The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

TO THE VNFAINED LOVERS of truthe, and constant Studentes of Noble Sciences, IOHN DEE of London, hartily wisheth grace from heauen, and most prospe∣rous successe in all their honest attemptes and exercises.

DIuine Plato, the great Master of many worthy Philosophers, and the constant auoucher, and pithy perswader of Vnum, Bo∣num, and Ens: in his Schole and Academie, sundry times (besides his ordinary Scholers) was visited of a certaine kinde of men, allured by the noble fame of Plato, and the great commendation of hys profound and profitable doctrine. But when such Hearers, after long harkening to him, perceaued, that the drift of his discourses issued out, to conclude, this Vnum, Bo∣num, and Ens, to be Spirituall, Infi∣nite, AEternall, Omnipotent, &c. Nothyng beyng alledged or expressed, How, worldly goods: how, worldly digni∣tie: how, health, Strēgth or Iustines of body: nor yet the meanes, how a merue••lous sensible and bodyly blysse and felicitie hereafter, might be atteyned: Straightway, the fantasies of those hearers, were dampt: their opinion of Plato, was clene chaun∣ged: yea his doctrine was by them despised: and his schole, no more of ••hem visi∣ted.

Which thing, his Scholer, Aristotle, narrowly cōsidering, founde the cause ther∣of, to be, For that they had no forwarnyng and information, in generall, whereto his doctrine tended.

For, so, might they haue had occasion, either to haue forborne his schole hauntyng: (if they, then, had misliked his Scope and purpose) or con∣stantly to haue continued therin: to their full satisfaction: if such finall scope & intent, had ben to their desire.

Wherfore, Aristotle, euer, after that, vsed in brief, to forewarne his owne Scholers and hearers, both of what matter, and also to what ende, he tooke in hand to speake, or teach. While I consider the diuerse trades of these two excellent Philosophers (and am most sure, both, that Plato right well, o∣therwise could teach:

and that Aristotle mought boldely, with his hearers, haue dealt in like sorte as Plato did) I am in no little pang of perplexitie: Bycause, that, which I mislike, is most easy for me to performe (and to haue Plato for my exāple.) And that, which I know to be most commendable: and (in this first bringyng, into common handling, the Artes Mathematicall) to be most necessary: is full of great difficultie and sundry daungers. Yet•• neither do I think it me••e, for so straunge mat∣ter (as now is ment to be published) and to so straunge an audience, to be bluntly, at first, put forth, without a peculiar Preface: Nor (Imitatyng Aristotle) well can I hope, that accordyng to the amplenes and dignitie of the State Mathematicall, I am able, either playnly to prescribe the materiall boundes: or precisely to expresse the chief purposes, and most wonderfull applications therof. And though I am sure, that such as did shrinke from Plato his s••hole, after they had perc••iued his fi∣nall

conclusion, would in these thinges haue ben his most diligent hearers) so infi∣nitely mought their desires, in fine and at length, by our Artes Mathematicall be sa∣tisfied) yet, by this my Praeface & forewarnyng, Aswell all such, may (to their great behofe) the soner, hither be allured: as also the Pythagoricall, and Platonicall perfect scholer, and the constant profound Philosopher, with more ease and spede, may (like the Bee,) gather, hereby, both wax and hony.

Wherfore, seyng I finde great occasion (for the causes alleged, and farder, in re∣spect of my Art Mathematike generall) to vse a certaine forewarnyng and Praeface, whose content shalbe, that mighty, most plesaunt, and frutefull Mathematicall Tree, with his chief armes and second (grifted) braunches:* 1.1 Both, what euery one is, and also, what commodity, in generall, is to be looked for, aswell of griff as stocke: And forasmuch as this enterprise is so great, that, to this our tyme, it neuer was (to my knowledge) by any achieued: And also it is most hard, in these our drery dayes, to such rare and straunge Artes, to wyn due and common credit:

Neuertheles, if, for my sincere endeuour to satisfie your honest expectation, you will but lend me your thākefull mynde a while: and, to such matter as, for this time, my penne (with spede) is hable to deliuer, apply your eye or eare attentifely: perchaunce, at once, and for the first salutyng, this Preface you will finde a lesson long enough. And ei∣ther you will, for a second (by this) be made much the apter: or shortly become, well hable your selues, of the lyons claw, to coniecture his royall symmetrie, and farder propertie. Now then, gentle, my frendes, and countrey men, Turne your eyes, and bend your myndes to that doctrine, which for our present purpose, my simple talent is hable to yeld you.

All thinges which are, & haue beyng, are found vnder a triple diuersitie generall.

For, either, they are demed Supernaturall, Naturall, or, of a third being. Thinges Supernaturall, are immateriall, simple, indiuisible, incorruptible, & vnchangeable. Things Naturall, are materiall, compounded, diuisible, corruptible, and chaungea∣ble. Thinges Supernaturall, are, of the minde onely, comprehended: Things Natu∣rall, of the sense exterior, ar hable to be perceiued. In thinges Naturall, probabilitie and coniecture hath place: But in things Supernaturall, chief demōstration, & most sure Science is to be had. By which properties & comparasons of these two, more easily may be described, the state, condition, nature and property of those thinges, which, we before termed of a third being: which, by a peculier name also, are called Thynges Mathematicall. For, these, beyng (in a maner) middle, betwene thinges su∣pernaturall and naturall: are not so absolute and excellent, as thinges supernatural: Nor yet so base and grosse, as things naturall: But are thinges immateriall: and ne∣uerthelesse, by materiall things hable somewhat to be signified. And though their particular Images, by Art, are agg••egable and diuisible: yet the generall Formes, notwithstandyng, are constant, vnchaungeable, vntrāsformable, and incorruptible. Neither of the sense, can they, at any tyme, be perceiued or iudged. Nor yet for all that, in the royall mynde of man, first conceiued. But, surmountyng the imperfectiō of coniecture, weenyng and opinion: and commyng short of high intellectuall cō∣ceptiō, are the Mercurial fruite of Dianoeticall discourse, in perfect imagination sub∣sistyng. A meruaylous newtralitie haue these thinges Mathematicall. and also a straunge participatiō betwene thinges supernaturall, immortall, intellectual, simple and indiuisible: and thynges naturall, mortall, sensible, compounded and diuisible. Probabilitie and sensible profe, may well serue in thinges naturall: and is commen∣dable: In Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor yet the testimony of sense, any whit credited: But onely a perfect demonstra∣tion, of truthes certaine, necessary, and inuincible: vniuersally and necessaryly con∣cluded:

is allowed as sufficient for an Argument exactly and purely Mathematical."

Of Mathematicall thinges, are two principall kinde•••• namely, Number,* 1.2 and Mag∣nitude. Number, we define, to be, a certayne Mathematicall Sūme, of Vnits.* 1.3 And, an Vnit, is that thing Mathematicall, Indiuisible, by participation of some likenes of whose property, any thing, which is in de••de, or is counted One, may resonably be called One. We account an Vnit, a thing Mathematicall, though it be no Number, and also indiuisible•• because, of it, materially, Number doth consist•• which, princi∣pally, is a thing Mathematicall. Magnitude is a thing Mathematicall,* 1.4 by participation of some likenes of whose nature, any thing is iudged long, broade, or thicke. A thicke Magnitude we call a Solide, or a Body. What Magnitude so euer, is Solide or" Thicke, is also broade, & long. A broade magnitude, we call a Superficies or a Plaine. Euery playne magnitude, hath also length. A long magnitude, we terme a Line. A Line is neither thicke nor broade, but onely long: Euery certayne Line, hath two endes: The endes of a line, are Pointes called.* 1.5 A Point, is a thing Mathematicall, indi∣uisible, which may haue a certayne determined situation. If a Poynt moue from a" determined situation, the way wherein it moued, is also a Line: mathematically produced. whereupon, of the auncient Mathematiciens, a Line is called the race or course of a Point.* 1.6 A Poynt we define, by the name of a thing Mathematicall: though it be no Magnitude, and indiuisible: because it is the propre ende, and bound of a Line: which is a true Magnitude.* 1.7 And Magnitude we may define to be that thing Mathematicall, which is diuisible for euer, in partes diuisible, long, broade or thicke. Therefore though a Poynt be no Magnitude, yet Terminatiuely we rec∣ken it a thing Mathematicall (as I sayd) by reason it is properly the end, and bound of a line.

Neither Number, nor Magnitude, haue any Materialitie. First, we will consider of Number, and of the Science Mathe••a••icall, to it appropriate, called Arithmetike: and afterward of Magnitude, and his Science, called Geometrie. But that name con∣tenteth me not: whereof a word or two hereafter shall be sayd. How Immateriall and free from all matter, Number is, who doth not perceaue? ye••, who doth not wonderfully wōder at it: For, neither pure Element, nor Aristoteles, Quinta Essentia, is hable to serue for Number, as his propre matter. Nor yet the puritie and simple∣nes of Substance Spirituall or Angelicall, will be found propre enough thereto. And therefore the great & godly Philosopher Anitius Boetius, sayd: Omnia quaecun{que} a primaua rerum na••••ra constructa sunt, Numerorum videntur ratione formata. Hoc enim fuit principale in animo Conditoris Exemplar. That is: All thinges (which from the very first originall be••ng of thinges, haue bene framed and made) do appeare to be Formed by the reason of Numbers. For this was the principall example or patterne in the minde of the Creator. O comfor∣table alluremen••, O rauishing perswasion, to deale with a Science, whose Subiect, i•• so Auncient, so pure, so excellent, so surmounting all creatures, so vsed of the Al∣mighty and incomprehensible wisdome of the Creator, in the distinct creation of all creatures•• in all their distinct partes, properties, natures, and vertues, by order, and most absolute number, brought, from Nothing, to the Formalitie of their being and state. By Numbers propertie therefore, of vs, by all possible meanes, (to the per∣fection of the Science) learned, we may both winde and draw our selues into the inward and deepe search and v••w, of all creatures distinct vertues, natures, proper∣ties, and Former: And also, farder, arise, clime, ascend, and mount vp (with Specula∣tiue winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all thinges Numerable: both visible and inuisible•• mortall and

immortall, Corporall and Spirituall. Part of this profound and diuine Science, had Ioachim the Prophesier atteyned vnto: by Numbers Formall, Naturall, and Rationall, forseyng, concludyng, and forshewyng great particular euents, long before their comming. His bookes yet remainyng, hereof, are good profe: And the noble Earle of Mirandula, (besides that,) a sufficient witnesse: that Ioachim, in his prophesies, proce∣ded by no other way, then by Numbers Formall. And this Earle hym selfe, in Rome, * 1.8 set vp 900. Conclusions, in all kinde of Sciences, openly to be disputed of: and among the rest, in his Conclusions Mathematicall, (in the eleuenth Conclusion) hath in Latin, this English sentence. By Numbers, a way is had, to the searchyng out, aud vnder∣standyng of euery thyng, hable to be knowen. For the verifying of which Conclusion, I pro∣mise to aunswere to the 74. Quaestions, vnder written, by the way of Numbers. Which Cō∣clusions, I omit here to rehearse: aswell auoidyng superfluous prolixitie: as, by∣cause Ioannes Picus, workes, are commonly had. But, in any case, I would wish that those Conclusions were red diligently, and perceiued of such, as are earnest Ob∣seruers and Considerers of the constant law of nūbers: which is planted in thyngs Naturall and Supernaturall: and is prescribed to all Creatures, inuiolably to be kept. For, so, besides many other thinges, in those Conclusions to be marked, it would apeare, how sincerely, & within my boundes, I disclose the wonderfull my∣steries, by numbers, to be atteyned vnto.

Of my former wordes, easy it is to be gathered, that Number hath a treble state: One, in the Creator: an other in euery Creature (in respect of his complete consti∣tution:) and the third, in Spirituall and Angelicall Myndes, and in the Soule of mā. In the first and third state, Number, is termed Number Numbryng. But in all Crea∣tures, otherwise, Number, is termed Nūber Numbred. And in our Soule, Nūber bea∣reth such a swaye, and hath such an affinitie therwith: that some of the old Philoso∣phers taught, Mans Soule, to be a Number mouyng it selfe. And in dede, in vs, though it be a very Accident: yet such an Accident it is, that before all Creatures it had per∣fect beyng, in the Creator, Sempiternally. Number Numbryng therfore, is the discre∣tion discerning, and distincting of thinges. But in God the Creator, This discre∣tion, in the beginnyng, produced orderly and distinctly all thinges. For his Num∣bryng, then, was his Creatyng of all thinges. And his Continuall Numbryng, of all thinges, is the Conseruation of them in being: And, where and when he will lacke an Vnit: there and then, that particular thyng shalbe Discreated. Here I stay. But our Seuerallyng, distinctyng, and Numbryng, createth nothyng•• but of Multitude con∣sidered, maketh certaine and distinct determination. And albeit these thynges be waighty and truthes of great importance, yet (by the infinite goodnes of the Al∣mighty Ternarie,) Artificiall Methods and easy wayes are made, by which the ze∣lous Philosopher, may wyn nere this Riuerish Ida•• this Mountayne of Contempla∣tion: and more then Contemplation. And also, though Number, be a thyng so Im∣materiall, so diuine, and aeternall: yet by degrees, by litle and litle, stretchyng forth, and applying some likenes of it, as first, to thinges Spirituall: and then, bryngyng it lower, to thynges sensibly perceiued: as of a momentanye sounde iterated: then to the least thynges that may be seen, numerable: And at length, (most grossely,) to a multitude of any corporall thynges seen, or felt: and so, of these grosse and sensible thynges, we are trayned to learne a certaine Image or likenes of numbers: and to vse Arte in them to our pleasure and proffit. So grosse is our conuersation, and dull is our apprehension: while mortall Sense, in vs, ruleth the common wealth of our litle world. Hereby we say, Three Lyons, are three: or a Ternarie. Three Egles, are three, or a Ternarie. Which* 1.9 Ternaries, are eche, the Vnion, knot, and Vniformitie, of three discrete and distinct Vnits. That is, we may in eche Ternarie, thrise, seuerally pointe, and shew a part, One, One, and One. Where, in Numbryng, we say One, two,

Three. But how farre, these visible Ones, do differre from our Indiuisible Vnits (in pure Arithmetike, principally considered) no man is ignorant. Yet from these grosse and materiall thynges, may we be led vpward, by degrees, so, informyng our rude Imagination, toward the coceiuyng of Numbers, absolutely (Not supposing, nor admixtyng any thyng created, Corporall or Spirituall, to support, conteyne, or represent those Numbers imagined:) that at length, we may be hable, to finde the number of our owne name, gloriously exemplified and registred in the booke of the Trinitie most blessed and aeternall.

But farder vnderstand, that vulgar Practisers, haue Numbers, otherwise, in sun∣dry Considerations: and extend their name farder, then to Numbers, whose least part is an Vnit. For the common Logist, Reckenmaster, or Arithmeticien, in hys v∣sing of Numbers: of an Vnit, imagineth lesse partes•• and calleth them Fractions. As of an Vnit, he maketh an halfe, and thus noteth it, ½ and so of other, (infinitely di∣uerse) partes of an Vnit, Yea and farder, hath, Fractions of Fractions. &c. And, foras∣much, as, Addition, Substraction, Multiplication, Diuision and Extraction of Rotes, are the chief, and sufficient partes of Arithmetike:* 1.10 which is, the Science that demonstra∣teth the properties, of Numbers, and all operatiōs, in numbers to be performed.

How often, therfore,* 1.11 these fiue sundry sortes of Operations, do, for the most part, of their exe∣cution, differre from the fiue operations of like generall property and name, in our Whole numbers practisable, So often, (for a more distinct doctrine) we, vulgarly account and name it, an other kynde of Arithmetike.

