The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara Part 6

[* _Note this Corollary._]

* This, my Demonstrati is more generall, then onely in Square Pyramis or Cone: Consider well. Thus, haue I, both Mathematically and Mechanically, ben very long in wordes: yet (I trust) nothing tedious to them, who, to these thinges, are well affected. And verily I am forced (auoiding prolixitie) to omit sundry such things, easie to be practised: which to the Mathematicien, would be a great Threasure: and to the Mechanicien, no small gaine.

[* The great Commodities following of these new Inuentions.]

* Now may you, +Betwene two lines giuen, finde two middle proportionals, in Continuall proportion: by the hollow Parallelipipedon, and the hollow Pyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre right lines may be found, proportionall in any proportion a.s.signed, of which, shal be produced a Parallelipipedon, aequall to the Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36.

proposition, of the 11. boke of _Euclide_. Now, all those thinges, which _Vitruuius_ in his Architecture, specified hable to be done, by dubbling of the Cube: Or, by finding of two middle proportionall lines, betwene two lines giuen, may easely be performed. Now, that Probleme, which I noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke of _Euclide_, is proued possible. Now, may any regular body, be Transformed into an other, &c. Now, any regular body: any Sphere, yea any Mixt Solid: and (that more is) Irregular Solides, may be made (in any proporti a.s.signed) like vnto the body, first giuen. Thus, of a _Manneken_, (as the _Dutch_ Painters terme it) in the same _Symmetrie_, may a Giant be made: and that, with any gesture, by the Manneken vsed: and contrarywise. Now, may you, of any Mould, or Modell of a s.h.i.+p, make one, of the same Mould (in any a.s.signed proportion) bigger or lesser.

Now, may you, of any * Gunne, or little peece of ordinauce, make an other, with the same _Symmetrie_ (in all pointes) as great, and as little, as you will. Marke that: and thinke on it. Infinitely, +may you apply this, so long sought for, and now so easily concluded: and withall, so willingly and frankly communicated to such, as faithfully deale with vertuous studies.+

[Such is the Fruite of the Mathematicall Sciences and Artes.]

Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte: and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he can, by order, Descend, to frame Naturall thinges, to wonderfull vses: and when he list, retire home into his owne Centre: and there, prepare more Meanes, to Ascend or Descend by: and, all, to the glory of G.o.d, and our honest delectation in earth.

Although, the Printer, hath looked for this Praeface, a day or two, yet could I not bring my pen from the paper, before I had giuen you comfortable warning, and brief instructions, of some of the Commodities, by _Statike_, hable to be reaped: In the rest, I will therfore, be as brief, as it is possible: and with all, describing them, somwhat accordingly. And that, you shall perceiue, by this, which in order commeth next. For, wheras, it is so ample and wonderfull, that, an whole yeare long, one might finde fruitfull matter therin, to speake of: and also in practise, is a Threasure endeles: yet will I glanse ouer it, with wordes very few.

This do I call +Anthropographie+. Which is an Art restored, and of my preferment to your Seruice. I pray you, thinke of it, as of one of the chief pointes, of Humane knowledge. Although it be, but now, first Cfirmed, with this new name: yet the matter, hath from the beginning, ben in consideration of all perfect Philosophers. +Anthropographie, is the description of the Number, Measure, Waight, figure, Situation, and colour of euery diuerse thing, conteyned in the perfect body of MAN: with certain knowledge of the Symmetrie, figure, waight, Characterization, and due locall motion, of any parcell of the sayd body, a.s.signed: and of Nubers, to the sayd parcell appertainyng.+ This, is the one part of the Definition, mete for this place: Sufficient to notifie, the particularitie, and excellency of the Arte: and why it is, here, ascribed to the Mathematicals. Yf the description of the heauenly part of the world, had a peculier Art, called _Astronomie:_ If the description of the earthly Globe, hath his peculier arte, called _Geographie_. If the Matching of both, hath his peculier Arte, called _Cosmographie:_ Which is the Descripti of the whole, and vniuersall frame of the world: Why should not the description of

[MAN is the Lesse World.]

