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+The Project Gutenberg eBook of The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara, by John Dee
+
+This eBook is for the use of anyone anywhere in the United States and
+most other parts of the world at no cost and with almost no restrictions
+whatsoever. You may copy it, give it away or re-use it under the terms
+of the Project Gutenberg License included with this eBook or online at
+www.gutenberg.org. If you are not located in the United States, you
+will have to check the laws of the country where you are located before
+using this eBook.
+
+Title: The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara
+
+Author: John Dee
+
+Release Date: July 13, 2007 [eBook #22062]
+[Most recently updated: June 28, 2021]
+
+Language: English
+
+Character set encoding: UTF-8
+
+Produced by: Louise Hope, David Starner, Suzanne Lybarger and the Online Distributed Proofreading Team
+
+*** START OF THE PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE ***
+
+
+
+
+[Transcriber’s Note:
+
+This e-text includes characters that will only display in UTF-8
+(Unicode) text readers, including a few words of Greek:
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+  ❧ ☞ [leaf symbol; pointing-finger symbol]
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+
+If any of these characters do not display properly-- in particular, if
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+of this plain-text file.
+
+Superscript letters are shown with ^: y^e, y^t.
+
+Marginal quotation marks are shown inline as “ and ”, approximating the
+beginning and end of the marked passage. In the original text, no
+quotation marks were printed inline.
+
+Paragraphs are broken up for sidenotes, with blank lines before and
+after. Original paragraph breaks are shown as two blank lines. Brackets
+within the body text are in the original.
+
+All sidenotes except the one beginning “This noble Earle” were printed
+in italics; markup has been omitted to reduce visual clutter.
+
+At least four sizes of text were used, often in combination with
+_italics_. The variants are shown here as:
+
+  +‡very large‡+
+  +larger+
+  =smaller=
+
+Further errors and anomalies are listed at the end of the text, along
+with those Euclid citations identified by number.]
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+
+  THE ELEMENTS
+  OF GEOMETRIE
+
+  of the most auncient
+  Philosopher
+  _EVCLIDE_
+  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 Sciẽces, what they are,
+  and wherunto commodious: where, also, are disclosed
+  certaine new Secrets Mathematicall and Mechanicall,
+  vntill these our daies, greatly missed._
+
+
+  Imprinted at London by _Iohn Daye_.
+
+
+
+
+❧ The Translator to the Reader.
+
+
+_There is (gentle Reader) nothing (the word of God onely set
+apart) which so much beautifieth and adorneth the soule and
+minde of mã, as doth the knowledge of good artes and sciences:
+as the knowledge of naturall and morall Philosophie. The one
+setteth before our eyes, the creatures of God, both in the
+heauens aboue, and in the earth beneath: in which as in a
+glasse, we beholde the exceding maiestie and wisedome of God,
+in adorning and beautifying them as we see: in geuing vnto them
+such wonderfull and manifolde proprieties, and naturall
+workinges, and that so diuersly and in such varietie: farther in
+maintaining and conseruing them continually, whereby to praise
+and adore him, as by S. Paule we are taught. The other teacheth
+vs rules and preceptes of vertue, how, in common life amongest
+men, we ought to walke vprightly: what dueties pertaine to our
+selues, what pertaine to the gouernment or good order both of an
+housholde, and also of a citie or common wealth. The reading
+likewise of histories, conduceth not a litle, to the adorning of
+the soule & minde of man, a studie of all men cõmended: by it
+are seene and knowen the artes and doinges of infinite wise men
+gone before vs. In histories are contained infinite examples of
+heroicall vertues to be of vs followed, and horrible examples of
+vices to be of vs eschewed. Many other artes also there are
+which beautifie the minde of man: but of all other none do more
+garnishe & beautifie it, then those artes which are called
+Mathematicall. Unto the knowledge of which no man can attaine,
+without the perfecte knowledge and instruction of the
+principles, groundes, and Elementes of Geometrie. But perfectly
+to be instructed in them, requireth diligent studie and reading
+of olde auncient authors. Amongest which, none for a beginner is
+to be preferred before the most auncient Philosopher _Euclide_
+of _Megara_. For of all others he hath in a true methode and
+iuste order, gathered together whatsoeuer any before him had of
+these Elementes written: inuenting also and adding many thinges
+of his owne: wherby he hath in due forme accomplished the arte:
+first geuing definitions, principles, & groundes, wherof he
+deduceth his Propositions or conclusions, in such wonderfull
+wise, that that which goeth before, is of necessitie required to
+the proufe of that which followeth. So that without the diligent
+studie of _Euclides_ Elementes, it is impossible to attaine vnto
+the perfecte knowledge of Geometrie, and consequently of any of
+the other Mathematicall sciences. Wherefore considering the want
+& lacke of such good authors hitherto in our Englishe tounge,
+lamenting also the negligence, and lacke of zeale to their
+countrey in those of our nation, to whom God hath geuen both
+knowledge, & also abilitie to translate into our tounge, and
+to publishe abroad such good authors, and bookes (the chiefe
+instrumentes of all learninges): seing moreouer that many good
+wittes both of gentlemen and of others of all degrees, much
+desirous and studious of these artes, and seeking for them as
+much as they can, sparing no paines, and yet frustrate of their
+intent, by no meanes attaining to that which they seeke: I haue
+for their sakes, with some charge & great trauaile, faithfully
+translated into our vulgare toũge, & set abroad in Print, this
+booke of _Euclide_. Whereunto I haue added easie and plaine
+declarations and examples by figures, of the definitions. In
+which booke also ye shall in due place finde manifolde
+additions, Scholies, Annotations, and Inuentions: which I haue
+gathered out of many of the most famous & chiefe Mathematiciẽs,
+both of old time, and in our age: as by diligent reading it in
+course, ye shall well perceaue. The fruite and gaine which I
+require for these my paines and trauaile, shall be nothing els,
+but onely that thou gentle reader, will gratefully accept the
+same: and that thou mayest thereby receaue some profite: and
+moreouer to excite and stirre vp others learned, to do the like,
+& to take paines in that behalfe. By meanes wherof, our Englishe
+tounge shall no lesse be enriched with good Authors, then are
+other straunge tounges: as the Dutch, French, Italian, and
+Spanishe: in which are red all good authors in a maner, found
+amongest the Grekes or Latines. Which is the chiefest cause,
+that amongest thẽ do florishe so many cunning and skilfull men,
+in the inuentions of straunge and wonderfull thinges, as in
+these our daies we see there do. Which fruite and gaine if I
+attaine vnto, it shall encourage me hereafter, in such like sort
+to translate, and set abroad some other good authors, both
+pertaining to religion (as partly I haue already done) and also
+pertaining to the Mathematicall Artes. Thus gentle
+reader farewell._ (?¿)
+
+
+    [Decoration]
+
+
+
+
+  ❧ TO THE VNFAINED LOVERS
+  of truthe, and constant Studentes of Noble
+  _Sciences, _IOHN DEE_ of London, hartily_
+  wisheth grace from heauen, and most prosperous
+  _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_, _Bonum_, 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_,
+_Bonum_, and _Ens_, to be Spirituall, Infinite, Æternall, Omnipotent,
+&c. Nothyng beyng alledged or expressed, How, worldly goods: how,
+worldly dignitie: how, health, Strẽgth or lustines of body: nor yet the
+meanes, how a merueilous sensible and bodyly blysse and felicitie
+hereafter, might be atteyned: Straightway, the fantasies of those
+hearers, were dampt: their opinion of _Plato_, was clene chaunged: yea
+his doctrine was by them despised: and his schole, no more of them
+visited. Which thing, his Scholer, _Aristotle_, narrowly cõsidering,
+founde the cause therof, 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 constantly to haue
+continued therin: to their full satisfaction: if such his 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, otherwise
+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 exãple.) 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 mete, for so
+straunge matter (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 schole,
+after they had perceiued his finall conclusion, would in these thinges
+haue ben his most diligent hearers (so infinitely mought their desires,
+in fine and at length, by our _Artes Mathematicall_ be satisfied) yet,
+by this my Præface & 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.
+
+
+    [The intent of this Preface.]
+
+Wherfore, seyng I finde great occasion (for the causes alleged, and
+farder, in respect of my _Art Mathematike generall_) to vse “a certaine
+forewarnyng and Præface, whose content shalbe, that mighty, most
+plesaunt, and frutefull _Mathematicall Tree_, with his chief armes and
+second (grifted) braunches: 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 thãkefull 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 either 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 chaungeable. Thinges
+Supernaturall, are, of the minde onely, comprehended: Things Naturall,
+of the sense exterior, ar hable to be perceiued. In thinges Naturall,
+probabilitie and coniecture hath place: But in things Supernaturall,
+chief demõstration, & 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 supernaturall 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 neuerthelesse, by materiall things
+hable somewhat to be signified. And though their particular Images, by
+Art, are aggregable and diuisible: yet the generall _Formes_,
+notwithstandyng, are constant, vnchaungeable, vntrãsformable, 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 imperfectiõ of coniecture, weenyng and
+opinion: and commyng short of high intellectuall cõceptiõ, are the
+Mercurial fruite of _Dianœticall_ discourse, in perfect imagination
+subsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_,
+and also a straunge participatiõ betwene thinges supernaturall,
+immortall, intellectual, simple and indiuisible: and thynges naturall,
+mortall, sensible, compounded and diuisible. Probabilitie and sensible
+prose, may well serue in thinges naturall: and is commendable: In
+Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor
+yet the testimony of sense, any whit credited: But onely a perfect
+demonstration, of truthes certaine, necessary, and inuincible:
+vniuersally and necessaryly concluded: is allowed as sufficient for “an
+Argument exactly and purely Mathematical.”
+
+
+    [Note the worde, Vnit, to expresse the Greke Monas,
+    & not Vnitie: as we haue all, commonly, till now, vsed.]
+
+Of _Mathematicall_ thinges, are two principall kindes: namely, _Number_,
+and _Magnitude_.
+
+    [Number.]
+
+_Number_, we define, to be, a certayne Mathematicall Sũme, of _Vnits_.
+And, an _Vnit_, is that thing Mathematicall, Indiuisible, by
+participation of some likenes of whose property, any thing, which is in
+deede, 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,
+principally, is a thing _Mathematicall_.
+
+    [Magnitude.]
+
+_Magnitude_ is a thing _Mathematicall_, 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:
+
+    [A point.]
+
+The endes of a line, are _Pointes_ called. A _Point_, is a thing
+_Mathematicall_, indiuisible, 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.]
+
+a _Line_ is called the race or course of a _Point_. 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_.
+
+    [Magnitude.]
+
+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
+recken 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 _Mathematicall_, to it appropriate, called _Arithmetike_: and
+afterward of _Magnitude_, and his Science, called _Geometrie_. But that
+name contenteth me not: whereof a word or two hereafter shall be sayd.
+How Immateriall and free from all matter, _Number_ is, who doth not
+perceaue? yea, who doth not wonderfully wõder 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 simplenes of
+Substance Spirituall or Angelicall, will be found propre enough thereto.
+And therefore the great & godly Philosopher _Anitius Boetius_, sayd:
+_Omnia quæcun[que] a primæua rerum natura constructa sunt, Numerorum
+videntur ratione formata. Hoc enim fuit principale in animo Conditoris
+Exemplar_. That is: +_All thinges (which from the very first originall
+being 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 comfortable allurement, O rauishing
+perswasion, to deale with a Science, whose Subiect, is so Auncient, so
+pure, so excellent, so surmounting all creatures, so vsed of the
+Almighty 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 perfection
+of the Science) learned, we may both winde and draw our selues into the
+inward and deepe search and vew, of all creatures distinct vertues,
+natures, properties, and _Formes_: And also, farder, arise, clime,
+ascend, and mount vp (with Speculatiue 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, proceded by no other way, then by Numbers Formall_. And
+this Earle hym selfe, in Rome,
+
+    [Ano. 1488.]
+
+* 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, and vnderstandyng of euery
+thyng, hable to be knowen. For the verifying of which Conclusion,
+I promise to aunswere to the 74. Questions, vnder written, by the way of
+Numbers_. Which Cõclusions, I omit here to rehearse: aswell auoidyng
+superfluous prolixitie: as, bycause _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 Obseruers and
+Considerers of the constant law of nũbers: 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 mysteries, 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 constitution:) and the third, in Spirituall and
+Angelicall Myndes, and in the Soule of mã. In the first and third state,
+_Number_, is termed _Number Numbryng_. But in all Creatures, otherwise,
+_Number_, is termed _Nũber Numbred_. And in our Soule, Nũber beareth
+such a swaye, and hath such an affinitie therwith: that some of the old
+_Philosophers_ 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 perfect beyng, in the Creator,
+Sempiternally. _Number Numbryng_ therfore, is the discretion discerning,
+and distincting of thinges. But in God the Creator, This discretion, in
+the beginnyng, produced orderly and distinctly all thinges. For his
+_Numbryng_, 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
+considered, maketh certaine and distinct determination. And albeit these
+thynges be waighty and truthes of great importance, yet (by the infinite
+goodnes of the Almighty _Ternarie_,) Artificiall Methods and easy wayes
+are made, by which the zelous Philosopher, may wyn nere this Riuerish
+_Ida_, this Mountayne of Contemplation: and more then Contemplation. And
+also, though _Number_, be a thyng so Immateriall, so diuine, and
+æternall: 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 * _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 cõceiuyng 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 æternall.
+
+
+But farder vnderstand, that vulgar Practisers, haue Numbers, otherwise,
+in sundry Considerations: and extend their name farder, then to Numbers,
+whose least part is an _Vnit_. For the common Logist, Reckenmaster, or
+Arithmeticien, in hys vsing 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 diuerse)
+partes of an _Vnit_. Yea and farder, hath, _Fractions of Fractions. &c_.
+And, forasmuch, as, _Addition_, _Substraction_, _Multiplication_,
+_Diuision_ and _Extraction of Rotes_, are the chief, and sufficient
+partes of _Arithmetike_:
+
+    [Arithmetike.]
+
+which is, the _Science that demonstrateth the properties, of Numbers,
+and all operatiõs, in numbers to be performed_:
+
+    [Note.]
+
+“How often, therfore, these fiue sundry sortes of Operations, do, for
+the most part, of their execution, 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:
+
+    [1.]
+
+the Consideration, doctrine, and working, in whole numbers onely: where,
+of an _Vnit_, is no lesse part to be allowed: is named (as it were) an
+_Arithmetike_ by it selfe. And so of the _Arithmetike of Fractions_.
+
+    [2.]
+
+In lyke sorte, the necessary, wonderfull and Secret doctrine of
+Proportion, and proportionalytie hath purchased vnto it selfe a peculier
+maner of handlyng and workyng: and so may seme an other forme of
+_Arithmetike_.
+
+    [3.]
+
+Moreouer, the _Astronomers_, for spede and more commodious calculation,
+haue deuised a peculier maner of orderyng nũbers, about theyr circular
+motions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes
+&c. which commonly is called the _Arithmetike_ of _Astronomical_ or
+_Phisicall Fractions_. That, haue I briefly noted, by the name of
+_Arithmetike Circular_. Bycause it is also vsed in circles, not
+_Astronomicall. &c._
+
+    [4.]
+
+Practise hath led _Numbers_ farder, and hath framed them, to take vpon
+them, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_
+and _Irrationalitie_. (For in pure _Arithmetike_, an _Vnit_, is the
+common Measure of all Numbers.) And, here, Nũbers are become, as Lynes,
+Playnes and Solides: some tymes _Rationall_, some tymes _Irrationall_.
+And haue propre and peculier characters, (as ²√. ³√. and so of other.
+Which is to signifie _Rote 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 compoundyng of Numbers: as some
+tyme, two, three, foure (or more) _Radicall_ nũbers, diuersly knit, by
+signes, of More & Lesse: as thus ²√12 + ³√15. Or thus ⁴√19 + ³√12 - ²√2.
+&c. And some tyme with whole numbers, or fractions of whole Number, amõg
+them: as 20 + ²√24. ³√16 + 33 - ²√10. ⁴√44 + 12¼ + ³√9. And so,
+infinitely, may hap the varietie. After this: Both the one and the other
+hath fractions incident: and so is this _Arithmetike_ greately enlarged,
+by diuerse exhibityng and vse of Compositions and mixtynges. Consider
+how, I (beyng desirous 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 spokẽ,
+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
+speculation and practise) and haue found out, and atteyned to the very
+chief perfection (almost) of _Numbers_ Practicall vse. Which thing, is
+well to be perceiued in that great Arithmeticall Arte of _Æquation_:
+commonly called the _Rule of Coss._ or _Algebra_. 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 *vse,
+
+    [* Anno. 1550.]
+
+of the famous Mathematicien, and my great frende, _Petrus Montaureus_:)
+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 Alpagus_ (most perfect in the Arabik tung) so
+translated. _Scientia faciendi Algiebar & Almachabel. i. Scientia
+inueniendi numerum ignotum, per additionem Numeri, & diuisionem &
+æquationem_. Which is to say: +_The Science of workyng Algiebar and
+Almachabel_+, that is, the +_Science of findyng an vnknowen number, by
+Addyng of a Number, & Diuision & æquation_+. Here haue you the name: and
+also the principall partes of the Rule, touched. To name it, _The rule,
+or Art of Æquation_, doth signifie the middle part and the State of the
+Rule. This Rule, hath his peculier Characters:
+
+    [5.]
+
+and the principal partes of _Arithmetike_, to it appertayning, do
+differre from the other _Arithmeticall operations_. This _Arithmetike,
+hath Nũbers_ Simple, Cõpound, Mixt: and Fractions, accordingly. This
+Rule, and _Arithmetike of Algiebar_, is so profound, so generall and so
+(in maner) conteyneth the whole power of Numbers Application practicall:
+that mans witt, can deale with nothyng, more proffitable about 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 testimony 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 misdoute 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
+forward and backward? How might they misse _Arithmeticall_ helpe in the
+Rules of Felowshyp: either without tyme, or with tyme? and betwene the
+Marchant & his Factor? The Rules of Bartering in wares onely: or part in
+wares, and part in money, would they gladly want? Our Marchant
+venturers, and Trauaylers ouer Sea, how could they order their doynges
+iustly and without losse, vnleast certaine and generall Rules for
+Exchaũge of money, 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 Marchant then to make an
+end) how ample & wonderfull is the Rule of False positions? especially
+as it is now, by two excellent Mathematiciens (of my familier
+acquayntance 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 Imagine 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 meruailously pleasured, if _Arithmetike_ be their guide?
+And the honorable Phisiciãs, 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_, _Auerrois_, _Arnoldus_, _Lullus_, and other haue published
+their positions, aswell in the quantities of the Degrees aboue
+Temperament, as in the Rules, concluding the new _Forme_ resulting: yet
+a more precise, commodious, and easy _Method_, is extant: by a
+Countreyman of ours
+
+    [R. B.]
+
+(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 also, for very good profe of
+Numbers vse, in this most subtile and frutefull, Philosophicall
+Conclusion,) I entend in the meane while, most briefly, and with my
+farder helpe, to communicate the pith therof vnto you.
+
+
+First describe a circle: whose diameter let be an inch. Diuide the
+Circumference into foure equall partes. Frõ 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. without the circumference of the
+circle: So that they shall be of 4. inches long (at the least) 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 _Colde_, one against the other. And
+likewise _Moyst_ and _Dry_, one against the other. And in the Circle
+write _Temperate_. Which _Temperature_ hath a good Latitude: as
+appeareth by the Complexion of man. And therefore we haue allowed vnto
+it, the foresayd Circle: and not a point Mathematicall or Physicall.
+
+
+    [* Take some part of Lullus counsayle in his booke
+    de Q. Essentia.]
+
+Now, when you haue two thinges Miscible, whose degrees are * 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 Qualities, or of one kinde (but of diuerse
+intentions and degrees) or a _Temperate_, and a Contrary, _The forme
+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_
+
+    [* Note.]
+
+(for the * _Temperament_ is counted none. And for it, you must put a
+Ciphre, if at any time, it be in mixture).
+
+                              HOTE
+                                +C
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +E
+                                |
+  MOIST                  A  TEMPERATE                B     DRYE
+    +------+------+------+------+------+------+------+------+
+                                |D
+                                |
+                                +
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +
+                              COLD
+
+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, accounted from _C_, toward _D_, ende
+iuste in the 2. degree of heate. Of the third maner, I will geue also an
+example: which let be this:
+
+    [Note.]
+
+I haue a liquid Medicine whose Qualitie of heate is in the 4. degree
+exalted: 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 frõ 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
+1½: Adde to 1. this 1½: you haue 2½. Or subtract from 4. this 1½: you
+haue likewise 2½ remayning. Which declareth, the _Forme resulting_, to
+be _Heate_, in the middle of the third degree.
+
+
+    [The Second Rule.]
+
+“But if the Quantities of two thinges Commixt, be diuerse, and the
+Intensions (of their Formes Miscible) be in diuerse degrees, and
+heigthes. (Whether those Formes be of one kinde, or of Contrary kindes,
+or of a Temperate and a Contrary, _What proportion is of the lesse
+quantitie 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
+quantitie_. As for example. Let two pound of Liquor be geuen, hote in
+the 4. degree: & one pound of Liquor be geuen, hote in the third
+degree.” I would gladly know the Forme resulting, in the Mixture of
+these two Liquors. Set downe your nũbers in order, thus.
+   ___________________________
+  |            |              |
+  |  {P}. _2._ |  _Hote. 4._  |
+  |            |              |
+  |  {P}. _1._ |  _Hote. 3._  |
+  |____________|______________|
+
+Now by the rule of Algiebar, haue I deuised a very easie, briefe, and
+generall maner of working in this case. Let vs first, suppose that
+_Middle Forme resulting_, to be 1{x}: as 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{x}: to
+the difference betwene 1{x} and 3: (the degree of the thing, in lesse
+quãtitie. And with all, 1{x}, being alwayes in a certaine middell,
+betwene the two heigthes or degrees). For the first difference, I set
+4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1.
+is to 2. so is 4-1{x} to 1{x}-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 the other.
+Let these Multiplications be made accordingly. And of the first and the
+fourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore,
+our Æquation is betwene 1{x}-3: and 8-2{x}. Which may be reduced,
+according to the Arte of Algiebar: as, here, adding 3. to eche part,
+geueth the Æquation, thus, 1{x}=11-2{x}. And yet againe, contracting, or
+Reducing it: Adde to eche part, 2{x}: Then haue you 3{x} æquall to 11:
+thus represented 3{x}=11. Wherefore, diuiding 11. by 3: the Quotient is
+3⅔: the _Valew_ of our 1{x}, _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 3⅔, resteth ⅔. Subtracte the same heigth of the
+Forme resulting, (which is 3⅔) frõ 4: then resteth ⅓: You see, that ⅔ is
+double to ⅓: as 2.{P}. is double to 1.{P}. So should it be: by the rule
+here geuen. Note. As you added to eche part of the Æquation, 3: so if ye
+first added to eche part 2{x}, it would stand, 3{x}-3=8. And now adding
+to eche part 3: you haue (as afore) 3{x}=11.
+    _________________________
+   |           |             | _
+   | {P}. _2._ |  _Hote. 4._ |   ⅓ _   _The forme_
+   |           |             |     _   _3⅔ resulting._
+   | {P}. _1._ |  _Hote. 3._ | _ ⅔
+   |___________|_____________|
+
+
+And though I, here, speake onely of two thyngs Miscible: and most
+commonly 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 Rules are sufficient: duely repeated and
+iterated.
+
+    [Note.]
+
+In procedyng first, with any two: and then, with the Forme Resulting,
+and an other: & so forth: For, the last worke, concludeth 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 definition, 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 varieties, by the
+foresayd two generall 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 particular: would haue larger
+discoursing, then is mete in this place to be dealt withall: And other
+(perchaunce) with a proude snuffe will disdaine this litle: 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 Qualities Naturall_, is so Matcht and maryed with
+the most simple, easie, and short way of the noble Rule of _Algiebar_.
+Who can remaine, therfore vnpersuaded, to loue, alow, 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
+hunderd thousand mens wittes, of the middle sorte, are hable to
+perfourme, or truely to conclude, with out helpe thereof.
+
+
+Now will we farder, 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
+
+    [Τακτικὴ.]
+
+Τακτικὴ. “That is, the Skill of Ordring Souldiers in Battell ray after
+the best maner to all purposes.” This Art so much dependeth vppon
+Numbers vse, and the Mathematicals, that _Ælianus_ (the best writer
+therof,) in his worke, to 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 _Æneas_: _Cyneas_ of
+_Thessaly_: _Pyrrhus Epirota_: and _Alexander_ his sonne: _Clearchus_:
+_Pausanias_: _Euangelus_: _Polybius_, familier frende to _Scipio_:
+_Eupolemus_: _Iphicrates_, _Possidonius_: and very many other worthy
+Capitaines, Philosophers and Princes of Immortall fame and memory: Whose
+fayrest floure of their garland (in this feat) was _Arithmetike_: and a
+litle perceiuerance, in _Geometricall_ Figures. But in many other cases
+doth _Arithmetike_ 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 company, to make the
+victuals, longer to serue the remanent, & for a certaine determined
+tyme: if nede so require. And so in sundry his other accountes,
+Reckeninges, Measurynges, and proportionynges, the wise, expert, and
+Circumspect Capitaine 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, the loyall, and
+Curteous
+
+    [☞]
+
+_Iohn_, late Earle of Warwicke. Who was a yong Gentleman, throughly
+knowne to very few. Albeit his lusty valiantnes, force, and Skill in
+Chiualrous feates and exercises: his humblenes, and frendelynes 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 achieue: 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, forecastinges and determinations, no twayne,
+(I thinke) beside my selfe, can so perfectly, and truely report. And
+therfore, in Conscience, I count it my part, for the honor, preferment,
+& procuring of vertue (thus, briefly) to haue put his Name, in the
+Register of _Fame Immortall_.
+
+
+To our purpose. This _Iohn_, by one of his actes (besides many other:
+both in England and Fraunce, by me, in him noted.) did disclose his
+harty loue to vertuous Sciences: and his noble intent, to excell in
+Martiall prowesse: When he, with humble 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 deuised) for
+ordring of all Companies, summes and Numbers of mẽ, (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 seem few, being many. To marche in battaile or Iornay: with many such
+feates, to Foughten field, Skarmoush, or Ambushe appartaining:
+
+    [This noble Earle, dyed Anno. 1554. skarse of 24. yeares
+    of age: hauing no issue by his wife: Daughter to the Duke
+    of Somerset.]