And by this reason: the Con∣sideration, [ 1] doctrine, and working, in whole numbers onely: where, of an Vnit, is no lesse part to be allowed: is named (as it were) an Ari••hmetike by it selfe. And so of the Arithmetike of Fractions. In lyke sorte, the necessary, wonderfull and Secret doc∣trine of Proportion, and proportionalytie hath purchased vnto it selfe a peculier [ 2] maner of handlyng and workyng: and so may seme an other forme of Arithmetike. Moreouer, the Astronomers, for spede and more commodious calculation, haue de∣uised [ 3] a peculier maner of orderyng nūbers, about theyr circular motions, by Sexa∣genes, and Sexagesmes. By Signes, Degrees and Minutes &c. which commonly is called the Arithmetike of Astronomical or Phisicall Fractions. That, haue I briefly no∣ted, by the name of Arithmetike Circular. By cause it is also vsed in circles, not Astro∣nomicall. &c. Practise hath led Numbers farder, and hath framed them, to take vpon [ 4] them, the shew of Magnitudes propertie: Which is Incommensurabilitie and Irratio∣nalitie. (For in pu••e Arithmetike, an Vnit, is the common Measure of all Numbers.) And, here, Nūbers are become, as Lynes, Playnes and Solides: some tymes Ratio∣nall, some tymes Irrationall•• And haue propre and peculier characters, (as √••.√••. and so of other. Which is to signifie Ro••e Square, Rote Cubik: and so forth:) & propre and peculier fashions in the fiue principall partes: Wherfore the practiser, estemeth this, a diuerse Arithmetike from the other. Practise bryngeth in, here, diuerse com∣poundyng of Numbers: as some tyme, two, three, foure (or more) Radicall nūbers, diuersly knit, by signes, o•• More & Lesse: as thus √•• 12 + √•• 15. Or ••hus √•••• 19 + √•• 12 - √•• 2•• &c. And some tyme with whole numbers, or fractions of whole Number, amōg them as 20 + √•• ••4•• √•• + 33 - √•• 10, √•••• 44 + 12 - +√••9. And so, infinitely, may hap the varie••ie. After this: Both the one and the other hath fractions incident: and so is this Arithmetike greately enlarged, by diuerse ex∣hibityng and vse of Compositions and mixtynges. Consider how•• I (beyng desi∣rous to deliuer the student from error and Cauillation) do giue to this Practise, the name of the Arithmetike of Radicall numbers: Not of Irrationall or Surd Numbers•• which other while, are Rationall: though they haue the Signe of a Rote before

them, which, Arithmetike of whole Numbers most vsuall, would say they had no such Roote: and so account them Surd Numbers: which generally spokē, is vntrue: as Euclides tenth booke may teach you. Therfore to call them, generally, Radicall Numbers, (by reason of the signe √. prefixed,) is a sure way: and a sufficient generall distinction from all other ordryng and vsing of Numbers: And yet (beside all this) Consider: the infinite desire of knowledge, and incredible power of mans Search and Capacitye: how, they, ioyntly haue waded farder (by mixtyng of spe∣culation and practise) and haue found out, and atteyned to the very chief perfec∣tion (almost) of Numbers Practicall vse. Which thing, is well to be perceiued in that great Arithmeticall Arte of AEquation: commonly called the Rule of Coss. or Alge∣bra. The Latines termed it, Regulam Rei & Census, that is, the Rule of the thyng and his value. With an apt name: comprehendyng the first and last pointes of the worke. And the vulgar names, both in Italian, Frenche and Spanish, depend (in namyng it,) vpon the signification of the Latin word, Res: A thing: vnleast they vse the name of Algebra. And therin (commonly) is a dubble error. The one, of them, which thinke it to be of Geber his inuentyng: the other of such as call it Algebra. For, first, though Geber for his great skill in Numbers, Geometry, Astronomy, and other maruailous Artes, mought haue semed hable to haue first deuised the sayd Rule: and also the name carryeth with it a very nere likenes of Geber his name: yet true it is, that a Greke Philosopher and Mathematicien, named Diophantus, before Geber his tyme, wrote 13. bookes therof (of which, six are yet extant: and I had them to * 1.12 vse, of the famous Mathematicien, and my great frende, Petrus Mon••au∣reus:) And secondly, the very name, is Algiebar, and not Algebra: as by the Arabien Auicen, may be proued: who hath these precise wordes in Latine, by Andreas Alpa∣gus (most perfect in the Arabik tung) so translated. Scientia faciendi Algiebar & Almachabel. i. Scientia inueniendi numerum ignotum, per additionem Numeri, & diuisio∣nem & aequationem. Which is to say: The Science of workyng Algiebar and Al∣machabel, that is, the Science of findyng an vnknowen number, by Addyng of a Number, & Diuision & aequation. Here haue you the name: and also the prin∣cipall partes of the Rule, touched. To name it, The rule, or Art of AEquation, doth sig∣ni••ie the middle part and the State of the Rule. This Rule, hath his peculier Cha∣racters: [ 5] and the principal partes of Arithmetike, to it appertayning, do differe from the other Arithmeticall operations. This Arithmetike, hath Nūbers Simple, Cōpound, Mixt: and Fractions, accordingly. This Rule, and Arithmetike of Algiebar, is so pro∣found, so generall and so (in maner) conteyneth the whole power of Numbers Application practicall: that mans witt, can deale with nothyng•• more proffitable a∣bout numbers: nor match, with a thyng, more mete for the diuine force of the Soule, (in humane Studies, affaires, or exercises) to be tryed in. Perchaunce you looked for, (long ere now,) to haue had some particular profe, or euident testimo∣ny of the vse, proffit and Commodity of Arithmetike vulgar, in the Common lyfe and trade of men. Therto, then, I will now frame my selfe: But herein great care I haue, least length of sundry profes, might make you deme, that either I did mis∣doute your zelous mynde to vertues schole: or els mistrust your hable witts, by some, to gesse much more. A profe then, foure, fiue, or six, such, will I bryng, as any reasonable man, therwith may be persuaded, to loue & honor, yea learne and exercise the excellent Science of Arithmetike.

And first: who, nerer at hand, can be a better witnesse of the frute receiued by Arithmetike, then all kynde of Marchants? Though not all, alike, either nede it, or vse it. How could they forbeare the vse and helpe of the Rule, called the Golden

Rule? Simple and Compounde•• both fo••ward and backward? How might they misse Arithmeticall helpe in the Rules of Felowshyp•• either without tyme, or with tyme•• and betwene the Marchant & his ••actor•• The Rul•••• of Ba••tering in wares onely•• or part in wares, and part in money, would they gladly want? Our Mar∣chant venturers, and Trauaylers ouer Sea, how could they order their doynges iustly and without losse, vnleast certa••ne and generall Rules for Exchaūge of mo∣ney, and Rechaunge, were, for their vse, deuised? The Rule of Alligation, in how sundry cases, doth it conclude for them, such precise verities, as neither by naturall witt, nor other experience, they, were hable, els, to know? And (with the Mar∣chant then to make an end) how ample & wonderfull is the Rule of False positi∣ons? especially as it is now, by two excellent Mathematiciens (of my familer ac∣quayntance in their life time) enlarged? I meane Gemma Frisius, and Simon Iacob. Who can either in brief conclude, the generall and Capitall Rules? or who can I∣magine the Myriades of sundry Cases, and particular examples, in Act and earnest, continually wrought, tried and concluded by the forenamed Rules, onely? How sundry other Arithmeticall practises, are commonly in Marchantes handes, and knowledge: They them selues, can, at large, testifie.

The Mintmaster, and Goldsmith, in their Mixture of Metals, either of diuerse kindes, or diuerse values: how are they, or may they, exactly be directed, and mer∣uailously pleasured, if Arithmetike be their guide? And the honorable Phisiciās, will gladly confesse them selues, much beholding to the Science of Arithmetike, and that sundry wayes: But chiefly in their Art of Graduation, and compounde Medicines. And though Galenus, Auer••ois, Arnoldus, Lullus, and other haue pu∣blished their positions, aswell in the quantities of the Degrees aboue Tempera∣ment, as in the Rules, concluding the new Forme resulting: yet a more precise, commodious, and easy Method, is extant:* 1.13 by a Countreyman of ours (aboue 200 yeares ago) inuented. And forasmuch as I am vncertaine, who hath the same: or when that litle Latin treatise, (as the Author writ it,) shall come to be Printed: (Both to declare the desire I haue to pleasure my Countrey, wherin I may: and al∣so, for very good profe of Numbers vse, in this most subtile and frutefull, Philoso∣phicall Conclusion,) I entend in the meane while, most briefly, and with my far∣der helpe, to communicate the pith therof vnto you.

First describe a circle: whose diameter let be an inch. Diuide the Circumfe∣rence into foure equall partes. Frō the Center, by those 4. sections, extend 4. right lines: eche of 4. inches and a halfe long: or of as many as you liste, aboue 4. with∣out the circumference of the circle: So that they shall be of 4. inches long (at the lea••t) without the Circle•• Make good euident markes, at euery inches end. If you list, you may subdiuide the inches againe into 10. or 12. smaller partes, equall. At the endes of the lines, write the names of the 4. principall elementall Qualities. Hote and Cold••, one against the other. And likewise Moyst and Dry, one against the other. And in the Circle write Temperate. Which Temperature hath a good La∣titude: as appeareth by the Complexion of man. And therefore we haue allow∣ed vnto it, the foresayd Circle: and not a point Mathematicall or Physicall.

Now, when you haue two thinges Miscible, whose degrees are * 1.14 truely knowen•• Of necessitie, either they are of one Quantitie and waight, or of diuerse. If they be of one Quantitie and waight: whether their formes, be Contrary Qua∣lities, or of one kinde (but of diuerse intentions and degrees) or a Temperate, and a Contrary, The form•• resulting of their Mixture, is in the Middle betwene the degrees of

the formes mixt. As for example, let A, be Moist in the first degree: and B, Dry in the third degree. Adde 1. and 3. that maketh 4: the halfe or middle of 4. is 2. This 2. is the middle, equally distant from A and B (for the* 1.15 Temperament is coun∣ted none. And for it, you must put a Ciphre, if at any time, it be in mixture).

[illustration]

Counting then from B, 2. degrees, toward A: you finde it to be Dry in the first degree: So is the Forme resulting of the Mixture of A, and B, in our example, I will geue you an other example. Suppose, you haue two thinges, as C, and D•• and of C, the Heate to be in the 4. degree: and of D, the Colde, to be remisse, euen vnto the Temperament. Now, for C, you take 4•• and for D, you take a Ciphre: which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is 2. Wherefore the Forme resulting of C, and D, is Hote in the second degree: for 2. degrees, accoun∣ted from C, toward D, ende iuste in the 2. degree of heate. Of the third ma∣ner, I will geue also an example: which let be this: I haue a liquid Medicine whose Qualitie of heate is in the 4. degree exalted:* 1.16 as was C, in the example foregoing: and an other liquid Medicine I haue: whose Qualitie, is heate, in the first degree. Of eche of these, I mixt a like quantitie•• Subtract here, the lesse frō the more: and the residue diuide into two equall partes: whereof, the one part, either added to the lesse, or subtracted from the higher degree, doth produce the degree of the

Forme resulting, by this mixture of C, and E. As, if from 4. ye abate 1. there resteth 3, the halfe of 3, is 1 1/••: Adde to 1•• this 1 1/••: you haue 2 1/••. Or subtract from 4. this 1 1/••: you haue likewise 2 ••/•• remayning. Which declareth, the Forme resul∣ting, to be Heate, in the middle of the third degree.

But if the Quantities of two thinges Commixt,* 1.17 be diuerse, and the Intensi∣ons (of their Formes Miscible) be in diuerse degree••, and heigthes. (Whether those Formes be of one kinde, or of Contrary kindes, or of a Temperate and a Contrary, What proportion •••• of th•• lesse quantiti•• to the greater, the same shall be of the difference, which is betwene the degree of the Forme resulting, and the degree of the greater quantitie of the thing miscible, to the difference, which is betwene the same degree of the Forme resulting, and the degree of the lesse quantiti••. As for example. Let two pound of Liquor be geuen, hote in the 4. degree: & one pound of Liqour be geuen, hote in the third degree, I would gladly know the Forme resulting, in the Mixture of these two Liquors.

Set downe your nūbers in order, thus.

[illustration]

Now by the rule of Algiebar, haue I deuised a very ••aste, briefe, and generall maner of working in this case. Let vs first, suppose that Middle Forme resulting, to be 1•••• at that Rule teacheth. And because (by our Rule, here geuen) as the waight of 1. is to 2: So is the difference betwene 4. (the degree of the greater quantitie) and 1•• to the difference betwene 1•• and 3•• (the degree of the thing, in lesse quāt••tie. And with all, 1••, being alwayes in a cer∣taine middell, betwene the two heigthes or degrees). For the first difference, I set—1••: and for the second, I set 1••—3. And, now againe, I say, as 1. is to 2. so is 4—1•• to 1••—3. Wherfore, of these foure proportionall numbers, the first and the fourth Multiplied, one by the other, do make as much, as the second and the third Multiplied the one by 〈◊〉〈◊〉 other? Let these Multiplications be made accor∣dingly. And of the first and the fourth, we haue 1••—3. and of the second & the third, 8—••••. Wherfore, our AEquation is betwene ••••—3: and 8—2••. Which may be reduced, according to the Arte of Algiebar as, here, adding 3. to eche part, ge••eth the AEquation, thus, 1••=11—2••. And yet againe, contracting, or Redu∣cing it: Adde to eche part, ••••? Then haue you 3•• equall to 11: thus represen∣ted 3••=11. Wherefore, diuiding 11 by 3: the Quotient is 3 ••/••: the Valew of our 1••, Coss, or Thing, first supposed. And that is the heigth, or Intension of the Forme resulting: which is, Heate, in two thirdes of the fourth degree: And here I set the shew of the worke in conclusion, thus. The proufe hereof is easie by subtracting 3. from 3 ••/•••• resteth

[illustration]

••/••. Subtracte the same heigth of the Forme resulting, (which is 3 ••/••) frō 4: then resteth ••/••: You see, that ••/•• is double to ••/••: as 2. ••. is double to 1. ••. So should it be: by the rule here geuen. Note. As you ad∣ded to eche part of the AEquation, 3: so if ye first added to eche part 2••, it would stand, 3••—3=8. And now adding to eche par•• 3•• you haue (as afore) 3••=11.

And though I, here, speake onely of two thyngs Miscible: and most common∣ly, mo then three, foure, fiue or six, (&c.) are to be Mixed: (and in one Compound

to be reduced•• & the Forme resultyng of the same, to serue the turne) yet these Ru∣les are sufficient: duely repeated and iterated.* 1.18 In procedyng first, with any two: and then, with the Fonne Resulting, and an other: & so forth: For, the last worke, con∣cludeth the Forme resultyng of them all: I nede nothing to speake, of the Mixture (here supposed) what it is. Common Philosophie hath defined it, saying, Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio. Euery word in the de∣finition, is of great importance. I nede not also spend any time, to shew, how, the other manner of distributing of degrees, doth agree to these Rules. Neither nede I of the farder vse belonging to the Crosse of Graduation (before described) in this place declare, vnto such as are capable of that, which I haue all ready sayd. Neither yet with examples specifie the Manifold varie••ies, by the foresayd two gene∣rall Rules, to be ordered. The witty and Studious, here, haue sufficient: And they which are not hable to atteine to this, without liuely teaching, and more in parti∣cular: would haue larger discoursing, then is mete in this place to be dealt w••thall: And other (perchaunce) with a proude snuffe will disdaine this litl•••• and would be vnthankefull for much more. I, therfore conclude: and wish such as haue modest and earnest Philosophicall mindes, to laude God highly for this: and to Meruayle, that the profoundest and subtilest point, concerning Mixture of Formes and Quali∣ties Naturall, is so Match•• and maryed with the most simple, easie, and short way of the noble Rule of Algiebar. Who can remaine, therfore vnpersuaded, to loue, a∣low, and honor the excellent Science of Arithmetike? For, here, you may perceiue that the litle finger of Arithmetike, is of more might and contriuing then a hun∣derd thousand mens wittes, of the middle sorte, are hable to perfourme, or truely to conclude, with out helpe thereof.

Now will we f••rder, by the wise and valiant Capitaine, be certified, what helpe he hath, by the Rules of Arithmetike: in one of the Artes to him appertaining:

And of the Grekes named 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.* 1.19 That is, the Skill of O••dring Souldiers in Battell ray after the best maner to all purposes.

This Art so much dependeth vppon Numbers vse, and the Mathematicals, that AElianus (the best writer therof,) in his worke, tö the Emperour Hadrianus, by his perfection, in the Mathematicals, (beyng greater, then other before him had,) thinketh his booke to passe all other the excellent workes, written of that Art, vnto his dayes. For, of it, had written AEneas: Gyneas of Thessaly: Pyrrhus Epirota: and Alexander his sonne: Clearchus: Pausanias: Euangelus: Polybi••s, familier frende to Scipio: Eupolemus: Iphicrates, Possidonius: and very many other worthy Capitaines, Philosophers and Princes of Immortall fame and me∣mory: Whose fayrest floure of their garland (in this feat) was Arithmetike: and a litle perceiuerance, in Geometricall Figures. But in many other cases doth Arith∣metike stand the Capitaine in great stede. As in proportionyng of vittayles, for the Army, either remaining at a stay: or suddenly to be encreased with a certaine number of Souldiers: and for a certain tyme. Or by good Art to diminish his com∣pany, to make the victuals, longer to serue the remanent, & for a certaine determi∣ned tyme: if nede so require. And so in sundry his other accountes, Recke∣ninges, Measurynges, and proportionynges, the wise, expert, and Circumspect ••a∣pitaine will affirme the Science of Arithmetike, to be one of his chief Counsaylors, directers and aiders. Which thing (by good meanes) was euident to the Noble, the Couragious,* 1.20 the loyall, and Curteous Iohn, late Earle of Warwicke. Who was a yong Gentleman, throughly knowne to very few. Albeit his lusty valiant∣nes, force, and Skill in Chiualrous feates and exerci••es: his humblenes, and frende∣lynes to all men, were thinges, openly, of the world perceiued. But what rotes (otherwise,) vertue had fastened in his brest, what Rules of godly and honorable

life he had framed to him selfe: what vices, (in some then liuing) notable, he tooke great care to eschew: what manly vertues, in other noble men, (florishing before his eyes,) he Sythingly aspired after: what prowesses he purposed and ment to a∣chieue: with what feats and Artes, he began to furnish and fraught him selfe, for the better seruice of his Kyng and Countrey, both in peace & warre. These (I say) his Heroicall Meditations, forecas••inges and determinations, no twayne, (I thinke) beside my selfe, can so perfectly, and truely report. And therfore, in Con∣science, I count it my part, for the honor, preferment, & procuring of vertue (thus, briefly) to haue put his Name, in the Register of Fa•••• Immortall.

