall_, _De aciebus instruendis._ bycause, there, is necessary the wisedome and foresight, to what purpose he so ordreth the men: and Skillfull hability, also, for any occasion, or purpose, to deuise and vse the aptest and most necessary order, array and figure of his Company and Summe of men.” By figure, I meane: as, either of a _Perfect Square_, _Triangle_, _Circle_, _Ouale_, _long square_, (of the Grekes it is called _Eteromekes_) _Rhombe_, _Rhomboïd_, _Lunular_, _Ryng_, _Serpentine_, and such other Geometricall figures: Which, in warres, haue ben, and are to be vsed: for commodiousnes, necessity, and auauntage &c. And no small skill ought he to haue, that should make true report, or nere the truth, of the numbers and Summes, of footemen or horsemen, in the Enemyes ordring. A farre of, to make an estimate, betwene nere termes of More and Lesse, is not a thyng very rife, among those that gladly would do it.

[I. D.
Frende, you will finde it hard, to performe my description
of this Feate. But by Chorographie, you may helpe your selfe
some what: where the Figures knowne (in Sides and Angles)
are not Regular: And where, Resolution into Triangles can
serue. &c. And yet you will finde it strange to deale thus
generally with Arithmeticall figures: and, that for Battayle
ray. Their contentes, differ so much from like Geometricall
Figures.]

Great pollicy may be vsed of the Capitaines, (at tymes fete, and in places conuenient) as to vse Figures, which make greatest shew, of so many as he hath: and vsing the aduauntage of the three kindes of vsuall spaces: (betwene footemen or horsemen) to take the largest: or when he would seme to haue few, (beyng many:) contrarywise, in Figure, and space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer is carefull to come nere the truth herein, besides the Iudgement of his expert eye, his skill of Ordering _Tacticall_, the helpe of his Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously framed for cariage and vse) He may wonderfully helpe him selfe, by perspectiue Glasses. In which, (I trust) our posterity will proue more skillfull and expert, and to greater purposes, then in these dayes, can (almost) be credited to be possible.

Thus haue I lightly passed ouer the Artificiall Feates, chiefly dependyng vpon vulgar _Geometrie_: & commonly and generally reckened vnder the name of _Geometrie_. But there are other (very many) _Methodicall Artes_, which, declyning from the purity, simplicitie, and Immateriality, of our Principall Science of _Magnitudes_: do yet neuertheles vse the great ayde, direction, and Method of the sayd principall Science, and haue propre names, and distinct: both from the Science of _Geometrie_, (from which they are deriued) and one from the other. As +Perspectiue, Astronomie, Musike, Cosmographie, Astrologie, Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie, Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture, Nauigation, Thaumaturgike+ and +Archemastrie+. I thinke it necessary, orderly, of these to giue some peculier descriptions: and withall, to touch some of their commodious vses, and so to make this Preface, to be a little swete, pleasant Nosegaye for you: to comfort your Spirites, beyng almost out of courage, and in despayre, (through brutish brute) Weenyng that _Geometrie_, had but serued for buildyng of an house, or a curious bridge, or the roufe of Westminster hall, or some witty pretty deuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That the thing is farre otherwise, then the world, (commonly) to this day, hath demed, by worde and worke, good profe wilbe made.