him, who is the Lesse world: and, fr the beginning, called _Microcosmus_ (that is. _The Lesse World._) And for whose sake, and seruice, all bodily creatures els, were created: Who, also, partic.i.p.ateth with Spirites, and Angels: and is made to the Image and similitude of _G.o.d_: haue his peculier Art? and be called the _Arte of Artes_: rather, then, either to want a name, or to haue to base and impropre a name? You must of sundry professions, borow or challenge home, peculier partes hereof: and farder procede: as, G.o.d, Nature, Reason and Experience shall informe you. The Anatomistes will restore to you, some part: The Physiognomistes, some: The Chyromantistes some. The Metaposcopistes, some: The excellent, _Albert Durer_, a good part: the Arte of Perspectiue, will somwhat, for the Eye, helpe forward: _Pythagoras_, _Hipocrates_, _Plato_, _Galenus_, _Meletius_, & many other (in certaine thinges) will be Contributaries. And farder, the Heauen, the Earth, and all other Creatures, will eche shew, and offer their Harmonious seruice, to fill vp, that, which wanteth hereof: and with your own Experience, concluding: you may Methodically register the whole, for the posteritie: Whereby, good profe will be had, of our Harmonious, and

[Micro Cosmus.]

Microcosmicall const.i.tution.

The outward Image, and vew hereof: to the Art of _Zographie_ and Painting, to Sculpture, and Architecture: (for Church, House, Fort, or s.h.i.+p) is most necessary and profitable: for that, it is the chiefe base and foundation of them.

[* Lib. 3. Cap. 1.]

Looke in * _Vitruuius_, whether I deale sincerely for your behoufe, or no. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Looke in the 27. and 28. Chapters, of the second booke, _De occulta Philosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther.

Remember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ so long agoe p.r.o.nounced: of so many a Philosopher repeated: and of the _Wisest_ attempted: And then, you will perceaue, how long agoe, you haue bene called to the Schole, where this Arte might be learned. Well. I am nothing affrayde, of the disdayne of some such, as thinke Sciences and Artes, to be but Seuen. Perhaps, those Such, may, with ignorance, and shame enough, come short of them Seuen also: and yet neuerthelesse they can not prescribe a certaine number of Artes: and in eche, certaine vnpa.s.sable boundes, to G.o.d, Nature, and mans Industrie. New Artes, dayly rise vp: and there was no such order taken, that,

All Artes, should in one age, or in one land, or of one man, be made knowen to the world. Let vs embrace the giftes of G.o.d, and wayes to wisedome, in this time of grace, from aboue, continually bestowed on them, who thankefully will receiue them: _Et bonis Omnia Cooperabuntur in bonum._

+Trochilike, is that Art Mathematicall, which demonstrateth the properties of all Circular motions, Simple and Compounde.+ And bycause the frute hereof, vulgarly receiued, is in Wheles, it hath the name of _Trochilike:_ as a man would say, _Whele Art_. By this art, a Whele may be geuen which shall moue ones about, in any tyme a.s.signed. Two Wheles may be giuen, whose turnynges about in one and the same tyme, (or equall tymes), shall haue, one to the other, any proportion appointed. By Wheles, may a straight line be described: Likewise, a Spirall line in plaine, Conicall Section lines, and other Irregular lines, at pleasure, may be drawen. These, and such like, are princ.i.p.all Conclusions of this Arte: and helpe forward many pleasant and profitable Mechanicall workes:

[Saw Milles.]

As Milles, to Saw great and very long Deale bordes, no man being by.

Such haue I seene in Germany: and in the Citie of Prage: in the kingdome of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight, Spring, Man or Beast, moued. Take in your hand, _Agricola De re Metallica:_ and then shall you (in all Mines) perceaue, how great nede is, of Whele worke. By Wheles, straunge workes and incredible, are done: as will, in other Artes hereafter, appeare. A wonderfull example of farther possibilitie, and present commoditie, was sene in my time, in a certaine Instrument: which by the Inuenter and Artificer (before) was solde for xx. Talentes of Golde: and then had (by misfortune) receaued some iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend the same, and presented it vnto the Emperour _Charles_ the fifth.

_Hieronymus Carda.n.u.s_, can be my witnesse, that therein, was one Whele, which moued, and that, in such rate, that, in 7000. yeares onely, his owne periode should be finished. A thing almost incredible: But how farre, I keepe me within my boundes: very many men (yet aliue) can tell.

+Helicosophie+, is nere Sister to _Trochilike:_ and is, +An Arte Mathematicall, which demonstrateth the designing of all Spirall lines in Plaine, on Cylinder, Cone, Sphaere, Conoid, and Sphaeroid, and their properties appertayning.+ The vse hereof, in _Architecture_, and diuerse Instrumentes and Engines, is most necessary. For, in many thinges, the Skrue worketh the feate, which, els, could not be performed. By helpe hereof,

[* Atheneus Lib. 5. cap. 8.]

it is * recorded, that, where all the power of the Citie of Syracusa, was not hable to moue a certaine s.h.i.+p (being on ground) mightie _Archimedes_, setting to, his Skruish Engine, caused _Hiero_ the king, by him self, at ease, to remoue her, as he would.