To our purpose. This Iohn, by one of his actes (besides many other: both in En∣gland and Fraunce, by me, in him noted.) did dislose his harty loue to vertuous Sciences: and his noble intent, to excell in Martiall prowesse: When he, with hum∣ble request, and instant Solliciting: got the best Rules (either in time past by Greke or Romaine, or in our time vsed: and new Stratagemes therin de••ised) for ordring of all Companies, summes and Number•• of mē, (Many or few) with one kinde of weapon, or mo, appointed: with Artillery, or without: on horsebacke, or on fote: to giue, or take onset: to seem many, being few•• to s••em few, being many. To marche in battaile or Iornay: with many such feate••, to Foughten f••••ld, Skarmoush, or Ambushe appartaining: And of all these, liuely desi••nementes (most curiously) to be in velame parchement described: with Notes & peculier markes, as the Arte requireth: and all these Rules•• and descriptions Arithmeticall,* 1.21 inclosed in a riche Case of Gold, he vsed to weare about his necke•• as his Iuell most precious, and Counsaylour most trusty. Thus, Arithmetik••, of him was shryned in gold: Of Numbers frute, he had good hope. Now, Numbers therfore innumerable, in Numbers prayse, his sh••••ne shall finde.

What nede I, (for farder ••rofe to you) of the Scholemasters of Iustice, to require testimony: how nedefull, how frutefull, how skillfull a thing Arithmetike is? I meane•• the Lawyers of all sortes. Vndoubtedly, the Ciuilians, can meruaylous∣ly declare: how, neither the Auncient Romaine lawes, without good knowledge of Numbers art, can be perceiued: Nor (Iustice in infinite Cases) without due pro∣portion, (narrowly considered,) is hable to be executed•• How Iustly, & with great knowledge of Arte, did Pap••••••••anus institute a law of partition, and allowance, be∣twene man and wife a•••••• a ••••••orce: But how Acc••rsius, Bald••s, Bartolus, Iason, Alex∣••nder, and finally Alciat•••• (bei••g o••••••rwise, notably well learned) do iumble, gesse, and erre, from the ••quity•• ••rt ••nd I••••ent of the lawmaker: Arithmetike can detect, and conuince: and clerely, make the truth to shine. Good Bartolus, tyred in the examining & proportioning of the matter: ••nd with Accursius Glosse, much cum∣bred: burst out, and sayd: Nulla est i•• 〈◊〉〈◊〉 libr••, 〈◊〉〈◊〉 glossa difficili••r: Cui••s c••mputatio∣nem nec Scholas••ici ••ec Doct••res intellig••••t. &c. That is•• In the whole booke, there is no ••losse harder then thi•••• Whose accoump•• or reckenyng, neither the Scho∣lers, nor the Doctours vnderstand. &c. What can they say of I••lianus law, Si ita S••ript••m. &c. Of the Testators will iustly performing, betwene the wife, Sonne and daughter? How can they perceiue the ••••••••••tie of Aphricanus, Arithmeticall Reckening, where he treateth of Lex Falcid••a•• How ••••n they deliuer him, from his Reprouers: and their maintainers: as I••••••••es, Accursius Hypolitus and Alciatus? How ••ustly and artificially, was African••s reckening made? Proportionating to the Sommes bequeathed, the Contributions of eche part? Namely, for the hundred presently receiued, 17 1/7. And for the hundred, receiued after ten monethes, 12 6/7: which make the 30: which were to be cōtributed by the legatari••s to the heire.

For, what proportion, 100 hath to 75: the same hath 17 1/7 to 12 6/7: Which is Ses∣quitertia: that is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law, require an expert Arithmeticien, to vnderstand the deepe Iudgemēt, & Iust de∣terminatiō of the Auncient Romaine Lawmakers. But much more expert ought he to be, who should be hable, to decide with aequitie, the infinite varietie of Cases, which do, or may happen, vnder euery one of those lawes and ordinances Ciuile. Hereby, easely, ye may now coniecture: that in the Canon law: and in the lawes of the Realme (which with vs, beare the chief Authoritie), Iustice and e∣quity might be greately preferred, and skilfully executed, through due skill of A∣rithmetike, and proportions appertainyng. The worthy Philosophers, and pru∣dent lawmakers (who haue written many bookes De Republica: How the best state of Common wealthes might be procured and mainteined,) haue very well deter∣mined of Iustice: (which, not onely, is the Base and foundacion of Common weales:

but also the totall perfection of all our workes, words, and thoughtes:) de∣fining it, to be that vertue,* 1.22 by which, to euery one, is rendred, that to him appertai∣neth. God challengeth this at our handes, to be honored as God:

to beloued, as a father: to be feared as a Lord & master. Our neighbours proportiō, is also pres∣cribed of the Almighty lawmaker: which is, to do to other, euen as we would be done vnto. These proportions, are in Iustice necessary•• in duety, commendable: and of Common wealthes, the life, strength, stay and florishing. Aristotle in his Ethikes (to fatch the sede of Iustice, and light of direction, to vse and execute the sam••) was fayne to fly to the perfection, and power of Numbers: for proportions Arithmeticall and Geometricall. Plato in his booke called Epinomis (which boke, is the Threasury of all his doctrine) where, his purpose is, to seke a Science, which, when a man had it, perfectly: he might seme, and so be, in dede, Wise. He, br••efly, of other Sciences discoursing, findeth them, not hable to bring it to passe: But of the Science of Numbers, he sayth. Illa, qua numerum mortalium generi d••••n, id profecto ef∣ficiet•• Deum antem aliquem, magis quam fortunam, ••d sa••••tem nostram, hoc m••nus nobis arbitror contulisse. &c. Nam ipsum ••onorum omnium Authorem, cur non maximi boni, Prudentiae dico, causam arbitramur? That Science, verely, which hath taught man∣kynde number, shall be able to bryng it to passe. And, I thinke, a certaine God, rather then fortune, to haue giuen vs this gift, for our blisse. For, why should we not Iudge him, who is the Author of all good things, to be also the cause of th•• greatest good thyng, namely, Wisedome? There, at length, he proueth Wisedome to be atteyned, by good Skill of N••mbers. With which great Testimony, and the manifold profes, and reasons, before expressed•• you may be sufficiently and fully persuaded: of the perfect Science of Arithmetike, to make this accounte: That of all Sciences,* 1.23 next to Theologie, it is most diuine, most pure, most ample and generall, most profounde, most subtile, most commodious and most necessary. Whose next Sister, is the Absolute Science of Magnitudes: of which (by the Direction and aide of him, whose Magnitude is Infinite, and of vs Incomprehensible) I now en∣tend, so to write, that both with the Multitude, and also with the Magnitude of Meruaylous and frutefull verities, you (my frendes and Countreymen) may be stird vp, and awaked, to behold what certaine Artes and Sciences, (to our vn∣speakable behofe) our heauenly father, hath for vs prepared, and reuealed, by sun∣dry Philosophers and Mathematiciens.

BOth, Number and Magnitude, haue a certaine Originall sede, (as it were) of an incredible property: and of man, neuer hable, Fully, to be declared. Of Number, an Vnit, and of Magnitude, a Poynte, doo seeme to be much like Origi∣nall

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pretence of iust content, and me••sure•• 〈◊〉〈◊〉 lan••es and groundes•• ••re•••• 〈◊〉〈◊〉, ••••••∣quietnes, murder, and warre did (full oft) ensue: ••ill•••• by Gods 〈◊〉〈◊〉 and 〈◊〉〈◊〉 in∣dustrie, The perfect Science of Lines, Plaines, and Solides (lik•• •• diuine 〈◊〉〈◊〉,) gaue vnto euery man, his owne. The people then, by this art pleasured, and great∣ly relieued, in their landes iust measuring: & other Philosophers•• writing Rules for land measuring•• betwene them both, thus, con••••rmed the name of G••••m••tria, that is, (according to the very etimologie of the word) Land measuring. Wherin, the p••o∣ple knew no farder, of Magnitudes vse, but in Plai••••••: and the Philosophers, of thē, had no feethearers, or Scholers•• ••a••der to disclose vnto, then of ••lat, plaine Geome∣trie. And though, these Philosophers, knew of farder vse, and best vnderstode the etymologye of the worde, yet this name G••••••etria, 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* 1.24 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, th••r∣fore, must nedes be had, for our Mathematicall Science of Magnitudes: which re∣gardeth neither clod, no•• turff•• neither hill, nor dale•• neither ••••••th nor heauen: but is absolute Megeth••logia: not creping on ground, and dissellng the eye, with pole perche, rod or lyne:* 1.25 but liftyng the hart aboue the heauens, by 〈◊〉〈◊〉 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 Meg••∣thica, 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 ••uidently be sene. Where (in La∣tin) thus it is: right well translated: 〈…〉〈…〉 quid Geometri•• 〈…〉〈…〉 loquuntur, qui in ipsa vers••••t••••. In English, 〈◊〉〈◊〉 o•• Verely (sayth Plat••) 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, Quod q••••ritur cognoscendi illius gratia, quod se••per est, 〈…〉〈…〉 & Interit. Geometria, eius quod est semper, Cognit•••• est. 〈…〉〈…〉 Veritatem, ••nimum: at{que} ita, ad Philosophandum 〈…〉〈…〉 qua, nunc, contra quam decet, ad inf••ri••r•• 〈…〉〈…〉 &c. 〈…〉〈…〉 praecipiendum est, vt qui praclarissimam hanc 〈…〉〈…〉 spernant. Nam & quoe pr••ter ipsius propositu••, 〈…〉〈…〉 sunt. &c. It must nedes be confessed (saith Plat••) That [〈◊〉〈◊〉] i•• learned, for the knowyng of that, which is 〈◊〉〈◊〉: 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 euer∣lastyng. It will lif•• 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 con••ert, toward heauenly thinges [〈…〉〈…〉] which now, otherwise then b••comme•••••• vs, we cast 〈…〉〈…〉 or inferior things &c. Chiefly, therfore, Comma••••dement must be giuen, that such as do ••nhabit this most honorable Citie, by no meanes, despise Geometrie. For euen those thinges [done by ••••] which, in manner, seame to be, beside the purpose of Geometrie: are of

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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 foūd (foreuer) in eche of our two Vniuersities, an excel∣lent 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 famou•• ••••••ersit••e 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 retur∣ning: Moreouer, of the former knowledge Geometricall, are growen the Skills of Geographie, Chorographie, Hydrographie, and S••••••rithmetrie.

Geographie teacheth wayes, by which, in sūdry formes, (as Spharike, Plaine or other), the Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, & such other things, vpō the outface of the earth∣ly Globe (either in the whole, or in some principall mēber and portion therof cō∣tayned) may be described and designed, in cōmensurations 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, ••arthquakes, 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 m••n•• trauaile••. To conclude, some, for one purpose: and some, for another, liketh, loueth, ge••••••th, and vse••••, Mappes, Chartes, & Geo∣graphicall Globes. Of whose vse, to speake su••••iciently, would require a booke peculi••r.

Chorographie 〈…〉〈…〉

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 ••les and chiefe

particular places of daungers, conteyned within the boundes, and Sea coastes de∣scribed: as, of Quicksandes, Bankes, Pittes, Rockes, Races, Countertides, Whorle∣pooles. &c.

This, dealeth with the Element of the water chiefly: as Geographie did principally take the Element of the Earthes description (with his apperte∣nances) to taske. And besides thys, Hydrographie, requireth a particular Register of certaine Landmarkes (where markes may be had) from the sea, well ha∣ble 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 or∣dinarily, 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 pro∣iection 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 Hy∣dographer, 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, cā 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 con∣teined. * 1.26 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. It differreth from the Feate Tacticall, De acic••us instruendis•• bycause,* 1.27 there, is necessary the wisedome and fore∣sight, 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 Etero∣mekes) Rhombe, Rhomboïd, Lunular, Ryng, Serpentine, and such other Geometricall figures:

Which, in warres, haue ben, and are to be vsed: for commodiousnes, ne∣cessity, 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 horse∣men, 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. Great pollicy may be vsed of the Capitaines,* 1.28 (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 ma∣ny:) contrary wise, 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 Geo∣metricall 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 Geome∣trie. 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 Per∣spectiue, Astronomie, Musike, Cosmographie, Astrologie, Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie, Me∣nadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Archi∣tecture, 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 Westmin∣ster 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 Geome∣tricall: At Perspectiue, we will begyn therfore. Perspectiue, is an Art Mathe∣maticall, which demonstrateth the maner, and properties, of all Ra∣diations 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 na∣turall 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 Ra∣diall em••nations. 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 thing••, in l••ght 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 roūd Globe or Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise iud∣geth

a plaine Square, to be roūd: 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, hol∣den, 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 shad∣dow. 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,* 1.29 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 neuerthe∣lesse 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 sa∣tisfying of the noble courrage, that longeth ardently for the wisedome of Causes Naturall: as to let him vnderstand, that, in London, he may wish his owne eyes, haue profe of that, which I haue sayd herein. A Gentleman, (which, for his good seruice,* 1.30 done to his Countrey, is famous and honorable: and for skill in the Ma∣thematicall Sciences, and Languages, is the Od man of this kind. &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 Phi∣losophie, which, without Perspectiue, can not be fully vnderstanded, nor perfectly at∣teined vnto. Nor, of Astronomie: which, without Perspectiue, can not well be groun∣ded: 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 v∣ses, both merueilous, and proffitable: that, both, it would hold me to long, to no•••• 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* 1.31•• 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 St••rtes: 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 di∣stance 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 1/•• 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 1/•• of the Mone. Wherfore the Sonne is bigger then the Mone, 7000 times, lesse, by 59 ••••/164 that is, precisely 6940 ••5/•••• 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 1/•• The nerest, that the Mone com∣meth to the earth, is Semidiameters 52 ¼ The distance of the Starry Skye is, frō vs, in Semidiameters of the earth 20081 1/•• 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 1/•• Twenty thousand nine and twenty and a quarter.* 1.32 So thicke is the heauenly Palace, that the Pla∣netes haue all their exercise in, and most meruailously perfourme the Commaūde∣ment 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 Semidia∣meter of the earth; coteineth of our common miles 3436 ••/•• 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 f••ute Astronomicall, as con∣cerning the bignesse, Dist••nces 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 glo∣ry 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 not so to ex∣pound 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 Ho∣ly 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 re∣flexion, reboundeth to our knowledge and perceiuerance, Beames, and Radiati∣ons•• representing the Image of his Infinite goodnes•• Omnipotēcy, 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 right wisenesse is risen aboue the Horizon, of our tempor••ll Hemispharie, and hath so abundantly streamed into our hartes, the direct beames of his goodnes, mercy, and grace: Whose heat All Creatures feele: Spirituall and Corpor••ll•• 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 Cre∣ated, 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 Al∣mighty Creator: and that, by all meanes, to vs possible. Nolite ignorare (saith Plat•• in Epinomis) Astronomiam, Sapientissimū quiddam esse. Be ye not ignorant, Astro∣nomie to be a thyng of excellent wisedome. Astronomie, was to vs, from the be∣ginning 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 Sea∣sons, 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. Leaui••g 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 per∣forme, 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 Consi∣deratiō of Sacred Prophesies, accomplished in due time, foretold: as for high My∣sticall 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 bru∣••ish Barbarousnes: without the wonderfull diligence and skill of this Arte: conti∣nually 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 Astron••∣micall ••taffe, in stede of a fest••e.