+
+And of all these, liuely designementes (most curiously) to be in velame
+parchement described: with Notes & peculier markes, as the Arte
+requireth: and all these Rules, and descriptions Arithmeticall, inclosed
+in a riche Case of Gold, he vsed to weare about his necke: as his Iuell
+most precious, and Counsaylour most trusty. Thus, _Arithmetike_, of him,
+was shryned in gold: Of _Numbers_ frute, he had good hope. Now, Numbers
+therfore innumerable, in _Numbers_ prayse, his shryne shall finde.
+
+
+What nede I, (for farder profe 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 meruaylously declare: how, neither the Auncient Romaine
+lawes, without good knowledge of _Numbers art_, can be perceiued: Nor
+(Iustice in infinite Cases) without due proportion, (narrowly
+considered,) is hable to be executed. How Iustly, & with great knowledge
+of Arte, did _Papinianus_ institute a law of partition, and allowance,
+betwene man and wife after a diuorce? But how _Accursius_, _Baldus_,
+_Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (being
+otherwise, notably well learned) do iumble, gesse, and erre, from the
+æquity, art and Intent 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: and with _Accursius_
+Glosse, much cumbred: burst out, and sayd: _Nulla est in toto libro, hac
+glossa difficilior: Cuius computationem nec Scholastici nec Doctores
+intelligunt. &c._ That is: +_In the whole booke, there is no Glosse
+harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor
+the Doctours vnderstand. &c._+ What can they say of _Iulianus_ law, _Si
+ita Scriptum. &c._ Of the Testators will iustly performing, betwene the
+wife, Sonne and daughter? How can they perceiue the æquitie of
+_Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _Lex
+Falcidia_? How can they deliuer him, from his Reprouers: and their
+maintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? How
+Iustly and artificially, was _Africanus_ 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
+cõtributed by the legataries to the heire. For, what proportion, 100
+hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that
+is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law,
+require an expert _Arithmeticien_, to vnderstand the deepe Iudgemẽt, &
+Iust determinatiõ of the Auncient Romaine Lawmakers. But much more
+expert ought he to be, who should be hable, to decide with æquitie, 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 equity might be
+greately preferred, and skilfully executed, through due skill of
+Arithmetike, and proportions appertainyng. The worthy Philosophers, and
+prudent lawmakers (who haue written many bookes _De Republica:_ How the
+best state of Common wealthes might be procured and mainteined,) haue
+very well determined 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:) defining it,
+
+    [Iustice.]
+
+“to be that vertue, by which, to euery one, is rendred, that to him
+appertaineth.” God challengeth this at our handes, to be honored as God:
+to be loued, as a father: to be feared as a Lord & master. Our
+neighbours proportiõ, is also prescribed 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 same) 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,
+briefly, of other Sciences discoursing, findeth them, not hable to bring
+it to passe: But of the Science of Numbers, he sayth. _Illa, quæ numerum
+mortalium generi dedit, id profecto efficiet. Deum autem aliquem, magis
+quam fortunam, ad salutem nostram, hoc munus nobis arbitror contulisse.
+&c. Nam ipsum bonorum omnium Authorem, cur non maximi boni, Prudentiæ
+dico, causam arbitramur? +That Science, verely, which hath taught
+mankynde 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 the greatest good thyng, namely,
+Wisedome?+_ There, at length, he proueth _Wisedome_ to be atteyned, by
+good Skill of _Numbers_. 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, 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 entend, 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 vnspeakable behofe) our heauenly father, hath for vs prepared, and
+reuealed, by sundry _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 Originall causes: But the diuersitie neuerthelesse, is
+great. We defined an _Vnit_, to be a thing Mathematicall Indiuisible:
+A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And
+farder, that a Point may haue a certaine determined Situation: that is,
+that we may assigne, and prescribe a Point, to be here, there, yonder.
+&c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to
+be tyed to any place, or seat: diuisible or indiuisible. Agayne, by
+reason, a Point may haue a Situation limited to him: a certaine motion,
+therfore (to a place, and from a place) is to a Point incident and
+appertainyng. But an _Vnit_, can not be imagined to haue any motion.
+A Point, by his motion, produceth, Mathematically, a line: (as we sayd
+before) which is the first kinde of Magnitudes, and most simple: An
+_Vnit_, can not produce any number. A Line, though it be produced of a
+Point moued, yet, it doth not consist of pointes: Number, though it be
+not produced of an _Vnit_, yet doth it Consist of vnits, as a materiall
+cause. But formally,
+
+    [Number.]
+
+Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting,
+is the workemanship of our minde: which, of distinct and discrete Vnits,
+maketh a Number: by vniformitie, resulting of a certaine multitude of
+Vnits. And so, euery number, may haue his least part, giuen: namely, an
+Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be
+giuẽ: by cause, infinitly, diuision therof, may be conceiued. All
+Magnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine,
+or Solid, of no Sense, can be perceiued, nor exactly by hãd (any way)
+represented: nor of Nature produced: But, as (by degrees) Number did
+come to our perceiuerance: So, by visible formes, we are holpen to
+imagine, what our Line Mathematicall, is. What our Point, is. So
+precise, are our Magnitudes, that one Line is no broader then an other:
+for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our
+Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we
+can haue Smaller, then either Arte or Nature can produce any: and
+Greater also, then all the world can comprehend. Our least Magnitudes,
+can be diuided into so many partes, as the greatest. As, a Line of an
+inch long, (with vs) may be diuided into as many partes, as may the
+diameter of the whole world, from East to West: or any way extended:
+What priuiledges, aboue all manual Arte, and Natures might, haue our two
+Sciences Mathematicall? to exhibite, and to deale with thinges of such
+power, liberty, simplicity, puritie, and perfection? And in them, so
+certainly, so orderly, so precisely to procede: as, excellent is that
+workemã Mechanicall Iudged, who nerest can approche to the representing
+of workes, Mathematically demonstrated?
+
+    [☞]
+
+And our two Sciences, remaining pure, and absolute, in their proper
+termes, and in their owne Matter: to haue, and allowe, onely such
+Demonstrations, as are plaine, certaine, vniuersall, and of an æternall
+veritye?
+
+    [Geometrie.]
+
+This Science of _Magnitude_, his properties, conditions, and
+appertenances: commonly, now is, and from the beginnyng, hath of all
+Philosophers, ben called _Geometrie_. But, veryly, with a name to base
+and scant, for a Science of such dignitie and amplenes. And, perchaunce,
+that name, by cõmon and secret consent, of all wisemen, hitherto hath
+ben suffred to remayne: that it might carry with it a perpetuall
+memorye, of the first and notablest benefite, by that Science, to common
+people shewed: Which was, when Boundes and meres of land and ground were
+lost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of
+_Nilus_, the greatest and longest riuer in the world) or, that ground
+bequeathed, were to be assigned: or, ground sold, were to be layd out:
+or (when disorder preuailed) that Commõs were distributed into
+seueralties. For, where, vpon these & such like occasiõs, Some by
+ignorãce, some by negligẽce, Some by fraude, and some by violence, did
+wrongfully limite, measure, encroach, or challenge (by pretence of iust
+content, and measure) those landes and groundes: great losse,
+disquietnes, murder, and warre did (full oft) ensue: Till, by Gods
+mercy, and mans Industrie, The perfect Science of Lines, Plaines, and
+Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. The
+people then, by this art pleasured, and greatly relieued, in their
+landes iust measuring: & other Philosophers, writing Rules for land
+measuring: betwene them both, thus, confirmed the name of _Geometria_,
+that is, (according to the very etimologie of the word) Land measuring.
+Wherin, the people knew no farder, of Magnitudes vse, but in Plaines:
+and the Philosophers, of thẽ, had no feet hearers, or Scholers: farder
+to disclose vnto, then of flat, plaine _Geometrie_. And though, these
+Philosophers, knew of farder vse, and best vnderstode the etymologye of
+the worde, yet this name _Geometria_, was of them applyed generally to
+all sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and
+_Pythagoras_: When they would precisely declare their owne doctrine.
+Then, was
+
+    [* Plato. 7. de Rep.]
+
+* _Geometria_, with them, _Studium quod circa planum versatur_. But,
+well you may perceiue by _Euclides Elementes_, that more ample is our
+Science, then to measure Plaines: and nothyng lesse therin is tought (of
+purpose) then how to measure Land. An other name, therfore, must nedes
+be had, for our Mathematicall Science of Magnitudes: which regardeth
+neither clod, nor turff: neither hill, nor dale: neither earth nor
+heauen: but is absolute _Megethologia_: not creping on ground, and
+dasseling the eye, with pole perche, rod or lyne: but “liftyng the hart
+aboue the heauens, by inuisible lines, and
+
+    [☞]
+
+immortall beames meteth with the reflexions, of the light
+incomprehensible: and so procureth Ioye, and perfection vnspeakable.” Of
+which true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_
+seemed to haue good taste, and iudgement: and (by the name of
+_Geometrie_) so noted it: and warned his Scholers therof: as, in hys
+seuenth _Dialog_, of the Common wealth, may euidently be sene. Where (in
+Latin) thus it is: right well translated: _Profecto, nobis hoc non
+negabunt, Quicun[que] vel paululum quid Geometriæ gustârunt, quin hæc
+Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa
+versantur._ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuer
+haue, (but euen very litle) tasted of Geometrie, will not denye vnto vs,
+this: but that this Science, is of an other condicion, quite contrary to
+that, which they that are exercised in it, do speake of it._+ And there
+it followeth, of our _Geometrie_, _Quòd quæritur cognoscendi illius
+gratia, quod semper est, non & eius quod oritur quando[que] & interit.
+Geometria, eius quod est semper, Cognitio est. Attollet igitur
+(ô Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum
+preparabit cogitationem, vt ad supera conuertamus: quæ, nunc, contra
+quàm decet, ad inferiora deijcimus. &c. Quàm maximè igitur præcipiendum
+est, vt qui præclarissimam hanc habitãt Civitatem, nullo modo,
+Geometriam spernant. Nam & quæ præter ipsius propositum, quodam modo
+esse videntur, haud exigua sunt. &c._ It must nedes be confessed (saith
+_Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, which
+is euer: and not of that, which, in tyme, both is bred and is brought to
+an ende. &c. Geometrie is the knowledge of that which is euerlastyng. It
+will lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by
+that meanes, it will prepare the Thought, to the Philosophicall loue of
+wisdome: that we may turne or conuert, toward heauenly thinges =[both
+mynde and thought]= which now, otherwise then becommeth vs, we cast down
+on base or inferior things. &c. Chiefly, therfore, Commaundement must be
+giuen, that such as do inhabit this most honorable Citie, by no meanes,
+despise Geometrie. For euen those thinges =[done by it]= which, in
+manner, seame to be, beside the purpose of Geometrie: are of no small
+importance. &c._+ And besides the manifold vses of _Geometrie_, in
+matters appertainyng to warre, he addeth more, of second vnpurposed
+frute, and commoditye, arrising by _Geometrie_: saying: _Scimus quin
+etiam, ad Disciplinas omnes facilius per discendas, interesse omnino,
+attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam,
+secundo loco discendam Iuuenibus statuamus._ That is. +_But, also, we
+know, that for the more easy learnyng of all Artes, it importeth much,
+whether one haue any knowledge in Geometrie, or no. &c. Let vs therfore
+make an ordinance or decree, that this Science, of young men shall be
+learned in the second place._+ This was _Diuine Plato_ his Iudgement,
+both of the purposed, chief, and perfect vse of _Geometrie_: and of his
+second, dependyng, deriuatiue commodities. And for vs, Christen men,
+a thousand thousand mo occasions are, to haue nede of the helpe of *
+
+    [I. D.
+    * Herein, I would gladly shake of, the earthly name,
+    of Geometrie.]
+
+_Megethologicall_ Contemplations: wherby, to trayne our Imaginations and
+Myndes, by litle and litle, to forsake and abandon, the grosse and
+corruptible Obiectes, of our vtward senses: and to apprehend, by sure
+doctrine demonstratiue, Things Mathematicall. And by them, readily to be
+holpen and conducted to conceiue, discourse, and conclude of things
+Intellectual, Spirituall, æternall, and such as concerne our Blisse
+euerlasting: which, otherwise (without Speciall priuiledge of
+Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally)
+is hable to reach vnto, or to Compasse. And, veryly, by my small Talent
+(from aboue) I am hable to proue and testifie, that the litterall Text,
+and order of our diuine Law, Oracles, and Mysteries, require more skill
+in Numbers, and Magnitudes: then (commonly) the expositors haue vttered:
+but rather onely (at the most) so warned: & shewed their own want
+therin. (To name any, is nedeles: and to note the places, is, here, no
+place: But if I be duely asked, my answere is ready.) And without the
+litterall, Grammaticall, Mathematicall or Naturall verities of such
+places, by good and certaine Arte, perceiued, no Spirituall sense
+(propre to those places, by Absolute _Theologie_) will thereon depend.
+
+    [☞]
+
+“No man, therfore, can doute, but toward the atteyning of knowledge
+incomparable, and Heauenly Wisedome: Mathematicall Speculations, both of
+Numbers and Magnitudes: are meanes, aydes, and guides: ready, certaine,
+and necessary.” From henceforth, in this my Preface, will I frame my
+talke, to _Plato_ his fugitiue Scholers: or, rather, to such, who well
+can, (and also wil,) vse their vtward senses, to the glory of God, the
+benefite of their Countrey, and their owne secret contentation, or
+honest preferment, on this earthly Scaffold. To them, I will orderly
+recite, describe & declare a great Number of Artes, from our two
+Mathematicall fountaines, deriued into the fieldes of _Nature_. Wherby,
+such Sedes, and Rotes, as lye depe hyd in the groũd of _Nature_, are
+refreshed, quickened, and prouoked to grow, shote vp, floure, and giue
+frute, infinite, and incredible. And these Artes, shalbe such, as vpon
+Magnitudes properties do depende, more, then vpon Number. And by good
+reason we may call them Artes, and Artes Mathematicall Deriuatiue: for
+(at this tyme) I Define
+
+    [An Arte.]
+
++An Arte, to be a Methodicall cõplete Doctrine, hauing abundancy of
+sufficient, and peculier matter to deale with, by the allowance of the
+Metaphisicall Philosopher: the knowledge whereof, to humaine state is
+necessarye.+ And that I account,
+
+    [Art Mathematicall Deriuatiue.]
+
++An Art Mathematicall deriuatiue, which by Mathematicall demonstratiue
+Method, in Nũbers, or Magnitudes, ordreth and confirmeth his doctrine,
+as much & as perfectly, as the matter subiect will admit.+ And for that,
+I entend to vse the name and propertie of a
+
+    [A Mechanitien.]
+
+_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke it
+good, (for distinction sake) to giue you also a brief description, what
+I meane therby. +A Mechanicien, or a Mechanicall workman is he, whose
+skill is, without knowledge of Mathematicall demonstration, perfectly to
+worke and finishe any sensible worke, by the Mathematicien principall or
+deriuatiue, demonstrated or demonstrable.+ Full well I know, that he
+which inuenteth, or maketh these demonstrations, is generally called _A
+speculatiue Mechanicien_: which differreth nothyng from a _Mechanicall
+Mathematicien_. So, in respect of diuerse actions, one man may haue the
+name of sundry artes: as, some tyme, of a Logicien, some tymes (in the
+same matter otherwise handled) of a Rethoricien. Of these trifles,
+I make, (as now, in respect of my Preface,) small account: to fyle thẽ
+for the fine handlyng of subtile curious disputers. In other places,
+they may commaunde me, to giue good reason: and yet, here, I will not be
+vnreasonable.
+
+
+    [+1.+]
+
+First, then, from the puritie, absolutenes, and Immaterialitie of
+Principall _Geometrie_, is that kinde of _Geometrie_ deriued, which
+vulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible
+magnitudes, their iust quãtities and contentes.+
+
+    [Geometrie vulgar.]
+
+This, teacheth to measure, either at hand: and the practiser, to be by
+the thing Measured: and so, by due applying of Cumpase, Rule, Squire,
+Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to
+the Length, Plaine, or Solide measured,
+
+    [1.]
+
+* to be certified, either of the length, perimetry, or distance lineall:
+and this is called, _Mecometrie_. Or
+
+    [2.]
+
+* to be certified of the content of any plaine Superficies: whether it
+be in ground Surueyed, Borde, or Glasse measured, or such like thing:
+which measuring, is named _Embadometrie_.
+
+    [3.]
+
+* Or els to vnderstand the Soliditie, and content of any bodily thing:
+as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels,
+small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale
+vessells, &c, the Measuring, commonly, hath a peculier name: and is
+called _Gaging_. And the generall name of these Solide measures, is
+_Stereometrie_.
+
+    [+2.+]
+
+Or els, this _vulgar Geometrie_, hath consideration to teach the
+practiser, how to measure things, with good distance betwene him and the
+thing measured: and to vnderstand thereby, either
+
+    [1.]
+
+* how Farre, a thing seene (on land or water) is from the measurer: and
+this may be called _Apomecometrie_:
+
+    [2.]
+
+Or, how High or depe, aboue or vnder the leuel of the measurers stãding,
+any thing is, which is sene on land or water, called _Hypsometrie_.
+
+    [3.]
+
+* Or, it informeth the measurer, how Broad any thing is, which is in the
+measurers vew: so it be on Land or Water, situated: and may be called
+_Platometrie_. Though I vse here to condition, the thing measured, to be
+on Land, or Water Situated:
+
+    [Note.]
+
+yet, know for certaine, that the sundry heigthe of Cloudes, blasing
+Starres, and of the Mone, may (by these meanes) haue their distances
+from the earth: and, of the blasing Starres and Mone, the Soliditie
+(aswell as distances) to be measured: But because, neither these things
+are vulgarly taught: nor of a common practiser so ready to be executed:
+I, rather, let such measures be reckened incident to some of our other
+Artes, dealing with thinges on high, more purposely, then this vulgar
+Land measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c._
+
+
+Of these Feates (farther applied) is Sprong the Feate of _Geodesie_, or
+Land Measuring: more cunningly to measure & Suruey Land, Woods, and
+Waters, a farre of. More cunningly, I say: But God knoweth (hitherto) in
+these Realmes of England and Ireland (whether through ignorance or
+fraude, I can not tell, in euery particular)
+
+    [Note.]
+
+how great wrong and iniurie hath (in my time) bene committed by vntrue
+measuring and surueying of Land or Woods, any way. And, this I am sure:
+that the Value of the difference, betwene the truth and such Surueyes,
+would haue bene hable to haue foũd (for euer) in eche of our two
+Vniuersities, an excellent Mathematicall Reader: to eche, allowing
+(yearly) a hundred Markes of lawfull money of this realme: which, in
+dede, would seme requisit, here, to be had (though by other wayes
+prouided for) as well, as, the famous Vniuersitie of Paris, hath two
+Mathematicall Readers: and eche, two hundreth French Crownes yearly, of
+the French Kinges magnificent liberalitie onely. Now, againe, to our
+purpose returning: Moreouer, of the former knowledge Geometricall, are
+growen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and
+_Stratarithmetrie_.
+
+
+“+‡Geographie‡+ teacheth wayes, by which, in sũdry formes, (as
+_Sphærike_, _Plaine_ or other), the Situation of Cities, Townes,
+Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, &
+such other things, vpõ the outface of the earthly Globe (either in the
+whole, or in some principall mẽber and portion therof cõtayned) may be
+described and designed, in cõmensurations Analogicall to Nature and
+veritie: and most aptly to our vew, may be represented.” Of this Arte
+how great pleasure, and how manifolde commodities do come vnto vs, daily
+and hourely: of most men, is perceaued. While, some, to beautifie their
+Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other
+some, for thinges past, as battels fought, earthquakes, heauenly
+fyringes, & such occurentes, in histories mentioned: therby liuely, as
+it were, to vewe the place, the region adioyning, the distance from vs:
+and such other circumstances. Some other, presently to vewe the large
+dominion of the Turke: the wide Empire of the Moschouite: and the litle
+morsell of ground, where Christendome (by profession) is certainly
+knowen. Litle, I say, in respecte of the rest. &c. Some, either for
+their owne iorneyes directing into farre landes: or to vnderstand of
+other mens trauailes. To conclude, some, for one purpose: and some, for
+an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, &
+Geographicall Globes. Of whose vse, to speake sufficiently, would
+require a booke peculier.
+
+
++‡Chorographie‡+ seemeth to be an vnderling, and a twig, of
+_Geographie_: and yet neuerthelesse, is in practise manifolde, and in
+vse very ample. “This teacheth Analogically to describe a small portion
+or circuite of ground, with the contentes: not regarding what
+commensuration it hath to the whole, or any parcell, without it,
+contained. But in the territory or parcell of ground which it taketh in
+hand to make description of, it leaueth out (or vndescribed) no notable,
+or odde thing, aboue the ground visible. Yea and sometimes, of thinges
+vnder ground, geueth some peculier marke: or warning: as of Mettall
+mines, Cole pittes, Stone quarries. &c.” Thus, a Dukedome, a Shiere,
+a Lordship, or lesse, may be described distinctly. But marueilous
+pleasant, and profitable it is, in the exhibiting to our eye, and
+commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true
+Symmetry: not approching to any of them: and out of Gunne shot. &c.
+Hereby, the _Architect_ may furnishe him selfe, with store of what
+patterns he liketh: to his great instruction: euen in those thinges
+which outwardly are proportioned: either simply in them selues: or
+respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also,
+terme this particular description of places, _Topographie_.
+
+
+“+‡Hydrographie‡+, deliuereth to our knowledge, on Globe or in Plaine,
+the perfect Analogicall description of the Ocean Sea coastes, through
+the whole world: or in the chiefe and principall partes thereof:” with
+the Iles and chiefe particular places of daungers, conteyned within the
+boundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes,
+Rockes, Races, Countertides, Whorlepooles. &c. This, dealeth with the
+Element of the water chiefly: as _Geographie_ did principally take the
+Element of the Earthes description (with his appertenances) to taske.
+And besides thys, _Hydrographie_, requireth a particular Register of
+certaine Landmarkes (where markes may be had) from the sea, well hable
+to be skried, in what point of the Seacumpase they appeare, and what
+apparent forme, Situation, and bignes they haue, in respecte of any
+daungerous place in the sea, or nere vnto it, assigned: And in all
+Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes,
+come and go, the _Hydrographer_ ought to recorde. The Soundinges
+likewise: and the Chanels wayes: their number, and depthes ordinarily,
+at ebbe and flud, ought the _Hydrographer_, by obseruation and diligence
+of _Measuring_, to haue certainly knowen. And many other pointes, are
+belonging to perfecte _Hydrographie_, and for to make a _Rutter_, by: of
+which, I nede not here speake: as of the describing, in any place, vpon
+Globe or Plaine, the 32. pointes of the Compase, truely: (wherof,
+scarsly foure, in England, haue right knowledge: bycause, the lines
+therof, are no straight lines, nor Circles.) Of making due proiection of
+a Sphere in plaine. Of the Variacion of the Compas, from true Northe:
+And such like matters (of great importance, all) I leaue to speake of,
+in this place: bycause, I may seame (al ready) to haue enlarged the
+boundes, and duety of an Hydographer, much more, then any man (to this
+day) hath noted, or prescribed. Yet am I well hable to proue, all these
+thinges, to appertaine, and also to be proper to the Hydrographer. The
+chief vse and ende of this Art, is the Art of Nauigation: but it hath
+other diuerse vses: euen by them to be enioyed, that neuer lacke sight
+of land.
+
+
++‡Stratarithmetrie‡+, is the Skill, (appertainyng to the warre,) by
+which a man can set in figure, analogicall to any _Geometricall_ figure
+appointed, any certaine number or summe of men: of such a figure
+capable: (by reason of the vsuall spaces betwene Souldiers allowed: and
+for that, of men, can be made no Fractions. Yet, neuertheles, he can
+order the giuen summe of men, for the greatest such figure, that of
+them, cã be ordred) and certifie, of the ouerplus: (if any be) and of
+the next certaine summe, which, with the ouerplus, will admit a figure
+exactly proportionall to the figure assigned. By which Skill, also, of
+any army or company of men: (the figure & sides of whose orderly
+standing, or array, is knowen) he is able to expresse the iust number of
+men, within that figure conteined: or (orderly) able to be conteined.
+
+    [* Note.]
+
+* And this figure, and sides therof, he is hable to know: either beyng
+by, and at hand: or a farre of. Thus farre, stretcheth the description
+and property of _Stratarithmetrie_: sufficient for this tyme and place.
+
+    [The difference betwene Stratarithmetrie and Tacticie.]
+
+“It differreth from the Feate _Tacticall_, _De aciebus instruendis._
+bycause, there, is necessary the wisedome and foresight, to what purpose
+he so ordreth the men: and Skillfull hability, also, for any occasion,
+or purpose, to deuise and vse the aptest and most necessary order, array
+and figure of his Company and Summe of men.” By figure, I meane: as,
+either of a _Perfect Square_, _Triangle_, _Circle_, _Ouale_, _long
+square_, (of the Grekes it is called _Eteromekes_) _Rhombe_, _Rhomboïd_,
+_Lunular_, _Ryng_, _Serpentine_, and such other Geometricall figures:
+Which, in warres, haue ben, and are to be vsed: for commodiousnes,
+necessity, and auauntage &c. And no small skill ought he to haue, that
+should make true report, or nere the truth, of the numbers and Summes,
+of footemen or horsemen, in the Enemyes ordring. A farre of, to make an
+estimate, betwene nere termes of More and Lesse, is not a thyng very
+rife, among those that gladly would do it.
+
+    [I. D.
+    Frende, you will finde it hard, to performe my description
+    of this Feate. But by Chorographie, you may helpe your selfe
+    some what: where the Figures knowne (in Sides and Angles)
+    are not Regular: And where, Resolution into Triangles can
+    serue. &c. And yet you will finde it strange to deale thus
+    generally with Arithmeticall figures: and, that for Battayle
+    ray. Their contentes, differ so much from like Geometricall
+    Figures.]
+
+Great pollicy may be vsed of the Capitaines, (at tymes fete, and in
+places conuenient) as to vse Figures, which make greatest shew, of so
+many as he hath: and vsing the aduauntage of the three kindes of vsuall
+spaces: (betwene footemen or horsemen) to take the largest: or when he
+would seme to haue few, (beyng many:) contrarywise, in Figure, and
+space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer
+is carefull to come nere the truth herein, besides the Iudgement of his
+expert eye, his skill of Ordering _Tacticall_, the helpe of his
+Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously
+framed for cariage and vse) He may wonderfully helpe him selfe, by
+perspectiue Glasses. In which, (I trust) our posterity will proue more
+skillfull and expert, and to greater purposes, then in these dayes, can
+(almost) be credited to be possible.