[Proclus. Pag. 18.]

Wherat, the King wondring: ?p? t??t?? t?? ???a?, pe?? pa?t??, ?????de?

?????t? p??e?te??. _From this day, forward_ (said the King) _Credit ought to be giuen to Archimedes, what soeuer he sayth._

+Pneumat.i.thmie demonstrateth by close hollow Geometricall Figures, (regular and irregular) the straunge properties (in motion or stay) of the Water, Ayre, Smoke, and Fire, in theyr ctinuitie, and as they are ioyned to the Elementes next them.+ This Arte, to the Naturall Philosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_ is not in the world. And that, all Nature, abhorreth it so much: that, contrary to ordinary law, the Elementes will moue or stand. As, Water to ascend: rather then betwene him and Ayre, s.p.a.ce or place should be left, more then (naturally) that qut.i.tie of Ayre requireth, or can fill.

Againe, Water to hang, and not descend: rather then by descending, to leaue Emptines at his backe. The like, is of Fire and Ayre: they will descend: when, either, their Ctinuitie should be dissolued: or their next Element forced from them. And as they will not be extended, to discontinuitie: So, will they not, nor yet of mans force, can be prest or pent, in s.p.a.ce, not sufficient and aunswerable to their bodily substance. Great force and violence will they vse, to enioy their naturall right and libertie.

[To go to the bottom of the Sea without daunger.]

Hereupon, two or three men together, by keping Ayre vnder a great Cauldron, and forcyng the same downe, orderly, may without harme descend to the Sea bottome: and continue there a tyme &c. Where, Note, how the thicker Element (as the Water) giueth place to the thynner (as, is the ayre:) and receiueth violence of the thinner, in maner. &c. Pumps and all maner of Bellowes, haue their ground of this Art: and many other straunge deuises. As, _Hydraulica_, Organes goyng by water. &c. Of this Feat, (called commonly _Pneumatica_,) goodly workes are extant, both in Greke, and Latin. With old and learned Schole men, it is called _Scientia de pleno & vacuo._

+Menadrie, is an Arte Mathematicall, which demonstrateth, how, aboue Natures vertue and power simple: Vertue and force may be multiplied: and so, to direct, to lift, to pull to, and to put or cast fro, any multiplied or simple, determined Vertue, Waight or Force: naturally, not, so, directible or moueable.+ Very much is this Art furdred by other Artes: as, in some pointes, by _Perspectiue_: in some, by _Statike_: in some, by _Trochilike_: and in other, by _Helicosophie_: and _Pneumat.i.thmie_. By this Art, all Cranes, Gybbettes, & Ingines to lift vp, or to force any thing, any maner way, are ordred: and the certaine cause of their force, is knowne: As, the force which one man hath with the d.u.c.h.e waghen Racke: therwith, to set vp agayne, a mighty waghen laden, being ouerthrowne. The force of the Crossebow Racke, is certainly, here, demonstrated. The reason, why one m, doth with a leauer, lift that, which Sixe men, with their handes onely, could not, so easily do. By this Arte, in our common Cranes in London, where powre is to Crane vp, the waight of 2000. pound: by two Wheles more (by good order added) Arte concludeth, that there may be Craned vp 200000. pound waight &c. So well knew _Archimedes_ this Arte: that he alone, with his deuises and engynes, (twise or thrise) spoyled and discomfited the whole Army and Hoste of the Romaines, besieging _Syracusa_,

[=Plutarchus in Marco Marcello.=]

_Marcus Marcellus the Consul_, being their Generall Capitaine.

[=Synesius in Epistolis.=]

Such huge Stones, so many, with such force, and so farre, did he with his engynes hayle among them, out of the Citie.

[=Polybius.=]

[=Plinius.=]

[=Quintilia.n.u.s.=]

[=T. Liuius.=]

And by Sea likewise: though their s.h.i.+ps might come to the walls of _Syracusa_, yet hee vtterly confounded the Romaine Nauye. What with his mighty Stones hurlyng:

[=* Athenaeus.=]

what with Pikes of * 18 fote long, made like shaftes: which he forced almost a quarter of a myle. What, with his catchyng hold of their Shyps, and hoysing them vp aboue the water, and suddenly letting them fall into the Sea againe:

[= * Galenus.=]