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 be∣twene the auncient Harmonistes, and Ca••onistes. 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 cares were made. But as Astronomie hath a more diuine Contemplation, and cō∣modity, then mortall eye can perceiue: So, is Musike to be considered, that the * 1.33 Minde may be pref••r••ed, 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, in the heauenly * 1.34 motions and distances, describe a meruallous Harmonie, of Pythagoras. Harpe [ 4] with ••ight stringes. Also, somwhat might be sayd of Mercurius * 1.35 two Harpes, eche of foure Stringes Elementall. And very straunge matter, might be alledged of the Harmo••••e, to our * 1.36 Spirituall part appropriate. As in Pt••lom••us third boke, in the fourth and sixth Chapters may appeare. * 1.37 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 (i•• a maner) 〈◊〉〈◊〉? In the * 1.38 T••ne of Mans voyce, and also * 1.39 the sound of Instrument, what might be sayd, of Harmonie: No common Musicien would lightly beleue. But of the sundry Mixture (as I may terme it) and concurse,* 1.40 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 di∣seases of the Minde, and body might be cured, or inferred. And we finde in Re∣corde, that Terpander, Arion, Ismenias, Orpheus, Amphion, Dauid, Pythagoras, Empedo∣cles, Asclepiades and Timotheus, by Harmonicall Consonācy, haue done, and brought to pas, thinges, more then meruailous, to here of. Of them then, making no far∣der 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 mea∣ning. 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 ho∣mologall 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 Correspō∣dent: So, as, the Heauenly Globe, may (in practise) be duely described vpon the Geographicall, and Hydrographicall Globe. And there, for vs to consider an AEquinoctiall Circle, an Ecliptike line, Colures, Poles, Sterr••s in their true Longitudes, Latitudes, Declinations, and Verticalitie: also Climes, and Parallels: and by an Ho∣rizon annexed, and reuolution of the earthly Globe (as the Heauen, is•• by the Pr••∣mouan••, caried about in 24•• aequall Houres) to learne the Risinges and Settinges of Sterres (of Virgill in his Georgikes: of Hes••od: of Hippocrates in his Medicinall Sphare, to Perdic•••• King of the Macedonians: of Diocles, to King Antigonus, and of other fa∣mous Philosophers prescribed) a thing necessary, for due manuring of the earth, for Nauigation, for the Alteration of mans body•• being, whole, Sicke, wounded, or bru∣sed. 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.* 1.41

Of Astrologie, here I make an Ar••e, 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 ma∣ny other great Artes and experiences: As with perfecte Perspectiue, Astronomie, Cosmographie, Naturall Philosophie of the 4. Elementes, the Arte of Graduation, and some good vnderstāding 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 mani∣folde 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, pleasu∣red, 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 Causa 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. Op••rtet igitur & horum principia sumamus, & causas omnium similiter. Principium igitur vy mouens, praecipuum{que} & omnium primum, Circulus ille est, in quo manifeste Solis latio, &c. And so forth. His Meteorologicall bookes, are full of argu∣mentes, 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 Genera∣tione & Corruptione, in the tenth Chapter. Quocirca & prima lati••, Or••us & Interi∣tus 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 Gene∣ration and Corruption, but the motion of the Zodiake: for, that, both, is con∣tinuall, 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 Philo∣sophers, Expert, Wise, and godly men, for that Conclusion: which, daily and houre∣ly, 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 Ma∣trix, of the thing produced: What shall be the heauenly Impression, the perfect and circumspecte Astrologien hath to Conclude. Not onely (by Apotelesmes) 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 but by Naturall and Mathematicall demonstration 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. Whereunto, what Sciences are requis••te (without exception) I partly haue here warned: And in my Brop••de•••••••• besides other matter there disclosed) I haue Mathematically furni∣shed vp the whole Method: To this our age, not so carefully handled by any, that

euer I saw, or heard of. I was, (for * 1.42 ••1. yeares ago) by certaine earnest disputati∣ons, of the Learned G••••ardus M••rc••t••••, and 〈◊〉〈◊〉 Goga••a, (and other,) therto so prouoked: and (by my constant and inuincible zeale to the veritie) in obseruations of Heauenly Influencies (to the Min••te 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 sel••ing for the Truth, will both finde & cōfesse, therin, to be the Veritie, of these my wordes: And also become a Reaso∣nable Reformer, of three Sortes of people: about these Influentiall Operations, greatly erring from the truth. Wherof, the one, is Light Beleuers, the other,* 1.43 Light Despisers, and the third Light Practisers. The first, & most cōmon Sort, thinke the Heauen and Sterres, to be answerable to any their doutes or de∣sires: [ 1] which is not so: and, in dede, they, to much, ouer reache. The Second sorte thinke no Influentiall vertue (frō the heauenly bodies) to beare any Sway in Ge∣neration [ 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 thē (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 wordly 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 Por••entine do••h: nor will allow these perfect, and incor∣ruptible 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 per∣chaunce they thinke, the Sea & Riuers (as the Thames) to be some quicke thing, and s•• to ebbe, a••d slow, run in and out, of them selues, at ••hei•• 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 Bay∣ard•• both in matter and maner, erreth: to the discredit of the Wary, and modest A∣strologien: and to the robbing of those most noble corporall Creatures, of their Na∣turall Vertue: being most mighty: most beneficiall to all elementall Generation, Corr••p••ion and the appa••••••nances•• and most Harmonious in thei•• Monarchie: For which thinges, being ••nowen, and modestly vsed: we might highly ••nd conti∣nually glorifie God, with the princely Prophet saying. The Heauens declare the Glorie of God: who made the Heauēs 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 ••••••••reth tal•••••• and night, to night declareth know∣ledge. Prayse him, all ye St••rr••s, and Light. Amen.

IN order, now foloweth, of Statike, somewhat to say, what we meane by ••hat name•• and what 〈…〉〈…〉 doth, on 〈◊〉〈◊〉 Art, depend. Statike, is an Arte Mathematicall, which demonstra••••th the causes of heauynes, and lightnes of all thynges: and of motions and properties, to hea∣uynes and lightnes belonging•• And for asmuch as by the Bilanx, or Ba∣lance (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 exa∣miner, 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 Ba∣lance: who hast peysed in thy hand, both Heauen and earth. We therfore war∣ned 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, fa∣mous, 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 commodi∣ties, 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 Con∣clusions, as foloweth.

1.

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

2.

If Solide Magnitudes, being of the same bignes, or quātitie, that any Liquor is, and hauyng also the same Waight: be let downe in∣to the same Liquor, they will settle downeward, so, that no parte of them, shall be aboue the Superficies of the Liquor: and yet neuer∣theles, 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 dow••e into the same Liquor •• is in Waight, aequall, to the waight of the whole Solid Magni••ude.

4.

Any Solide Magnitude, Lighter then a Liquor, forced downe

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, aequall to the Solid Magnitude.

6.

If any Solide Magnitude, Lighter then a Liquor,* 1.44 be let downe into the same Liquor, the waight of the same Magnitude, will be, to the Waight of the Liquor. (Which is aequall in quantitie to the whole Magnitude,) in that proportion, that the parte, of the Mag∣nitude 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 mē allowed: to much trusting to Authority: and false Suppositions. As, Of any two bodyes, the heauyer, to moue downward faster then the lighter. This error, is not first by me, Noted:* 1.45 but by one Iohn Baptist de Be∣nedictis. The chief of his propositions, is this: which seemeth a Paradox.

If there be two bodyes of one forme, and of one kynde, aequall in quantitie or vnaequall, they will moue by aequall space,* 1.46 in aequall tyme: So that both theyr mouynges be in ayre, or both in water: or in any one Middle.

Hereupon, in the feate of Gunnyng, certaine good discourses (otherwise) may receiue great amendement, and furderance.* 1.47 In the entended purpose, also, allowing somwhat to the imperfection of Nature:* 1.48 not aunswerable to the preci∣senes of demonstration. Moreouer, by the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth, the Fire, may be nerely, knowen, how light or hea∣uy they are (Naturally) in their ••••••gned partes: or in the whole. And then, to thinges Elementall, turning your practise: you may deale for the proportion of the Elementes, in the thinges Compounded. Then, to the proportions of the Hu∣mours in Man: their waightes: and the waight of his bones, and flesh. &c. Than, by waight, to haue consideration of the Force of man, any maner of way: in whole or in part. Then, may you, of Ships water drawing, diuersly, in the Sea and in fresh water, haue pleasant consideration: and of waying vp of any thing, sonken in Sea or in fresh water &c. And (to lift vp your head a loft:) by waight, you may, as precisely, as by any instrument els, measure the Diameters of Sonne and Mone. &c. Frende, I pray you, way these thinges, with the iust Balance of Reason. And you will finde Meruailes vpon Meruailes: And esteme one Drop of Truth (yea in Naturall Philosophie) more worth, then whole Libraries of Opinions, vndemon∣strated: or not aunswering to Natures Law, and your experience. Leauing these

thinges, thus: I will giue you two or three, light practises, to great purpose•• and so finish my Annotation Staticall. In Mathematicall matters, by the Mechaniciens ayde, we will behold, here, the Commodity of waight. Make a Cube, of any one Vniforme:* 1.49 and through like heauy stuffe: of the same Stuffe, make a Sphaere or Globe, precisely, of a Diameter aequall to the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne, Lead, Siluer. &c. (being, as I sayd, of like na∣ture, condition, and like waight throughout.) And you may, by Say Balance, haue prepared a great number of the smallest waightes: which, by those Balance can be discerned or tryed: and so, haue proceded to make you a perfect Pyle, com∣pany & Number of waightes: to the waight of six, eight, or twelue pound waight: most diligently tryed, all. And of euery one, the Content knowen, in your least waight, that is wayable. [They that can not haue these waightes of precisenes: may, by Sand, Vniforme, and well dusted, make them a number of waightes, some∣what nere precisenes: by halfing euer the Sand: they shall, at length, come to a least common waight. Therein, I leaue the farder matter, to their discretion, whom nede shall pinche.] The Venetians consideration of waight, may seme precise enough:* 1.50 by eight descentes progressionall, * 1.51 halfing, from a grayne. Your Cube, Sphaere, apt Balance, and conuenient waightes, being ready: fall to worke. ••. First, way your Cube. Note the Number of the waight. Way, after that, your Sphaere. Note likewise, the Nūber of the waight. If you now find the waight of your Cube, to be to the waight of the Sphaere, as 21. is to 11: Then you see, how the Mechani∣cien and Experimenter, without Geometrie and Demonstration, are (as nerely in effect) tought the proportion of the Cube to the Sphere: as I haue demonstrated it, in the end of the twelfth boke of Euclide. Often, try with the same Cube and Sphaere. Then, chaunge, your Sphaere and Cube, to an other matter: or to an other bignes: till you haue made a perfect vniuersall Experience of it. Possible it is, that you shall wynne to nerer termes, in the proportion.

When you haue found this one certaine Drop of Naturall veritie, procede on, to Inferre, and duely to make assay, of matter depending. As, bycause it is well de∣monstrated, that a Cylinder, whose heith, and Diameter of his base, is aequall to the Diameter of the Sphaere, is Sesquialter to the same Sphaere (that is, as 3. to 2:) To the number of the waight of the Sphaere, adde halfe so much, as it is: and so haue you the number of the waight of that Cylinder. Which is also Compre∣hended of our former Cube: So, that the base of that Cylinder, is a Circle descri∣bed in the Square, which is the base of our Cube. But the Cube and the Cy∣linder, being both of one heith, haue their Bases in the same proportion, in the which, they are, one to an other, in their Massines or Soliditie. But, before, we haue two numbers, expressing their Massines, Solidities, and Quantities, by waight: wherfore, we haue * 1.52 the proportion of the Square, to the Circle, inscribed in the same Square. And so are we fallen into the knowledge sensible, and Expe∣rimentall of Archimedes great Secret: of him, by great trauaile of minde, sought and found. Wherfore, to any Circle giuen, you can giue a Square aequall: * 1.53 as I haue taught, in my Annotation, vpon the first proposition of the twelfth boke, And likewise, to any Square giuen, you may giue a Circle aequall: * 1.54If you describe a Circle, which shall be in that proportion, to your Circle inscribed, as the Square is to the same Circle: This, you may do, by my Annotations, vpon the second pro∣position of the twelfth boke of Euclide, in my third Probleme there. Your dili∣gence may come to a proportion, of the Square to the Circle inscribed, nerer the truth, then is the proportion of 14. to 11. And consider, that you may begyn at the Circle and Square, and so come to conclude of the Sphaere, & the Cube, what

their proportion is: as now, you came from the Sph••ere to the Circle. For, of Sil∣uer, or Gold, or Latton Lamyns or plates (thorough one hole drawē, as the maner is) if you make a Square figure•• & way it: and then, describing theron, the Circle in¦scribed: & cut of, & file away, precisely (to the Circle) the ouerplus of the Square: you shall then, waying your Circle, see, whether the waight of the Square, be to your Circle, as 14. to 11. As I haue Noted, in the beginning of Euclides twelfth boke. &c. after this resort to my last proposition, vpon the last of the twelfth. And there, helpe your selfe, to the end. And, here, Note this,* 1.55 by the way. That we may Square the Circle, without hauing knowledge of the proportion, of the Cir∣cumference to the Diameter: as you haue here perceiued. And otherwayes also, I can demonstrate it. So that, many haue cumberd them selues superfluously, by trauailing in that point first, which was not of necessitie, first: and also very in∣tricate. And easily, you may, (and that diuersly) come to the knowledge of the Circumference: the Circles Quantitie, being first knowen. Which thing, I leaue to your consideration: making hast to despatch an other Magistrall Probleme: and to bring it, nerer to your knowledge, and readier dealing with, then the world (be∣fore this day,) had it for you, that I can tell of. And that is, A Mechanicall Dubblyng of the Cube: &c.* 1.56 Which may, thus, be done: Make of Copper plates, or Tyn plates, a foursquare vpright Pyramis, or a Cone: perfectly fashioned in the holow, within. Wherin, let great diligence be vsed, to ap∣proche (as nere as may be) to the Mathematicall perfection of those figures. At their bases, let them be all open: euery where, els, most close, and iust to. From the vertex, to the Circumference of the base of the Cone: & to the sides of the base of the Pyramis: Let 4. straight lines be drawen, in the inside of the Cone and Pyramis:* 1.57 makyng at their fall, on the perimeters of the bases, equall angles on both sides them selues, with the sayd perimeters. These 4. lines (in the Pyra∣mis: and as many, in the Cone) diuide: one, in 12. aequall partes: and an other, in 24. an other, in 60, and an other, in 100. (reckenyng vp from the vertex.) Or vse other numbers of diuision,* 1.58 as experience shall reach you•• Then,* 1.59 set your Cone or Pyramis, with the vertex downward, perpendicularly, in respect of the Base. (Though it be otherwayes, it hindreth nothyng.) So let thē most stedily be stayed. Now, if there be a Cube, which you wold haue Dubbled. Make you a prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or Cubi•• coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so proportiō in respect of your Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the waight of them, in wa••••, 3. or 4. times: at the least: what stuff so euer they be made of•• Let not your Solid angle, at the vertex, be to sharpe: but that the water may come with ease, to the very vertex, of your hollow Cone or Pyramis. Put one of your Solid Cubes in a Balance apt:* 1.60 take the waight therof ex∣actly in water. Powre that water, (without losse) into the hollow Pyramis or Cone, quietly. Marke in your lines, what numbers the water Cutteth: Take the waight of the same Cube againe•• in the same kinde of water, which you had be∣fore: put that* 1.61 also, into the Pyramis or Cone, where you did put the first. Marke now againe, in what number or place of the lines, the water Cutteth them. Two

wayes you may conclude your purpose: it is to wete, either by numbers or lines. By numbers: as, if you diuide the side of your Fundamentall Cube into so many aequall partes, as it is capable of, conueniently, with your ease, and pre∣cisenes of the diuision. For, as the number of your first and lesse line (in your hollow Pyramis or Cone,) is to the second or greater (both being counted from the vertex) so shall the number of the side of your Fundamentall Cube, be to the nūber belonging to the Radicall side, of the Cube, dubble to your Fun∣damentall Cube: Which being multiplied Cubik wise, will sone shew it selfe, whe∣ther it be dubble or no, to the Cubik number of your Fundamentall Cube. By lines, thus: As your lesse and first line, (in your hollow Pyramis or Cone,) is to the second or greater, so let the Radical side of your Fundamētall Cube, be to a fourth proportionall line, by the 12. proposition, of the sixth boke of Euclide. Which fourth line, shall be the Rote Cubik, or Radicall side of the Cube, dubble to your Fundamentall Cube: which is the thing we desired. For this, may I (with ioy) say, EYPHKA, EYPHKA, EYPHKA: thanking the holy and glorious Trinity: hauing greater cause therto, then * 1.62 Archimedes had (for finding the fraude vsed in the Kinges Crowne, of Gold): as all men may easily Iudge: by the diuersitie of the frute following of the one, and the other. Where I spake before, of a hollow Cu∣bik Coffen: the like vse, is of it: and without waight. Thus. Fill it with water, preci∣sely full, and poure that water into your Pyramis or Cone. And here note the lines cutting in your Pyramis or Cone. Againe, fill your coffen, like as you did before. Put that Water, also, to the first•• Marke the second cutting of your lines. Now, as you proceded before, so must you here procede. * 1.63 And if the Cube, which you should Double, be neuer so great: you haue, thus, the proportion (in small) be∣twene your two litle Cubes: And then, the side, of that great Cube (to be doubled) being the third, will haue the fourth, found, to it proportionall: by the 12. of the sixth of Eu••lide.