+
+
+Thus haue I lightly passed ouer the Artificiall Feates, chiefly
+dependyng vpon vulgar _Geometrie_: & commonly and generally reckened
+vnder the name of _Geometrie_. But there are other (very many)
+_Methodicall Artes_, which, declyning from the purity, simplicitie, and
+Immateriality, of our Principall Science of _Magnitudes_: do yet
+neuertheles vse the great ayde, direction, and Method of the sayd
+principall Science, and haue propre names, and distinct: both from the
+Science of _Geometrie_, (from which they are deriued) and one from the
+other. As +Perspectiue, Astronomie, Musike, Cosmographie, Astrologie,
+Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie,
+Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture,
+Nauigation, Thaumaturgike+ and +Archemastrie+. I thinke it necessary,
+orderly, of these to giue some peculier descriptions: and withall, to
+touch some of their commodious vses, and so to make this Preface, to be
+a little swete, pleasant Nosegaye for you: to comfort your Spirites,
+beyng almost out of courage, and in despayre, (through brutish brute)
+Weenyng that _Geometrie_, had but serued for buildyng of an house, or a
+curious bridge, or the roufe of Westminster hall, or some witty pretty
+deuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That the
+thing is farre otherwise, then the world, (commonly) to this day, hath
+demed, by worde and worke, good profe wilbe made.
+
+
+Among these Artes, by good reason, +‡Perspectiue‡+ ought to be had, ere
+of _Astronomicall Apparences_, perfect knowledge can be atteyned. And
+bycause of the prerogatiue of _Light_, beyng the first of _Gods
+Creatures_: and the eye, the light of our body, and his Sense most
+mighty, and his organ most Artificiall and _Geometricall_: At
+_Perspectiue_, we will begyn therfore. +Perspectiue, is an Art
+Mathematicall, which demonstrateth the maner, and properties, of all
+Radiations Direct, Broken, and Reflected.+ This Description, or
+Notation, is brief: but it reacheth so farre, as the world is wyde. It
+concerneth all Creatures, all Actions, and passions, by Emanation of
+beames perfourmed. Beames, or naturall lines, (here) I meane, not of
+light onely, or of colour (though they, to eye, giue shew, witnes, and
+profe, wherby to ground the Arte vpon) but also of other _Formes_, both
+_Substantiall_, and _Accidentall_, the certaine and determined actiue
+Radiall emanations. By this Art (omitting to speake of the highest
+pointes) we may vse our eyes, and the light, with greater pleasure: and
+perfecter Iudgement: both of things, in light seen, & of other: which by
+like order of Lightes Radiations, worke and produce their effectes. We
+may be ashamed to be ignorant of the cause, why so sundry wayes our eye
+is deceiued, and abused: as, while the eye weeneth a roũd Globe or
+Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise
+iudgeth a plaine Square, to be roũd: supposeth walles parallels, to
+approche, a farre of: rofe and floure parallels, the one to bend
+downward, the other to rise vpward, at a little distance from you.
+Againe, of thinges being in like swiftnes of mouing, to thinke the
+nerer, to moue faster: and the farder, much slower. Nay, of two thinges,
+wherof the one (incomparably) doth moue swifter then the other, to deme
+the slower to moue very swift, & the other to stand: what an error is
+this, of our eye? Of the Raynbow, both of his Colours, of the order of
+the colours, of the bignes of it, the place and heith of it, (&c) to
+know the causes demonstratiue, is it not pleasant, is it not necessary?
+of two or three Sonnes appearing: of Blasing Sterres: and such like
+thinges: by naturall causes, brought to passe, (and yet neuertheles, of
+farder matter, Significatiue) is it not commodious for man to know the
+very true cause, & occasion Naturall? Yea, rather, is it not, greatly,
+against the Souerainty of Mans nature, to be so ouershot and abused,
+with thinges (at hand) before his eyes? as with a Pecockes tayle, and a
+Doues necke: or a whole ore, in water, holden, to seme broken. Thynges,
+farre of, to seeme nere: and nere, to seme farre of. Small thinges, to
+seme great: and great, to seme small. One man, to seme an Army. Or a man
+to be curstly affrayed of his owne shaddow. Yea, so much, to feare,
+that, if you, being (alone) nere a certaine glasse, and proffer, with
+dagger or sword, to foyne at the glasse, you shall suddenly be moued to
+giue backe (in maner) by reason of an Image,
+
+    [☞ A marueilous Glasse.]
+
+appearing in the ayre, betwene you & the glasse, with like hand, sword
+or dagger, & with like quicknes, foyning at your very eye, likewise as
+you do at the Glasse. Straunge, this is, to heare of: but more
+meruailous to behold, then these my wordes can signifie. And
+neuerthelesse by demonstration Opticall, the order and cause therof, is
+certified: euen so, as the effect is consequent. Yea, thus much more,
+dare I take vpon me, toward the satisfying of the noble courrage, that
+longeth ardently for the wisedome of Causes Naturall: as to let him
+vnderstand, that, in London, he may with his owne eyes, haue profe of
+that, which I haue sayd herein. A Gentleman, (which, for his good
+seruice, done to his Countrey, is famous and honorable:
+
+    [S. W. P.]
+
+and for skill in the Mathematicall Sciences, and Languages, is the Od
+man of this land. &c.) euen he, is hable: and (I am sure) will, very
+willingly, let the Glasse, and profe be sene: and so I (here) request
+him: for the encrease of wisedome, in the honorable: and for the
+stopping of the mouthes malicious: and repressing the arrogancy of the
+ignorant. Ye may easily gesse, what I meane. This Art of _Perspectiue_,
+is of that excellency, and may be led, to the certifying, and executing
+of such thinges, as no man would easily beleue: without Actuall profe
+perceiued. I speake nothing of _Naturall Philosophie_, which, without
+_Perspectiue_, can not be fully vnderstanded, nor perfectly atteined
+vnto. Nor, of _Astronomie_: which, without _Perspectiue_, can not well
+be grounded: Nor _Astrologie_, naturally Verified, and auouched. That
+part hereof, which dealeth with Glasses (which name, Glasse, is a
+generall name, in this Arte, for any thing, from which, a Beame
+reboundeth) is called _Catoptrike_: and hath so many vses, both
+merueilous, and proffitable: that, both, it would hold me to long, to
+note therin the principall conclusions, all ready knowne: And also
+(perchaunce) some thinges, might lacke due credite with you: And I,
+therby, to leese my labor: and you, to slip into light Iudgement,
+
+    [* ☞]
+
+Before you haue learned sufficiently the powre of Nature and Arte.
+
+
+Now, to procede: +‡Astronomie‡, is an Arte Mathematicall, which
+demonstrateth the distance, magnitudes, and all naturall motions,
+apparences, and passions propre to the Planets and fixed Sterres: for
+any time past, present and to come: in respect of a certaine Horizon, or
+without respect of any Horizon.+ By this Arte we are certified of the
+distance of the Starry Skye, and of eche _Planete_ from the Centre of
+the Earth: and of the greatnes of any Fixed starre sene, or _Planete_,
+in respect of the Earthes greatnes. As, we are sure (by this Arte) that
+the Solidity, Massines and Body of the _Sonne_, conteineth the quantitie
+of the whole Earth and Sea, a hundred thre score and two times, lesse by
+⅛ one eight parte of the earth. But the Body of the whole earthly globe
+and Sea, is bigger then the body of the Mone, three and forty times
+lesse by ⅛ of the Mone. Wherfore the _Sonne_ is bigger then the _Mone_,
+7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then
+the _Mone_. And yet the vnskillfull man, would iudge them a like bigge.
+Wherfore, of Necessity, the one is much farder from vs, then the other.
+The _Sonne_, when he is fardest from the earth (which, now, in our age,
+is, when he is in the 8. degree, of Cancer) is, 1179 Semidiameters of
+the Earth, distante. And the _Mone_ when she is fardest from the earth,
+is 68 Semidiameters of the earth and ⅓ The nerest, that the _Mone_
+commeth to the earth, is Semidiameters 52¼ The distance of the Starry
+Skye is, frõ vs, in Semidiameters of the earth 20081½ Twenty thousand
+fourescore, one, and almost a halfe. Subtract from this, the _Mones_
+nerest distance, from the Earth: and therof remaineth Semidiameters of
+the earth 20029¼ Twenty thousand nine and twenty and a quarter.
+
+    [Note.]
+
+So thicke is the heauenly Palace, that the _Planetes_ haue all their
+exercise in, and most meruailously perfourme the Commaũdement and Charge
+to them giuen by the omnipotent Maiestie of the king of kings. This is
+that, which in _Genesis_ is called _Ha Rakia_. Consider it well. The
+Semidiameter of the earth, cõteineth of our common miles 3436 4/11 three
+thousand, foure hundred thirty six and foure eleuenth partes of one
+myle: Such as the whole earth and Sea, round about, is 21600. One and
+twenty thousand six hundred of our myles. Allowyng for euery degree of
+the greatest circle, thre score myles. Now if you way well with your
+selfe but this litle parcell of frute _Astronomicall_, as concerning the
+bignesse, Distances of _Sonne_, _Mone_, _Sterry Sky_, and the huge
+massines of _Ha Rakia_, will you not finde your Consciences moued, with
+the kingly Prophet, to sing the confession of Gods Glory, and say, +_The
+Heauens declare the glory of God, and the Firmament =[Ha Rakia]= sheweth
+forth the workes of his handes_+. And so forth, for those fiue first
+staues, of that kingly Psalme. Well, well, It is time for some to lay
+hold on wisedome, and to Iudge truly of thinges: and notso to expound
+the Holy word, all by Allegories: as to Neglect the wisedome, powre and
+Goodnes of God, in, and by his Creatures, and Creation to be seen and
+learned. By parables and Analogies of whose natures and properties, the
+course of the Holy Scripture, also, declareth to vs very many Mysteries.
+The whole Frame of Gods Creatures, (which is the whole world,) is to vs,
+a bright glasse: from which, by reflexion, reboundeth to our knowledge
+and perceiuerance, Beames, and Radiations: representing the Image of his
+Infinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taught
+and persuaded to Glorifie our Creator, as God: and be thankefull
+therfore. Could the Heathenistes finde these vses, of these most pure,
+beawtifull, and Mighty Corporall Creatures: and shall we, after that the
+true _Sonne_ of rightwisenesse is risen aboue the _Horizon_, of our
+temporall _Hemisphærie_, and hath so abundantly streamed into our
+hartes, the direct beames of his goodnes, mercy, and grace: Whose heat
+All Creatures feele: Spirituall and Corporall: Visible and Inuisible.
+Shall we (I say) looke vpon the _Heauen_, _Sterres_, and _Planets_, as
+an Oxe and an Asse doth: no furder carefull or inquisitiue, what they
+are: why were they Created, How do they execute that they were Created
+for? Seing, All Creatures, were for our sake created: and both we, and
+they, Created, chiefly to glorifie the Almighty Creator: and that, by
+all meanes, to vs possible. _Nolite ignorare_ (saith _Plato in
+Epinomis_) _Astronomiam, Sapientissimũ quiddam esse._ +_Be ye not
+ignorant, Astronomie to be a thyng of excellent wisedome._+
+_Astronomie_, was to vs, from the beginning commended, and in maner
+commaunded by God him selfe. In asmuch as he made the _Sonne_, _Mone_,
+and _Sterres_, to be to vs, for _Signes_, and knowledge of Seasons, and
+for Distinctions of Dayes, and yeares. Many wordes nede not. But I wish,
+euery man should way this word, _Signes_. And besides that, conferre it
+also with the tenth Chapter of _Hieremie_. And though Some thinke, that
+there, they haue found a rod: Yet Modest Reason, will be indifferent
+Iudge, who ought to be beaten therwith, in respect of our purpose.
+Leauing that: I pray you vnderstand this: that without great diligence
+of Obseruation, examination and Calculation, their periods and courses
+(wherby _Distinction_ of Seasons, yeares, and New Mones might precisely
+be knowne) could not exactely be certified. Which thing to performe, is
+that _Art_, which we here haue Defined to be _Astronomie_. Wherby, we
+may haue the distinct Course of Times, dayes, yeares, and Ages: aswell
+for Consideratiõ of Sacred Prophesies, accomplished in due time,
+foretold: as for high Mysticall Solemnities holding: And for all other
+humaine affaires, Conditions, and couenantes, vpon certaine time,
+betwene man and man: with many other great vses: Wherin, (verely), would
+be great incertainty, Confusion, vntruth, and brutish Barbarousnes:
+without the wonderfull diligence and skill of this Arte: continually
+learning, and determining Times, and periodes of Time, by the Record of
+the heauenly booke, wherin all times are written: and to be read with an
+_Astronomicall staffe_, in stede of a festue.
+
+
++‡Musike‡+, of Motion, hath his Originall cause: Therfore, after the
+motions most swift, and most Slow, which are in the Firmament, of Nature
+performed: and vnder the _Astronomers Consideration_: now I will Speake
+of an other kinde of _Motion_, producing sound, audible, and of Man
+numerable. _Musike_ I call here that _Science_, which of the Grekes is
+called _Harmonice_. Not medling with the Controuersie betwene the
+auncient _Harmonistes_, and _Canonistes_. +Musike is a Mathematicall
+Science, which teacheth, by sense and reason, perfectly to iudge, and
+order the diuersities of soundes, hye and low.+ _Astronomie_ and
+_Musike_ are Sisters, saith _Plato_. As, for _Astronomie_, the eyes: So,
+for _Harmonious Motion_, the eares were made. But as _Astronomie_ hath a
+more diuine Contemplation, and cõmodity, then mortall eye can perceiue:
+So, is _Musike_ to be considered,
+
+    [1.]
+
+that the * Minde may be preferred, before the eare. And from audible
+sound, we ought to ascende, to the examination: which numbers are
+_Harmonious_, and which not. And why, either, the one are: or the other
+are not. I could at large,
+
+    [2.]
+
+in the heauenly * motions and distances, describe a meruailous Harmonie,
+of _Pythagoras_ Harpe
+
+    [3.]
+
+with eight stringes. Also, somwhat might be sayd of _Mercurius_ * two
+Harpes,
+
+    [4.]
+
+eche of foure Stringes Elementall. And very straunge matter, might be
+alledged of the _Harmonie_,
+
+    [5.]
+
+to our * Spirituall part appropriate. As in _Ptolomaus_ third boke, in
+the fourth and sixth Chapters may appeare. *
+
+    [6.]
+
+And what is the cause of the apt bonde, and frendly felowship, of the
+Intellectuall and Mentall part of vs, with our grosse & corruptible
+body: but a certaine Meane, and _Harmonious Spiritualitie, with both
+participatyng, & of both (in a maner) resultynge In
+
+    [7.]
+
+the * Tune of Mans voyce, and also
+
+    [8.]
+
+* the sound of Instrument_, what might be sayd, of _Harmonie_: No common
+Musicien would lightly beleue.
+
+    [I. D.
+    Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7.
+    chapters. Where you shall haue some occasion farder to thinke
+    of Musike, than commonly is thought.]
+
+But of the sundry Mixture (as I may terme it) and concurse, diuerse
+collation, and Application of these _Harmonies_: as of thre, foure,
+fiue, or mo: Maruailous haue the effectes ben: and yet may be founde,
+and produced the like: with some proportionall consideration for our
+time, and being: in respect of the State, of the thinges then: in which,
+and by which, the wondrous effectes were wrought. _Democritus_ and
+_Theophrastus_ affirmed, that, by _Musike_, griefes and diseases of the
+Minde, and body might be cured, or inferred. And we finde in Recorde,
+that _Terpander_, _Arion_, _Ismenias_, _Orpheus_, _Amphion_, _Dauid_,
+_Pythagoras_, _Empedocles_, _Asclepiades_ and _Timotheus_, by
+_Harmonicall_ Consonãcy, haue done, and brought to pas, thinges, more
+then meruailous, to here of. Of them then, making no farder discourse,
+in this place: Sure I am, that Common _Musike_, commonly vsed, is found
+to the _Musiciens_ and Hearers, to be so Commodious and pleasant, That
+if I would say and dispute, but thus much: That it were to be otherwise
+vsed, then it is, I should finde more repreeuers, then I could finde
+priuy, or skilfull of my meaning. In thinges therfore euident, and
+better knowen, then I can expresse: and so allowed and liked of, (as I
+would wish, some other thinges, had the like hap) I will spare to
+enlarge my lines any farder, but consequently follow my purpose.
+
+
++‡Of Cosmographie‡+, I appointed briefly in this place, to geue you some
+intelligence. +Cosmographie, is the whole and perfect description of the
+heauenly, and also elementall parte of the world, and their homologall
+application, and mutuall collation necessarie.+ This Art, requireth
+_Astronomie_, _Geographie_, _Hydrographie_ and _Musike_. Therfore, it is
+no small Arte, nor so simple, as in common practise, it is (slightly)
+considered. This matcheth Heauen, and the Earth, in one frame, and aptly
+applieth parts Correspõdent: So, as, the Heauenly Globe, may (in
+practise) be duely described vpon the Geographicall, and Hydrographicall
+Globe. And there, for vs to consider an _Æquonoctiall Circle_, _an
+Ecliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes,
+Latitudes, Declinations, and Verticalitie: also Climes, and Parallels:
+and by an _Horizon_ annexed, and reuolution of the earthly Globe (as the
+Heauen, is, by the _Primouant_, caried about in 24. æquall Houres) to
+learne the Risinges and Settinges of Sterres (of _Virgill_ in his
+_Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sphære_,
+to Perdicca King of the Macedonians: of _Diocles_, to King _Antigonus_,
+and of other famous _Philosophers_ prescribed) a thing necessary, for
+due manuring of the earth, for _Nauigation_, for the Alteration of mans
+body: being, whole, Sicke, wounded, or brused. By the Reuolution, also,
+or mouing of the Globe Cosmographicall, the Rising and Setting of the
+Sonne: the Lengthes, of dayes and nightes: the Houres and times (both
+night and day) are knowne: with very many other pleasant and necessary
+vses: Wherof, some are knowne: but better remaine, for such to know and
+vse:
+
+    [☞]
+
+who of a sparke of true fire, can make a wonderfull bonfire, by applying
+of due matter, duely.
+
+
++‡Of Astrologie‡+, here I make an Arte, seuerall from _Astronomie_: not
+by new deuise, but by good reason and authoritie: for, +Astrologie, is
+an Arte Mathematicall, which reasonably demonstrateth the operations and
+effectes, of the naturall beames, of light, and secrete influence: of
+the Sterres and Planets: in euery element and elementall body: at all
+times, in any Horizon assigned.+ This Arte is furnished with many other
+great Artes and experiences: As with perfecte _Perspectiue_,
+_Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4.
+Elementes, the Arte of Graduation, and some good vnderstãding in
+_Musike_: and yet moreouer, with an other great Arte, hereafter
+following, though I, here, set this before, for some considerations me
+mouing. Sufficient (you see) is the stuffe, to make this rare and
+secrete Arte, of: and hard enough to frame to the Conclusion
+Syllogisticall. Yet both the manifolde and continuall trauailes of the
+most auncient and wise Philosophers, for the atteyning of this Arte: and
+by examples of effectes, to confirme the same: hath left vnto vs
+sufficient proufe and witnesse: and we, also, daily may perceaue, That
+mans body, and all other Elementall bodies, are altered, disposed,
+ordred, pleasured, and displeasured, by the Influentiall working of the
+_Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth
+_Aristotle_, in the first of his _Meteorologicall_ bookes, in the second
+Chapter: _Est autem necessariò Mundus iste, supernis lationibus ferè
+continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causà prima
+putanda omnibus est, vnde motus principium existit._ That is: +_This
+=[Elementall]= World is of necessitie, almost, next adioyning, to the
+heauenly motions: That, from thence, all his vertue or force may be
+gouerned. For, that is to be thought the first Cause vnto all: from
+which, the beginning of motion, is._+ And againe, in the tenth Chapter.
+_Oportet igitur & horum principia sumamus, & causas omnium similiter.
+Principium igitur vt mouens, præcipuum[que] & omnium primum, Circulus
+ille est, in quo manifeste Solis latio, &c._ And so forth. His
+_Meteorologicall_ bookes, are full of argumentes, and effectuall
+demonstrations, of the vertue, operation, and power of the heauenly
+bodies, in and vpon the fower Elementes, and other bodies, of them
+(either perfectly, or vnperfectly) composed. And in his second booke,
+_De Generatione & Corruptione_, in the tenth Chapter. _Quocirca & prima
+latio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea
+nam[que] & continua est, & duobus motibus fit:_ In Englishe, thus.
++_Wherefore the vppermost motion, is not the cause of Generation and
+Corruption, but the motion of the Zodiake: for, that, both, is
+continuall, and is caused of two mouinges._+ And in his second booke,
+and second Chapter of hys _Physikes_. _Homo nam[que] generat hominem,
+at[que] Sol._ +_For Man (sayth he) and the Sonne, are cause of mans
+generation._+ Authorities may be brought, very many: both of 1000. 2000.
+yea and 3000. yeares Antiquitie: of great _Philosophers_, _Expert_,
+_Wise_, and godly men, for that Conclusion: which, daily and hourely, we
+men, may discerne and perceaue by sense and reason: All beastes do
+feele, and simply shew, by their actions and passions, outward and
+inward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally,
+the Elementes, and all thinges of the Elementes composed, do geue
+Testimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions,
+vertues, and naturall motions, depend of the Actiuitie of the heauenly
+motions and Influences. Whereby, beside the specificall order and forme,
+due to euery seede: and beside the Nature, propre to the Indiuiduall
+Matrix, of the thing produced: What shall be the heauenly Impression,
+the perfect and circumspecte Astrologien hath to Conclude._+ Not onely
+(by _Apotelesmes_) τὸ ὁτὶ]. but by Naturall and Mathematicall
+demonstration τὸ διότι. Whereunto, what Sciences are requisite (without
+exception) I partly haue here warned: And in my _Propædeumes_ (besides
+other matter there disclosed) I haue Mathematically furnished vp the
+whole Method: To this our age, not so carefully handled by any, that
+euer I saw, or heard of. I was,
+
+    [* Anno. 1548 and 1549. in Louayn.]
+
+(for * 21. yeares ago) by certaine earnest disputations, of the Learned
+_Gerardus Mercator_, and _Antonius Gogaua_, (and other,) therto so
+prouoked: and (by my constant and inuincible zeale to the veritie) in
+obseruations of Heauenly Influencies (to the Minute of time,) than, so
+diligent: And chiefly by the Supernaturall influence, from the Starre of
+Iacob, so directed: That any Modest and Sober Student, carefully and
+diligently seking for the Truth, will both finde & cõfesse, therin, to
+be the Veritie, of these my wordes: And also become a Reasonable
+Reformer, of three Sortes of people: about these Influentiall
+Operations, greatly erring from the truth.
+
+    [Note.]
+
+Wherof, the one, is +Light Beleuers+, the other, +Light Despisers+, and
+the third +Light Practisers+. The first, & most cõmon Sort, thinke the
+Heauen and Sterres, to be answerable to any their doutes or desires:
+
+    [1.]
+
+which is not so: and, in dede, they, to much, ouer reache. The Second
+sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare
+any Sway in Generation
+
+    [2.]
+
+and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and
+_Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so
+farre in distance, so manifold in their motions, so constant in their
+periodes. &c.) they assigne a sleight, simple office or two, and so
+allow vnto thẽ (according to their capacities) as much vertue, and power
+Influentiall, as to the Signe of the _Sunne_, _Mone_, and seuen Sterres,
+hanged vp (for Signes) in London, for distinction of houses, & such
+grosse helpes, in our worldly affaires: And they vnderstand not (or will
+not vnderstand) of the other workinges, and vertues of the Heauenly
+_Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husband
+man: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, as
+the Porpentine doth: nor will allow these perfect, and incorruptible
+mighty bodies, so much vertuall Radiation, & Force, as they see in a
+litle peece of a _Magnes stone_: which, at great distance, sheweth his
+operation. And perchaunce they thinke, the Sea & Riuers (as the Thames)
+to be some quicke thing, and so to ebbe, and flow, run in and out, of
+them selues, at their owne fantasies. God helpe, God helpe. Surely,
+these men, come to short: and either are to dull: or willfully blind:
+or, perhaps, to malicious. The third man, is the common and vulgare
+_Astrologien_, or Practiser: who, being not duely, artificially, and
+perfectly
+
+    [3.]
+
+furnished: yet, either for vaine glory, or gayne: or like a simple dolt,
+& blinde Bayard, both in matter and maner, erreth: to the discredit of
+the _Wary_, and modest _Astrologien_: and to the robbing of those most
+noble corporall Creatures, of their Naturall Vertue: being most mighty:
+most beneficiall to all elementall Generation, Corruption and the
+appartenances: and most Harmonious in their Monarchie: For which
+thinges, being knowen, and modestly vsed: we might highly, and
+continually glorifie God, with the princely Prophet, saying. +_The
+Heauens declare the Glorie of God: who made the Heauẽs in his wisedome:
+who made the Sonne, for to haue dominion of the day: the Mone and
+Sterres to haue dominion of the nyght: whereby, Day to day vttereth
+talke: and night, to night declareth knowledge. Prayse him, all ye
+Sterres, and Light. Amen._+
+
+
+In order, now foloweth, of +‡Statike‡+, somewhat to say, what we meane
+by that name: and what commodity, doth, on such Art, depend. +Statike,
+is an Arte Mathematicall, which demonstrateth the causes of heauynes,
+and lightnes of all thynges: and of motions and properties, to heauynes
+and lightnes, belonging.+ And for asmuch as, by the Bilanx, or Balance
+(as the chief sensible Instrument,) Experience of these demonstrations
+may be had: we call this Art, _Statike:_ that is, _the Experimentes of
+the Balance_. Oh, that men wist, what proffit, (all maner of wayes) by
+this Arte might grow, to the hable examiner, and diligent practiser.
