The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara Part 2

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 cmon and secret consent, of all wis.e.m.e.n, 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 a.s.signed: or, ground sold, were to be layd out: or (when disorder preuailed) that Comms were distributed into seueralties. For, where, vpon these & such like occasis, Some by ignorce, some by neglig?ce, 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 G.o.ds 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 th?, 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 da.s.seling the eye, with pole perche, rod or lyne: but "liftyng the hart aboue the heauens, by inuisible lines, and

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, n.o.bis hoc non negabunt, Quicun[que] vel paululum quid Geometriae gustarunt, quin haec Scientia, contra, omnino se habeat, quam 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_, _Qud quaeritur cognoscendi illius gratia, quod semper est, non & eius quod oritur quando[que] & interit.

Geometria, eius quod est semper, Cognitio est. Attollet igitur (o Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum preparabit cogitationem, vt ad supera conuertamus: quae, nunc, contra quam decet, ad inferiora deijcimus. &c. Quam maxime igitur praecipiendum est, vt qui praeclarissimam hanc habitt Civitatem, nullo modo, Geometriam spernant. Nam & quae praeter 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, aeternall, and such as concerne our Blisse euerlasting: which, otherwise (without Speciall priuiledge of Illumination, or Reuelation fr heauen) No mortall mans wyt (naturally) is hable to reach vnto, or to Compa.s.se. 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.

"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 G.o.d, 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 groud 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 cplete 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 Nubers, 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 princ.i.p.all 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 th?

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 Princ.i.p.all _Geometrie_, is that kinde of _Geometrie_ deriued, which vulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible magnitudes, their iust qut.i.ties 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 c.u.mpase, 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 Gla.s.se 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 fas.h.i.+ons. 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 stding, 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 Sp.r.o.ng the Feate of _Geodesie_, or Land Measuring: more cunningly to measure & Suruey Land, Woods, and Waters, a farre of. More cunningly, I say: But G.o.d 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 foud (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 sudry formes, (as _Sphaerike_, _Plaine_ or other), the Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, & such other things, vp the outface of the earthly Globe (either in the whole, or in some princ.i.p.all m?ber and portion therof ctayned) may be described and designed, in cmensurations a.n.a.logicall 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 circ.u.mstances. 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 a.n.a.logically 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 s.h.i.+ere, a Lords.h.i.+p, 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 a.n.a.logicall description of the Ocean Sea coastes, through the whole world: or in the chiefe and princ.i.p.all 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 princ.i.p.ally 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 Seac.u.mpase they appeare, and what apparent forme, Situation, and bignes they haue, in respecte of any daungerous place in the sea, or nere vnto it, a.s.signed: 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, a.n.a.logicall to any _Geometricall_ figure appointed, any certaine number or summe of men: of such a figure capable: (by reason of the vsuall s.p.a.ces 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, c 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 a.s.signed. 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.]