* 1.64Note, that all this while, I forget not my first Proposition Staticall, here rehear∣sed: that, the Supersicies of the water, is Sphaericall. Wherein, vse your discretion: to the first line, adding a small heare breadth, more: and to the second, halfe a heare breadth more, to his length. For, you will easily perceaue, that the difference can be no greater, in any Pyramis or Cone, of you to be handled. Which you shall thus trye. For ••inding the swelling of the water aboue leuell.

Square the Semidiame∣ter,* 1.65 from the Centre of the earth, to your first Waters Superficies. Square then, halfe the Subtendent of that watry Superficies (which Subtendent must haue the equall partes of his measure, all one, with those of the Semidiameter of the earth to your watry Superficies): Subtracte this square, from the first: Of the residue, take the Rote Square. That Ro••e, Subtracte from your first Semidiameter of the earth to your watry Superficies: that, which remaineth, is the heith of the water, in the middle, aboue the leuell.

Which, you will finde, to be a thing insensible. And though it were greatly sensible,* 1.66 yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides twelfth booke, noted: you may reduce all, to a true Leuell. But, farther diligence, of you is to be vsed, against accidentall causes of the waters swelling: as by hauing (somwhat) with a moy••t Sponge, before, made moyst your hollow Pyramis or Cone, will preuent an accidentall cause of Swelling, &c. Experience will teach you abundantly: with great ease, pleasure, and cōmoditie.

* 1.67Thus, may you Double the Cube Mechanically, Treble it, and so forth, in any proportion. Now will I Abridge your paine, cost, and Care herein. Without all preparing of your Fundamentall Cubes: you may (alike) worke this Conclusion. For, that, was rather a kinde of Experimentall demōstration, then the shortest way:

and all, vpon one Mathematicall Demonstration depending.

Take water (as much as conueniently will serue your turne: as I warned before of your Funda∣mentall Cubes bignes) Way it precisely. Put that water, into your Pyramis or Cone. Of the same kinde of water, then take againe, the same waight you had before: put that likewise into the Pyramis or Cone. For, in eche time, your mar∣king of the lines, how the Water doth cut them, shall geue you the proportion be∣twen the Radicall sides, of any two Cubes, wherof the one is Double to the other: working as before I haue taught you:* 1.68

sauing that for you Fundamentall Cube his Radicall side: here, you may take a right line, at pleasure.

Yet farther proceding with our droppe of Naturall truth: you may (now) geue Cubes, one to the other, in any proportiō geuē:* 1.69 Rationall or Ir∣rationall: on this maner. Make a hollow Parallelipipedon of Copper or Tinne: with one Base wāting, or open: as in our Cubike Coffen. Frō the bottome of that Parallelipipedon, raise vp, many perpendiculars, in euery of his fower sides. Now if any proportion be assigned you, in right lines:

Cut one of your perpendiculars (or a line equall to it, or lesse then it) likewise: by the 10. of the sixth of Euclide. And those two partes, set in two sundry lines of those perpendiculars (or you may set them both, in one line) making their beginninges, to be, at the base: and so their lengthes to extend vpward. Now, set your hollow Parallelipipedon, vpright, perpendicularly, steadie. Poure in water, handsomly, to the heith of your shorter line. Poure that water, into the hollow Pyramis or Cone. Marke the place of the rising. Settle your hollow Parallelipipedon againe. Poure water into it: vnto the heith of the second line, exactly. Poure that water* 1.70 duely into the hollow Pyramis or Cone: Marke now againe, where the water cutteth the same line which you marked before. For, there, as the first marked line, is to the se∣cond: So shall the two Radicall sides be, one to the other, of any two Cubes: which, in their Soliditie, shall haue the same proportion, which was at the first as∣signed: were it Rationall or Irrationall.

Thus, in sundry waies you may furnishe your selfe with such straunge and pro∣fitable matter: which, long hath bene wished for. And though it be Naturally done and Mechanically: yet hath it a good Demonstration Mathematicall.* 1.71 Which is this•• Alwaies, you haue two Like Pyramids: or two Like Cones, in the proporti∣ons assigned: and like Pyramids or Cones, are in proportion, one to the other, in the proportion of their Homologall sides (or lines) tripled. Wherefore, if to the first, and second lines, found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in continuall proportion: that fourth line, shall be to the first, as the greater Pyramis or Cone, is to the lesse: by the 33•• of the eleuenth of Euclide. If Pyramis to Pyramis,* 1.72 or Cone to Cone, be double, then shall * 1.73 Line to Line, be also double, &c. But, as our first line, is to the second, so is the Radicall side of our Fundamentall Cube, to the Radicall side of the Cube to be made, or to be dou∣bled: and therefore, to those twaine also, a third and a fourth line, in continuall proportion, ioyned: will geue the fourth line in that proportion to the first, as our fourth Pyramidall, or Conike line, was to his first: but that was double, or tre∣ble, &c. as the Pyramids or Cones were, one to an other (as we haue proued) ther∣fore, this fourth, shalbe also double or treble to the first, as the Pyramids or Cones were one to an other: But our made Cube, is described of the second in proporti∣on, of the fower proportionall lines: therfore* 1.74 as the fourth line, is to the first, so is that Cube, to the first Cube: and we haue proued the fourth line, to be to the first, as the Pyramis or Cone, is to the Pyramis o•• Cone: Wherefore the Cube is

to the Cube,* 1.75 as Pyramis is to Pyramis, or Cone is to Cone. But we * 1.76 Suppose Py∣ramis to Pyramis, or Cone to Cone, to be double or treble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was to be demonstrated. And of the Paralle∣lipipedō, it is euidēt, that the water Solide Parallelipipedons, are one to the other, as their heithes are, seing they haue one base. Wherfore the Pyramids or Cones, made of those water Parallelipipedons, are one to the other, as the lines are (one to the other) betwene which, our proportion was assigned. But the Cubes made of lines, after the proportiō of the Pyramidal or Conik homologall lines, are one to the other, as the Pyramides or Cones are, one to the other (as we before did proue) therfore, the Cubes made, shalbe one to the other, as the lines assigned, are one to the other: Which was to be demonstrated. Note.* 1.77This, my Demonstratiō 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) no∣thing tedious to them, who, to these thinges, are well a••fected. 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. * 1.78 Now may you, Betwene two lines giuen, finde two middle proportionals, in Continuall proportion: by the hollow Paralleli∣pipedon, and the hollow Pyramis, or Cone. Now, any Parallelipipedon rectangle being giuen: thre right lines may be found, proportionall in any propor∣tion assigned, of which, shal be produced a Parallelipipedon, aequall to the Paralle∣lipipedon 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 propor∣tionall lines, betwene two lines giuen, may easely be performed. Now, that Pro∣bleme, 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 proportiō assigned) 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 Manne∣ken vsed: and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make one, of the same Mould (in any assigned proportion) bigger or lesser. Now, may you, of any* 1.79 Gunne, or little peece of ordinaūce, make an other, with the same Symm••tri•• (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 communi∣cated to such, as faithfully deale with vertuous studies. Thus, can the Mathematicall minde,* 1.80 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 God, 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 com∣meth

next. For, wheras, it ••s so ample and wonderful, that, an whole yeare long, one might finde fruitfull matter therin, to speake of: and also in pr••ctise, is a Threa∣sure 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 Cōfirmed, with this new name: yet the matter, hath from the beginning, ben in consideration of all perfect Philosophers. Anthropographie, is the description of the Num∣ber, Measure, Waight, figure, Situation, and colour of euery diuerse thing, conteyned in the perfect body of MAN: with certain know∣ledge of the Symmetrie, figure, waight, Characterization, and due locall motion, of any parcell of the sayd body, a••signed•• and of Nū∣bers, to the sayd parcell appertainyng. This, is the one part of the Desini∣tion, 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 •• ••eculier Art, called Astronomie: If the de∣scription of the earthly Globe, hath h•••• ••••culier arte, called Geographie. If the Mat∣ching of both, hath his peculier Arte, called Cosmographie: Which is the Descriptiō of the whole, and vniuersall frame of the world: Why should not the description of him, who is the Lesse world: and, frō the beginning, called Microcosmus (that is. The Lesse World.* 1.81) And for whose sake, and seruice, all bodily creatures els, were created: Who, also, participateth with Spirites, and Angels: and is made to the I∣mage and similitude of God: 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, God, Nature, Reason and Experience shall informe you. The Anatomistes will restore to you, some part: The Physiognomistes, some: The Chy∣romantistes 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 Con∣tributaries. 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 Microcosmicall constitution.* 1.82 The outward Image, and vew hereof: to the Art of Zographie and Painting, to Sculpture, and Architecture: (for Church, House, Fort, or Ship) is most necessary and profitable: for that, it is the chiefe base and foundation of them. Looke in * 1.83 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. Consi∣der the Arke of Noe. And by that, wade farther. Remember the Delphicall Oracle NOSCE TEIPSVM (Knowe thy selfe) so long agoe pronounced: 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 igno∣rance, 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 vnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly rise vp: and there was no such order taken,* 1.84 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 God, 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 assigned. 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 Irre∣gular lines, at pleasure, may be drawen. These, and such like, are principall Con∣clusions of this Arte: and helpe forward many pleasant and profitable Mechani∣call workes: As Milles,* 1.85 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 ••ere 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 won∣derfull 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 iniu∣rie and hurt: And one Ianellus of Cremona did mend the same, and presented it vn∣to the Emperour Charles the fifth. Hieronymus Cardanus, 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 Mathema∣ticall, 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 di∣uerse Instrumentes and Engines, is most necessary. For, in many thinges, the Skrue worketh the feate, which, els, could not be performed. By helpe hereof, it is * 1.86 recorded, that, where all the power of the Citie of Syracusa, was not hable to moue a certaine Ship (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. Wherat, the King wondring:* 1.87 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. From this day, forward (said the King) Credit ought to be giuen to Archimedes, what so••uer he sayth.

Pneumatithmie demonstrateth by close hollow Geometri∣call Figures, (regular and irregular) the straunge properties (in mo∣tion or stay) of the Water, Ayre, Smoke, and Fire, in theyr cōtinuitie,

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 ordi∣nary law, the Elementes will moue or stand. As, Water to ascend: rather then be∣twene him and Ayre, Spac•• or place should be left, more then (naturally) that quā∣titie 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 Cōtinuitie 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 space, not sufficient and aunswerable to their bodily substance. Great force and violence will they vse, to enioy their naturall right and libertie.* 1.88 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 de∣uises. As Hydraulica, Organes goyng by water. &c. Of this Feat, (called common∣ly 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 o•• 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 Pneumatithmie. By this Art, all Granes, 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 Duche waghen Racke: therwith, to set vp agayne, a mighty waghen laden, being ouerthrowne. The force of the Crossebow Racke, is certain∣ly, here, demonstrated. The reason, why one mā, doth with a leauer, lift that, which Sixe men, with their handes onely, could no••, so easily do. By this Arte, in out 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 discomfi∣ted the whole•• Army and Hoste of the Romaines, besieging Syracusa, Marcus Mar∣cellas the Consul,* 1.89 being their Generall Capitaine. Such huge Stones, so many, with such force, and so farre, did he with his ••ngynes hayle among them, out of the Citie. And by Sea likewise: though their Ships might come to the walls of Syra∣cusa, yet hee vtterly confounded the Romaine N••uye. What with his mighty Stones hurlyng: what with Pikes of* 1.90 18 fote long, made like sliaftes: which he for∣ced 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 a∣gaine: what with his* 1.91 Burning Glasses•• by which he fired their other Shippes a far-of: what, with his other pollicies, deuises, and engines, he so manfully acquit him selfe: that all the Force, courage, and pollicie of the Romaines (for a great season)

could nothing preuaile, for the winning of Syracusa. Wherupon, the Romanes named Archimedes, Briareus, and Centimanus. Zonaras maketh mention of one Pro∣clus, who so well had perceiued Archimedes Arte of Menadrie•• and had so well in∣uented of his owne, that with his Burning Glasses,* 1.92 being placed vpon the walles of Bysance, he multiplied so the heate of the Sunne, and directed the beames of the same against his enemies Nauie with such force, and so sodeinly (like lighte∣ning) that he burned and destroyed both man and ship. And Dion specifieth of Priscus, a Geometricien in Bysance, who inuented and vsed sondry Engins, of Force multiplied: Which was cause, that the Emperour Senerus pardoned him, his life, af∣ter he had wonne Bysance: Bycause he honored the Arte, wytt, and rare industrie of Priscus. But nothing inferior to the inuention of these engines of Force, was the inuention of Gunnes.* 1.93 Which, from an English man, had the occasion and order of first inuenting: though in an other land, and by other men, it was first executed. And they that should see the record, where the occasion and order generall, of

Gunning, is first discoursed of, would thinke: that, small thinges, flight, and cōmon: comming to wise mens consideration, and industrious mens handling, may grow to be of force incredible.

Hypogeiodie, is an Arte Mathematicall, demonstratyng, how, vnder the Sphaericall Superficies of the earth, at any depth, to any perpendicular line assigned (whose distance from the perpendicular of the entrance: and the Azimuth, likewise, in respect of the said en∣trance, is knowen) certaine way may be praescribed and gone: And how, any way aboue the Superficies of the earth designed, may vn∣der earth, at any depth limited, be kept: goyng alwayes, perpendi∣cularly, vnder the way, on earth designed: And, contrarywise, Any way, (straight or croked,) vnder the earth, beyng giuen: vppon the vtface, or Superficies of the earth, to Lyne out the same: So, as, from the Centre of the earth, perpendiculars drawen to the Sphaericall Superficies of the earth, shall precisely fall in the Correspondent pointes of those two wayes. This, with all other Cases and cir∣cumstances herein, and appertenances, this Arte demonstrateth. This Arte, is very ample in varietie of Conclusions•• and very profitable sundry wayes to the Common Wealth. The occasion of my Inuenting this Arte, was at the request of two Gentlemen, who had a certaine worke (of gaine) vnder ground: and their groundes did ioyne ouer the worke: and by reason of the crokednes, diuers depthes, and heithes of the way vnder ground, they were in doubt, and at controuersie, vnder whose ground, as then, the worke was: The name onely (be∣fore this) was of me published, De Itin••r•• Subterranco: The rest, be at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole, and for secrete passage•• vnder ground, betwene place and place (as this land hath diuerse) and for other purposes, any man may easily perceaue, both the great fruite of this Arte, and also in this Arte, the great aide of Geometrie.

Hydragogie, demonstrateth the possible leading of Water, by Natures lawe, and by artificiall helpe, from any head (being a Spring, standing, or running Water) to any other place assigned.

Long, hath this Arte bene in vse: and much thereof written: and very marueilous workes therein, performed•• as may ye•• appeare, in Italy: by the Ruynes remaining of the Aqueductes. In other places, of Riuers leading through the Maine land, Nauigable many a Mile. And in other places, of the marueilous forcinges of Wa∣ter to Ascend which all, declare the great skill, to be required of him, who should in this Arte be perfecte, for all occasions of waters possible leading. To speake of the allowance of the Fall, for euery hundred foote: or of the Ventills (if the wa∣ters labour be farre, and great) I neede not: Seing, at hand (about vs) many expert men can sufficiently testi••ie, in effecte, the order: though the Demonstration of the Necessiti•• thereof, they know not: Not yet, if they should be led, vp and downe, and about Mountaines, from the head of the Spring: and then, a place be∣ing assigned: and of them, to be demaunded, how low or high, that last place is, in respecte of the head, from which (so crokedly, and vp and downe) they be come: Perhaps, they would not, or could not, very redily, or nerely assoyle that question. Geometrie therefore, is necessary to Hydragogie. Of the sundry wayes to force wa∣ter to ascend, eyther by Tympane, Kettell mills, Skrue, Ctesibike, or such like: in Vi∣truuius, Agricola, (and other,) fully, the maner may appeare. And so, thereby, also be most ••uident, how the Artes, of Pneumatithmie, Helicosophie, Statik••, Trochilike, and Menadrie, come to the furniture of this, in Speculation, and to the Commo∣ditie of the Common Wealth, in practise.