+“Thou onely, knowest all thinges precisely (O God) who hast made weight
+and Balance, thy Iudgement: who hast created all thinges in _Number,
+Waight, and Measure_: and hast wayed the mountaines and hils in a
+Balance: who hast peysed in thy hand, both Heauen and earth. We therfore
+warned by the Sacred word, to Consider thy Creatures: and by that
+consideration, to wynne a glyms (as it were,) or shaddow of
+perceiuerance, that thy wisedome, might, and goodnes is infinite, and
+vnspeakable, in thy Creatures declared: And being farder aduertised, by
+thy mercifull goodnes, that, three principall wayes, were, of the, vsed
+in Creation of all thy Creatures, namely, _Number_, _Waight_ and
+_Measure_, And for as much as, of _Number_ and _Measure_, the two Artes
+(auncient, famous, and to humaine vses most necessary,) are, all ready,
+sufficiently knowen and extant: This third key, we beseche thee (through
+thy accustomed goodnes,) that it may come to the nedefull and sufficient
+knowledge, of such thy Seruauntes, as in thy workemanship, would gladly
+finde, thy true occasions (purposely of the vsed) whereby we should
+glorifie thy name, and shew forth (to the weaklinges in faith) thy
+wondrous wisedome and Goodnes. Amen.”
+
+
+Meruaile nothing at this pang (godly frend, you Gentle and zelous
+Student.) An other day, perchaunce, you will perceiue, what occasion
+moued me. Here, as now, I will giue you some ground, and withall some
+shew, of certaine commodities, by this Arte arising. And bycause this
+Arte is rare, my wordes and practises might be to darke: vnleast you had
+some light, holden before the matter: and that, best will be, in giuing
+you, out of _Archimedes_ demonstrations, a few principal Conclusions, as
+foloweth.
+
+  +1.+
+
+  +The Superficies of euery Liquor, by it selfe consistyng, and in
+  quyet, is Sphæricall: the centre whereof, is the same, which is the
+  centre of the Earth.+
+
+  +2.+
+
+  +If Solide Magnitudes, being of the same bignes, or quãtitie, that
+  any Liquor is, and hauyng also the same Waight: be let downe into
+  the same Liquor, they will settle downeward, so, that no parte of
+  them, shall be aboue the Superficies of the Liquor: and yet
+  neuertheles, they will not sinke vtterly downe, or drowne.+
+
+  +3.+
+
+  +If any Solide Magnitude beyng Lighter then a Liquor, be let downe
+  into the same Liquor, it will settle downe, so farre into the same
+  Liquor, that so great a quantitie of that Liquor, as is the parte of
+  the Solid Magnitude, settled downe into the same Liquor: is in
+  Waight, æquall, to the waight of the whole Solid Magnitude.+
+
+  +4.+
+
+  +Any Solide Magnitude, Lighter then a Liquor, forced downe into the
+  same Liquor, will moue vpward, with so great a power, by how much,
+  the Liquor hauyng æquall quantitie to the whole Magnitude, is
+  heauyer then the same Magnitude.+
+
+  +5.+
+
+  +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into
+  the same Liquor, will sinke downe vtterly: And wilbe in that Liquor,
+  Lighter by so much, as is the waight or heauynes of the Liquor,
+  hauing bygnes or quantitie, æquall to the Solid Magnitude.+
+
+  +6.+
+
+      [I. D.
+      The Cutting of a Sphære according to any proportion assigned
+      may by this proposition be done Mechanically by tempering
+      Liquor to a certayne waight in respect of the waight of the
+      Sphære therein Swymming.]
+
+  +If any Solide Magnitude, Lighter then a Liquor, be let downe into
+  the same Liquor, the waight of the same Magnitude, will be, to the
+  Waight of the Liquor. (Which is æquall in quantitie to the whole
+  Magnitude,) in that proportion, that the parte, of the Magnitude
+  settled downe, is to the whole Magnitude.+
+
+
+By these verities, great Errors may be reformed, in Opinion of the
+Naturall Motion of thinges, Light and Heauy. Which errors, are in
+Naturall Philosophie (almost) of all mẽ allowed: to much trusting to
+Authority: and false Suppositions. As, +Of any two bodyes, the heauyer,
+to moue downward faster then the lighter.+
+
+    [A common error, noted.]
+
+This error, is not first by me, Noted: but by one _Iohn Baptist de
+Benedictis_. The chief of his propositions, is this: which seemeth a
+Paradox.
+
+
++If there be two bodyes of one forme, and of one kynde, æquall in
+quantitie or vnæquall,
+
+    [A paradox.]
+
+they will moue by æquall space, in æquall tyme: So that both theyr
+mouynges be in ayre, or both in water: or in any one Middle.+
+
+
+Hereupon, in the feate of +Gunnyng+,
+
+    [N. T.]
+
+certaine good discourses (otherwise) may receiue great amendement, and
+furderance.
+
+    [The wonderfull vse of these Propositions.]
+
+In the entended purpose, also, allowing somwhat to the imperfection of
+Nature: not aunswerable to the precisenes of demonstration. Moreouer, by
+the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth,
+the Fire, may be nerely, knowen, how light or heauy they are (Naturally)
+in their assigned 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 Humours 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, vndemonstrated: 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.
+
+    [The practise Staticall, to know the proportion, betwene
+    the Cube, and the Sphære.]
+
+Make a Cube, of any one Vniforme: and through like heauy stuffe: of the
+same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall to
+the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne,
+Lead, Siluer. &c. (being, as I sayd, of like nature, 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,
+company & 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, somewhat 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: by eight descentes progressionall, * halfing, from a grayne.
+
+    [I. D.
+    * For, so, haue you .256. partes of a Graine.]
+
+Your Cube, Sphære, apt Balance, and conuenient waightes, being ready:
+fall to worke.❉. First, way your Cube. Note the Number of the waight.
+Way, after that, your Sphære. Note likewise, the Nũber of the waight. If
+you now find the waight of your Cube, to be to the waight of the Sphære,
+as 21. is to 11: Then you see, how the Mechanicien 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 Sphære. Then, chaunge, your Sphære 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 demonstrated, that a Cylinder, whose heith, and
+Diameter of his base, is æquall to the Diameter of the Sphære, is
+Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of
+the waight of the Sphære, adde halfe so much, as it is: and so haue you
+the number of the waight of that Cylinder. Which is also Comprehended of
+our former Cube: So, that the base of that Cylinder, is a Circle
+described in the Square, which is the base of our Cube. But the Cube and
+the Cylinder, 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,
+
+    [* =The proportion of the Square to the Circle inscribed.=]
+
+we haue * the proportion of the Square, to the Circle, inscribed in the
+same Square. And so are we fallen into the knowledge sensible, and
+Experimentall of _Archimedes_ great Secret: of him, by great trauaile of
+minde, sought and found. Wherfore, to any Circle giuen, you can giue a
+Square æquall:
+
+    [* =The Squaring of the Circle, Mechanically.=]
+
+* 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
+æquall:
+
+    [* =To any Square geuen, to geue a Circle, equall.=]
+
+* If 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 proposition of the twelfth boke of
+_Euclide_, in my third Probleme there. Your diligence 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 Sphære, & the Cube,
+what their proportion is: as now, you came from the Sphære to the
+Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough
+one hole drawẽ, as the maner is) if you make a Square figure & way it:
+and then, describing theron, the Circle inscribed: & 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, by the way.
+
+    [Note Squaring of the Circle without knowledge of the
+    proportion betwene Circumference and Diameter.]
+
+That we may Square the Circle, without hauing knowledge of the
+proportion, of the Circumference 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 intricate. 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 (before this day,) had it for you, that I can tell of. And that
+is, _A Mechanicall Dubblyng of the Cube: &c._ Which may, thus, be done:
+
+    [To Dubble the Cube redily: by Art Mechanicall: depending
+    vppon Demonstration Mathematicall.]
+
++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 approche (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:+
+
+    [=I. D.=
+    =The 4. sides of this Pyramis must be 4. Isosceles
+    Triangles alike and æquall.=]
+
++Let 4. straight lines be drawen, in the inside of the Cone and Pyramis:
+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
+Pyramis: and as many, in the Cone) diuide: one, in 12. æquall 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, as experience
+shall teach you.+
+
+    [=I. D.=
+    =* In all workinges with this Pyramis or Cone, Let their
+    Situations be in all Pointes and Conditions, alike,
+    or all one: while you are about one Worke. Els you
+    will erre.=]
+
++Then, * set your Cone or Pyramis, with the vertex downward,
+perpendicularly, in respect of the Base. (Though it be otherwayes, it
+hindreth nothyng.) So let thẽ 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
+Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so
+proportiõ in respect of your Pyramis or Cone, that the Pyramis or Cone,
+will be hable to conteine the waight of them, in water, 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: take the waight therof exactly 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 before:
+
+    [=I. D.=
+    =* Consider well whan you must put your waters togyther:
+    and whan, you must empty your first water, out of your
+    Pyramis or Cone. Els you will erre.=]
+
+put that* 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 æquall partes, as it is capable of,
+conueniently, with your ease, and precisenes 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
+nũber belonging to the Radicall side, of the Cube, dubble to your
+Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it
+selfe, whether 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 Fundamẽtall 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.
+
+    [☞ God be thanked for this Inuention, & the fruite ensuing.]
+
+For this, may I (with ioy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking the
+holy and glorious Trinity: hauing greater cause therto, then
+
+    [* Vitruuius. Lib. 9. Cap. 3.]
+
+* _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
+Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it
+with water, precisely 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.
+
+    [* Note.]
+
+* And if the Cube, which you should Double, be neuer so great: you haue,
+thus, the proportion (in small) betwene 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
+Euclide.
+
+
+    [Note, as concerning the Sphæricall Superficies of the Water.]
+
+Note, that all this while, I forget not my first Proposition Staticall,
+here rehearsed: that, the Superficies of the water, is Sphæricall.
+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 finding the swelling of the water aboue leuell._
+
+    [☞]
+
+“Square the Semidiameter, 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 Rote, 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, *
+
+    [* Note.]
+
+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 moyst 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 cõmoditie.
+
+
+Thus, may you Double the Cube Mechanically, Treble it, and so forth, in
+any proportion.
+
+    [Note this Abridgement of Dubbling the Cube. &c.]
+
+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 demõstration,
+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 Fundamentall 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 marking of the lines, how
+the Water doth cut them, shall geue you the proportion betwen the
+Radicall sides, of any two Cubes, wherof the one is Double to the other:
+working as before I haue taught you:
+
+    [* ☞ Note.]
+
+* 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:
+
+    [To giue Cubes one to the other in any proportion,
+    Rationall or Irrationall.]
+
++you may (now) geue Cubes, one to the other, in any proportiõ geuẽ:
+Rationall or Irrationall+: on this maner. Make a hollow Parallelipipedon
+of Copper or Tinne: with one Base wãting, or open: as in our Cubike
+Coffen. Frõ 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.
+
+    [* Emptying the first.]
+
+Poure that water * 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 second: 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
+assigned: were it Rationall or Irrationall.
+
+
+Thus, in sundry waies you may furnishe your selfe with such straunge and
+profitable matter: which, long hath bene wished for. And though it be
+Naturally done and Mechanically: yet hath it a good Demonstration
+Mathematicall.
+
+    [=The demonstrations of this Dubbling of the Cube, and of the
+rest.=]
+
+Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones,
+in the proportions 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, or Cone to Cone, be double,
+
+    [I. D.
+    = * Hereby, helpe your self to become a præcise practiser.
+    And so consider, how, nothing at all, you are hindred
+    (sensibly) by the Conuexitie of the water.=]
+
+then shall * 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 doubled: 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 treble, &c. as the Pyramids or
+Cones were, one to an other (as we haue proued) therfore, 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
+proportion, of the fower proportionall lines:
+
+    [= * By the 33. of the eleuenth booke of Euclide.=]
+
+therfore * 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 or Cone: Wherefore the Cube is to
+the Cube, as Pyramis is to Pyramis, or Cone is to Cone.
+
+    [I. D.
+    = * And your diligence in practise, can so (in waight of
+    water) performe it: Therefore, now, you are able to geue
+    good reason of your whole doing.=]
+
+But we * Suppose Pyramis 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 Parallelipipedõ, it is euidẽt, 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 proportiõ 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.
+
+    [* _Note this Corollary._]
+
+* This, my Demonstratiõ is more generall, then onely in Square Pyramis
+or Cone: Consider well. Thus, haue I, both Mathematically and
+Mechanically, ben very long in wordes: yet (I trust) nothing tedious to
+them, who, to these thinges, are well affected. And verily I am forced
+(auoiding prolixitie) to omit sundry such things, easie to be practised:
+which to the Mathematicien, would be a great Threasure: and to the
+Mechanicien, no small gaine.
+
+    [* The great Commodities following of these new Inuentions.]
+
+* Now may you, +Betwene two lines giuen, finde two middle proportionals,
+in Continuall proportion: by the hollow Parallelipipedon, and the hollow
+Pyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre
+right lines may be found, proportionall in any proportion assigned, of
+which, shal be produced a Parallelipipedon, æquall to the
+Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36.
+proposition, of the 11. boke of _Euclide_. Now, all those thinges, which
+_Vitruuius_ in his Architecture, specified hable to be done, by dubbling
+of the Cube: Or, by finding of two middle proportionall lines, betwene
+two lines giuen, may easely be performed. Now, that Probleme, which I
+noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke
+of _Euclide_, is proued possible. Now, may any regular body, be
+Transformed into an other, &c. Now, any regular body: any Sphere, yea
+any Mixt Solid: and (that more is) Irregular Solides, may be made (in
+any proportiõ 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 Manneken 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 * Gunne, or little peece of ordinaũce, make an
+other, with the same _Symmetrie_ (in all pointes) as great, and as
+little, as you will. Marke that: and thinke on it. Infinitely, +may you
+apply this, so long sought for, and now so easily concluded: and
+withall, so willingly and frankly communicated to such, as faithfully
+deale with vertuous studies.+
+
+    [Such is the Fruite of the Mathematicall Sciences and Artes.]
+
+Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte:
+and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he
+can, by order, Descend, to frame Naturall thinges, to wonderfull vses:
+and when he list, retire home into his owne Centre: and there, prepare
+more Meanes, to Ascend or Descend by: and, all, to the glory of God, and
+our honest delectation in earth.
+
+
+Although, the Printer, hath looked for this Præface, a day or two, yet
+could I not bring my pen from the paper, before I had giuen you
+comfortable warning, and brief instructions, of some of the Commodities,
+by _Statike_, hable to be reaped: In the rest, I will therfore, be as
+brief, as it is possible: and with all, describing them, somwhat
+accordingly. And that, you shall perceiue, by this, which in order
+commeth next. For, wheras, it is so ample and wonderfull, that, an whole
+yeare long, one might finde fruitfull matter therin, to speake of: and
+also in practise, is a Threasure endeles: yet will I glanse ouer it,
+with wordes very few.
+
+
+This do I call +‡Anthropographie‡+. Which is an Art restored, and of my
+preferment to your Seruice. I pray you, thinke of it, as of one of the
+chief pointes, of Humane knowledge. Although it be, but now, first
+Cõfirmed, with this new name: yet the matter, hath from the beginning,
+ben in consideration of all perfect Philosophers. +Anthropographie, is
+the description of the Number, Measure, Waight, figure, Situation, and
+colour of euery diuerse thing, conteyned in the perfect body of MAN:
+with certain knowledge of the Symmetrie, figure, waight,
+Characterization, and due locall motion, of any parcell of the sayd
+body, assigned: and of Nũbers, to the sayd parcell appertainyng.+ This,
+is the one part of the Definition, mete for this place: Sufficient to
+notifie, the particularitie, and excellency of the Arte: and why it is,
+here, ascribed to the Mathematicals. Yf the description of the heauenly
+part of the world, had a peculier Art, called _Astronomie:_ If the
+description of the earthly Globe, hath his peculier arte, called
+_Geographie_. If the Matching of both, hath his peculier Arte, called
+_Cosmographie:_ Which is the Descriptiõ of the whole, and vniuersall
+frame of the world: Why should not the description of
+
+    [MAN is the Lesse World.]
+
+him, who is the Lesse world: and, frõ the beginning, called
+_Microcosmus_ (that is. _The Lesse World._) And for whose sake, and
+seruice, all bodily creatures els, were created: Who, also,
+participateth with Spirites, and Angels: and is made to the Image 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 Chyromantistes some. The
+Metaposcopistes, some: The excellent, _Albert Durer_, a good part: the
+Arte of Perspectiue, will somwhat, for the Eye, helpe forward:
+_Pythagoras_, _Hipocrates_, _Plato_, _Galenus_, _Meletius_, & many other
+(in certaine thinges) will be Contributaries. And farder, the Heauen,
+the Earth, and all other Creatures, will eche shew, and offer their
+Harmonious seruice, to fill vp, that, which wanteth hereof: and with
+your own Experience, concluding: you may Methodically register the
+whole, for the posteritie: Whereby, good profe will be had, of our
+Harmonious, and
+
+    [Micro Cosmus.]
+
+Microcosmicall constitution.
+
+    [* ☞]
+
+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.
+
+    [* Lib. 3. Cap. 1.]
+
+Looke in * _Vitruuius_, whether I deale sincerely for your behoufe, or
+no. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Looke
+in the 27. and 28. Chapters, of the second booke, _De occulta
+Philosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther.
+Remember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ so
+long agoe 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 ignorance, and
+shame enough, come short of them Seuen also: and yet neuerthelesse they
+can not prescribe a certaine number of Artes: and in eche, certaine
+vnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly
+rise vp: and there was no such order taken, that,
+
+    [☞]
+
+All Artes, should in one age, or in one land, or of one man, be made
+knowen to the world. Let vs embrace the giftes of 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 Irregular lines, at pleasure,
+may be drawen. These, and such like, are principall Conclusions of this
+Arte: and helpe forward many pleasant and profitable Mechanicall workes:
+
+    [Saw Milles.]
+
+As Milles, to Saw great and very long Deale bordes, no man being by.
+Such haue I seene in Germany: and in the Citie of Prage: in the kingdome
+of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all
+maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight,
+Spring, Man or Beast, moued. Take in your hand, _Agricola De re
+Metallica:_ and then shall you (in all Mines) perceaue, how great nede
+is, of Whele worke. By Wheles, straunge workes and incredible, are done:
+as will, in other Artes hereafter, appeare. A wonderfull example of
+farther possibilitie, and present commoditie, was sene in my time, in a
+certaine Instrument: which by the Inuenter and Artificer (before) was
+solde for xx. Talentes of Golde: and then had (by misfortune) receaued
+some iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend the
+same, and presented it vnto the Emperour _Charles_ the fifth.
+_Hieronymus 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
+Mathematicall, which demonstrateth the designing of all Spirall lines in
+Plaine, on Cylinder, Cone, Sphære, Conoid, and Sphæroid, and their
+properties appertayning.+ The vse hereof, in _Architecture_, and diuerse
+Instrumentes and Engines, is most necessary. For, in many thinges, the
+Skrue worketh the feate, which, els, could not be performed. By helpe
+hereof,
+
+    [* Atheneus Lib. 5. cap. 8.]
+
+it is * recorded, that, where all the power of the Citie of Syracusa,
+was not hable to moue a certaine 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.
+
+    [Proclus. Pag. 18.]
+
+Wherat, the King wondring: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει
+λέγοντι πιϛευτεόν. _From this day, forward_ (said the King) _Credit
+ought to be giuen to Archimedes, what soeuer he sayth._
+
+
++‡Pneumatithmie‡ demonstrateth by close hollow Geometricall Figures,
+(regular and irregular) the straunge properties (in motion or stay) of
+the Water, Ayre, Smoke, and Fire, in theyr cõtinuitie, and as they are
+ioyned to the Elementes next them.+ This Arte, to the Naturall
+Philosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_
+is not in the world. And that, all Nature, abhorreth it so much: that,
+contrary to ordinary law, the Elementes will moue or stand. As, Water to
+ascend: rather then betwene him and Ayre, Space or place should be left,
+more then (naturally) that quãtitie 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 Cõtinuitie 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.
+
+    [To go to the bottom of the Sea without daunger.]
+
+Hereupon, two or three men together, by keping Ayre vnder a great
+Cauldron, and forcyng the same downe, orderly, may without harme descend
+to the Sea bottome: and continue there a tyme &c. Where, Note, how the
+thicker Element (as the Water) giueth place to the thynner (as, is the
+ayre:) and receiueth violence of the thinner, in maner. &c. Pumps and
+all maner of Bellowes, haue their ground of this Art: and many other
+straunge deuises. As, _Hydraulica_, Organes goyng by water. &c. Of this
+Feat, (called commonly _Pneumatica_,) goodly workes are extant, both in
+Greke, and Latin. With old and learned Schole men, it is called
+_Scientia de pleno & vacuo._
+
+
++‡Menadrie‡, is an Arte Mathematicall, which demonstrateth, how, aboue
+Natures vertue and power simple: Vertue and force may be multiplied: and
+so, to direct, to lift, to pull to, and to put or cast fro, any
+multiplied or simple, determined Vertue, Waight or Force: naturally,
+not, so, directible or moueable.+ Very much is this Art furdred by other
+Artes: as, in some pointes, by _Perspectiue_: in some, by _Statike_: in
+some, by _Trochilike_: and in other, by _Helicosophie_: and
+_Pneumatithmie_. By this Art, all Cranes, Gybbettes, & Ingines to lift
+vp, or to force any thing, any maner way, are ordred: and the certaine
+cause of their force, is knowne: As, the force which one man hath with
+the Duche waghen Racke: therwith, to set vp agayne, a mighty waghen
+laden, being ouerthrowne. The force of the Crossebow Racke, is
+certainly, here, demonstrated. The reason, why one mã, doth with a
+leauer, lift that, which Sixe men, with their handes onely, could not,
+so easily do. By this Arte, in our common Cranes in London, where powre
+is to Crane vp, the waight of 2000. pound: by two Wheles more (by good
+order added) Arte concludeth, that there may be Craned vp 200000. pound
+waight &c. So well knew _Archimedes_ this Arte: that he alone, with his
+deuises and engynes, (twise or thrise) spoyled and discomfited the whole
+Army and Hoste of the Romaines, besieging _Syracusa_,
+
+    [=Plutarchus in Marco Marcello.=]
+
+_Marcus Marcellus the Consul_, being their Generall Capitaine.
+
+    [=Synesius in Epistolis.=]
+
+Such huge Stones, so many, with such force, and so farre, did he with
+his engynes hayle among them, out of the Citie.
+
+    [=Polybius.=]
+
+    [=Plinius.=]
+
+    [=Quintilianus.=]
+
+    [=T. Liuius.=]
+
+And by Sea likewise: though their Ships might come to the walls of
+_Syracusa_, yet hee vtterly confounded the Romaine Nauye. What with his
+mighty Stones hurlyng:
+
+    [=* Athenæus.=]
+
+what with Pikes of * 18 fote long, made like shaftes: which he forced
+almost a quarter of a myle. What, with his catchyng hold of their Shyps,
+and hoysing them vp aboue the water, and suddenly letting them fall into
+the Sea againe:
+
+    [= * Galenus.=]
+
+    [=Anthemius.=]
+
+what with his * 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 _Proclus_,
+who so well had perceiued _Archimedes_ Arte of _Menadrie_, and had so
+well inuented of his owne, that with his Burning Glasses,
+
+    [Burning Glasses.]
+
+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 lightening) 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 Seuerus_ pardoned
+him, his life, after 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.
+
+    [Gunnes.]
+
+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, slight, and cõmon: 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
+Sphæricall 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 entrance, is knowen)
+certaine way may be præscribed and gone: And how, any way aboue the
+Superficies of the earth designed, may vnder earth, at any depth
+limited, be kept: goyng alwayes, perpendicularly, 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 Sphæricall Superficies of the earth, shall
+precisely fall in the Correspondent pointes of those two wayes. This,
+with all other Cases and circumstances 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
+(before this) was of me published, _De Itinere Subterraneo_: The rest,
+be at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole,
+and for secrete passages 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 yet 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 Water 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 waters
+labour be farre, and great) I neede not: Seing, at hand (about vs) many
+expert men can sufficiently testifie, in effecte, the order: though the
+Demonstration of the Necessitie thereof, they know not: Nor yet, if they
+should be led, vp and downe, and about Mountaines, from the head of the
+Spring: and then, a place being 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
+water to ascend, eyther by _Tympane_, _Kettell mills_, _Skrue_,
+_Ctesibike_, or such like: in _Vitruuius_, _Agricola_, (and other,)
+fully, the maner may appeare. And so, thereby, also be most euident, how
+the Artes, of _Pneumatithmie_, _Helicosophie_, _Statike_, _Trochilike_,
+and _Menadrie_, come to the furniture of this, in Speculation, and to
+the Commoditie of the Common Wealth, in practise.
+
+
++‡Horometrie‡, is an Arte Mathematicall, which demõstrateth, how, at all
+times appointed, the precise vsuall denominatiõ of time, may be knowen,
+for any place assigned.+ These wordes, are smoth and plaine easie
+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, _Horologiographia_: 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:
+
+    [4. Reg. 20.]
+
+in * King _Achaz_ 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, Sphæricall, Phænomenall,
+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) theron, to describe (all maner of wayes) how, vsuall howers, may
+be (by the _Sunnes_ shadow) 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 Æquinoctiall Diall, &c. At all howers
+(the Sunne shining) the Signe and Degree 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. Bur 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 Water, 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 reformation,
+according to the heauenly Æquinoctiall Motion: besides the inæqualitie
+of their owne Operation. There remayneth (without parabolicall meaning
+herein) among the Philosophers,
+
+    [A perpetuall Motion.]
+
+a more excellent, more commodious, and more marueilous way, then all
+these: of hauing the motion of the Primouant (or first æquinoctiall
+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
+demonstrateth, how, the Intersection of all visuall Pyramides, made by
+any playne assigned, (the Centre, distance, and lightes, beyng
+determined) may be, by lynes, and due propre colours, represented.+ A
+notable Arte, is this: and would require a whole Volume, to declare the
+property thereof: and the Commodities ensuyng. Great skill of
+_Geometrie_, _Arithmetike_, _Perspectiue_, and _Anthropographie_, with
+many other particular Artes, hath the _Zographer_, 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 (commonly 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 foule, 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 Goldsmith, 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 commodious and meruailous: what shall be thought of _Zographie_,
+the Scholemaster of Picture, 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 Sisters germaine: and
+both, right profitable, in a Commõ wealth. and of _Sculpture_, aswell 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_, _Cutter_, _Grauer_, _Founder_,
+or _Paynter (&c)_ know their Arte, to be commodious.
+
+
+    [An objection.]
+
++‡Architecture‡+, to many may seme not worthy, or not mete, 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 also did
+demonstratiuely procede in their faculty, by Number or Magnitude. First,
+
+    [The Answer.]