Among these Artes, by good reason, +‡Perspectiue‡+ ought to be had, ere of _Astronomicall Apparences_, perfect knowledge can be atteyned. And bycause of the prerogatiue of _Light_, beyng the first of _Gods Creatures_: and the eye, the light of our body, and his Sense most mighty, and his organ most Artificiall and _Geometricall_: At _Perspectiue_, we will begyn therfore. +Perspectiue, is an Art Mathematicall, which demonstrateth the maner, and properties, of all Radiations Direct, Broken, and Reflected.+ This Description, or Notation, is brief: but it reacheth so farre, as the world is wyde. It concerneth all Creatures, all Actions, and passions, by Emanation of beames perfourmed. Beames, or naturall lines, (here) I meane, not of light onely, or of colour (though they, to eye, giue shew, witnes, and profe, wherby to ground the Arte vpon) but also of other _Formes_, both _Substantiall_, and _Accidentall_, the certaine and determined actiue Radiall emanations. By this Art (omitting to speake of the highest pointes) we may vse our eyes, and the light, with greater pleasure: and perfecter Iudgement: both of things, in light seen, & of other: which by like order of Lightes Radiations, worke and produce their effectes. We may be ashamed to be ignorant of the cause, why so sundry wayes our eye is deceiued, and abused: as, while the eye weeneth a roũd Globe or Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise iudgeth a plaine Square, to be roũd: supposeth walles parallels, to approche, a farre of: rofe and floure parallels, the one to bend downward, the other to rise vpward, at a little distance from you. Againe, of thinges being in like swiftnes of mouing, to thinke the nerer, to moue faster: and the farder, much slower. Nay, of two thinges, wherof the one (incomparably) doth moue swifter then the other, to deme the slower to moue very swift, & the other to stand: what an error is this, of our eye? Of the Raynbow, both of his Colours, of the order of the colours, of the bignes of it, the place and heith of it, (&c) to know the causes demonstratiue, is it not pleasant, is it not necessary? of two or three Sonnes appearing: of Blasing Sterres: and such like thinges: by naturall causes, brought to passe, (and yet neuertheles, of farder matter, Significatiue) is it not commodious for man to know the very true cause, & occasion Naturall? Yea, rather, is it not, greatly, against the Souerainty of Mans nature, to be so ouershot and abused, with thinges (at hand) before his eyes? as with a Pecockes tayle, and a Doues necke: or a whole ore, in water, holden, to seme broken. Thynges, farre of, to seeme nere: and nere, to seme farre of. Small thinges, to seme great: and great, to seme small. One man, to seme an Army. Or a man to be curstly affrayed of his owne shaddow. Yea, so much, to feare, that, if you, being (alone) nere a certaine glasse, and proffer, with dagger or sword, to foyne at the glasse, you shall suddenly be moued to giue backe (in maner) by reason of an Image,

[☞ A marueilous Glasse.]

appearing in the ayre, betwene you & the glasse, with like hand, sword or dagger, & with like quicknes, foyning at your very eye, likewise as you do at the Glasse. Straunge, this is, to heare of: but more meruailous to behold, then these my wordes can signifie. And neuerthelesse by demonstration Opticall, the order and cause therof, is certified: euen so, as the effect is consequent. Yea, thus much more, dare I take vpon me, toward the satisfying of the noble courrage, that longeth ardently for the wisedome of Causes Naturall: as to let him vnderstand, that, in London, he may with his owne eyes, haue profe of that, which I haue sayd herein. A Gentleman, (which, for his good seruice, done to his Countrey, is famous and honorable:

[S. W. P.]

and for skill in the Mathematicall Sciences, and Languages, is the Od man of this land. &c.) euen he, is hable: and (I am sure) will, very willingly, let the Glasse, and profe be sene: and so I (here) request him: for the encrease of wisedome, in the honorable: and for the stopping of the mouthes malicious: and repressing the arrogancy of the ignorant. Ye may easily gesse, what I meane. This Art of _Perspectiue_, is of that excellency, and may be led, to the certifying, and executing of such thinges, as no man would easily beleue: without Actuall profe perceiued. I speake nothing of _Naturall Philosophie_, which, without _Perspectiue_, can not be fully vnderstanded, nor perfectly atteined vnto. Nor, of _Astronomie_: which, without _Perspectiue_, can not well be grounded: Nor _Astrologie_, naturally Verified, and auouched. That part hereof, which dealeth with Glasses (which name, Glasse, is a generall name, in this Arte, for any thing, from which, a Beame reboundeth) is called _Catoptrike_: and hath so many vses, both merueilous, and proffitable: that, both, it would hold me to long, to note therin the principall conclusions, all ready knowne: And also (perchaunce) some thinges, might lacke due credite with you: And I, therby, to leese my labor: and you, to slip into light Iudgement,

[* ☞]

Before you haue learned sufficiently the powre of Nature and Arte.