Horometrie, is an Arte Mathematicall, which demōstrateth, how, at all times appointed, the precise vsuall denominatiō of time, may be knowen, for any place assigned. These wordes, are smoth and plaine eas••e Englishe, but the reach of their meaning, is farther, then you woulde lightly imagine. Some part of this Arte, was called in olde time, Gnomonice: and of late, Ho••ologiographia:) and in Englishe, may be termed, Dialling. Auncient is the vse, and more auncient, is the Inuention. The vse, doth well appeare to haue bene (at the least) aboue two thousand and three hundred yeare agoe: in * 1.94 King Acha•• Diall, then, by the Sunne, shewing the distinction of time. By Sunne, Mone, and Sterres, this Dialling may be performed, and the precise Time of day or night knowen. But the demonstratiue delineation of these Dialls, of all sortes, requireth good skill, both of Astronomie, and Geometrie Elementall, Sphaericall, Phae∣nomenall, and Conikall. Then, to vse the groundes of the Arte, for any regular Superficies, in any place offred: and (in any possible apt position therof) th••ron, to describe (all maner of wayes) how, vsuall howers, may be (by the Sunnes sha∣dow) truely determined: will be found no sleight Painters worke. So to Paint, and prescribe the Sunnes Motion, to the breadth of a heare. In this Feate (in my youth) I Inuented a way, How in any Horizontall, Murall, or AEquino∣ctiall Diall, &c. At all howers (the Sunne shining) the Signe and De∣gree ascendent, may be knowen. Which is a thing very necessary for the Rising of those fixed Sterres: whose Operation in the Ayre, is of great might, euidently. I speake no further, of the vse hereof. But forasmuch as, Mans affaires require knowledge of Times & Momentes, when, neither Sunne, Mone, or Sterre, can be sene: Therefore, by Industrie Mechanicall, was inuented, first, how, by Wa∣ter, running orderly, the Time and howers might be knowen: whereof, the famous Ctesibius, was Inuentor: a man, of Vitruuius, to the Skie (iustly) extolled. Then, after that, by Sand running, were howers measured: Then, by Trochilike with waight: And of late time, by Trochilike with Spring: without waight. All these,

by Sunne or Sterres direction (in certaine time) require ouersight and reformati∣on, according to the heauenly AEquinoctiall Motion: besides the inaequalitie of their owne Operation. There remayneth (without parabolicall meaning herein) among the Philosophers,* 1.95 a more excellent, more commodious, and more maruei∣lous way, then all these: of hauing the motion of the Primouant (or first ••quino∣ctiall motion,) by Nature and Arte•• Imitated: which you shall (by furder search in waightier studyes) hereafter, vnderstand more of. And so, it is tyme to finish this Annotation, of Tymes distinction, vsed in our common, and priuate affaires: The commoditie wherof, no man would want, that can tell, how to bestow his tyme.

Zographie, is an Arte Mathematicall, which teacheth and de∣monstrateth, how, the Intersection of all visuall Pyramides, made by any playne assigned, (the Centre, distance, and lightes, beyng de∣termined) may be, by lynes, and due propre colours, represented. A notable Arte, is this and would require a whole Volume, to declare the proper∣ty thereof: and the Commodities ensuyng. Great skill of Geometrie, Arithme∣tike, Perspectiue, and Anthropographie, with many other particular Art••s, hath the Zo∣grapher, nede of, for his perfection. For, the most excellent Painter, (who is but the propre Mechanicien, & Imitator sensible, of the Zographer) hath atteined to such perfection, that Sense of Man and beast, haue iudged thinges painted, to be things naturall, and not artificiall: aliue, and not dead. This Mechanicall Zographer (com∣monly called the Painter) is meruailous in his skill: and seemeth to haue a certaine diuine power: As, of frendes absent, to make a frendly, present comfort: yea, and of frendes dead, to giue a continuall, silent presence: not onely with vs, but with our posteritie, for many Ages. And so procedyng, Consider, How, in Winter, he can shew you, the liuely vew of Sommers Ioy, and riches: and in Sommer, exhibite the countenance of Winters dolefull State, and nakednes. Cities, Townes, Fortes, Woodes, Armyes, yea whole Kingdomes (be they neuer so farre, or greate) can he, with ease, bring with him, home (to any mans Iudgement) as Paternes liuely, of the thinges rehearsed. In one little house, can he, enclose (with great pleasure of the beholders,) the portrayture liuely, of all visible Creatures, either on earth, or in the earth, liuing: or in the waters lying, Creping, slyding, or swimming: or of any ••oule, or fly, in the ayre flying. Nay, in respect of the Starres, the Skie, the Cloudes: yea, in the shew of the very light it selfe (that Diuine Creature) can he match our eyes Iudgement, most nerely. What a thing is this? thinges not yet being, he can represent so, as, at their being, the Picture shall seame (in maner) to haue Created them. To what Artificer, is not Picture, a great pleasure and Commoditie•• Which of them all, will refuse the Direction and ayde of Picture? The Architect, the Gold∣smith, and the Arras Weauer: of Picture, make great account. Our liuely Herbals, our portraitures of birdes, beastes, and fishes: and our curious Anatomies, which way, are they most perfectly made, or with most pleasure, of vs beholden? Is it not, by Picture onely? And if Picture, by the Industry of the Painter, be thus commo∣dious and meruailous: what shall be thought of Zographie, the Scholemaster of Pi∣cture, and chief gouernor? Though I mencion not Sculpture, in my Table of Artes Mathematicall: yet may all men perceiue, How, that Picture and Sculpture, are Si∣sters germaine: and both, right profitable, in a Commō wealth. and of Sculpture, as∣well as of Picture, excellent Artificers haue written great bokes in commendation. Witnesse I take, of Georgio Vasari, Pittore Aretino: of Pomponius Gauricus•• and other. To these two Artes, (with other,) is a certaine od Arte, called Althalmasat, much beholdyng: more, then the common Sculptor, Entayler, Keruer, Cut••er, Grauer, Foun∣der,

or Paynter (&c) know their Arte, to be commodious.

Architecture, to many may seme not worthy, or not mete,* 1.96 to be reckned among the Artes Mathematicall•• To whom, I thinke good, to giue some account of my so doyng. Not worthy, (will they say,) bycause it is but for building, of a house, Pallace, Church, Forte, or such like, grosse workes. And you, also, defined the Artes Mathematicall, to be such, as dealed with no Materiall or corruptible thing: and al∣••o did demonstrat••uely procede in their faculty, by Number or Magnitude. First, you see, that I count, here, Architecture, among those Artes Mathematicall,* 1.97 which are Deriued from the Principals: and you know, that such, may deale with Na∣turall thinges and sensib•••• ••a••••er. Of which, some draw nerer, to the Simple and absolute Mathematicall Speculation, then other do.

And though, the Architect* 1.98 procureth, enformeth, & directeth, the Mechanicien, to handworke, & the building actuall, of house, Castell, or Pallace, and is chief Iudge of the same: yet, with him selfe (as chief Master and Architect,) remaineth the Demonstratiue reason and cause, of the Mechaniciens worke in Lyne, plaine, and Solid: by Geometricall, A∣rithmeticall, Opticall, Musi••all, Astronomicall, Cosmographicall (& to be brief) by all the former Deriued Artes Mathematicall, and other Naturall Artes, hable to be confir∣med and stablished.

If this be so••then, may you thinke, that Architecture, hath good and due allowance, in this honest Company of Artes Mathematicall Deriuatiue, I will, herein, craue Iudgement of two most perfect Architect••s: the one, being Vi∣truuius, the Romaine: who did write ten bookes thereof, to the Emperour Augu∣stus (in whose daies our Heauenly Archemaster, was borne): and the other, Leo Baptista Albertus, a Florentine: who also published ten bookes therof. Archi∣tectura (sayth Vitruuius) est Scientia pluribus disciplinis & varijs eruditionibus ornata•• cuius Iudicio probantur omni••, qua ab cateris Artificibus perficiuntur opera. That is. Architecture, is a Science garnished with many doctrines & diuerse instructions: by whose Iudgement, all workes, by other workmen finished, are Iudged. It followeth. Ea nascitur ex Fabric••. & Ratiocinatione. &c. Ratiocinatio aute•• est, quae, res fabricatas, Solertia ac ratione proportionis, demonstrare at{que} explicare potest. Architecture, groweth of Framing, and Reasoning. &c. Rea∣soning, is that, which of thinges framed, with forecast, and proportion: can make demonstration, and manifest declaration. Againe. Cùm, in omnibus enim re∣bus, tùm maxime etiam in Architectura, haec duo insunt: qoud significatur, & qoud signi∣fica••. Significatur proposita res, de qua dicitur: hanc autem Significat Demonstratio, rati∣onibus doctrinarum explicata. Forasmuch as, in all thinges: therefore chiefly in Architecture, these two thinges are: the thing signified: and that which sig∣nifieth. The thing propounded, whereof we speake, is the thing Signified. But Demonstration, expressed with the reasons of diuerse doctrines, doth signi∣fie the same thing. After that•• Vt literatus sit, peritus Graphidos, eruditus Geometriae, & Optices non ignarus: instructus Arithmetica: historias complures nouerit, Philosophos diligenter audiuerit: Musicam sciuerit: Medicinae non sit ignarus, responsa Iurisperitorū nouerit: Astrologiam, Caeli{que} rationes cognitas habeat. An Architect (sayth he) ought to vnderstand Languages, to be skilfull of Painting, well instructed in Geome∣tri••, not ignorant of Perspectiue, furnished with Arithmetike, haue knowledge of many histories, and diligently haue heard Philosophers, haue skill of Mu∣sike, not ignorant of Physike, know the aunsweres of Lawyers, and haue Astro∣nomie,

and the courses Caelestiall, in good knowledge. He geueth reason, or∣derly, wherefore all these Artes, Doctrines, and Instructions, are requisite in an ex∣cellent Architect. And (for breuitie) omitting the Latin text, thus he hath. Secondly, it is behofefull for an Architect to haue the knowledge of Painting•• that he may the more easilie fashion out, in patternes painted, the forme of what worke he liketh. And Geometrie, geueth to Architecture many helpes: and first teacheth the Vse of the Rule, and the Cumpasse: wherby (chiefly and easilie) the descriptions of Buildinges, are despatched in Groundplats: and the directions of Squires, Leuells, and Lines. Likewise, by Perspectiue, the Lightes of the hea∣uen, are well led, in the buildinges: from certaine quarters of the world. By Arithmetike, the charges of Buildinges are summed together: the measures are expressed, and the hard questions of Symmetries, are by Geometricall Meanes and Methods discoursed on. &c. Besides this, of the Nature of thinges (which in Greke is called 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉) Philosophie doth make declaration. Which, it is necessary, for an Architect, with diligence to haue learned: because it hath ma∣ny and diuers naturall questions: as specially, in Aqueductes. For in their courses, leadinges about, in the leuell ground, and in the mountinges, the natu∣rall Spirites or breathes are ingendred diuers wayes: The hindrances, which they cause, no man can helpe, but he, which out of Philosophie, hath learned the originall causes of thinges. Likewise, who soeuer shall read C••esibius, or Ar∣chimedes bookes, (and of others, who haue written such Rules) can not thinke, as they do: vnlesse he shall haue receaued of Philosophers, instructions in these thinges. And Musike he must nedes know: that he may haue vnderstanding, both of Regular and Mathematicall Musike: that he may temper well his Ba∣listes, Catapultes, and Scorpions. &c. Moreouer, the Brasen Vessels, which in Theatres, are placed by Mathematicall order, in ambries, vnder the steppes: and the diuersities of the soundes (which y^e Grecians call 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉) are ordred according to Musicall Symphonies & Harmonies: being distributed in y^e Circuites, by Di∣atessaron, Diapente, and Diapason. That the conuenient voyce, of the players sound, whē it came to these preparations, made in order, there being increased: with y^t increasing, might come more cleare & pleasant, to y^e eares of the lokers on. &c. And of Astronomie, is knowē y^e East, West, South, and North. The fashion of the heauen, the AEquinox, the Solsticie, and the course of the sterres. Which thinges, vnleast one know: he can not perceiue, any thyng at all, the reason of Ho∣rologies. Seyng therfore this ample Science, is garnished, beautified and stored, with so many and sundry skils and knowledges: I thinke, that none can iustly ac∣count them selues Architectes, of the suddeyne. But they onely, who from their childes yeares, ascendyng by these degrees of knowledges, beyng fostered vp with the atteynyng of many Languages and Artes, haue wonne to the high Taber∣nacle of Archicture. &c. And to whom Nature hath giuen such quicke Circum∣spection, sharpnes of witt, and Memorie, that they may be very absolutely skill∣full in Geometrie, Astronomie, Musike, and the rest of the Artes Mathemati∣call:

Such surmount and p••sse the callyng, and state, of Architectes: and are be∣come Mathematiciens.* 1.99 &c. And they are found, seldome. As, in tymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes: Apollonius Pergeus: Eratosthenes Cyreneus: Archimedes, and Scopas, Syracusians. Who also, left to theyr posteritie, many Engines and Gnomonicall workes: by numbers and natu∣rall meanes, inuented and declared.

Thus much, and the same wordes (in sense) in one onely Chapter of this Incō∣parable Architect Vitrunius,* 1.100 shall you finde. And if you should, but take his boke in your hand, and slightly loke thorough it, you would say straight way: This is Geo∣metrie, Arithmetike, Astronomie, Musike, Anthropographie, Hydragogie, Horometrie &c. and (to cōclude) the Storehouse of all workmāship. Now, let vs listen to our other Iudge, our Florentine, Leo Baptista: and narrowly consider, how he doth determine of Architecture. Sed ante{que} vltra progrediar. &c. But before I procede any further (sayth he) I thinke, that I ought to expresse, what man I would haue to bee al∣lowed an Architect. For, I will not bryng in place a Carpenter: as though you might Compare him to the Chief Master•• of other Artes. For the hand of the Carpenter, is the Architectes Instrument.* 1.101

But I will appoint the Architect to be that man, who hath the skill, (by a certaine and meruailous meanes and way,) both in minde and Imagination to determine: and also in worke to finish: what workes so euer, by motion of waight, and cuppling and framyng together of bo∣dyes, may most aptly be Commodious for the worthiest. Vses of Man. And that he may be able to performe these thinges, he hath nede of atteynyng and knowledge of the best, and most worthy thynges. &c.

The whole Feate of Architecture in buildyng, confisteth in Lineamentes, and in Framyng. And the whole power and skill of Lineamentes, tendeth to this: that the right and absolute way may he had, of Coaptyng and ioyning Lines and angles: by which, the face of the buil∣dyng or frame, may be comprehended and concluded. And it is the property of Lineamentes, to prescribe vnto buildynges, and euery part of them, an apt place, & certaine nūber: a worthy maner, and a semely order: that, so, y^e whole ••orme and figure of the buildyng, may rest in the very Lineamentes. &c. And we may prescribe in mynde and imagination the whole formes,* 1.102 all materiall stuffe be∣yng secluded. Which point we shall atteyne, by Notyng and forepointyng the an∣gles, and lines, by a ••ure and certaine direction and connexion. Seyng then, these thinges, are thus: Lineamente,* 1.103 shalbe the ••ertaine and constant prescribyng, conceiued in mynde: made in lines and angles: and finished with a learned minde and wyt.

We thanke you Master Baptist, that you haue so aptly brought your Arte, and phrase therof, to haue some Mathematicall perfection:* 1.104 by certaine or∣der, nūber, forme, figure, and Symmetri•• mentall: all naturall & sensible stuffe set a∣part.

Now, then, it is euident, (Gentle reader) how aptely and worthely, I haue prefer••ed Architecture, to be bred and fostered vp in the Dominion of the per••les Princesse, Mathematica: and to be a naturall Subiect of hers. And the name of Architectur••, is of the principalitie, which this Science hath, aboue all other Artes. And Plato affirmeth, the Architect to be Master ouer all, that make any worke. Wherupon, he is neither Smith, not Builder: nor, separately, any Artificer: but the

Hed, the Prouost, the Directer, and Iudge of all Artificiall workes, and all Artifi∣cers. For, the true Architect, is hable to teach, Demonstrate, distribute, des••ribe, and Iudge all workes wrought. And he, onely, searcheth out the causes and reasons of all Artificiall thynges. Thus excellent, is Architecture: though few (in our dayes) at∣teyne: ••ereto: yet may not the Arte, be otherwise thought on, then in very dede it is worthy. Nor we may not, of auncient Artes, make new and imperfect Definiti∣ons in our dayes: for scarsitie of Artificers: No more, than we may pynche in, the Definitions of Wisedome, or Honestie, or of Frendeshyp or of Iustice. No more will I consent, to Diminish any whit, of the perfection and dignitie, (by iust cause) al∣lowed to absolute Architecture. Vnder the Direction of this Arte, are thre prin∣cipall, necessary Mechanicall Artes. Namely, Howsing, Fortification, and Naupegie. Howsing, I vnderstand, both for Diuine Seruice, and Mans common vsage: publike, and priuate. Of Fortification and Naupegie, straunge matter might be told you: But perchaunce, some will be tyred, with this Bederoll, all ready rehearsed: and other some, w•••••• nycely nip my grosse and homely discoursing with you: made in post hast: for feare you should wante this true and frendly warnyng, and tast giuyng, of the Power Mathematicall. Lyfe is short, and vncertaine: Tymes are perilouse: &c. And still the Printer awayting, for my pen staying: All these thinges, with farder matter of Ingratefulnes, giue me occasion to passe away, to the other Artes remainyng, with all spede possible.