+
+you see, that I count, here, _Architecture_, among those _Artes
+Mathematicall_, which are Deriued from the Principals: and you know,
+that such, may deale with Naturall thinges, and sensible matter. Of
+which, “some draw nerer, to the Simple and absolute Mathematicall
+Speculation, then other do.
+
+    [☞]
+
+And though, the _Architect_ 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_, _Arithmeticall_, _Opticall_, _Musicall_,
+_Astronomicall_, _Cosmographicall_” (& to be brief) by all the former
+Deriued _Artes Mathematicall_, and other Naturall Artes, hable to be
+confirmed 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 _Architectes_: the one, being _Vitruuius_, the Romaine: who
+did write ten bookes thereof, to the Emperour _Augustus_ (in whose daies
+our Heauenly Archemaster, was borne): and the other, _Leo Baptista
+Albertus_, a Florentine: who also published ten bookes therof.
+_Architectura_ (sayth _Vitruuius_) _est Scientia pluribus disciplinis &
+varijs eruditionibus ornata: cuius Iudicio probantur omnia, quæ ab
+cæteris 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 Fabrica, & Ratiocinatione. &c. Ratiocinatio
+autem est, quæ, res fabricatas, Solertia ac ratione proportionis,
+demonstrare at[que] explicare potest. +Architecture, groweth of Framing,
+and Reasoning. &c. Reasoning, is that, which of thinges framed, with
+forecast, and proportion: can make demonstration, and manifest
+declaration.+_ Againe. _Cùm, in omnibus enim rebus, tùm maximè etiam in
+Architectura, hæc duo insunt: quod significatur, & quod significat.
+Significatur proposita res, de qua dicitur: hanc autem Significat
+Demonstratio, rationibus doctrinarum explicata. +Forasmuch as, in all
+thinges: therefore chiefly in Architecture, these two thinges are: the
+thing signified: and that which signifieth. The thing propounded,
+whereof we speake, is the thing Signified. But Demonstration, expressed
+with the reasons of diuerse doctrines, doth signifie the same thing.+_
+After that. _Vt literatus sit, peritus Graphidos, eruditus Geometriæ, &
+Optices non ignarus: instructus Arithmetica: historias complures
+nouerit, Philosophos diligenter audiuerit: Musicam sciuerit: Medicinæ
+non sit ignarus, responsa Iurisperitorũ nouerit: Astrologiam, Cæli[que]
+rationes cognitas habeat. +An Architect+_ (sayth he) +_ought to
+vnderstand Languages, to be skilfull of Painting, well instructed in
+Geometrie, not ignorant of Perspectiue, furnished with Arithmetike, haue
+knowledge of many histories, and diligently haue heard Philosophers,
+haue skill of Musike, not ignorant of Physike, know the aunsweres of
+Lawyers, and haue Astronomie, and the courses Cælestiall, in good
+knowledge._+ He geueth reason, orderly, wherefore all these Artes,
+Doctrines, and Instructions, are requisite in an excellent _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 heauen, 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 φυσιολογία) Philosophie doth make declaration. Which,
+it is necessary, for an Architect, with diligence to haue learned:
+because it hath many and diuers naturall questions: as specially, in
+Aqueductes. For in their courses, leadinges about, in the leuell ground,
+and in the mountinges, the naturall 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 Ctesibius, or Archimedes
+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 Balistes, 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 ηχεῖα) are ordred according to Musicall Symphonies &
+Harmonies: being distributed in y^e Circuites, by Diatessaron, Diapente,
+and Diapason. That the conuenient voyce, of the players sound, whẽ 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 knowẽ y^e East, West, South, and
+North. The fashion of the heauen, the Æquinox, the Solsticie, and the
+course of the sterres. Which thinges, vnleast one know: he can not
+perceiue, any thyng at all, the reason of Horologies. Seyng therfore
+this ample Science, is garnished, beautified and stored, with so many
+and sundry skils and knowledges: I thinke, that none can iustly account
+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 Tabernacle of Architecture. &c. And to whom Nature hath giuen such
+quicke Circumspection, sharpnes of witt, and Memorie, that they may be
+very absolutely skillfull in Geometrie, Astronomie, Musike, and the rest
+of the Artes Mathematicall: Such, surmount and passe the callyng, and
+state, of Architectes:
+
+    [A Mathematicien.]
+
+and are become Mathematiciens. &c. And they are found, seldome. As, in
+tymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes:
+Apollonius Pergęus: Eratosthenes Cyreneus: Archimedes, and Scopas,
+Syracusians. Who also, left to theyr posteritie, many Engines and
+Gnomonicall workes: by numbers and naturall meanes, inuented and
+declared._+
+
+
+Thus much, and the same wordes (in sense) in one onely Chapter of this
+Incõparable _Architect Vitruuius_, shall you finde. And if you should,
+but take his boke in your hand, and slightly loke thorough it, you would
+say straight way:
+
+    [Vitruuius.]
+
+This is _Geometrie_, _Arithmetike_, _Astronomie_, _Musike_,
+_Anthropographie_, _Hydragogie_, _Horometrie_. _&c_. and (to cõclude)
+the Storehouse of all workmãship. 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 allowed an Architect. For, I will
+not bryng in place a Carpenter: as though you might Compare him to the
+Chief Masters of other Artes. For the hand of the Carpenter, is the
+Architectes Instrument._+
+
+    [VVho is an Architect.]
+
++_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 bodyes,
+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, consisteth in Lineamentes, and in Framyng. And
+the whole power and skill of Lineamentes, tendeth to this: that the
+right and absolute way may be had, of Coaptyng and ioyning Lines and
+angles: by which, the face of the buildyng 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 nũber:
+a worthy maner, and a semely order: that, so, y^e whole forme and figure
+of the buildyng, may rest in the very Lineamentes. &c. And we may
+prescribe in mynde and imagination the whole formes, *
+
+    [* The Immaterialitie of perfect Architecture.]
+
+all material stuffe beyng secluded. Which point we shall atteyne, by
+Notyng and forepointyng the angles, and lines, by a sure and certaine
+direction and connexion. Seyng then, these thinges, are thus:_+
+
+    [What, Lineament is.]
+
++_Lineamente, shalbe the certaine 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:
+
+    [Note.]
+
+by certaine order, nũber, forme, figure, and _Symmetrie_ mentall:” all
+naturall & sensible stuffe set a part. Now, then, it is euident, (Gentle
+reader) how aptely and worthely, I haue preferred _Architecture_, to be
+bred and fostered vp in the Dominion of the pereles _Princesse_,
+_Mathematica_: and to be a naturall Subiect of hers. And the name of
+_Architecture_, 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, nor
+Builder: nor, separately, any Artificer: but the Hed, the Prouost, the
+Directer, and Iudge of all Artificiall workes, and all Artificers. For,
+the true _Architect_, is hable to teach, Demonstrate, distribute,
+describe, 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) atteyne thereto: 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 Definitions 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) allowed to absolute
+_Architecture_. Vnder the Direction of this Arte, are thre principall,
+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, will 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 Directiõ, & in the shortest time, a sufficient Ship,
+betwene any two places (in passage Nauigable,) assigned: may be
+cõducted: 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 _Horometrie_. 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 Astronomers Ryng, The Astronomers staffe, The
+Astrolabe vniuersall. An Hydrographicall Globe. Charts Hydrographicall,
+true, (not with parallell Meridians). The Common Sea Compas: The Compas
+of variacion: The Proportionall, and Paradoxall Compasses
+
+    [Anno. 1559.]
+
+(of me Inuented, for our two Moscouy Master Pilotes, at the request of
+the Company) Clockes with spryng: houre, halfe houre, and three houre
+Sandglasses: & sundry other Instrumẽtes: 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
+Longitude & 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, appertayneth expert meanes, to be certified
+euer, of the Ships way. &c. And by foreseing 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,)
+_Mutationes, opportunitates[que] temporum presentire, non minus rei
+militari, quàm Agriculturæ, Nauigationi[que] conuenit. +To foresee the
+alterations and opportunities of tymes, is conuenient, no lesse to the
+Art of Warre, then to Husbandry and Nauigation.+_ And besides such
+cunnyng meanes, more euident tokens in Sonne and Mone, ought of hym to
+be knowen: such as (the Philosophicall Poëte) _Virgilius_ teacheth, in
+hys _Georgikes_. Where he sayth,
+
+    [Sidenote: Georgic. 1.]
+
+  _Sol quo[que] & exoriens & quum se condet in vndas,
+  Signa dabit, Solem certissima signa sequuntur. &c.
+    -------- Nam sæpe videmus,
+  Ipsius in vultu varios errare colores.
+  Cæruleus, pluuiam denunciat, igneus Euros.
+  Sin maculæ incipient rutilo immiscerier igni,
+  Omnia tum pariter vento, nimbis[que] videbis
+  Feruere: non illa quisquam me nocte per altum
+  Ire, ne[que] a terra moueat conuellere funem. &c.
+  Sol tibi signa dabit. Solem quis dicere falsum
+  Audeat? -------- &c._
+
+And so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and
+Beastes, and of many thynges els, a certaine Sympathicall forewarnyng
+may be had: sometymes to great pleasure and proffit, both on Sea and
+Land. Sufficiently, for my present purpose, it doth appeare, by the
+premisses, how _Mathematicall_, the _Arte_ of _Nauigation_, is: and how
+it nedeth and also vseth other _Mathematicall Artes_: And now, if I
+would go about to speake of the manifold Commodities, commyng to this
+Land, and others, by Shypps and _Nauigation_, you might thinke, that I
+catch at occasions, to vse many wordes, where no nede is.
+
+
+Yet, this one thyng may I, (iustly) say. In _Nauigation_, none ought to
+haue greater care, to be skillfull, then our English Pylotes. And
+perchaunce, Some, would more attempt: And other Some, more willingly
+would be aydyng, it they wist certainely, What Priuiledge, God had
+endued this Iland with, by reason of Situation, most commodious for
+_Nauigation_, to Places most Famous & Riche. And though,
+
+    [* Anno. 1567 S. H. G.]
+
+(of * Late) a young Gentleman, a Courragious Capitaine, was in a great
+readynes, with good hope, and great causes of persuasion, to haue
+ventured, for a Discouerye, (either _Westerly_, by _Cape de Paramantia_:
+or _Esterly_, aboue _Noua Zemla_, and the _Cyremisses_) and was, at the
+very nere tyme of Attemptyng, called and employed otherwise (both then,
+and since,) in great good seruice to his Countrey, as the Irish Rebels
+haue * tasted:
+
+    [* Anno. 1569]
+
+Yet, I say, (though the same Gentleman, doo not hereafter, deale
+therewith) Some one, or other, should listen to the Matter: and by good
+aduise, and discrete Circumspection, by little, and little, wynne to the
+sufficient knowledge of that +Trade+ and +Voyage+: Which, now, I would
+be sory, (through Carelesnesse, want of Skill, and Courrage,) should
+remayne Vnknowne and vnheard of. Seyng, also, we are herein, halfe
+Challenged, by the learned, by halfe request, published. Therof, verely,
+might grow Commoditye, to this Land chiefly, and to the rest of the
+Christen Common wealth, farre passing all riches and worldly Threasure.
+
+
++‡Thaumaturgike‡, is that Art Mathematicall, which giueth certaine order
+to make straunge workes, of the sense to be perceiued, and of men
+greatly to be wondred at.+ By sundry meanes, this _Wonder-worke_ is
+wrought. Some, by _Pneumatithmie_. As the workes of _Ctesibius_ and
+_Hero_, Some by waight. wherof _Timæus_ speaketh. Some, by Stringes
+strayned, or Springs, therwith Imitating liuely Motions. Some, by other
+meanes, as the Images of Mercurie: and the brasen hed, made by _Albertus
+Magnus_, which dyd seme to speake. _Boethius_ was excellent in these
+feates. To whom, _Cassiodorus_ writyng, sayth. +_Your purpose 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,
+
+    [* Anno. 1551]
+
+* 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 thẽ selues: misdeaming of the meanes. Therfore sayd
+_Claudius Cælestinus_.
+
+    [De his quæ Mundo mirabiliter eueniunt. cap. 8.]
+
+_Hodie magnæ literaturæ viros & magna reputationis videmus, opera quedam
+quasi miranda, supra Naturã putare: de quibus in Perspectiua doctus
+causam faciliter reddidisset._ That is. +_Now a dayes, we see some men,
+yea of great learnyng and reputation, to Iudge certain workes as
+meruaylous, aboue the power of Nature: Of which workes, one that were
+skillfull in Perspectiue might easely haue giuen the Cause._+ Of
+_Archimedes Sphære_, _Cicero_ witnesseth.
+
+    [Tusc. 1.]
+
+Which is very straunge to thinke on. +_For when Archimedes_+ (sayth he)
++_did fasten in a Sphære, the mouynges of the Sonne, Mone, and of the
+fiue other Planets, he did, as the God, which (in Timæus of Plato) did
+make the world. That, one turnyng, should rule motions most vnlike in
+slownes, and swiftnes._+ But a greater cause of meruayling we haue by
+_Claudianus_ report hereof. Who affirmeth this _Archimedes worke_, to
+haue ben of Glasse. And discourseth of it more at large: which I omit.
+The Doue of wood, which the _Mathematicien Archytas_ did make to flye,
+is by _Agellius_ spoken of. Of _Dædalus_ straunge Images, _Plato_
+reporteth. _Homere_ of _Vulcans Selfmouers_, (by secret wheles) leaueth
+in writyng. _Aristotle_, in hys _Politikes_, of both, maketh mention.
+Meruaylous was the workemanshyp, of late dayes, performed by good skill
+of _Trochilike. &c._ For in Noremberge, A flye of Iern, beyng let out of
+the Artificers hand, did (as it were) fly about by the gestes, at the
+table, and at length, as though it were weary, retourne to his masters
+hand agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of
+the same Towne, a mighty way, and that a loft in the Ayre, toward the
+Emperour comming thether: and followed hym, beyng come to the gate of
+the towne. *
+
+    [* ☞]
+
+Thus, you see, what, Arte Mathematicall can performe, when Skill, will,
+Industry, and Hability, are duely applyed to profe.
+
+
+    [A Digression.]
+
+And for these, and such like marueilous Actes and Feates, Naturally,
+Mathematically, and Mechanically, wrought and contriued:
+
+    [Apologeticall.]
+
+ought any honest Student, and Modest Christian Philosopher, be counted,
+& called a +Coniurer+? Shall the folly of Idiotes, 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 pure 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 Æternall, 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 mã 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 peruerter 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, æternally? 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 trauailing,
+onely for good learninges sake: And that, in all maner of wethers: in
+all maner 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 bereuing life: (as God knoweth): with
+lodginges, oft times, to small ease: and somtime to lesse securitie. And
+for much more (then all this) done & suffred, for Learning 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 aduise of Lady Philosophie, & Queene Theologie): but at
+length, to haue catched, and drawen vp, * a Frog?
+
+    [* A prouerb. Fayre fisht, and caught 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, them selues:
+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 giltlesse: 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 vnnaturall 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 iudgemẽt) am I become,
+worse, then when I begã? Worse, thẽ 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 Wisedome?
+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 dungeon 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 Æternall: and to cleaue vnto the
+Author of Death euerlasting? that Murderous Tyrant, 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 Iniuries, that I do: or rather, greater. Pacient _Socrates_,
+his _Apologie_ will testifie: _Apuleius_ 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 Malicious
+Ignorant against him. _Ioannes Trithemius_, his _Apologie_ will
+specifie, how he had occasion to make publike Protestation: as well by
+reason of the Rude Simple: as also, in respect of such, as were counted
+to be of the wisest sort of men. “Many could I recite: But I deferre the
+precise and determined handling of this matter: being loth to detect the
+Folly & Mallice of my Natiue Countrey men. *
+
+    [* ☞]
+
+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, chiefly. The first, I may name, _Vaine pratling busie
+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 or two, and so returne to my Præface.
+
+    [1.]
+
+_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.
+
+    [2.]
+
+_Fonde 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.
+
+    [3.]
+
+_Imperfectly zelous_, to you, 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
+ædified. Nor your faire pretense, by such rashe ragged Rhetorike, 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 detractione parcite linguæ. +Cause thy toung to refraine frõ
+euill. Refraine your toung from slaunder.+_ Though your tounges be
+sharpned, Serpent like, & Adders poyson lye in your lippes:
+
+    [Psal. 140.]
+
+yet take heede, and thinke, betimes, with your selfe, _Vir linguosus non
+stabilietur in terra. Virum violentum venabitur malum, donec
+præcipitetur._ For, sure I am, _Quia faciet Dominus Iudicium afflicti:
+& vindictam pauperum._
+
+
+Thus, I require you, my assured frendes, and Countrey men (you
+Mathematiciens, 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 Magnus_, layeth _Moses_ and _Daniel_,
+before the eyes of those, which count all such Studies Philosophicall
+(as mine hath bene) to be vngodly, or vnprofitable. Waye well
+_S. Stephen_ his witnesse of _Moses_.
+
+    [Act. 7. C.]
+
+_Eruditus est Moses omni Sapientia Ægyptiorũ: & erat potens in verbis &
+operibus suis. +Moses was instructed in all maner of wisedome of the
+Ægyptians: 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 _Magicien_. 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 wonders, 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 mention of _Moses_ Philosophie: with good liking of it: And
+_Basilius Magnus_ also, auoucheth it, to haue bene to _Moses_ profitable
+(and therefore, I say, to the Church of God, necessary). But as
+cõcerning _Moses_ wonders, done before King _Pharao_: God, him selfe,
+sayd: _Vide vt omnia ostenta, quæ posui in manu tua, 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_,
+
+    [Lib. 30. Cap. 1.]
+
+of vayne fraudulent _Magike_, saying: _Est & alia Magices Factio,
+a Mose, Iamne, & Iotape, Iudæis pendens: sed multis millibus annorum
+post Zoroastrem. &c._
+
+    [1.]
+
+Let all such, therefore, who, in Iudgement and Skill of Philosophie, are
+farre Inferior to _Plinie_, “take good heede, least they ouershoote them
+selues rashly,” in
+
+    [☞]
+
+Iudging of _Philosophers straunge Actes_: and the Meanes, how they are
+done.
+
+    [2.]
+
+But, much more, ought they 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.
+
+    [3.]
+
+And (to conclude) most of all, let them be ashamed of Man, and afraide
+of the dreadfull and Iuste Iudge: both Folishly or Maliciously to
+deuise: and then, deuilishly to father their new fond Monsters 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 experience sensible, all worthy conclusions
+by all the Artes Mathematicall purposed, & by true Naturall Philosophie
+concluded: & both addeth to them a farder scope, in the termes of the
+same Artes, & also by hys propre Method, and in peculier termes,
+procedeth, with helpe of the foresayd Artes, to the performance of
+complet Experiẽces, which of no particular Art, are hable (Formally) to
+be challenged.+ If you remember, how we considered _Architecture_, in
+respect of all common handworkes: some light may you haue, therby, to
+vnderstand the Souerainty 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 Science_+. _Nicolaus Cusanus_ termeth it so, in hys
+_Experimentes Statikall_, And an other _Philosopher_,
+
+    [R. B.]
+
+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 _Experience_ complete and
+absolute: and other Artes, with their Argumentes, and Demonstrations,
+persuade: and in wordes, proue very well their Conclusions. *
+
+    [☞]
+
+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 Opticall 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 fantasticall Imagination: as some Sophister, might, _Cum suis
+Insolubilibus_, make a florish: 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 discharged my
+Duety toward God: for my small Talent, at hys most mercyfull handes
+receiued. 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) requested me, (an old forworne Mathematicien) to take pen
+in hand: (through the confidence they reposed in my long experience: and
+tryed sincerity) for the declaryng and reportyng somewhat, of the frute
+and commodity, by the +Artes Mathematicall, to be atteyned vnto+: euen
+they, Sore agaynst their willes, are forced, for sundry causes, to
+satisfie the workemans request, in endyng forthwith: He, so feareth
+this, so new an attempt, & so costly: And in matter so slenderly
+(hetherto) 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.
+
+
+    [2.]
+
+Neither are their Studies, hereby, any whit hindred. No more, then the
+Italian _Vniuersities_, as _Academia Bononiensis_, _Ferrariensis_,
+_Florentina_, _Mediolanensis_, _Patauina_, _Papiensis_, _Perusina_,
+_Pisana_, _Romana_, _Senensis_, or any one of them, finde them selues,
+any deale, disgraced, or their Studies any thing hindred, by _Frater
+Lucas de Burgo_, or by _Nicolaus Tartalea_, who in vulgar Italian
+language, haue published, not onely _Euclides Geometrie_, but of
+_Archimedes_ somewhat: and in Arithmetike and Practicall Geometrie, very
+large volumes, all in their vulgar speche. Nor in Germany haue the
+famous _Vniuersities_, any thing bene discontent with _Albertus
+Durerus_, his Geometricall Institutions in Dutch: or with _Gulielmus
+Xylander_, his learned translation of the first sixe bookes of
+_Euclide_, out of the Greke into the high Dutch. Nor with _Gualterus H.
+Riffius_, his Geometricall Volume: very diligently translated into the
+high Dutch tounge, and published. Nor yet the _Vniuersities_ of Spaine,
+or Portugall, thinke their reputation to be decayed: or suppose any
+their Studies to be hindred by the Excellent _P. Nonnius_, his
+Mathematicall workes, in vulgare speche by him put forth. Haue you not,
+likewise, in the French tounge, the whole Mathematicall Quadriuie? and
+yet neither Paris, Orleance, or any of the other Vniuersities of
+Fraunce, at any time, with the Translaters, or Publishers offended: or
+any mans Studie thereby hindred?
+
+
+    [3.]
+
+And surely, the Common and Vulgar Scholer (much more, the Gramarian)
+before his comming to the _Vniuersitie_, shall (or may) be, now
+(according to _Plato_ his Counsell) sufficiently instructed in
+_Arithmetike_ and _Geometrie_, for the better and easier learning of all
+maner of _Philosophie_, _Academicall_, or _Peripateticall_. And by that
+meanes, goe more cherefully, more skilfully, and spedily forwarde, in
+his Studies, there to be learned. And, so, in lesse time, profite more,
+then (otherwise) he should, or could do.
+
+
+    [4.]
+
+Also many good and pregnant Englishe wittes, of young Gentlemen, and of
+other, who neuer intend to meddle with the profound search and Studie of
+Philosophie (in the _Vniuersities_ to be learned) may neuerthelesse,
+now, with more ease and libertie, haue good occasion, vertuously to
+occupie the sharpnesse of their wittes: where, els (perchance)
+otherwise, they would in fond exercises, spend (or rather leese) their
+time: neither seruing God: nor furdering the Weale, common or priuate.
+
+
+    [5.]
+
+And great Comfort, with good hope, may the _Vniuersities_ haue, by
+reason of this _Englishe_ +Geometrie, and Mathematicall Præface+, 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 Commodities are ensuing (as I
+haue 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: or els, not so fully credited:
+“Well, may all men coniecture, that farre greater ayde, and better
+furniture, to winne to the Perfection of all Philosophie,
+
+    [Vniuersities.]
+
+may in the Vniuersities be had: being the Storehouses & Threasory of all
+Sciences,
+
+    [☞]
+
+and all Artes, necessary for the best, and most noble State of Common
+Wealthes.”
+
+    [6.]
+
+Besides this, how many a Common Artificer, is there, in these Realmes of
+England and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who,
+with their owne Skill and experience, already had, will be hable (by
+these good helpes and informations) to finde out, and deuise, new
+workes, straunge Engines, and Instrumentes: for sundry purposes in the
+Common Wealth? or for priuate pleasure? and for the better maintayning
+of their owne estate? I will not (therefore) fight against myne owne
+shadowe. For, no man (I am sure) will open his mouth against this
+Enterprise. No mã (I say) who either hath Charitie toward his brother
+(and would be glad of his furtherance in vertuous knowledge): or that
+hath any care & zeale for the bettering of the Cõmon state of this
+Realme. Neither any, that make accompt, what the wiser sort of men (Sage
+and Stayed) do thinke of them. To none (therefore) will I make any
+_Apologie,_ for a vertuous acte doing: and for cõmending, 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
+Goodnes, by his Powre, and in his wisedome,
+
+    [☞]
+
+hath Created all thynges, in Number, Waight, and Measure_+: So, to vs,
+of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne the
+sufficient and necessary knowledge of the foresayd hys 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 matter (in my Mynde) as I determined: hopyng of
+conuenient laysure: Yet. if vertuous zeale, and honest Intent 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 pleasure you. And that you
+may the easier perceiue, and better remember, the principall pointes,
+whereof my Preface treateth,
+
+    [The Ground platt of this Præface in a Table.]
+
+I will giue you the +Groundplatt+ of my whole discourse, in a Table
+annexed: from the first to the last, somewhat Methodically 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 pleasure you), Consider the rockish huge
+      mountaines, and the perilous vnbeaten wayes, which
+        (both night and day, for the while) it hath
+          toyled 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 House
+                     At Mortlake._
+
+                      _Anno. 1570. February. 9._
+
+
+  [Decoration]
+
+
+
+
+  [Transcriber’s Note:
+
+  The “Groundplat” was printed in the form of a stemma, or tree, on an
+  oversized fold-out page. The layout was impossible to reproduce for
+  this e-text, so the information has been rearranged in nested-list
+  form. Size markings (see note at beginning of e-text) are relative
+  within each paragraph.]
+
+
+      _J. DEE_
+
+  +‡Here haue you (according to my promisse) the Groundplat of‡+
+    +my MATHEMATICALL Præface: annexed to _Euclide_ (now first)+
+       published in our Englishe tounge. An. 1570. Febr. 3.
+
+
++‡Sciences, and Artes Mathematicall,‡ are, either+
+
+  +‡Principall,‡ which are two, onely,+
+
+    +Arithmetike.+
+
+      +‡Simple‡+, Which dealeth with Numbers onely: and demonstrateth
+        all their properties and appertenances: where, an Vnit, is
+        Indiuisible.
+      +‡Mixt‡+, Which with aide of Geometrie principall, demonstrateth
+        some Arithmeticall Conclusion, or Purpose.
+
+    +Geometrie.+
+
+      +‡Simple‡+, Which dealeth with Magnitudes, onely: and
+        demonstrateth all their properties, passions, and
+        appertenances: whose Point, is Indiuisible.
+      +‡Mixt‡+, Which with aide of Arithmetike principall,
+        demonstrateth some Geometricall purpose, as +EVCLIDES
+        ELEMENTES+.