Now, to procede: +‡Astronomie‡, is an Arte Mathematicall, which demonstrateth the distance, magnitudes, and all naturall motions, apparences, and passions propre to the Planets and fixed Sterres: for any time past, present and to come: in respect of a certaine Horizon, or without respect of any Horizon.+ By this Arte we are certified of the distance of the Starry Skye, and of eche _Planete_ from the Centre of the Earth: and of the greatnes of any Fixed starre sene, or _Planete_, in respect of the Earthes greatnes. As, we are sure (by this Arte) that the Solidity, Massines and Body of the _Sonne_, conteineth the quantitie of the whole Earth and Sea, a hundred thre score and two times, lesse by ⅛ one eight parte of the earth. But the Body of the whole earthly globe and Sea, is bigger then the body of the Mone, three and forty times lesse by ⅛ of the Mone. Wherfore the _Sonne_ is bigger then the _Mone_, 7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then the _Mone_. And yet the vnskillfull man, would iudge them a like bigge. Wherfore, of Necessity, the one is much farder from vs, then the other. The _Sonne_, when he is fardest from the earth (which, now, in our age, is, when he is in the 8. degree, of Cancer) is, 1179 Semidiameters of the Earth, distante. And the _Mone_ when she is fardest from the earth, is 68 Semidiameters of the earth and ⅓ The nerest, that the _Mone_ commeth to the earth, is Semidiameters 52¼ The distance of the Starry Skye is, frõ vs, in Semidiameters of the earth 20081½ Twenty thousand fourescore, one, and almost a halfe. Subtract from this, the _Mones_ nerest distance, from the Earth: and therof remaineth Semidiameters of the earth 20029¼ Twenty thousand nine and twenty and a quarter.

[Note.]

So thicke is the heauenly Palace, that the _Planetes_ haue all their exercise in, and most meruailously perfourme the Commaũdement and Charge to them giuen by the omnipotent Maiestie of the king of kings. This is that, which in _Genesis_ is called _Ha Rakia_. Consider it well. The Semidiameter of the earth, cõteineth of our common miles 3436 4/11 three thousand, foure hundred thirty six and foure eleuenth partes of one myle: Such as the whole earth and Sea, round about, is 21600. One and twenty thousand six hundred of our myles. Allowyng for euery degree of the greatest circle, thre score myles. Now if you way well with your selfe but this litle parcell of frute _Astronomicall_, as concerning the bignesse, Distances of _Sonne_, _Mone_, _Sterry Sky_, and the huge massines of _Ha Rakia_, will you not finde your Consciences moued, with the kingly Prophet, to sing the confession of Gods Glory, and say, +_The Heauens declare the glory of God, and the Firmament =[Ha Rakia]= sheweth forth the workes of his handes_+. And so forth, for those fiue first staues, of that kingly Psalme. Well, well, It is time for some to lay hold on wisedome, and to Iudge truly of thinges: and notso to expound the Holy word, all by Allegories: as to Neglect the wisedome, powre and Goodnes of God, in, and by his Creatures, and Creation to be seen and learned. By parables and Analogies of whose natures and properties, the course of the Holy Scripture, also, declareth to vs very many Mysteries. The whole Frame of Gods Creatures, (which is the whole world,) is to vs, a bright glasse: from which, by reflexion, reboundeth to our knowledge and perceiuerance, Beames, and Radiations: representing the Image of his Infinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taught and persuaded to Glorifie our Creator, as God: and be thankefull therfore. Could the Heathenistes finde these vses, of these most pure, beawtifull, and Mighty Corporall Creatures: and shall we, after that the true _Sonne_ of rightwisenesse is risen aboue the _Horizon_, of our temporall _Hemisphærie_, and hath so abundantly streamed into our hartes, the direct beames of his goodnes, mercy, and grace: Whose heat All Creatures feele: Spirituall and Corporall: Visible and Inuisible. Shall we (I say) looke vpon the _Heauen_, _Sterres_, and _Planets_, as an Oxe and an Asse doth: no furder carefull or inquisitiue, what they are: why were they Created, How do they execute that they were Created for? Seing, All Creatures, were for our sake created: and both we, and they, Created, chiefly to glorifie the Almighty Creator: and that, by all meanes, to vs possible. _Nolite ignorare_ (saith _Plato in Epinomis_) _Astronomiam, Sapientissimũ quiddam esse._ +_Be ye not ignorant, Astronomie to be a thyng of excellent wisedome._+ _Astronomie_, was to vs, from the beginning commended, and in maner commaunded by God him selfe. In asmuch as he made the _Sonne_, _Mone_, and _Sterres_, to be to vs, for _Signes_, and knowledge of Seasons, and for Distinctions of Dayes, and yeares. Many wordes nede not. But I wish, euery man should way this word, _Signes_. And besides that, conferre it also with the tenth Chapter of _Hieremie_. And though Some thinke, that there, they haue found a rod: Yet Modest Reason, will be indifferent Iudge, who ought to be beaten therwith, in respect of our purpose. Leauing that: I pray you vnderstand this: that without great diligence of Obseruation, examination and Calculation, their periods and courses (wherby _Distinction_ of Seasons, yeares, and New Mones might precisely be knowne) could not exactely be certified. Which thing to performe, is that _Art_, which we here haue Defined to be _Astronomie_. Wherby, we may haue the distinct Course of Times, dayes, yeares, and Ages: aswell for Consideratiõ of Sacred Prophesies, accomplished in due time, foretold: as for high Mysticall Solemnities holding: And for all other humaine affaires, Conditions, and couenantes, vpon certaine time, betwene man and man: with many other great vses: Wherin, (verely), would be great incertainty, Confusion, vntruth, and brutish Barbarousnes: without the wonderfull diligence and skill of this Arte: continually learning, and determining Times, and periodes of Time, by the Record of the heauenly booke, wherin all times are written: and to be read with an _Astronomicall staffe_, in stede of a festue.