THe Arte of Nauigation, demonstrateth how, by the shortest good way, by the aptest Directiō, & in the shortest time, a sufficient Ship, betwene any two places (in passage Nauigable,) assigned: may be cōducted: and in all stormes, & naturall disturbances chauncyng, how, to vse the best possible meanes, whereby to recouer the place first assigned. What nede, the Master Pilote, hath of other Artes, here before recited, it is easie to know: as, of Hydrographie, Astronomie, Astrologie, and Horome∣trie. Presupposing continually, the common Base, and foundacion of all: namely Arithmetike and Geometrie. So that, he be hable to vnderstand, and Iudge his own necessary Instrumentes, and furniture Necessary: Whether they be perfectly made or no: and also can, (if nede be) make them, hym selfe. As Quadrantes, The Astro∣nomers Ryng, The Astronomers staffe, The Astrolabe vniuersall. An Hydrogra∣phicall Globe. Charts Hydrographicall, true, (not with parallell Meridians). The Common Sea Compas: The Compas of variacion: The Proportionall, and Para∣doxall Compasses (of me Inuented,* 1.105 for our two Moscouy Master Pilotes, at the re∣quest of the Company) Clockes with spryng: houre, halfe houre, and three houre Sandglasses: & sundry other Instrumētes: And also, be hable, on Globe, or Playne to describe the Paradoxall Compasse: and duely to vse the same, to all maner of purposes, whereto it was inuented. And also, be hable to Calculate the Planetes places for all tymes.

Moreouer, with Sonne Mone or Sterre (or without) be hable to define the Lon∣gitude & Latitude of the place, which he is in: So that, the Longitude & Latitude of the place, from which he sayled, be giuen: or by him, be knowne. whereto, apper∣tayneth expert meanes, to be certified euer, of the Ships way. &c. And by forese∣ing the Rising, Settyng, Nonestedyng, or Midnightyng of certaine tempestuous fixed Sterres: or their Coniunctions, and Anglynges with the Planetes, &c. he ought to haue expert coniecture of Stormes, Tempestes, and Spoutes: and such lyke Meteorologicall effectes, daungerous on Sea. For (as Plato sayth,) Mu••ationes••

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Some by waight. wherof Ti•••••••••• speaketh. Some, by Stringes strayned, or Springs, therwith Imitating liuely Motions. Some, by other meanes, as the Images of Mer∣curie: and the brasen hed, made by Albertus Magn••••, which dyd seme to speake. B••••∣thius was excellent in these feates. To whom, Cassiod•••••••• writyng, sayth. Your pur∣pose is to know profound thynges: and to shew meruayles. By the disposition of your Arte, Metals do low: Diomedes of brasse, doth blow a Trumpet loude: a brasen Serpent hisseth: byrdes made, sing swetely. Small thynges we rehearse of you, who can Imitate the heauen. &c. Of the straunge Selfmouyng, which, at Saint Denys, by Paris, * 1.106 I saw, ones or twise (Orontius beyng then with me, in Company) it were to straunge to tell. But some haue written it. And yet, (I hope) it is there, of other to be sene. And by Perspectiue also straunge thinges, are done. As partly (before) I gaue you to vnderstand in Perspectiue. As, to see in the Ayre, a loft, the lyuely Image of an other man, either walkyng to and fro: or standyng still. Likewise, to come into an house, and there to see the liuely shew of Gold, Siluer or precious stones: and commyng to take them in your hand, to finde nought but Ayre. Hereby, haue some men (in all other matters counted wise) fouly ouershot thē selues: misdeaming of the meanes.* 1.107 Therfore sayd Claudius Caelestinus. Hodie mag∣na liter••••••rae vi•• ••s 〈◊〉〈◊〉 magna 〈…〉〈…〉, opera qu••da•••• quasi miranda, supra Natura 〈◊〉〈◊〉 de qu••••••s in 〈…〉〈…〉 sa••iliter reddidisse••. That is. Now a dayes, 〈◊〉〈◊〉 see 〈…〉〈…〉, y••a of great learnyng and reputation, to Iudge certain workes 〈◊〉〈◊〉 ••eruaylous, aboue the power of Nature: Of which workes, one that were skillfull in Perspectiue might easely haue giuen the Cause. Of Archimedes Sphaere, Cicero witnesseth.* 1.108 Which is very straunge to thinke on. For when Archi∣medes (sayth he) did fasten in a Sphere, the mo••ynges of the Sonne, Mone, and of the fi••e other Planets, he did, as the God, which (in Timaeus of Plato) did make the world. That, on•• 〈◊〉〈◊〉, should rule motions most vnlike in slownes, and swiftnes. But a greater cause of meruayling we haue by Claudianus report hereof. Who affirme in this Archimedes w••rke, to haue ••en of Glasse. And discourseth of it more at large: which I omit. The Doue of wood, which the Mathematicien Ar∣c••y••a•• did make to flye, is by Agellius spoken of. Of Dadalus straunge Images, Plat•• reporteth. H••mere of Vul••ans Selfmouers, (by secret wheles) leaueth in writyng. Ari∣stotle, in hys Politikes, of both, maketh mention. Mer••aylous was the workeman∣shyp, of la••e dayes, performed by good skill of Tr••chi••ike. &c. For in Noremberge, A slye of Iern, heyng l••t out of the Artifi••ers ••••nd, did (as it were) fly about by the g••stes, at the table, and at leng••h, as though •••• were weary, retourne to his masters ha•••• agayne. Moreouer, an Artificiall E••le, was ordred, to fly out of the same Towne, a mighty way, and that a lo••t in the Ayre, toward the Emperour comming thether: and followed hym, beyng come to the gate of the town••.* 1.109 Thus, you see, what•• Arte Mathematicall an p••rforme, when Skill, will, Industry, and Habili∣ty, are duely applyed to pro••e.

* 1.110ANd for these, and such like marueilous Actes and Feates, Naturally, Mathe∣matically, and Mechanically, wrought and contriued: ought any honest Student, and Modest Christian Philosopher, be counted, & called a Coniurer? Shall the folly of Idio••••s, and the Mallice of the Scornfull, so much preuaile, that He, who seeketh no worldly gaine or glory at their handes•• But onely, of God, the threasor of heauenly wisedome, & knowledge of p••re veritie: Shall he (I say) in the meane

space, be robbed and spoiled of his honest name and fame? He that seketh (by S. Paules aduertisement) in the Creatures Properties, and wonderfull vertues, to finde iuste cause, to glorifie the AEternall; and Almightie Creator by: Shall that man, be (in hugger mugger) condemned, as a Companion of the Helhoundes, and a Caller, and Coniurer of wicked and damned Spirites? He that bewaileth his great want of time, sufficient (to his contentation) for learning of Godly wisdome, and Godly Verities in: and onely therin setteth all his delight: Will that mā leese and abuse his time, in dealing with the Chiefe enemie of Christ our Redemer: the deadly foe of all mankinde: the subtile and impudent per••erter of Godly Veritie: the Hypocriticall Crocodile: the Enuious Basiliske, continually desirous, in the twinke of an eye, to destroy all Mankinde, both in Body and Soule, aeternally? Surely (for my part, somewhat to say herein) I haue not learned to make so brutish, and so wicked a Bargaine. Should I, for my xx. or xxv. yeares Studie: for two or three thousand Markes spending: seuen or eight thousand Miles going and trauai∣ling, onely for good learninges sake: And that, in all maner of wethers: in all ma∣ner of waies and passages: both early and late: in daunger of violence by man: in daunger of destruction by wilde beastes: in hunger: in thirst: in perilous heates by day, with toyle on foote: in daungerous dampes of colde, by night, almost be∣reuing life: (as God knoweth): with lodginges, oft times, to small ease: and som∣time to lesse securitie. And for much more (then all this) done & suffred, for Lear∣ning and attaining of Wisedome: Should I (I pray you) for all this, no otherwise, nor more warily: or (by Gods mercifulnes) no more luckily, haue fished, with so large, and costly, a Nette, so long time in drawing (and that with the helpe and ad∣uise of Lady Philosophie, & Queene Theologie): but at length, to haue catched, and drawen vp,* 1.111 a Frog? Nay, a Deuill? For, so, doth the Common peuish Pratler Imagine and Iangle: And, so, doth the Malicious skorner, secretly wishe, & brauely and boldly face down, behinde my backe. Ah, what a miserable thing, is this kinde of Men? How great is the blindnes & boldnes, of the Multitude, in thinges aboue their Capacitie? What a Land: what a People: what Maners: what Times are these? Are they become Deuils, themselues: and, by false witnesse bearing against their Neighbour, would they also, become Murderers? Doth God, so long geue them respite, to reclaime them selues in, from this horrible slaundering of the gilt∣lesse: contrary to their owne Consciences: and yet will they not cease? Doth the Innocent, forbeare the calling of them, Iuridically to aunswere him, according to the rigour of the Lawes: and will they despise his Charitable pacience? As they, against him, by name, do forge, fable, rage, and raise slaunder, by Worde & Print: Will they prouoke him, by worde and Print, likewise, to Note their Names to the World: with their particular deuises, fables, beastly Imaginations, and vnchristen∣like slaunders? Well: Well. O (you such) my vnkinde Countrey men. O vn∣naturall Countrey men. O vnthankfull Countrey men. O Brainsicke, Rashe, Spitefull, and Disdainfull Countrey men. Why oppresse you me, thus violently, with your slaundering of me: Contrary to Veritie: and contrary to your owne Consciences? And I, to this hower, neither by worde, deede, or thought, haue bene, any way, hurtfull, damageable, or iniurious to you, or yours? Haue I, so long, so dearly, so farre, so carefully, so painfully, so daungerously sought & trauailed for the learning of Wisedome, & atteyning of Vertue: And in the end (in your iudge∣mēt) am I become, worse, then when I begā? Worse, thē a Mad man? A dangerous Member in the Common Wealth: and no Member of the Church of Christ? Call you this, to be Learned? Call you this, to be a Philosopher? and a louer of Wise∣dome? To forsake the straight heauenly way: and to wallow in the broad way of

damnation? To forsake the light of heauenly Wisedome: and to lurke in the dun∣geon of the Prince of darkenesse? To forsake the Veritie of God, & his Creatures: and to fawne vpon the Impudent, Craftie, Obstinate Lier, and continuall disgracer of Gods Veritie, to the vttermost of his power? To forsake the Life & Blisse AEter∣nall: and to cleaue vnto the Author of Death euerlasting? that Murderous Ty∣rant, most gredily awaiting the Pray of Mans Soule? Well: I thanke God and our Lorde Iesus Christ, for the Comfort which I haue by the Examples of other men, before my time: To whom, neither in godlines of life, nor in perfection of learning, I am worthy to be compared: and yet, they sustained the very like Iniu∣ries, that I do: or rather, greater. Pacient Socrates, his Apologie will testifie: Apu∣leius his Apologies, will declare the Brutishnesse of the Multitude. Ioannes Picus, Earle of Mirandula, his Apologie will teach you, of the Raging slaunder of the Ma∣licious Ignorant against him. Ioannes Trithemius, his Apologie will specifie, how he had occasion to make publike Protestation: as well by reason of the Rude Sim∣ple: as also, in respect of such, as were counted to be of the wisest sort of men.

Ma∣ny could I recite: But I deferre the precise and determined handling of this mat∣ter: being loth to detect the Folly & Mallice of my Natiue Countrey men.* 1.112 Who, so hardly, can disgest or like any extraordinary course of Philosophicall Studies: not falling within the Cumpasse of their Capacitie: or where they are not made priuie of the true and secrete cause, of such wonderfull Philosophicall Feates.

These men, are of fower sortes, chie••ly. The first, I may name, Vaine pratling bu∣sie bodies: The second, Fond Frendes: The third, Imperfectly zelous: and the fourth, Malicious Ignorant. To eche of these (briefly, and in charitie) I will say a word [ 1] or two, and so returne to my Praeface. Vaine pratling busie bodies, vse your idle assemblies, and conferences, otherwise, then in talke of matter, either aboue your Capacities, for hardnesse: or contrary to your Consciences, in Veritie. Fonde [ 2] Frendes, leaue of, so to commend your vnacquainted frend, vpon blinde affection: As, because he knoweth more, then the common Student: that, therfore, he must needes be skilfull, and a doer, in such matter and maner, as you terme Coniuring. Weening, thereby, you aduaunce his fame: and that you make other men, great marueilers of your hap, to haue such a learned frend. Cease to ascribe Impietie, where you pretend Amitie. For, if your tounges were true, then were that your frend, Vntrue, both to God, and his Soueraigne. Such Frendes and Fondlinges, I shake of, and renounce you: Shake you of, your Folly. Imperfectly zelous, to you, [ 3] do I say: that (perhaps) well, do you Meane: But farre you misse the Marke: If a Lambe you will kill, to feede the flocke with his bloud. Sheepe, with Lambes bloud, haue no naturall sustenaunce: No more, is Christes flocke, with horrible slaunders, duely aedified. Nor your faire pretense, by such rashe ragged Rheto∣rike, any whit, well graced. But such, as so vse me, will finde a fowle Cracke in their Credite. Speake that you know: And know, as you ought: Know not, by Heare say, when life lieth in daunger. Search to the quicke, & let Charitie be your guide. [ 4] Malicious Ignorant, what shall I say to thee? Prohibe linguam tuam a malo. A de∣tractione parcite linguae. Cause thy toung to refraine frō euill. Refraine your toung from slaunder. Though your tounges be sharpned, Serpent like, & Adders poy∣son lye in your lippes:* 1.113 yet take heede, and thinke, betimes, with your selfe, Vir lin∣guosus non stabilietur in terra. Virum violentum venabitur malum, donec praecipitetur. For, sure I am, Quia faciet Dominus Iudicium afflicti: & vindictam pauperum.

Thus, I require you, my assured frendes, and Countrey men (you Mathemati∣ciens, Mechaniciens, and Philosophers, Charitable and discrete) to deale in my

behalf, with the light & vntrue tounged, my enuious Aduersaries, or Fond frends. And farther, I would wishe, that at leysor, you would consider, how Basilius Mag∣nus, layeth Moses and Daniel, before the eyes of those, which count all such Stu∣dies Philosophicall (as mine hath bene) to be vngodly, or vnprofitable. Waye well S. Stephen his witnesse of Moses.* 1.114 Eruditus est Moses omni Sapientia AEgyptiorū: & erat potens in verbis & operibus suis. Moses was instructed in all maner of wise∣dome of the AEgyptians: and he was of power both in his wordes, and workes. You see this Philosophicall Power & Wisedome, which Moses had, to be nothing misliked of the Holy Ghost. Yet Plinius hath recorded, Moses to be a wicked Magi∣cien. And that (of force) must be, either for this Philosophicall wisedome, learned, before his calling to the leading of the Children of Israel: or for those his won∣ders, wrought before King Pharao, after he had the conducting of the Israelites. As concerning the first, you perceaue, how S. Stephen, at his Martyrdome (being full of the Holy Ghost) in his Recapitulation of the olde Testament, hath made men∣tion of Moses Philosophie: with good liking of it: And Basilius Magnus also, auou∣cheth it, to haue bene to Moses profitable (and therefore, I say, to the Church of God, necessary). But as cōcerning Moses wonders, done before King Pharao: God, him selfe, sayd: Vide vt omnia ostenta, quae posui in manutua, facias coram Pharaone. See that thou do all those wonders before Pharao, which I haue put in thy hand. Thus, you euidently perceaue, how rashly, Plinius hath slaundered Moses,* 1.115 of vayne fra••dulent Magike, saying: Est & alia Magices Factio, a Mose, Iamne, & Iotape, Iu∣daeis pendens: sed multis millibus annorum post Zoroastrem. &c.

Let all such, there∣fore, [ 1] who, in Iudgement and Skill of Philosophie, are farre Inferior to Plinie, take good heede, least they ouershoote them selues rashly,* 1.116 in Iudging of Philosophers straunge Act••s•• and the Meanes, how they are done.

