The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara Part 3

* 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_, _Rhombod_, _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 hors.e.m.e.n, 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 s.p.a.ces: (betwene footemen or hors.e.m.e.n) to take the largest: or when he would seme to haue few, (beyng many:) contrarywise, in Figure, and s.p.a.ce. 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 Gla.s.ses. 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 pa.s.sed 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 Princ.i.p.all Science of _Magnitudes_: do yet neuertheles vse the great ayde, direction, and Method of the sayd princ.i.p.all 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, Pneumat.i.thmie, 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 _G.o.ds 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 pa.s.sions, 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 roud Globe or Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise iudgeth a plaine Square, to be roud: 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 pa.s.se, (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 Pec.o.c.kes 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 gla.s.se, and proffer, with dagger or sword, to foyne at the gla.s.se, you shall suddenly be moued to giue backe (in maner) by reason of an Image,

[? A marueilous Gla.s.se.]

appearing in the ayre, betwene you & the gla.s.se, with like hand, sword or dagger, & with like quicknes, foyning at your very eye, likewise as you do at the Gla.s.se. 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 n.o.ble 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 Gla.s.se, 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 Gla.s.ses (which name, Gla.s.se, 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 princ.i.p.all 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 pa.s.sions 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, Ma.s.sines and Body of the _Sonne_, conteineth the quant.i.tie of the whole Earth and Sea, a hundred thre score and two times, lesse by ? 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 ? 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 ? The nerest, that the _Mone_ commeth to the earth, is Semidiameters 52 The distance of the Starry Skye is, fr vs, in Semidiameters of the earth 20081 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 20029 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 Commaudement 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, cteineth 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 ma.s.sines of _Ha Rakia_, will you not finde your Consciences moued, with the kingly Prophet, to sing the confession of G.o.ds Glory, and say, +_The Heauens declare the glory of G.o.d, 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 G.o.d, in, and by his Creatures, and Creation to be seen and learned. By parables and a.n.a.logies of whose natures and properties, the course of the Holy Scripture, also, declareth to vs very many Mysteries.

The whole Frame of G.o.ds Creatures, (which is the whole world,) is to vs, a bright gla.s.se: from which, by reflexion, reboundeth to our knowledge and perceiuerance, Beames, and Radiations: representing the Image of his Infinite goodnes, Omnipot?cy, and wisedome. And we therby, are taught and persuaded to Glorifie our Creator, as G.o.d: 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 _Hemisphaerie_, 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 a.s.se 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, Sapientissimu 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 G.o.d 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 Considerati 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 cmodity, 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 felows.h.i.+p, of the Intellectuall and Mentall part of vs, with our grosse & corruptible body: but a certaine Meane, and _Harmonious Spiritualitie, with both partic.i.p.atyng, & 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_ Consoncy, 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 h.o.m.ologall 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 Correspdent: So, as, the Heauenly Globe, may (in practise) be duely described vpon the Geographicall, and Hydrographicall Globe. And there, for vs to consider an _aequonoctiall Circle_, _an Ecliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes, Lat.i.tudes, 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. aequall Houres) to learne the Risinges and Settinges of Sterres (of _Virgill_ in his _Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sphaere_, 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 a.s.signed.+ 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 vnderstding 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 necessari Mundus iste, supernis lationibus fere continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causa prima putanda omnibus est, vnde motus principium exist.i.t._ 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, praecipuum[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_. _h.o.m.o 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 G.o.dly 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 pa.s.sions, 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 circ.u.mspecte Astrologien hath to Conclude._+ Not onely (by _Apotelesmes_) t? ?t?]. but by Naturall and Mathematicall demonstration t? d??t?. Whereunto, what Sciences are requisite (without exception) I partly haue here warned: And in my _Propaedeumes_ (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.]