+‡Musike‡+, of Motion, hath his Originall cause: Therfore, after the motions most swift, and most Slow, which are in the Firmament, of Nature performed: and vnder the _Astronomers Consideration_: now I will Speake of an other kinde of _Motion_, producing sound, audible, and of Man numerable. _Musike_ I call here that _Science_, which of the Grekes is called _Harmonice_. Not medling with the Controuersie betwene the auncient _Harmonistes_, and _Canonistes_. +Musike is a Mathematicall Science, which teacheth, by sense and reason, perfectly to iudge, and order the diuersities of soundes, hye and low.+ _Astronomie_ and _Musike_ are Sisters, saith _Plato_. As, for _Astronomie_, the eyes: So, for _Harmonious Motion_, the eares were made. But as _Astronomie_ hath a more diuine Contemplation, and cõmodity, then mortall eye can perceiue: So, is _Musike_ to be considered,

[1.]

that the * Minde may be preferred, before the eare. And from audible sound, we ought to ascende, to the examination: which numbers are _Harmonious_, and which not. And why, either, the one are: or the other are not. I could at large,

[2.]

in the heauenly * motions and distances, describe a meruailous Harmonie, of _Pythagoras_ Harpe

[3.]

with eight stringes. Also, somwhat might be sayd of _Mercurius_ * two Harpes,

[4.]

eche of foure Stringes Elementall. And very straunge matter, might be alledged of the _Harmonie_,

[5.]

to our * Spirituall part appropriate. As in _Ptolomaus_ third boke, in the fourth and sixth Chapters may appeare. *

[6.]

And what is the cause of the apt bonde, and frendly felowship, of the Intellectuall and Mentall part of vs, with our grosse & corruptible body: but a certaine Meane, and _Harmonious Spiritualitie, with both participatyng, & of both (in a maner) resultynge In

[7.]

the * Tune of Mans voyce, and also

[8.]

* the sound of Instrument_, what might be sayd, of _Harmonie_: No common Musicien would lightly beleue.

[I. D.
Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7.
chapters. Where you shall haue some occasion farder to thinke
of Musike, than commonly is thought.]

But of the sundry Mixture (as I may terme it) and concurse, diuerse collation, and Application of these _Harmonies_: as of thre, foure, fiue, or mo: Maruailous haue the effectes ben: and yet may be founde, and produced the like: with some proportionall consideration for our time, and being: in respect of the State, of the thinges then: in which, and by which, the wondrous effectes were wrought. _Democritus_ and _Theophrastus_ affirmed, that, by _Musike_, griefes and diseases of the Minde, and body might be cured, or inferred. And we finde in Recorde, that _Terpander_, _Arion_, _Ismenias_, _Orpheus_, _Amphion_, _Dauid_, _Pythagoras_, _Empedocles_, _Asclepiades_ and _Timotheus_, by _Harmonicall_ Consonãcy, haue done, and brought to pas, thinges, more then meruailous, to here of. Of them then, making no farder discourse, in this place: Sure I am, that Common _Musike_, commonly vsed, is found to the _Musiciens_ and Hearers, to be so Commodious and pleasant, That if I would say and dispute, but thus much: That it were to be otherwise vsed, then it is, I should finde more repreeuers, then I could finde priuy, or skilfull of my meaning. In thinges therfore euident, and better knowen, then I can expresse: and so allowed and liked of, (as I would wish, some other thinges, had the like hap) I will spare to enlarge my lines any farder, but consequently follow my purpose.

+‡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.]