But, much more, ought they [ 2] to beware of forging, deuising, and imagining monstrous feates, and wonderfull workes, when and where, no such were done: no, not any sparke or likelihode, of such, as they, without all shame, do report. And (to conclude) most of all, let [ 3] them be ashamed of Man, and afraide of the dreadfull and Iuste Iudge: both Fo∣lishly or Maliciously to deuise: and then, deuilishly to father their new fond Mon∣sters on me: Innocent, in hand and hart: for trespacing either against the lawe of God, or Man, in any my Studies or Exercises, Philosophicall, or Mathematicall: As in due time, I hope, will be more manifest.

NOw end I, with Archemastrie. Which name, is not so new, as this Arte is rare. For an other Arte, vnder this, a degree (for skill and power) hath bene indued with this English name before. And yet, this, may serue for our purpose, sufficiently, at this present. This Arte, teacheth to bryng to actuall ex∣perience sensible, all worthy conclusions by all the Artes Mathema∣ticall purposed, & by true Naturall Philosophie concluded: & both addeth to them a farder scope, in the termes of the same Artes, & al∣so by hys propre Method, and in peculier termes, procedeth, with helpe of the foresayd Artes, to the performance of complet Expe∣riēces, which of no particular Art, are hable (Formally) to be challen∣ged. If you remember, how we considered Architecture, in respect of all com∣mon handworkes: some light may you haue, therby, to vnderstand the Souerain∣ty and propertie of this Science. Science I may call it, rather, then an Arte: for the excellency and Mastershyp it hath, ouer so many, and so mighty Artes and

Sciences. And bycause it procedeth by Experiences, and searcheth forth the causes of Conclusions, by Experiences: and also putteth the Conclusions them selues, in Experience, it is named of some, Scientia Experimentalis. The Experimentall Sci∣ence. Nicolaus Cusanus termeth it so, in hys Experimentes Statikall, And an other Philosopher,* 1.117 of this land Natiue (the floure of whose worthy fame, can neuer dye nor wither) did write therof largely, at the request of Clement the sixt. The Arte carrieth with it, a wonderfull Credit: By reason, it certefieth, sensibly, fully, and completely to the vtmost power of Nature, and Arte. This Arte, certifieth by Ex∣perience complete and absolute: and other Artes, with their Argumentes, and De∣monstrations, persuade: and in wordes, proue very well their Conclusions. * 1.118 But wordes, and Argumentes, are no sensible certifying: nor the full and finall frute of Sciences practisable. And though some Artes, haue in them, Experiences, yet they are not complete, and brought to the vttermost, they may be stretched vnto, and applyed sensibly. As for example: the Naturall Philosopher disputeth and maketh goodly shew of reason: And the Astronomer, and the Optical Mechanicien, put some thynges in Experience: but neither, all, that they may: nor yet sufficiently, and to the vtmost, those, which they do, There, then, the Archemaster steppeth in, and leadeth forth on, the Experiences, by order of his doctrine Experimentall, to the chief and finall power of Naturall and Mathematicall Artes. Of two or three men, in whom, this Description of Archemastry was Experimentally, verified, I haue read and hard: and good record, is of their such perfection. So that, this Art, is no fan∣tasticall Imagination: as some Sophister, might, Cum suis Insolubilibus, make a slo∣rish: and dassell your Imagination: and dash your honest desire and Courage, from beleuing these thinges, so vnheard of, so meruaylous, & of such Importance. Well: as you will. I haue forewarned you. I haue done the part of a frende: I haue dischar∣ged my Duety toward God: for my small Talent, at hys most mercyfull handes re∣ceiued. To this Science, doth the Science Alnirangiat, great Seruice. Muse nothyng of this name. I chaunge not the name, so vsed, and in Print published by other: beyng a name, propre to the Science. Vnder this, commeth Ars Sintrillia, by Artephius, briefly written. But the chief Science, of the Archemaster, (in this world) as yet knowen, is an other (as it were) OPTICAL Science: wherof, the name shall be told (God willyng) when I shall haue some, (more iust) occasion, therof, to Discourse.

Here, I must end, thus abruptly (Gentle frende, and vnfayned louer of honest and necessary verities.) For, they, who haue (for your sake, and vertues cause) re∣quested me, (an old forworne Mathematicien) to take pen in hand: (through the confidence they reposed in my long experience: and tryed sincerity) for the decla∣ryng and reportyng somewhat, of the frute and commodity, by the Artes Ma∣thematicall, to be atteyned vnto: euen they, Sore agaynst their willes, are forced, for sundry causes, to satiffie the workemans request, in endyng forthwith: He, so feareth this, so new an attempt, & so costly: And in matter so slenderly (he∣therto) among the common Sorte of Studentes, considered or estemed.

And where I was willed, somewhat to alledge, why, in our vulgare Speche, this part of the Principall Science of Geometrie, called Euclides Geometricall Elementes, is published, to your handlyng: being vnlatined people, and not Vniuersitie Scholers: Verily, I thinke it nedelesse.

[ 1] For, the Honour, and Estimation of the Vniuersities, and Graduates, is, hereby, nothing diminished. Seing, from, and by their Nurse Children, you receaue all this Benefite: how great soeuer it be.

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And su••ely, the Common and Vulgar Scholer (much more, the Grama••••an) [ 3] before his comming to the Vniuersitie, shall (or may) be, now (according to Plato his Counsell) su••••iciently ins••ructed in Arithmetike, and Geometrie, for the better and ea••ier learn••ng of all maner of ••hilosophie, Academicall, or Perip••••e••icall. And by that meanes, goe more cherefu••••y, more skil••ully, and spedily fo••warde, in his Studies, there to be learned. And, so, in lesse time, profite more, then (otherw••se) he should•• or could do.

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And great Comfort, with good hope, may the Vniuersities haue, by reason of [ 5] this Englishe Geometrie and Mathematicall Praeface, that they (hereafter) shall be the more regarded, esteemed, and resorted vnto. For, when it shall be knowen and reported, that of the Mathematicall Sciences onely, such great Commo∣dities are enfu••ng (as Thaue specified): and that in dede, some of you vnlatined Studentes, can be good witnesse, of such rare fruite by you enioyed (thereby): as either, before this, was not heard of 〈…〉〈…〉 fully credited:

Well, may all men coniecture, that farre greater 〈…〉〈…〉, to winne to the Perfection of all Philosophie,* 1.119 may in 〈…〉〈…〉 the Storehouses & Threa∣sory of all Sciences, and 〈…〉〈…〉,* 1.120 and most noble State of Common Wealthes.

Besides this, how ma〈…〉〈…〉 here, in these Realmes of [ 6] England and Ireland, tha•• 〈…〉〈…〉 & Cumpasse: Who, with their owne Skill and expe〈…〉〈…〉ble (by these good helpes and informations) to find 〈…〉〈…〉s, straunge Engines, and In∣strumentes: for sundry purp〈…〉〈…〉 Wealth? or for priuate plea∣sure? and for the better maintayni•••• 〈…〉〈…〉 owne estate? I will not (therefore)

fight against myne owne shadowe. For, no man (I am s••re) will open his mouth against this Enterprise. No mā (I say) who either hath Charitie toward his brother (and would be glad of his further••nce in vertuous knowledge): o•• that hath any care & zeale for the bettering of the Cōmon state of this Realme. Neither any, that make accompt, what the wiser sort of men (Sage and Stayed) do thinke of them. To ••one (therefore) will I make any Ap••logie, for a vertuous acte doing: and for cōmending, or setting forth Profitable Artes to English men, in the English toung.

But, vnto God our Creator, let vs all be thankefull: for that, As he, of his Good∣nes, by his Powre, and in his wisedome, hath Created all thynges, in Number, Waight,* 1.121 and Measure: So, to vs, of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne the sufficient and necessary knowledge of the foresayd hy•• three principall Instrumentes: Which Meanes, I haue abundantly proued vnto you, to be the Sciences and Artes Mathematicall.

And though I haue ben pinched with straightnes of tyme: that no way, I could so pen downe the m••tter (in my Mynde), as I determined•• hopyng of conuenient laysure: Yet, if vertuous zeale, and honest I••tent prouoke and bryng you to the readyng and examinyng of this Compendious treatise, I do not doute, but, as the veritie therof (accordyng to our purpose) will be euident vnto you: So the pith and force therof, will persuade you: and the wonderfull frute therof, highly plea∣sure you. And that you may the easier perceiue, and better remember, the prin∣cipall pointes, whereof my Preface treateth, I will giue you the Ground platt of my whole discourse,* 1.122 in a Table annexed from the first to the last, somewhat Me∣thodically contriued.

If Hast, hath caused my poore pen, any where, to stumble: You will, (I am sure) in part of recompence•• (for my earnest and sincere good will to plea∣sure you). Conside•• the rockish huge mountaines, and the perilous vnbeaten wayes which (both night and day, for the while) it hath ••oyled and labored through, to bryng you this good Newes and Comfortable profe of Vertues frute.

So, I Commit you vnto Gods Mercyfull direction, for the rest: hartely besechyng hym, to prosper your Studyes, and honest Intentes: to his Glory & the Commodity of our Countrey. Amen.

Written at my poore H••••se At Mortlake. Anno. 1570•• February ••.

Notes

• * 1.1

  The intent of this Preface.
• * 1.2

  Number.
• * 1.3

  Note the worde, Vnit, to expresse the Greke Mona••, & not Vnitie: as we hau•• all, commonly, till now, vsed.
• * 1.4

  Magnitude.
• * 1.5

  A point.
• * 1.6

  A Line.
• * 1.7

  Magnitude.
• * 1.8

  Ano. 1488.
• * 1.9

  ☞
• * 1.10

  Arithmetike.
• * 1.11

  Note.
• * 1.12

  Anno. 1550.
• * 1.13

  R. B.
• * 1.14

  Take some part of Lullus counsayle in his booke de Q. Essentia.
• * 1.15

  No••e.
• * 1.16

  Note.
• * 1.17

  The Se∣cond Rule.
• * 1.18

  Note.
• * 1.19

  〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.
• * 1.20

  ☞
• * 1.21

  This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age•• hauing no is∣sue by his wife: Daugh∣ter to the Duke of So∣merset.
• * 1.22

  Iustice.
• * 1.23

  ☞
• * 1.24

  Plato. 7. de Rep.
• * 1.25

  ☞
• * 1.26

  Note.
• * 1.27

  The diffe∣rence be∣twene Stra∣taruhme∣trie and Tacticie.
• * 1.28

  I.D. F••end•••• you will finde it hard, to perform•• my descripti••n of ••his F••ate. But by Ch••r••graphie•• you may helpe your self•• some ••hat: wher•• th•• Figures knowne (in Sid•••• ••nd Angles) are not Regular: And where•• Resolu∣tion into Triangles can s••••u••. &c. And yet you will finde it strange to deale thus gener••lly with A∣rithmeticall figures: and, that for Bat∣tayle ••ay. Their co••∣tent•••••• differ so much from like Geometr••∣call Figur••s.
• * 1.29

  A marueilous Glasse. ☞
• * 1.30

  S.W.P.
• * 1.31

  ☞
• * 1.32

  Note.
• * 1.33

  1.
• * 1.34

  2.
• * 1.35

  3.
• * 1.36

  5.
• * 1.37

  6.
• * 1.38

  7.
• * 1.39

  8.
• * 1.40

  I.D. Read in A∣ristotle his 8. booke of Politikes: the 5, 6, and 7. chapters. Where you shall haue some occasion farder to thinke of Mu¦sike, than commonly is thought.
• * 1.41

  ☜
• * 1.42

  Anno. 1548 and 1549. in Louayn.
• * 1.43

  Note.
• * 1.44

  I.D. The Cutting of a Sphare according to any proportion as∣signed may by this proposition be done Mechanically by tempering L••∣quor to a certayne waight in respect of the waight of the Sphare 〈◊〉〈◊〉 Swy••••••ng.
• * 1.45

  A common error•• noted.
• * 1.46

  A paradox.
• * 1.47

  N. T.
• * 1.48

  The wonder∣full vse of these Proposi∣tions.
• * 1.49

  The practise Staticall, to know the pro∣portion, be∣twene the Cube, and the Sphare.
• * 1.50

  I. D.
• * 1.51

  For, so, haue you. 256. partes of a Graine.
• * 1.52

  The proportion of the Square to the Circle ins••ribed.
• * 1.53

  The Squa••ing of the Cir••le. M••••ha∣ni••ally.
• * 1.54

  To any Squir•• g••uen•• to 〈…〉〈…〉.
• * 1.55

  Note Squaring of the Circle without know∣ledge of the proportion be∣twene Cir∣cumference and Diame∣ter.
• * 1.56

  To Dubble the Cube re∣dily: by Art Mechanicall: depending vp∣pon Demon∣stration Ma∣thematicall.
• * 1.57

  I. D. The 4. sides of this Pyrami•• must be 4. Isosceles Triangles •• lik•• and ••quall.
• * 1.58

  I. D.
• * 1.59

  In all work••nge•• with this Pyramis or Cone. Let their Situation•• be in all Pointe•• and Condi∣tions, a like, o•• all one: while you are about ••ne worke. Els you will 〈◊〉〈◊〉.
• * 1.60

  I. D.
• * 1.61

  Consider well whan you must put your wate•••• togyther: and whan, you must emp∣ty you•• first water•• out of your Pyrami•• or Cone. El•• you will 〈◊〉〈◊〉.
• * 1.62

  Vitruuius. Lib. 9. Cap. 3.

  ☞ God b•• than∣ked ••or this Inuention•• & the frui••e ••n∣suing.
• * 1.63

  Note.
• * 1.64

  Note, as con∣cerning the Sphaericall Super••icies of the water.
• * 1.65

  ☞
• * 1.66

  Note.
• * 1.67

  Note this A∣bridge••ent of Dubbling 〈◊〉〈◊〉 Cube. ••••.
• * 1.68

  Note. * ☜
• * 1.69

  To giue Cubes one to the o∣ther in any proportion, Rationall or Irrationall.
• * 1.70

  Empty∣ing the first.
• * 1.71

  The demonstrations of this Dubbling of the Cube, and o•• the rest.
• * 1.72

  I.D.
• * 1.73

  Here 〈…〉〈…〉 of the water.
• * 1.74

  By the 33. of the e∣leuenth books of Euclide.
• * 1.75

  I.D.
• * 1.76

  And your diligence in pra••••ise, can ••o (in waight of wa∣te•••• pe••forme it: Therefore. now, you ar•• able to ••eue good reason of your whole doing.
• * 1.77

  Note this Corollary.
• * 1.78

  The great Commodities following of these new In∣uentions.
• * 1.79

  ☞
• * 1.80

  Such is the Fruite of the Mathemati∣call Sciences and Artes.
• * 1.81

  MAN is the Lesse World.
• * 1.82

  ☜ Microcos∣mus.
• * 1.83

  Lib. 3. Cap. 1.
• * 1.84

  ☞
• * 1.85

  Saw Milles.
• * 1.86

  Atheneus Lib. 5. cap. 8.
• * 1.87

  Proclus. Pag. 18.
• * 1.88

  To go to the bottom of the Sea without daunger.
• * 1.89

  Plut••••••bus in Mar∣co M••rcello.

  Sy••asius in Episto∣lis.

  Polybius.

  Plinius.

  Quint••lianus.

  T. Liuius.
• * 1.90

  Athena••s.
• * 1.91

  Gale••us. Anthemius.
• * 1.92

  Burning Glasses.
• * 1.93

  Gunnes.
• * 1.94

  4. Reg. 20.
• * 1.95

  A perpetuall Motion.
• * 1.96

  An obiection.
• * 1.97

  The Answer.
• * 1.98

  ☜
• * 1.99

  A Mathe∣maticien.
• * 1.100

  Vitrunius.
• * 1.101

  VVho is an Architect.
• * 1.102

  The Im∣materialitie of perfect Ar∣chitecture.
• * 1.103

  What, Linea∣ment is.
• * 1.104

  Note.
• * 1.105

  Anno. 1559.
• * 1.106

  Anno. 1551
• * 1.107

  De his quae Mundo mi∣rabiliter eue∣niunt. cap. 8.
• * 1.108

  Tusc. ••.
• * 1.109

  ☞
• * 1.110

  A Digression. Apologeticall.
• * 1.111

  A prouerb. Fayre fisht, and caught a Frog.
• * 1.112

  ☞
• * 1.113

  Psal. 140.
• * 1.114

  Act. 7. C.
• * 1.115

  Lib. 30. Cap. 1.
• * 1.116

  ☜
• * 1.117

  R. B.
• * 1.118

  ☞
• * 1.119

  Vniuersities.
• * 1.120

  ☜
• * 1.121

  ☞
• * 1.122

  The Ground platt of this Praeface in a Table.