+
+    +‡The vse‡ whereof, is either,+
+
+      In thinges Supernaturall, æternall, & Diuine: By Application,
+        _Ascending_.
+      In thinges Mathematicall: without farther Application.
+      In thinges Naturall: both Substãtiall, & Accidentall, Visible,
+        & Inuisible. &c. By Application: _Descending_.
+
+    The like Vses and Applications are, (though in a degree lower) in
+    the +Artes Mathematicall Deriuatiue+.
+
+  +‡Deriuatiue‡ frõ the Principalls: of which, some haue+
+
+    +‡The names of‡ the Principalls: as,+
+
+      +_Arithmetike_, vulgar: which considereth+
+
+        --Arithmetike of most vsuall whole numbers: And of Fractions to
+        them appertaining.
+        --Arithmetike of Proportions.
+        --Arithmetike Circular.
+        --Arithmetike of Radicall Nũbers: Simple, Compound, Mixt: And of
+        their Fractions.
+        --Arithmetike of Cossike Nũbers: with their Fractions: And the
+        great Arte of Algiebar.
+
+      +_Geometrie_, vulgar: which teacheth Measuring+
+
+        +‡At hand‡+
+
+          All Lengthes.--+Mecometrie.+
+          All Plaines: As, Land, Borde, Glasse, &c.--+Embadometrie.+
+          All Solids: As, Timber, Stone, Vessels, &c.--+Stereometrie.+
+
+        +‡With distãce‡+ from the thing Measured, as,
+
+          +‡How farre‡+, from the Measurer, any thing is: of him
+            sene, on Land or Water: called +Apomecometrie+.
+          +‡How high or deepe‡+, from the leuell of the Measurers
+            standing, any thing is: Seene of hym, on Land or Water:
+            called +Hypsometrie+.
+          +‡How broad‡+, a thing is, which is in the Measurers view:
+            so it be situated on Land or Water: called
+            +Platometrie+.
+
+        +‡Of which‡ are growen the Feates & Artes of+
+
+          +Geodesie+: more cunningly to Measure and Suruey Landes,
+            Woods, Waters. &c.
+          +Geographie.+
+          +Chorographie.+
+          +Hydrographie.+
+          +Stratarithmetrie.+
+
+    +‡Propre names‡ as+,
+
+      +Perspectiue,+--Which demonstrateth the maners and properties
+      of all Radiations: Directe, Broken, and Reflected.
+
+      +Astronomie,+--Which demonstrateth the Distances, Magnitudes,
+      and all Naturall motions, Apparences, and Passions, proper to
+      the Planets and fixed Starres: for any time, past, present, and
+      to come: in respecte of a certaine Horizon, or without respecte
+      of any Horizon.
+
+      +Musike,+--Which demonstrateth by reason, and teacheth by
+      sense, perfectly to iudge and order the diuersitie of Soundes,
+      hie or low.
+
+      +Cosmographie,+--Which, wholy and perfectly maketh description
+      of the Heauenlym and also Elementall part of the World: and of
+      these partes, maketh homologall application, and mutuall
+      collation necessary.
+
+      +Astrologie,+--Which reasonably demonstrateth the operations
+      and effectes of the naturall beames of light, and secrete
+      Influence of the Planets, and fixed Starres, in euery Element
+      and Elementall body: at all times, in any Horizon assigned.
+
+      +Statike,+--Which demonstrateth the causes of heauines and
+      lightnes of all thinges: and of the motions and properties to
+      heauines and lightnes belonging.
+
+      +Anthropographie,+ Which describeth the Nũber, Measure, Waight,
+      Figure, Situation, and colour of euery diuers thing contained in
+      the perfecte body of MAN: and geueth certaine knowledge of the
+      Figure, Symmetrie, Waight, Characterization, & due Locall motion
+      of any percell of the said body assigned: and of numbers to the
+      said percell appertaining.
+
+      +Trochilike,+--Which demonstrateth the properties of all
+      Circular motions: Simple and Compound.
+
+      +Helicosophie,+--Which demonstrateth the designing of all
+      Spirall lines: in Plaine, on Cylinder, Cone, Sphære, Conoïd, and
+      Sphæroid: and their properties.
+
+      +Pneumatithmie,+--Which demonstrateth by close hollow
+      Geometricall figures (Regular and Irregular) the straunge
+      properties (in motion or stay) of the Water, Ayre, Smoke, and
+      Fire, in their Continuitie, and as they are ioyned to the
+      Elementes next them.
+
+      +Menadrie,+--Which demonstrateth, how, aboue Natures Vertue,
+      and power simple: Vertue and force, may be multiplied: and so
+      to directe, to lift, to pull to, and to put or cast fro, any
+      multiplied, or simple determined Vertue, Waight, or Force:
+      naturally, not, so, directible, or moueable.
+
+      +Hypogeiodie,+--Which demonstrateth, how, vnder the Sphæricall
+      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 respecte of the sayd
+      entrance, is knowen) certaine way, may be prescribed and gone,
+      &c.
+
+      +Hydragogie,+--Which demonstrateth the possible leading of
+      water by Natures law, and by artificiall helpe, from any head
+      (being Spring, standing, or running water) to any other place
+      assigned.
+
+      +Horometrie,+--Which demonstrateth, how, at all times
+      appointed, the precise, vsuall denomination of time, may be
+      knowen, for any place assigned.
+
+      +Zographie,+--Which demonstrateth and teacheth, how, the
+      Intersection of all visuall Pyramids, made by any plaine
+      assigned (the Center, distance, and lightes being determined)
+      may be, by lines, and proper colours represented.
+
+      +Architecture,+--Which is a Science garnished with many
+      doctrines, and diuers Instructions: by whose iudgement, all
+      workes by other workmen finished, are iudged.
+
+      +Nauigation,+--Which demonstrateth, how, by the Shortest good
+      way, by the aptest direction, and in the shortest time:
+      a sufficient Shippe, betwene any two places (in passage
+      nauigable) assigned, may be conducted: and in all stormes and
+      naturall disturbances chauncing, how to vse the best possible
+      meanes, to recouer the place first assigned.
+
+      +Thaumaturgike,+--Which geueth certaine order to make straunge
+      workes, of the sense to be perceiued: and of men greatly to be
+      wondred at.
+
+      +Archemastrie,+--Which teacheth to bring to actuall experience
+      sensible, all worthy conclusions, by all the Artes Mathematicall
+      purposed: and by true Naturall philosophie, concluded: And both
+      addeth to them a farder Scope, in the termes of the same Artes:
+      and also, by his proper Method, and in peculiar termes,
+      procedeth, with helpe of the forsayd Artes, to the performance
+      of complete Experiences: which, of no particular Arte, are hable
+      (Formally) to be challenged.
+
+
++¶ Imprinted by _Iohn Day_.+
+
+An. 1570. Feb. 25.
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+Errors and Anomalies:
+
+Unless otherwise noted, spelling and punctuation are unchanged.
+Errors are listed below, with the original form, if changed, shown in
+[brackets]. Unusual words include “fatch” (probably used as a variant
+of “fetch”) and the mathematical terms “sexagene” and “sexagesme”.
+
+  How, worldly goods: how, worldly dignitie
+    [_“o” in second “worldly” invisible_]
+  his most diligent hearers (so infinitely mought  [hearers) so]
+  the boundes, and duety of an Hydrographer  [Hydographer]
+  of the Grekes it is called _Eteromekes_
+    [_text unchanged: correct form is “Heteromekes”_]
+  τὸ ὁτὶ  [_accent unchanged_]
+  in our worldly affaires  [wordly]
+  fall to worke.❉.
+    [_Some text readers may not display the oversized-asterisk symbol._]
+  _Emptying the first._  [Emptyting]
+  Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει λέγοντι πιϛευτεόν
+    [ἡμήρας ... πιϛευτέομ]
+  of the suddeyne  [snddeyne]
+  that the right and absolute way may be had  [he had]
+  Georgic I: [_The quoted segments, each ending in “&c.”, are
+    438-439; 451-457; 463-464._]
+
+Additional Notes:
+
+  The Greek letter η (eta) was consistently printed as if it were the
+    ou-ligature ȣ.
+  The Latin “-que” was written as an abbreviation resembling “-q´;”.
+    It is shown here as [que].
+
+  Mathematical symbols seen in the section accompanying the diagrams
+  could not be reproduced. The following substitutions were made:
+    --The curly “P” used for “Pounds” is shown as {P}.
+    --The “potestas” symbol, used to represent “x” (the unknown),
+    is shown as {x}.
+    --All roots were expressed as the “root” sign √ combined with
+    symbols for the power of 2 (doubled for power of 4, or fourth root)
+    and 3. They are shown as ²√ ³√ ⁴√.
+
+Euclid:
+
+The following Propositions were identified by number.
+
+6.12: (How) to find a fourth (line) proportional to three given straight
+lines.
+
+11.34: In equal parallelepipedal solids the bases are reciprocally
+proportional to the heights; and those parallelepipedal solids in which
+the bases are reciprocally proportional to the heights are equal.
+
+11.36: If three straight lines are proportional, then the
+parallelepipedal solid formed out of the three equals the
+parallelepipedal solid on the mean which is equilateral, but equiangular
+with the aforesaid solid.
+
+12.1: Similar polygons inscribed in circles are to one another as the
+squares on their diameters.
+
+12.2: Circles are to one another as the squares on their diameters.
+
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+<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara, by John Dee</div>
+<div style='display:block; margin:1em 0'>
+This eBook is for the use of anyone anywhere in the United States and
+most other parts of the world at no cost and with almost no restrictions
+whatsoever. You may copy it, give it away or re-use it under the terms
+of the Project Gutenberg License included with this eBook or online
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+country where you are located before using this eBook.
+</div>
+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara</div>
+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: John Dee</div>
+<div style='display:block; margin:1em 0'>Release Date: July 13, 2007 [eBook #22062]<br />
+[Most recently updated: June 28, 2021]</div>
+<div style='display:block; margin:1em 0'>Language: English</div>
+<div style='display:block; margin:1em 0'>Character set encoding: UTF-8</div>
+<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Louise Hope, David Starner, Suzanne Lybarger and the Online Distributed Proofreading Team</div>
+<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE ***</div>
+
+<div class="mynote">
+
+<p>The variation in text sizes is as in the original.</p>
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+<p>In the printed book, all sidenotes were shown in the outer margin.
+For this e-text, they have been divided. Text sidenotes&mdash;including
+single numerals&mdash;are in the right margin; symbols are on the left,
+along with page numbers. The continuous quotation marks could not be
+reproduced, so they are shown as oversized open quotes “ and close
+quotes ” in the left margin.</p>
+
+<p>The original book did not have numbered pages. Instead it labeled all
+recto (right-side) pages, grouped into six eight-page signatures. The
+signatures themselves have an unusual sequence: <img src="images/finger20.gif" alt="pointer" />&nbsp;(pointing finger);
+*&nbsp;(asterisk); a; b; c; A. The verso (left-side) pages were
+unmarked; they are identified here as&nbsp;||. The <a href="#groundplat">final page</a> was an unnumbered foldout, about twice the
+size of a normal page.</p>
+
+<p>Unless otherwise noted, spelling and punctuation are unchanged.
+A&nbsp;few specific notes are given <a href="#textnotes">at the end of
+the text</a>, along with those passages of <a href="#euclid">Euclid</a> identified by number.</p>
+
+</div>
+
+<p>&nbsp;</p>
+
+<!-- png 01 -->
+
+<p class="illustration">
+<img src="images/titlepage.png" width="417" height="641"
+alt="text linked below" />
+<br />
+<br />
+<a href="#title_text">Title Page Text</a>
+</p>
+
+<div class="main">
+
+<!-- png 02 -->
+
+<hr class="spacer" />
+
+<span class="pagenum"><img src="images/finger20.gif" alt="pointer" />.ii</span>
+<!-- png 03 -->
+<h5><span class="leaf">❧</span> The Translator to the Reader.</h5>
+
+
+<p><span class="capletter">
+<img src="images/capT.png" width="151" height="153"
+alt="T(There)" /></span>
+<i>Here is (gentle Reader) nothing (the word of God onely set apart)
+which so much beautifieth and adorneth the soule and minde of mã, as
+doth the knowledge of good artes and sciences: as the knowledge of
+naturall and morall Philosophie. The one setteth before our eyes, the
+creatures of God, both in the heauens aboue, and in the earth beneath:
+in which as in a glasse, we beholde the exceding maiestie and wisedome
+of God, in adorning and beautifying them as we see: in geuing vnto them
+such wonderfull and manifolde proprieties, and naturall workinges, and
+that so diuersly and in such varietie: farther in maintaining and
+conseruing them continually, whereby to praise and adore him, as by
+S.&nbsp;Paule we are taught. The other teacheth vs rules and preceptes
+of vertue, how, in common life amongest men, we ought to walke
+vprightly: what dueties pertaine to our selues, what pertaine to the
+gouernment or good order both of an housholde, and also of a citie or
+common wealth. The reading likewise of histories, conduceth not a litle,
+to the adorning of the soule &amp; minde of man, a&nbsp;studie of all
+men cõmended: by it are seene and knowen the artes and doinges of
+infinite wise men gone before vs. In histories are contained infinite
+examples of heroicall vertues to be of vs followed, and horrible
+examples of vices to be of vs eschewed. Many other artes also there are
+which beautifie the minde of man: but of all other none do more garnishe
+&amp; beautifie it, then those artes which are called Mathematicall.
+Unto the knowledge of which no man can attaine, without the perfecte
+knowledge and instruction of the principles, groundes, and Elementes of
+Geometrie. But perfectly
+<span class="pagenum verso">||</span>
+<!-- png 04 -->
+to be instructed in them, requireth diligent studie and reading of olde
+auncient authors. Amongest which, none for a beginner is to be preferred
+before the most auncient Philosopher </i>Euclide<i> of </i>Megara<i>.
+For of all others he hath in a true methode and iuste order, gathered
+together whatsoeuer any before him had of these Elementes written:
+inuenting also and adding many thinges of his owne: wherby he hath in
+due forme accomplished the arte: first geuing definitions, principles,
+&amp; groundes, wherof he deduceth his Propositions or conclusions, in
+such wonderfull wise, that that which goeth before, is of necessitie
+required to the proufe of that which followeth. So that without the
+diligent studie of </i>Euclides<i> Elementes, it is impossible to
+attaine vnto the perfecte knowledge of Geometrie, and consequently of
+any of the other Mathematicall sciences. Wherefore considering the want
+&amp; lacke of such good authors hitherto in our Englishe tounge,
+lamenting also the negligence, and lacke of zeale to their countrey in
+those of our nation, to whom God hath geuen both knowledge, &amp; also
+abilitie to translate into our tounge, and to publishe abroad such good
+authors, and bookes (the chiefe instrumentes of all learninges): seing
+moreouer that many good wittes both of gentlemen and of others of all
+degrees, much desirous and studious of these artes, and seeking for them
+as much as they can, sparing no paines, and yet frustrate of their
+intent, by no meanes attaining to that which they seeke: I&nbsp;haue for
+their sakes, with some charge &amp; great trauaile, faithfully
+translated into our vulgare toũge, &amp; set abroad in Print, this booke
+of </i>Euclide<i>. Whereunto I haue added easie and plaine declarations
+and examples by figures, of the definitions. In which booke also ye
+shall in due place finde manifolde additions, Scholies, Annotations, and
+Inuentions: which I haue gathered out of many of the most famous &amp;
+chiefe Mathematiciẽs, both of old time, and in our age: as by diligent
+reading it in course, ye shall
+<span class="pagenum"><img src="images/finger20.gif" alt="pointer" /><ins class="correction" title="text reads ‘.ii’">.iij</ins></span>
+<!-- png 05 -->
+well perceaue. The fruite and gaine which I require for these my paines
+and trauaile, shall be nothing els, but onely that thou gentle reader,
+will gratefully accept the same: and that thou mayest thereby receaue
+some profite: and moreouer to excite and stirre vp others learned, to do
+the like, &amp; to take paines in that behalfe. By meanes wherof, our
+Englishe tounge shall no lesse be enriched with good Authors, then are
+other straunge tounges: as the Dutch, French, Italian, and Spanishe: in
+which are red all good authors in a maner, found amongest the Grekes or
+Latines. Which is the chiefest cause, that amongest thẽ do florishe so
+many cunning and skilfull men, in the inuentions of straunge and
+wonderfull thinges, as in these our daies we see there do. Which fruite
+and gaine if I attaine vnto, it shall encourage me hereafter, in such
+like sort to translate, and set abroad some other good authors, both
+pertaining to religion (as partly I haue already done)<br />
+and also pertaining to the Mathematicall Artes.<br />
+Thus gentle reader farewell.</i></p>
+<p>(?¿)</p>
+
+
+<p class="illustration">
+<img src="images/dec_iii.png" width="132" height="136"
+alt="decoration" />
+</p>
+
+
+<!-- png 06 -->
+
+<hr class="spacer" />
+
+<span class="pagenum verso">[<img src="images/finger20.gif" alt="pointer" />.iiij]</span>
+<!-- png 07 -->
+
+<h5><span class="leaf">❧</span>
+<span class="larger">TO THE VNFAINED LOVERS<br />
+of truthe, and constant Studentes of Noble<br />
+<i>Sciences, </i>IOHN DEE<i> of London, hartily</i></span><br />
+wisheth grace from heauen, and most prosperous<br />
+<i>successe in all their honest attemptes and</i><br />
+exercises.</h5>
+
+
+<p><span class="capletter">
+<img src="images/capD.png" width="175" height="179"
+alt="D(Divine)" /></span>
+Iuine <i>Plato</i>, the great Master of many worthy Philosophers, and
+the constant auoucher, and pithy perswader of <i>Vnum</i>, <i>Bonum</i>,
+and <i>Ens</i>: 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 <i>Plato</i>, 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 <i>Vnum</i>, <i>Bonum</i>, and <i>Ens</i>, to be
+Spirituall, Infinite, Æternall, Omnipotent, &amp;c. Nothyng beyng
+alledged or expressed, How, worldly goods: how, <ins class="correction" title="‘o’ invisible">worldly</ins> dignitie: how,
+health, Strẽgth or lustines of body: nor yet the meanes, how a
+merueilous sensible and bodyly blysse and felicitie hereafter, might be
+atteyned: Straightway, the fantasies of those hearers, were dampt: their
+opinion of <i>Plato</i>, was clene chaunged: yea his doctrine was by
+them despised: and his schole, no more of them visited. Which thing, his
+Scholer, <i>Aristotle</i>, narrowly cõsidering, founde the cause therof,
+to be, <span class="sidequote begin">“</span>For that they had no
+forwarnyng and information, in generall,<span class="sidequote
+end">”</span> 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 constantly to haue continued
+therin: to their full satisfaction: if such his finall scope &amp;
+intent, had ben to their desire. Wherfore, <i>Aristotle</i>, euer, after
+that, vsed in brief, to forewarne his owne Scholers and hearers, <span
+class="sidequote">“</span>both of what matter, and also to what ende,
+he tooke in hand to speake, or teach.<span class="sidequote">”</span>
+While I consider the diuerse trades of these two excellent Philosophers
+(and am most sure, both, that <i>Plato</i> right well, otherwise could
+teach: and that <i>Aristotle</i> mought boldely, with his hearers, haue
+dealt in like sorte as <i>Plato</i> did) I&nbsp;am in no little pang of
+perplexitie: Bycause, that, which I mislike, is most easy for me to
+performe (and to haue <i>Plato</i> for my exãple.) And that, which I
+know to be most commendable: and (in this first bringyng, into common
+handling, the <i>Artes Mathematicall</i>) to be most necessary: is full
+of great difficultie and sundry daungers. Yet, neither do I think it
+mete, for so straunge matter (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 <i>Aristotle</i>) well can I hope, that
+accordyng to the amplenes and dignitie of the <i>State
+Mathematicall</i>, I&nbsp;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 <i>Plato</i> his schole, after they had perceiued his
+finall
+<span class="pagenum verso">||</span>
+<!-- png 08 -->
+conclusion, would in these thinges haue ben his most diligent <ins class="correction" title="text has ‘hearers) so’">hearers (so</ins>
+infinitely mought their desires, in fine and at length, by our <i>Artes
+Mathematicall</i> be satisfied) yet, by this my Præface &amp;
+forewarnyng, Aswell all such, may (to their great behofe) the soner,
+hither be allured: as also the <i>Pythagoricall</i>, and
+<i>Platonicall</i> perfect scholer, and the constant profound
+Philosopher, with more ease and spede, may (like the Bee,) gather,
+hereby, both wax and hony.</p>
+
+<p>Wherfore, seyng I finde great occasion (for the causes alleged, and
+farder, in respect of my <i>Art Mathematike generall</i>) to vse <span
+class="sidequote">“</span>a certaine forewarnyng and Præface, whose
+content shalbe,
+<span class="sidenote">
+The intent of this Preface.</span>
+that mighty, most plesaunt, and frutefull <i>Mathematicall Tree</i>,
+with his chief armes and second (grifted) braunches: 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:<span class="sidequote">”</span> Neuertheles, if, for my sincere endeuour to
+satisfie your honest expectation, you will but lend me your thãkefull
+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 either 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.</p>
+
+<p>All thinges which are, &amp; 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, &amp; vnchangeable. Things Naturall,
+are materiall, compounded, diuisible, corruptible, and chaungeable.
+Thinges Supernaturall, are, of the minde onely, comprehended: Things
+Naturall, of the sense exterior, ar hable to be perceiued. In thinges
+Naturall, probabilitie and coniecture hath place: But in things
+Supernaturall, chief demõstration, &amp; most sure Science is to be had.
+By which properties &amp; 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 <i>Thynges Mathematicall</i>. For, these, beyng (in a
+maner) middle, betwene thinges supernaturall 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
+neuerthelesse, by materiall things hable somewhat to be signified. And
+though their particular Images, by Art, are aggregable and diuisible:
+yet the generall <i>Formes</i>, notwithstandyng, are constant,
+vnchaungeable, vntrãsformable, 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
+imperfectiõ of coniecture, weenyng and opinion: and commyng short of
+high intellectuall cõceptiõ, are the Mercurial fruite of
+<i>Dianœticall</i> discourse, in perfect imagination subsistyng.
+A&nbsp;meruaylous newtralitie haue these thinges <i>Mathematicall</i>,
+and also a straunge participatiõ betwene thinges supernaturall,
+immortall, intellectual, simple and indiuisible: and thynges naturall,
+mortall, sensible, compounded and diuisible. Probabilitie and sensible
+prose, may well serue in thinges naturall: and is commendable: In
+Mathematicall reasoninges, a&nbsp;probable Argument, is nothyng
+regarded: nor yet the testimony of sense, any whit credited: But onely a
+perfect demonstration, of truthes certaine, necessary, and inuincible:
+vniuersally and necessaryly concluded:
+<span class="pagenum">*.i</span>
+<!-- png 09 -->
+is allowed as sufficient for <span class="sidequote">“</span>an
+Argument exactly and purely Mathematical.<span class="sidequote">”</span></p>
+
+<p>Of <i>Mathematicall</i> thinges, are two principall kindes: namely,
+<i>Number</i>, and <i>Magnitude</i>.
+<span class="sidenote">
+Number.</span>
+<i>Number</i>, we define, to be, a&nbsp;certayne Mathematicall Sũme, of
+<i>Vnits</i>.
+<span class="sidenote smaller">
+Note the worde, Vnit, to expresse the Greke Monas, &amp; not Vnitie: as
+we haue all, commonly, till now, vsed.</span>
+And, an <i>Vnit</i>, is that thing Mathematicall, Indiuisible, by
+participation of some likenes of whose property, any thing, which is in
+deede, or is counted One, may resonably be called One. We account an
+<i>Vnit</i>, a&nbsp;thing <i>Mathematicall</i>, though it be no Number,
+and also indiuisible: because, of it, materially, Number doth consist:
+which, principally, is a thing <i>Mathematicall</i>.
+<span class="sidenote">
+Magnitude.</span>
+<i>Magnitude</i> is a thing <i>Mathematicall</i>, by participation of
+some likenes of whose nature, any thing is iudged long, broade, or
+thicke. <span class="sidequote">“</span>A&nbsp;thicke <i>Magnitude</i>
+we call a <i>Solide</i>, or a <i>Body</i>. What <i>Magnitude</i> so
+euer, is Solide or Thicke, is also broade, &amp; long. A&nbsp;broade
+magnitude, we call a <i>Superficies</i> or a Plaine. Euery playne
+magnitude, hath also length. A&nbsp;long magnitude, we terme a
+<i>Line</i>. A&nbsp;<i>Line</i> is neither thicke nor broade, but onely
+long: Euery certayne Line, hath two endes:
+<span class="sidenote">
+A point.</span>
+The endes of a line, are <i>Pointes</i> called. A&amp;nbsp;<i>Point</i>,
+is a thing <i>Mathematicall</i>, indiuisible, which may haue a certayne
+determined situation.<span class="sidequote">”</span> If a Poynt moue
+from a determined situation, the way wherein it moued, is also a
+<i>Line</i>: mathematically produced, whereupon, of the auncient
+Mathematiciens,
+<span class="sidenote">
+A Line.</span>
+a&nbsp;<i>Line</i> is called the race or course of a <i>Point</i>.
+A&nbsp;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 <i>Magnitude</i>.
+<span class="sidenote">
+Magnitude.</span>
+And <i>Magnitude</i> we may define to be that thing
+<i>Mathematicall</i>, which is diuisible for euer, in partes diuisible,
+long, broade or thicke. Therefore though a Poynt be no <i>Magnitude</i>,
+yet <i>Terminatiuely</i>, we recken it a thing <i>Mathematicall</i> (as
+I sayd) by reason it is properly the end, and bound of a line. Neither
+<i>Number</i>, nor <i>Magnitude</i>, haue any Materialitie. First, we
+will consider of <i>Number</i>, and of the Science <i>Mathematicall</i>,
+to it appropriate, called <i>Arithmetike</i>: and afterward of
+<i>Magnitude</i>, and his Science, called <i>Geometrie</i>. But that
+name contenteth me not: whereof a word or two hereafter shall be sayd.
+How Immateriall and free from all matter, <i>Number</i> is, who doth not
+perceaue? yea, who doth not wonderfully wõder at&nbsp;it? For, neither
+pure <i>Element</i>, nor <i>Aristoteles, Quinta Essentia</i>, is hable
+to serue for Number, as his propre matter. Nor yet the puritie and
+simplenes of Substance Spirituall or Angelicall, will be found propre
+enough thereto. And therefore the great &amp; godly Philosopher
+<i>Anitius Boetius</i>, sayd: <i>Omnia quæcunq<sup>ue</sup>
+a&nbsp;primæua rerum natura constructa sunt, Numerorum videntur ratione
+formata. Hoc enim fuit principale in animo Conditoris Exemplar</i>. That
+is: <span class="larger"><i>All thinges (which from the very first
+originall being 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</i>.</span> O&nbsp;comfortable
+allurement, O&nbsp;rauishing perswasion, to deale with a Science, whose
+Subiect, is so Auncient, so pure, so excellent, so surmounting all
+creatures, so vsed of the Almighty 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 <i>Nothing</i>, to the <i>Formalitie</i>
+of their being and state. By <i>Numbers</i> propertie therefore, of vs,
+by all possible meanes, (to the perfection of the Science) learned, we
+may both winde and draw our selues into the inward and deepe search and
+vew, of all creatures distinct vertues, natures, properties, and
+<i>Formes</i>: And also, farder, arise, clime, ascend, and mount vp
+(with Speculatiue winges) in spirit, to behold in the Glas of Creation,
+the <i>Forme of Formes</i>, the <i>Exemplar Number</i> of all thinges
+<i>Numerable</i>: both visible and inuisible, mortall and
+<span class="pagenum verso">||</span>
+<!-- png 10 -->
+immortall, Corporall and Spirituall. Part of this profound and diuine
+Science, had <i>Ioachim</i> the Prophesier atteyned vnto: by <i>Numbers
+Formall, Naturall</i>, and <i>Rationall</i>, forseyng, concludyng, and
+forshewyng great particular euents, long before their comming. His
+bookes yet remainyng, hereof, are good profe: And the noble Earle of
+<i>Mirandula</i>, (besides that,) a&nbsp;sufficient witnesse: that
+<i>Ioachim, in his prophesies, proceded by no other way, then by Numbers
+Formall</i>. And this Earle hym selfe, in Rome,
+<span class="sidenote">
+Ano. 1488.</span>
+*set vp 900. Conclusions, in all kinde of Sciences, openly to be
+disputed of: and among the rest, in his Conclusions
+<i>Mathematicall</i>, (in the eleuenth Conclusion) hath in Latin, this
+English sentence. <i>By Numbers, a&nbsp;way is had, to the searchyng
+out, and vnderstandyng of euery thyng, hable to be knowen. For the
+verifying of which Conclusion, I&nbsp;promise to aunswere to the 74.
+Questions, vnder written, by the way of Numbers</i>. Which Cõclusions,
+I&nbsp;omit here to rehearse: aswell auoidyng superfluous prolixitie:
+as, bycause <i>Ioannes Picus, workes</i>, are commonly had. But, in any
+case, I&nbsp;would wish that those Conclusions were red diligently, and
+perceiued of such, as are earnest Obseruers and Considerers of the
+constant law of nũbers: 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, &amp; within my boundes,
+I&nbsp;disclose the wonderfull mysteries, by numbers, to be atteyned
+vnto.</p>
+
+<p>Of my former wordes, easy it is to be gathered, that <i>Number</i>
+hath a treble state: One, in the Creator: an other in euery Creature (in
+respect of his complete constitution:) and the third, in Spirituall and
+Angelicall Myndes, and in the Soule of&nbsp;mã. In the first and third
+state, <i>Number</i>, is termed <i>Number Numbryng</i>. But in all
+Creatures, otherwise, <i>Number</i>, is termed <i>Nũber Numbred</i>. And
+in our Soule, Nũber beareth such a swaye, and hath such an affinitie
+therwith: that some of the old <i>Philosophers</i> taught, <i>Mans
+Soule, to be a Number mouyng it selfe</i>. And in dede, in vs, though it
+be a very Accident: yet such an Accident it is, that before all
+Creatures it had perfect beyng, in the Creator, Sempiternally. <i>Number
+Numbryng</i> therfore, is the discretion discerning, and distincting of
+thinges. But in God the Creator, This discretion, in the beginnyng,
+produced orderly and distinctly all thinges. For his <i>Numbryng</i>,
+then, was his Creatyng of all thinges. And his Continuall
+<i>Numbryng</i>, of all thinges, is the Conseruation of them in being:
+And, where and when he will lacke an <i>Vnit</i>: there and then, that
+particular thyng shalbe <i>Discreated</i>. Here I stay. But our
+Seuerallyng, distinctyng, and <i>Numbryng</i>, createth nothyng: but of
+Multitude considered, maketh certaine and distinct determination. And
+albeit these thynges be waighty and truthes of great importance, yet (by
+the infinite goodnes of the Almighty <i>Ternarie</i>,) Artificiall
+Methods and easy wayes are made, by which the zelous Philosopher, may
+wyn nere this Riuerish <i>Ida</i>, this Mountayne of Contemplation: and
+more then Contemplation. And also, though <i>Number</i>, be a thyng so
+Immateriall, so diuine, and æternall: 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 <i>Ternarie</i>. Three
+Egles, are three, or a <i>Ternarie</i>.
+<span class="sidefinger">
+&nbsp;</span>
+Which* <i>Ternaries</i>, are eche, the <i>Vnion</i>, <i>knot</i>, and
+<i>Vniformitie</i>, of three discrete and distinct <i>Vnits</i>. That
+is, we may in eche <i>Ternarie</i>, thrise, seuerally pointe, and shew a
+part, <i>One</i>, <i>One</i>, and <i>One</i>. Where, in Numbryng, we say
+One, two,
+<span class="pagenum">*.ij </span>
+<!-- png 11 -->
+Three. But how farre, these visible Ones, do differre from our
+Indiuisible Vnits (in pure <i>Arithmetike</i>, 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 cõceiuyng of <i>Numbers</i>, absolutely (:Not supposing, nor
+admixtyng any thyng created, Corporall or Spirituall, to support,
+conteyne, or represent those <i>Numbers</i> 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 <i>Trinitie</i> most
+blessed and æternall.</p>
+
+<p>But farder vnderstand, that vulgar Practisers, haue Numbers,
+otherwise, in sundry Considerations: and extend their name farder, then
+to Numbers, whose least part is an <i>Vnit</i>. For the common Logist,
+Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit,
+imagineth lesse partes: and calleth them <i>Fractions</i>. As of an
+<i>Vnit</i>, he maketh an halfe, and thus noteth it, ½. and so of other,
+(infinitely diuerse) partes of an <i>Vnit</i>. Yea and farder, hath,
+<i>Fractions of Fractions. &amp;c</i>. And, forasmuch, as,
+<i>Addition</i>, <i>Substraction</i>, <i>Multiplication</i>,
+<i>Diuision</i> and <i>Extraction of Rotes</i>, are the chief, and
+sufficient partes of <i>Arithmetike</i>:
+<span class="sidenote">
+Arithmetike.</span>
+which is, the <i>Science that demonstrateth the properties, of Numbers,
+and all operatiõs, in numbers to be performed</i>:
+<span class="sidenote">
+Note.</span>
+<span class="sidequote">“</span>How often, therfore, these fiue sundry
+sortes of Operations, do, for the most part, of their execution,
+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
+<i>Arithmetike</i>.<span class="sidequote">”</span> And by this
+reason:
+<span class="sidenote">
+1.</span>
+the Consideration, doctrine, and working, in whole numbers onely: where,
+of an <i>Vnit</i>, is no lesse part to be allowed: is named (as it were)
+an <i>Arithmetike</i> by it selfe. And so of the <i>Arithmetike of
+Fractions</i>.
+<span class="sidenote">
+2.</span>
+In lyke sorte, the necessary, wonderfull and Secret doctrine of
+Proportion, and proportionalytie hath purchased vnto it selfe a peculier
+maner of handlyng and workyng: and so may seme an other forme of
+<i>Arithmetike</i>.
+<span class="sidenote">
+3.</span>
+Moreouer, the <i>Astronomers</i>, for spede and more commodious
+calculation, haue deuised a peculier maner of orderyng nũbers, about
+theyr circular motions, by Sexagenes, and Sexagesmes. By Signes, Degrees
+and Minutes &amp;c. which commonly is called the <i>Arithmetike</i> of
+<i>Astronomical</i> or <i>Phisicall Fractions</i>. That, haue I briefly
+noted, by the name of <i>Arithmetike Circular</i>. Bycause it is also
+vsed in circles, not <i>Astronomicall. &amp;c.</i>
+<span class="sidenote">
+4.</span>
+Practise hath led <i>Numbers</i> farder, and hath framed them, to take
+vpon them, the shew of <i>Magnitudes</i> propertie: Which is
+<i>Incommensurabilitie</i> and <i>Irrationalitie</i>. (For in pure
+<i>Arithmetike</i>, an <i>Vnit</i>, is the common Measure of all
+Numbers.) And, here, Nũbers are become, as Lynes, Playnes and Solides:
+some tymes <i>Rationall</i>, some tymes <i>Irrationall</i>. And haue
+propre and peculier characters, (as <sup>2</sup>√. <sup>3</sup>√. and so
+of other.<a class="tag" name="endtag1" id="endtag1" href="#endnote1">A</a>
+Which is to signifie <i>Rote Square, Rote Cubik: and so forth</i>:)
+&amp; propre and peculier fashions in the fiue principall partes:
+Wherfore the practiser, estemeth this, a&nbsp;diuerse <i>Arithmetike</i>
+from the other. Practise bryngeth in, here, diuerse compoundyng of
+Numbers: as some tyme, two, three, foure (or more) <i>Radicall</i>
+nũbers, diuersly knit, by signes, of More &amp; Lesse: as thus
+<sup>2</sup>√12 + <sup>3</sup>√15. Or thus <sup>4</sup>√19 +
+<sup>3</sup>√12 - <sup>2</sup>√2. &amp;c. And some tyme with whole
+numbers, or fractions of whole Number, amõg them: as 20 +
+<sup>2</sup>√24. <sup>3</sup>√16 + 33 - <sup>2</sup>√10. <sup>4</sup>√44
++ 12¼ + <sup>3</sup>√9. And so, infinitely, may hap the varietie. After
+this: Both the one and the other hath fractions incident: and so is this
+<i>Arithmetike</i> greately enlarged, by diuerse exhibityng and vse of
+Compositions and mixtynges. Consider how, I&nbsp;(beyng desirous to
+deliuer the student from error and Cauillation) do giue to this
+<i>Practise</i>, the name of the <i>Arithmetike of Radicall numbers</i>:
+Not, of <i>Irrationall</i> or <i>Surd Numbers</i>: which other while,
+are Rationall: though they haue the Signe of a Rote before
+<span class="pagenum verso">||</span>
+<!-- png 12 -->
+them, which, <i>Arithmetike</i> of whole Numbers most vsuall, would say
+they had no such Roote: and so account them <i>Surd Numbers</i>: which,
+generally spokẽ, is vntrue: as <i>Euclides</i> tenth booke may teach
+you. Therfore to call them, generally, <i>Radicall Numbers</i>, (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 speculation and practise) and haue found
+out, and atteyned to the very chief perfection (almost) of
+<i>Numbers</i> Practicall vse. Which thing, is well to be perceiued in
+that great Arithmeticall Arte of <i>Æquation</i>: commonly called the
+<i>Rule of Coss.</i> or <i>Algebra</i>. The Latines termed it,
+<i>Regulam Rei &amp; Census</i>, that is, the <span class="larger"><i>Rule of the thyng and his value</i></span>. 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, <i>Res</i>: <span class="larger"><i>A thing</i></span>: vnleast they vse the name of
+<i>Algebra</i>. And therin (commonly) is a dubble error. The one, of
+them, which thinke it to be of <i>Geber</i> his inuentyng: the other of
+such as call it <i>Algebra</i>. For, first, though <i>Geber</i> 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 <i>Geber</i> his name:
+yet true it is, that a <i>Greke</i> Philosopher and Mathematicien, named
+<i>Diophantus</i>, before <i>Geber</i> his tyme, wrote 13. bookes therof
+(of which, six are yet extant: and I had them to *vse,
+<span class="sidenote">
+* Anno. 1550.</span>
+of the famous Mathematicien, and my great frende, <i>Petrus
+Montaureus</i>:) And secondly, the very name, is <i>Algiebar</i>, and
+not <i>Algebra</i>: as by the Arabien <i>Auicen</i>, may be proued: who
+hath these precise wordes in Latine, by <i>Andreas Alpagus</i> (most
+perfect in the Arabik tung) so translated. <i>Scientia faciendi Algiebar
+&amp; Almachabel. i. Scientia inueniendi numerum ignotum, per additionem
+Numeri, &amp; diuisionem &amp; æquationem</i>. Which is to say: <span
+class="larger"><i>The Science of workyng Algiebar and
+Almachabel</i></span>, that is, the <span class="larger"><i>Science of
+findyng an vnknowen number, by Addyng of a Number, &amp; Diuision &amp;
+æquation</i></span>. Here haue you the name: and also the principall
+partes of the Rule, touched. To name it, <i>The rule, or Art of
+Æquation</i>, doth signifie the middle part and the State of the Rule.
+This Rule, hath his peculier Characters:
+<span class="sidenote">
+5.</span>
+and the principal partes of <i>Arithmetike</i>, to it appertayning, do
+differre from the other <i>Arithmeticall operations</i>. This
+<i>Arithmetike, hath Nũbers</i> Simple, Cõpound, Mixt: and Fractions,
+accordingly. This Rule, and <i>Arithmetike of Algiebar</i>, is so
+profound, so generall and so (in maner) conteyneth the whole power of
+Numbers Application practicall: that mans witt, can deale with nothyng,
+more proffitable about 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 testimony of the vse, proffit
+and Commodity of Arithmetike vulgar, in the Common lyfe and trade of
+men. Therto, then, I&nbsp;will now frame my selfe: But herein great care
+I haue, least length of sundry profes, might make you deme, that either
+I did misdoute your zelous mynde to vertues schole: or els mistrust your
+hable witts, by some, to gesse much more. A&nbsp;profe then, foure,
+fiue, or six, such, will I bryng, as any reasonable man, therwith may be
+persuaded, to loue &amp; honor, yea learne and exercise the excellent
+Science of <i>Arithmetike</i>.</p>
+
+<p>And first: who, nerer at hand, can be a better witnesse of the frute
+receiued by <i>Arithmetike</i>, 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
+<span class="pagenum"> *.iij </span>
+<!-- png 13 -->
+Rule? Simple and Compounde: both forward and backward? How might they
+misse <i>Arithmeticall</i> helpe in the Rules of Felowshyp: either
+without tyme, or with tyme? and betwene the Marchant &amp; his Factor?
+The Rules of Bartering in wares onely: or part in wares, and part in
+money, would they gladly want? Our Marchant venturers, and Trauaylers
+ouer Sea, how could they order their doynges iustly and without losse,
+vnleast certaine and generall Rules for Exchaũge of money, 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 Marchant then to make an end) how ample &amp;
+wonderfull is the Rule of False positions? especially as it is now, by
+two excellent Mathematiciens (of my familier acquayntance in their life
+time) enlarged? I&nbsp;meane <i>Gemma Frisius</i>, and <i>Simon
+Iacob</i>. Who can either in brief conclude, the generall and Capitall
+Rules? or who can Imagine 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 <i>Arithmeticall
+practises</i>, are commonly in Marchantes handes, and knowledge: They
+them selues, can, at large, testifie.</p>
+
+<p>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 meruailously pleasured, if <i>Arithmetike</i> be their
+guide? And the honorable Phisiciãs, will gladly confesse them selues,
+much beholding to the Science of <i>Arithmetike</i>, and that sundry
+wayes: But chiefly in their Art of Graduation, and compounde Medicines.
+And though <i>Galenus</i>, <i>Auerrois</i>, <i>Arnoldus</i>,
+<i>Lullus</i>, and other haue published their positions, aswell in the
+quantities of the Degrees aboue Temperament, as in the Rules, concluding
+the new <i>Forme</i> resulting: yet a more precise, commodious, and easy
+<i>Method</i>, is extant: by a Countreyman of ours
+<span class="sidenote">
+R. B.</span>
+(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 also, for very good profe of
+Numbers vse, in this most subtile and frutefull, Philosophicall
+Conclusion,) I&nbsp;entend in the meane while, most briefly, and with my
+farder helpe, to communicate the pith therof vnto you.</p>
+
+<p>First describe a circle: whose diameter let be an inch. Diuide the
+Circumference into foure equall partes. Frõ 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. without the circumference of the
+circle: So that they shall be of 4. inches long (at the least) 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. <i>Hote</i> and <i>Colde</i>, one against the
+other. And likewise <i>Moyst</i> and <i>Dry</i>, one against the other.
+And in the Circle write <i>Temperate</i>. Which <i>Temperature</i> hath
+a good Latitude: as appeareth by the Complexion of man. And therefore we
+haue allowed vnto it, the foresayd Circle: and not a point Mathematicall
+or Physicall.<a class="tag" name="endtag2" id="endtag2" href="#endnote2">B</a></p>
+
+<p class="illustration">
+<img src="images/fourways.png" width="411" height="405"
+alt="diagram: see end of text for alternative" />
+</p>
+
+<span class="sidenote">
+* <i>Take some part of Lullus counsayle in his booke de Q.
+Essentia.</i></span>
+
+<p>Now, when you haue two thinges Miscible, whose degrees are * 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 Qualities, or of one kinde (but of diuerse
+intentions and degrees) or a <i>Temperate</i>, and a Contrary, <i>The
+forme resulting of their Mixture, is in the Middle betwene the degrees
+of
+<span class="pagenum verso">||</span>
+<!-- png 14 -->
+the formes mixt</i>. As for example, let <i>A</i>, be <i>Moist</i> in
+the first degree: and <i>B</i>, <i>Dry</i> 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 <i>A</i> and <i>B</i> (for the
+<span class="sidenote">
+* <i>Note.</i></span>
+*<i>Temperament</i> is counted none. And for it, you must put a Ciphre,
+if at any time, it be in mixture). Counting then from <i>B</i>, 2.
+degrees, toward <i>A</i>: you finde it to be <i>Dry</i> in the first
+degree: So is the <i>Forme resulting</i> of the Mixture of <i>A</i>, and
+<i>B</i>, in our example. I&nbsp;will geue you an other example.
+Suppose, you haue two thinges, as <i>C</i>, and <i>D</i>: and of
+<i>C</i>, the Heate to be in the 4. degree: and of <i>D</i>, the Colde,
+to be remisse, euen vnto the <i>Temperament</i>. Now, for <i>C</i>, you
+take 4: and for <i>D</i>, you take a Ciphre: which, added vnto 4,
+yeldeth onely 4. The middle, or halfe, whereof, is 2. Wherefore the
+<i>Forme resulting</i> of <i>C</i>, and <i>D</i>, is Hote in the second
+degree: for, 2. degrees, accounted from <i>C</i>, toward <i>D</i>, ende
+iuste in the 2. degree of heate. Of the third maner, I&nbsp;will geue
+also an example: which let be this:
+<span class="sidenote">
+Note.</span>
+I&nbsp;haue a liquid Medicine whose Qualitie of heate is in the 4.
+degree exalted: as was <i>C</i>, in the example foregoing: and an other
+liquid Medicine I haue: whose Qualitie, is heate, in the first degree.
+Of eche of these, I&nbsp;mixt a like quantitie: Subtract here, the lesse
+frõ 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
+<span class="pagenum"> *.iiij </span>
+<!-- png 15 -->
+Forme resulting, by this mixture of <i>C</i>, and <i>E</i>. As, if from
+4. ye abate 1. there resteth 3. the halfe of 3. is 1½: Adde to 1. this
+1½: you haue 2½. Or subtract from 4. this 1½: you haue likewise 2½
+remayning. Which declareth, the <i>Forme resulting</i>, to be
+<i>Heate</i>, in the middle of the third degree.</p>
+
+<p>
+<span class="sidequote">“</span>
+<span class="sidenote">
+The Second Rule.</span>
+But if the Quantities of two thinges Commixt, be diuerse, and the
+Intensions (of their Formes Miscible) be in diuerse degrees, and
+heigthes. (Whether those Formes be of one kinde, or of Contrary kindes,
+or of a Temperate and a Contrary, <i>What proportion is of the lesse
+quantitie 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
+quantitie</i>. As for example. Let two pound of Liquor be geuen, hote in
+the 4. degree: &amp; one pound of Liquor be geuen, hote in the third
+degree.<span class="sidequote">”</span> I&nbsp;would gladly know the
+Forme resulting, in the Mixture of these two Liquors. Set downe your
+nũbers in order, thus.
+<span class="figfloat">
+<img src="images/diagram1.png" width="168" height="89"
+alt="diagram: see end of text for alternative" />
+</span>
+Now by the rule of Algiebar, haue I deuised a very easie, briefe, and
+generall maner of working in this case. Let vs first, suppose that
+<i>Middle Forme resulting</i>, to be 1<span class="xmath">X</span>: as
+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<span class="xmath">X</span>: to the
+difference betwene 1<span class="xmath">X</span> and 3: (the degree of
+the thing, in lesse quãtitie. And with all, 1<span class="xmath">X</span>, being alwayes in a certaine middell, betwene the two
+heigthes or degrees). For the first difference, I&nbsp;set 4-1<span
+class="xmath">X</span>: and for the second, I&nbsp;set 1<span class="xmath">X</span>-3. And, now againe, I&nbsp;say, as 1. is to 2. so is
+4-1<span class="xmath">X</span> to 1<span class="xmath">X</span>-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 the other. Let these Multiplications be made
+accordingly. And of the first and the fourth, we haue 1<span class="xmath">X</span>-3. and of the second &amp; the third, 8-2<span class="xmath">X</span>. Wherfore, our Æquation is betwene 1<span class="xmath">X</span>-3: and 8-2<span class="xmath">X</span>. Which may be
+reduced, according to the Arte of Algiebar: as, here, adding 3. to eche
+part, geueth the Æquation, thus, 1<span class="xmath">X</span>=11-2<span class="xmath">X</span>. And yet againe,
+contracting, or Reducing it: Adde to eche part, 2<span class="xmath">X</span>: Then haue you 3<span class="xmath">X</span> æquall
+to 11: thus represented 3<span class="xmath">X</span>=11. Wherefore,
+diuiding 11. by 3: the Quotient is 3⅔: the <i>Valew</i> of our 1<span
+class="xmath">X</span>, <i>Coss</i>, or <i>Thing</i>, first supposed.
+And that is the heigth, or Intension of the <i>Forme resulting:</i>
+which is, <i>Heate</i>, 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 3⅔,
+<span class="figfloat">
+<img src="images/diagram2.png" width="437" height="130"
+alt="diagram: see end of text for alternative" />
+</span>
+resteth ⅔. Subtracte the same heigth of the Forme resulting, (which is
+3⅔) frõ&nbsp;4: then resteth&nbsp;⅓: You see, that ⅔ is double to ⅓: as
+2.<span class="pmath">P</span>. is double to 1.<span class="pmath">P</span>. So should it be: by the rule here geuen. Note. As you
+added to eche part of the Æquation, 3: so if ye first added to eche part
+2<span class="xmath">X</span>, it would stand, 3<span class="xmath">X</span>-3=8. And now adding to eche part 3: you haue (as afore)
+3<span class="xmath">X</span>=11.</p>
+
+<p>And though I, here, speake onely of two thyngs Miscible: and most
+commonly mo then three, foure, fiue or six, (&amp;c.) are to be Mixed:
+(and in one Compound
+<span class="pagenum verso">||</span>
+<!-- png 16 -->
+to be reduced: &amp; the Forme resultyng of the same, to serue the
+turne) yet these Rules are sufficient: duely repeated and iterated.
+<span class="sidenote">
+Note.</span>
+In procedyng first, with any two: and then, with the Forme Resulting,
+and an other: &amp; so forth: For, the last worke, concludeth the Forme
+resultyng of them all: I&nbsp;nede nothing to speake, of the Mixture
+(here supposed) what it is. Common Philosophie hath defined it, saying,
+<i>Mixtio est miscibilium, alteratorum, per minima coniunctorum,
+Vnio</i>. Euery word in the definition, is of great importance.
+I&nbsp;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
+varieties, by the foresayd two generall 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 particular: would
+haue larger discoursing, then is mete in this place to be dealt withall:
+And other (perchaunce) with a proude snuffe will disdaine this litle:
+and would be vnthankefull for much more. I,&nbsp;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 <i>Mixture of Formes and Qualities Naturall</i>, is so
+Matcht and maryed with the most simple, easie, and short way of the
+noble Rule of <i>Algiebar</i>. Who can remaine, therfore vnpersuaded, to
+loue, alow, and honor the excellent Science of <i>Arithmetike</i>? For,
+here, you may perceiue that the litle finger of <i>Arithmetike</i>, is
+of more might and contriuing, then a hunderd thousand mens wittes, of
+the middle sorte, are hable to perfourme, or truely to conclude, with
+out helpe thereof.</p>
+
+<p>Now will we farder, by the wise and valiant Capitaine, be certified,
+what helpe he hath, by the Rules of <i>Arithmetike</i>: in one of the
+Artes to him appertaining: And of the Grekes named
+<span class="sidenote">
+<span class="greek" title="Taktikê">Τακτικὴ</span>.</span>
+<span class="greek" title="Taktikê">Τακτικὴ</span>. <span class="sidequote">“</span>That is, the Skill of Ordring Souldiers in Battell
+ray after the best maner to all purposes.<span class="sidequote">”</span> This Art so much dependeth vppon Numbers vse, and
+the Mathematicals, that <i>Ælianus</i> (the best writer therof,) in his
+worke, to the <i>Emperour Hadrianus</i>, 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 <i>Æneas</i>: <i>Cyneas</i> of
+<i>Thessaly</i>: <i>Pyrrhus Epirota</i>: and <i>Alexander</i> his sonne:
+<i>Clearchus</i>: <i>Pausanias</i>: <i>Euangelus</i>: <i>Polybius</i>,
+familier frende to <i>Scipio</i>: <i>Eupolemus</i>: <i>Iphicrates</i>,
+<i>Possidonius</i>: and very many other worthy Capitaines, Philosophers
+and Princes of Immortall fame and memory: Whose fayrest floure of their
+garland (in this feat) was <i>Arithmetike</i>: and a litle
+perceiuerance, in <i>Geometricall</i> Figures. But in many other cases
+doth <i>Arithmetike</i> 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 company, to make the
+victuals, longer to serue the remanent, &amp; for a certaine determined
+tyme: if nede so require. And so in sundry his other accountes,
+Reckeninges, Measurynges, and proportionynges, the wise, expert, and
+Circumspect Capitaine will affirme the Science of <i>Arithmetike</i>, to
+be one of his chief Counsaylors, directers and aiders. Which thing (by
+good meanes) was euident to the Noble, the Couragious, the loyall, and
+Curteous
+<span class="sidefinger">
+&nbsp;</span>
+<i>Iohn</i>, late Earle of Warwicke. Who was a yong Gentleman, throughly
+knowne to very few. Albeit his lusty valiantnes, force, and Skill in
+Chiualrous feates and exercises: his humblenes, and frendelynes 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
+<span class="pagenum">a.j </span>
+<!-- png 17 -->
+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 achieue: 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 &amp; warre. These (I&nbsp;say)
+his Heroicall Meditations, forecastinges and determinations, no twayne,
+(I&nbsp;thinke) beside my selfe, can so perfectly, and truely report.
+And therfore, in Conscience, I&nbsp;count it my part, for the honor,
+preferment, &amp; procuring of vertue (thus, briefly) to haue put his
+Name, in the Register of <i>Fame Immortall</i>.</p>
+
+<p>To our purpose. This <i>Iohn</i>, by one of his actes (besides many
+other: both in England and Fraunce, by me, in him noted.) did disclose
+his harty loue to vertuous Sciences: and his noble intent, to excell in
+Martiall prowesse: When he, with humble 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 deuised) for
+ordring of all Companies, summes and Numbers of mẽ, (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 seem few, being many. To marche in battaile or Iornay: with many such
+feates, to Foughten field, Skarmoush, or Ambushe appartaining:
+<span class="sidenote plain">
+This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age: hauing
+no issue by his wife: Daughter to the Duke of Somerset.</span>
+And of all these, liuely designementes (most curiously) to be in velame
+parchement described: with Notes &amp; peculier markes, as the Arte
+requireth: and all these Rules, and descriptions Arithmeticall, inclosed
+in a riche Case of Gold, he vsed to weare about his necke: as his Iuell
+most precious, and Counsaylour most trusty. Thus, <i>Arithmetike</i>, of
+him, was shryned in gold: Of <i>Numbers</i> frute, he had good hope.
+Now, Numbers therfore innumerable, in <i>Numbers</i> prayse, his shryne
+shall finde.</p>
+
+<p>What nede I, (for farder profe to you) of the Scholemasters of
+Iustice, to require testimony: how nedefull, how frutefull, how
+skillfull a thing <i>Arithmetike</i> is? I&nbsp;meane, the Lawyers of
+all sortes. Vndoubtedly, the Ciuilians, can meruaylously declare: how,
+neither the Auncient Romaine lawes, without good knowledge of <i>Numbers
+art</i>, can be perceiued: Nor (Iustice in infinite Cases) without due
+proportion, (narrowly considered,) is hable to be executed. How Iustly,
+&amp; with great knowledge of Arte, did <i>Papinianus</i> institute a
+law of partition, and allowance, betwene man and wife after a diuorce?
+But how <i>Accursius</i>, <i>Baldus</i>, <i>Bartolus</i>, <i>Iason</i>,
+<i>Alexander</i>, and finally <i>Alciatus</i>, (being otherwise, notably
+well learned) do iumble, gesse, and erre, from the æquity, art and
+Intent of the lawmaker: <i>Arithmetike</i> can detect, and conuince: and
+clerely, make the truth to shine. Good <i>Bartolus</i>, tyred in the
+examining &amp; proportioning of the matter: and with <i>Accursius</i>
+Glosse, much cumbred: burst out, and sayd: <i>Nulla est in toto libro,
+hac glossa difficilior: Cuius computationem nec Scholastici nec Doctores
+intelligunt. &amp;c.</i> That is: <span class="larger"><i>In the whole
+booke, there is no Glosse harder then this: Whose accoumpt or reckenyng,
+neither the Scholers, nor the Doctours vnderstand. &amp;c.</i></span>
+What can they say of <i>Iulianus</i> law, <i>Si ita Scriptum.
+&amp;c.</i> Of the Testators will iustly performing, betwene the wife,
+Sonne and daughter? How can they perceiue the æquitie of
+<i>Aphricanus</i>, <i>Arithmeticall</i> Reckening, where he treateth of
+<i>Lex Falcidia</i>? How can they deliuer him, from his Reprouers: and
+their maintainers: as <i>Ioannes</i>, <i>Accursius Hypolitus</i> and
+<i>Alciatus</i>? How Iustly and artificially, was <i>Africanus</i>
+reckening made? Proportionating to the Sommes bequeathed, the
+Contributions of eche part? Namely, for the hundred presently receiued,
+17&nbsp;1/7. And for the hundred, receiued after ten monethes,
+12&nbsp;6/7: which make the 30: which were to be cõtributed by the
+legataries to the heire.
+<span class="pagenum verso">||</span>
+<!-- png 18 -->
+For, what proportion, 100 hath to 75: the same hath 17&nbsp;1/7 to
+12&nbsp;6/7: Which is Sesquitertia: that is, as 4, to 3. which make 7.
+Wonderfull many places, in the Ciuile law, require an expert
+<i>Arithmeticien</i>, to vnderstand the deepe Iudgemẽt, &amp; Iust
+determinatiõ of the Auncient Romaine Lawmakers. But much more expert
+ought he to be, who should be hable, to decide with æquitie, 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 equity might be
+greately preferred, and skilfully executed, through due skill of
+Arithmetike, and proportions appertainyng. The worthy Philosophers, and
+prudent lawmakers (who haue written many bookes <i>De Republica:</i> How
+the best state of Common wealthes might be procured and mainteined,)
+haue very well determined 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:) defining it,
+<span class="sidenote">
+Iustice.</span>
+<span class="sidequote">“</span>to be that vertue, by which, to euery
+one, is rendred, that to him appertaineth.<span class="sidequote">”</span> God challengeth this at our handes, to be honored
+as God: to be loued, as a father: to be feared as a Lord &amp; master.
+Our neighbours proportiõ, is also prescribed 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.
+<i>Aristotle</i> in his <i>Ethikes</i> (to fatch the sede of Iustice,
+and light of direction, to vse and execute the same) was fayne to fly to
+the perfection, and power of Numbers: for proportions Arithmeticall and
+Geometricall. <i>Plato</i> in his booke called <i>Epinomis</i> (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, <i>Wise</i>. He, briefly, of other Sciences discoursing,
+findeth them, not hable to bring it to passe: But of the Science of
+Numbers, he sayth. <i>Illa, quæ numerum mortalium generi dedit, id
+profecto efficiet. Deum autem aliquem, magis quam fortunam, ad salutem
+nostram, hoc munus nobis arbitror contulisse. &amp;c. Nam ipsum bonorum
+omnium Authorem, cur non maximi boni, Prudentiæ dico, causam arbitramur?
+<span class="larger">That Science, verely, which hath taught mankynde
+number, shall be able to bryng it to passe. And, I&nbsp;thinke,
+a&nbsp;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 the greatest good thyng,
+namely, Wisedome?</span></i> There, at length, he proueth
+<i>Wisedome</i> to be atteyned, by good Skill of <i>Numbers</i>. With
+which great Testimony, and the manifold profes, and reasons, before
+expressed, you may be sufficiently and fully persuaded: of the perfect
+Science of <i>Arithmetike</i>, to make this accounte: That
+<span class="sidefinger">
+&nbsp;</span>
+of all Sciences, next to <i>Theologie</i>, 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
+<i>Magnitudes</i>: of which (by the Direction and aide of him, whose
+<i>Magnitude</i> is Infinite, and of vs Incomprehensible) I&nbsp;now
+entend, so to write, that both with the <i>Multitude</i>, and also with
+the <i>Magnitude</i> 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 vnspeakable behofe) our heauenly
+father, hath for vs prepared, and reuealed, by sundry
+<i>Philosophers</i> and <i>Mathematiciens</i>.</p>
+
+<p>Both, <i>Number</i> and <i>Magnitude</i>, haue a certaine Originall
+sede, (as it were) of an incredible property: and of man, neuer hable,
+Fully, to be declared. Of <i>Number</i>, an Vnit, and of
+<i>Magnitude</i>, a&nbsp;Poynte, doo seeme to be much like Originall
+<span class="pagenum">a.ij </span>
+<!-- png 19 -->
+causes: But the diuersitie neuerthelesse, is great. We defined an
+<i>Vnit</i>, to be a thing Mathematicall Indiuisible: A&nbsp;Point,
+likewise, we sayd to be a Mathematicall thing Indiuisible. And farder,
+that a Point may haue a certaine determined Situation: that is, that we
+may assigne, and prescribe a Point, to be here, there, yonder. &amp;c.
+Herein, (behold) our Vnit is free, and can abyde no bondage, or to be
+tyed to any place, or seat: diuisible or indiuisible. Agayne, by reason,
+a&nbsp;Point may haue a Situation limited to him: a&nbsp;certaine
+motion, therfore (to a place, and from a place) is to a Point incident
+and appertainyng. But an <i>Vnit</i>, can not be imagined to haue any
+motion. A&nbsp;Point, by his motion, produceth, Mathematically,
+a&nbsp;line: (as we sayd before) which is the first kinde of Magnitudes,
+and most simple: An <i>Vnit</i>, can not produce any number.
+A&nbsp;Line, though it be produced of a Point moued, yet, it doth not
+consist of pointes: Number, though it be not produced of an <i>Vnit</i>,
+yet doth it Consist of vnits, as a materiall cause. But formally,
+<span class="sidenote">
+Number.</span>
+Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting,
+is the workemanship of our minde: which, of distinct and discrete Vnits,
+maketh a Number: by vniformitie, resulting of a certaine multitude of
+Vnits. And so, euery number, may haue his least part, giuen: namely, an
+Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be
+giuẽ: by cause, infinitly, diuision therof, may be conceiued. All
+Magnitude, is either a Line, a&nbsp;Plaine, or a Solid. Which Line,
+Plaine, or Solid, of no Sense, can be perceiued, nor exactly by hãd (any
+way) represented: nor of Nature produced: But, as (by degrees) Number
+did come to our perceiuerance: So, by visible formes, we are holpen to
+imagine, what our Line Mathematicall, is. What our Point, is. So
+precise, are our Magnitudes, that one Line is no broader then an other:
+for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our
+Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we
+can haue Smaller, then either Arte or Nature can produce any: and
+Greater also, then all the world can comprehend. Our least Magnitudes,
+can be diuided into so many partes, as the greatest. As, a&nbsp;Line of
+an inch long, (with vs) may be diuided into as many partes, as may the
+diameter of the whole world, from East to West: or any way extended:
+What priuiledges, aboue all manual Arte, and Natures might, haue our two
+Sciences Mathematicall? to exhibite, and to deale with thinges of such
+power, liberty, simplicity, puritie, and perfection? And in them, so
+certainly, so orderly, so precisely to procede: as, excellent is that
+workemã Mechanicall Iudged, who nerest can approche to the representing
+of workes, Mathematically demonstrated?
+<span class="sidefinger">
+&nbsp;</span>
+And our two Sciences, remaining pure, and absolute, in their proper
+termes, and in their owne Matter: to haue, and allowe, onely such
+Demonstrations, as are plaine, certaine, vniuersall, and of an æternall
+veritye?
+<span class="sidenote">
+Geometrie.</span>
+This Science of <i>Magnitude</i>, his properties, conditions, and
+appertenances: commonly, now is, and from the beginnyng, hath of all
+Philosophers, ben called <i>Geometrie</i>. But, veryly, with a name to
+base and scant, for a Science of such dignitie and amplenes. And,
+perchaunce, that name, by cõmon and secret consent, of all wisemen,
+hitherto hath ben suffred to remayne: that it might carry with it a
+perpetuall memorye, of the first and notablest benefite, by that
+Science, to common people shewed: Which was, when Boundes and meres of
+land and ground were lost, and confounded (as in <i>Egypt</i>, yearely,
+with the ouerflowyng of <i>Nilus</i>, the greatest and longest riuer in
+the world) or, that ground bequeathed, were to be assigned: or, ground
+sold, were to be layd out: or (when disorder preuailed) that Commõs were
+distributed into seueralties. For, where, vpon these &amp; such like
+occasiõs, Some by ignorãce, some by negligẽce, Some by fraude, and some
+by violence, did wrongfully limite, measure, encroach, or challenge (by
+<span class="pagenum verso">||</span>
+<!-- png 20 -->
+pretence of iust content, and measure) those landes and groundes: great
+losse, disquietnes, murder, and warre did (full oft) ensue: Till, by
+Gods mercy, and mans Industrie, The perfect Science of Lines, Plaines,
+and Solides (like a diuine Iusticier,) gaue vnto euery man, his owne.
+The people then, by this art pleasured, and greatly relieued, in their
+landes iust measuring: &amp; other Philosophers, writing Rules for land
+measuring: betwene them both, thus, confirmed the name of
+<i>Geometria</i>, that is, (according to the very etimologie of the
+word) Land measuring. Wherin, the people knew no farder, of Magnitudes
+vse, but in Plaines: and the Philosophers, of thẽ, had no feet hearers,
+or Scholers: farder to disclose vnto, then of flat, plaine
+<i>Geometrie</i>. And though, these Philosophers, knew of farder vse,
+and best vnderstode the etymologye of the worde, yet this name
+<i>Geometria</i>, was of them applyed generally to all sortes of
+Magnitudes: vnleast, otherwhile, of <i>Plato</i>, and <i>Pythagoras</i>:
+When they would precisely declare their owne doctrine. Then, was
+<span class="sidenote">
+* <i>Plato. 7. de Rep.</i></span>
+*<i>Geometria</i>, with them, <i>Studium quod circa planum versatur</i>.
+But, well you may perceiue by <i>Euclides Elementes</i>, that more ample
+is our Science, then to measure Plaines: and nothyng lesse therin is
+tought (of purpose) then how to measure Land. An other name, therfore,
+must nedes be had, for our Mathematicall Science of Magnitudes: which
+regardeth neither clod, nor turff: neither hill, nor dale: neither earth
+nor heauen: but is absolute <i>Megethologia</i>: not creping on ground,
+and dasseling the eye, with pole perche, rod or lyne: but <span class="sidequote">“</span>liftyng the hart aboue the heauens, by inuisible
+lines, and
+<span class="sidefinger">
+&nbsp;</span>
+immortall beames meteth with the reflexions, of the light
+incomprehensible: and so procureth Ioye, and perfection
+vnspeakable.<span class="sidequote">”</span> Of which true vse of our
+<i>Megethica</i>, or <i>Megethologia</i>, <i>Diuine Plato</i> seemed to
+haue good taste, and iudgement: and (by the name of <i>Geometrie</i>) so
+noted it: and warned his Scholers therof: as, in hys seuenth
+<i>Dialog</i>, of the Common wealth, may euidently be sene. Where (in
+Latin) thus it is: right well translated: <i>Profecto, nobis hoc non
+negabunt, Quicunq<sup>ue</sup> vel paululum quid Geometriæ gustârunt,
+quin hæc Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui
+in ipsa versantur.</i> In English, thus. <span class="larger"><i>Verely</i></span> (sayth <i>Plato</i>) <span class="larger"><i>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.</i></span> And there it followeth, of our
+<i>Geometrie</i>, <i>Quòd quæritur cognoscendi illius gratia, quod
+semper est, non &amp; eius quod oritur quandoq<sup>ue</sup> &amp;
+interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur
+(ô&nbsp;Generose vir) ad Veritatem, animum: atq<sup>ue</sup> ita, ad
+Philosophandum preparabit cogitationem, vt ad supera conuertamus: quæ,
+nunc, contra quàm decet, ad inferiora deijcimus. &amp;c. Quàm maximè
+igitur præcipiendum est, vt qui præclarissimam hanc habitãt Civitatem,
+nullo modo, Geometriam spernant. Nam &amp; quæ præter ipsius propositum,
+quodam modo esse videntur, haud exigua sunt. &amp;c.</i> It must nedes
+be confessed (saith <i>Plato</i>) <span class="larger"><i>That</i></span> <span class="smaller">[<i>Geometrie</i>]</span> <span class="larger"><i>is
+learned, for the knowyng of that, which is euer: and not of that, which,
+in tyme, both is bred and is brought to an ende. &amp;c. Geometrie is
+the knowledge of that which is euerlastyng. It will lift vp therfore
+(O&nbsp;Gentle Syr) our mynde to the Veritie: and by that meanes, it
+will prepare the Thought, to the Philosophicall loue of wisdome: that we
+may turne or conuert, toward heauenly thinges</i></span> <span class="smaller">[<i>both mynde and thought</i>]</span> <span class="larger"><i>which now, otherwise then becommeth vs, we cast down on base
+or inferior things. &amp;c. Chiefly, therfore, Commaundement must be
+giuen, that such as do inhabit this most honorable Citie, by no meanes,
+despise Geometrie. For euen those thinges</i></span> <span class="smaller">[<i>done by it</i>]</span> <span class="larger"><i>which, in
+manner, seame to be, beside the purpose of Geometrie: are of</i></span>
+<span class="pagenum">a.iij </span>
+<!-- png 21 -->
+<span class="larger"><i>no small importance. &amp;c.</i></span> And
+besides the manifold vses of <i>Geometrie</i>, in matters appertainyng
+to warre, he addeth more, of second vnpurposed frute, and commoditye,
+arrising by <i>Geometrie</i>: saying: <i>Scimus quin etiam, ad
+Disciplinas omnes facilius per discendas, interesse omnino, attigerit ne
+Geometriam aliquis, an non. &amp;c. Hanc ergo Doctrinam, secundo loco
+discendam Iuuenibus statuamus.</i> That is. <span class="larger"><i>But, also, we know, that for the more easy learnyng of all
+Artes, it importeth much, whether one haue any knowledge in Geometrie,
+or no. &amp;c. Let vs therfore make an ordinance or decree, that this
+Science, of young men shall be learned in the second place.</i></span>
+This was <i>Diuine Plato</i> his Iudgement, both of the purposed, chief,
+and perfect vse of <i>Geometrie</i>: and of his second, dependyng,
+deriuatiue commodities. And for vs, Christen men, a&nbsp;thousand
+thousand mo occasions are, to haue nede of the helpe of*
+<span class="sidenote">
+I.&nbsp;D.<br />
+* Herein, I&nbsp;would gladly shake of, the earthly name, of
+Geometrie.</span>
+<i>Megethologicall</i> Contemplations: wherby, to trayne our
+Imaginations and Myndes, by litle and litle, to forsake and abandon, the
+grosse and corruptible Obiectes, of our vtward senses: and to apprehend,
+by sure doctrine demonstratiue, Things Mathematicall. And by them,
+readily to be holpen and conducted to conceiue, discourse, and conclude
+of things Intellectual, Spirituall, æternall, and such as concerne our
+Blisse euerlasting: which, otherwise (without Speciall priuiledge of
+Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally)
+is hable to reach vnto, or to Compasse. And, veryly, by my small Talent
+(from aboue) I&nbsp;am hable to proue and testifie, that the litterall
+Text, and order of our diuine Law, Oracles, and Mysteries, require more
+skill in Numbers, and Magnitudes: then (commonly) the expositors haue
+vttered: but rather onely (at the most) so warned: &amp; shewed their
+own want therin. (To name any, is nedeles: and to note the places, is,
+here, no place: But if I be duely asked, my answere is ready.) And
+without the litterall, Grammaticall, Mathematicall or Naturall verities
+of such places, by good and certaine Arte, perceiued, no Spirituall
+sense (propre to those places, by Absolute <i>Theologie</i>) will
+thereon depend.
+<span class="sidefinger">
+&nbsp;</span>
+<span class="sidequote">“</span>No man, therfore, can doute, but
+toward the atteyning of knowledge incomparable, and Heauenly Wisedome:
+Mathematicall Speculations, both of Numbers and Magnitudes: are meanes,
+aydes, and guides: ready, certaine, and necessary.<span class="sidequote">”</span> From henceforth, in this my Preface, will I frame
+my talke, to <i>Plato</i> his fugitiue Scholers: or, rather, to such,
+who well can, (and also wil,) vse their vtward senses, to the glory of
+God, the benefite of their Countrey, and their owne secret contentation,
+or honest preferment, on this earthly Scaffold. To them, I&nbsp;will
+orderly recite, describe &amp; declare a great Number of Artes, from our
+two Mathematicall fountaines, deriued into the fieldes of <i>Nature</i>.
+Wherby, such Sedes, and Rotes, as lye depe hyd in the groũd of
+<i>Nature</i>, are refreshed, quickened, and prouoked to grow, shote vp,
+floure, and giue frute, infinite, and incredible. And these Artes,
+shalbe such, as vpon Magnitudes properties do depende, more, then vpon
+Number. And by good reason we may call them Artes, and Artes
+Mathematicall Deriuatiue: for (at this tyme) I&nbsp;Define
+<span class="sidenote">
+An Arte.</span>
+<span class="larger">An Arte, to be a Methodicall cõplete Doctrine,
+hauing abundancy of sufficient, and peculier matter to deale with, by
+the allowance of the Metaphisicall Philosopher: the knowledge whereof,
+to humaine state is necessarye.</span> And that I account,
+<span class="sidenote">
+Art Mathematicall Deriuatiue.</span>
+<span class="larger">An Art Mathematicall deriuatiue, which by
+Mathematicall demonstratiue Method, in Nũbers, or Magnitudes, ordreth
+and confirmeth his doctrine, as much &amp; as perfectly, as the matter
+subiect will admit.</span> And for that,
+<span class="pagenum verso">||</span>
+<!-- png 22 -->
+I entend to vse the name and propertie of a
+<span class="sidenote">
+A Mechanitien.</span>
+<i>Mechanicien</i>, otherwise, then (hitherto) it hath ben vsed,
+I&nbsp;thinke it good, (for distinction sake) to giue you also a brief
+description, what I meane therby. <span class="larger">A Mechanicien,
+or a Mechanicall workman is he, whose skill is, without knowledge of
+Mathematicall demonstration, perfectly to worke and finishe any sensible
+worke, by the Mathematicien principall or deriuatiue, demonstrated or
+demonstrable.</span> Full well I know, that he which inuenteth, or
+maketh these demonstrations, is generally called <i>A speculatiue
+Mechanicien</i>: which differreth nothyng from a <i>Mechanicall
+Mathematicien</i>. So, in respect of diuerse actions, one man may haue
+the name of sundry artes: as, some tyme, of a Logicien, some tymes (in
+the same matter otherwise handled) of a Rethoricien. Of these trifles,
+I&nbsp;make, (as now, in respect of my Preface,) small account: to fyle
+thẽ for the fine handlyng of subtile curious disputers. In other places,
+they may commaunde me, to giue good reason: and yet, here, I&nbsp;will
+not be vnreasonable.</p>
+
+<p>
+<span class="sidenote larger">1.</span>
+First, then, from the puritie, absolutenes, and Immaterialitie of
+Principall <i>Geometrie</i>, is that kinde of <i>Geometrie</i> deriued,
+which vulgarly is counted <i>Geometrie</i>: and is the <span class="larger">Arte of Measuring sensible magnitudes, their iust quãtities and
+contentes.</span>
+<span class="sidenote">
+Geometrie vulgar.</span>
+This, teacheth to measure, either at hand: and the practiser, to be by
+the thing Measured: and so, by due applying of Cumpase, Rule, Squire,
+Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to
+the Length, Plaine, or Solide measured,
+<span class="sidenote">
+1.</span>
+*to be certified, either of the length, perimetry, or distance lineall:
+and this is called, <i>Mecometrie</i>. Or
+<span class="sidenote">
+2.</span>
+*to be certified of the content of any plaine Superficies: whether it be
+in ground Surueyed, Borde, or Glasse measured, or such like thing: which
+measuring, is named <i>Embadometrie</i>.
+<span class="sidenote">
+3.</span>
+*Or els to vnderstand the Soliditie, and content of any bodily thing: as
+of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels,
+small &amp; great, of all fashions. Where, of Wine, Oyle, Beere, or Ale
+vessells, &amp;c, the Measuring, commonly, hath a peculier name: and is
+called <i>Gaging</i>. And the generall name of these Solide measures, is
+<i>Stereometrie</i>.
+<span class="sidenote larger">2.</span>
+Or els, this <i>vulgar Geometrie</i>, hath consideration to teach the
+practiser, how to measure things, with good distance betwene him and the
+thing measured: and to vnderstand thereby, either
+<span class="sidenote">
+1.</span>
+*how Farre, a&nbsp;thing seene (on land or water) is from the measurer:
+and this may be called <i>Apomecometrie</i>:
+<span class="sidenote">
+2.</span>
+Or, how High or depe, aboue or vnder the leuel of the measurers stãding,
+any thing is, which is sene on land or water, called <i>Hypsometrie</i>.
+<span class="sidenote">
+3.</span>
+*Or, it informeth the measurer, how Broad any thing is, which is in the
+measurers vew: so it be on Land or Water, situated: and may be called
+<i>Platometrie</i>. Though I vse here to condition, the thing measured,
+to be on Land, or Water Situated:
+<span class="sidenote">
+Note.</span>
+yet, know for certaine, that the sundry heigthe of Cloudes, blasing
+Starres, and of the Mone, may (by these meanes) haue their distances
+from the earth: and, of the blasing Starres and Mone, the Soliditie
+(aswell as distances) to be measured: But because, neither these things
+are vulgarly taught: nor of a common practiser so ready to be executed:
+I, rather, let such measures be reckened incident to some of our other
+Artes, dealing with thinges on high, more purposely, then this vulgar
+Land measuring Geometrie doth: as in <i>Perspectiue</i> and
+<i>Astronomie, &amp;c.</i></p>
+
+<p><span class="dropcap">O</span>F these Feates (farther applied) is
+Sprong the Feate of <i>Geodesie</i>, or Land Measuring: more cunningly
+to measure &amp; Suruey Land, Woods, and Waters, a&nbsp;farre of. More
+cunningly, I&nbsp;say: But God knoweth (hitherto) in these Realmes of
+England and Ireland (whether through ignorance or fraude, I&nbsp;can not
+tell, in euery particular)
+<span class="sidenote">
+Note.</span>
+how great wrong and iniurie hath (in my time) bene committed
+<span class="pagenum">a.iij </span>
+<!-- png 23 -->
+by vntrue measuring and surueying of Land or Woods, any way. And, this I
+am sure: that the Value of the difference, betwene the truth and such
+Surueyes, would haue bene hable to haue foũd (for euer) in eche of our
+two Vniuersities, an excellent Mathematicall Reader: to eche, allowing
+(yearly) a&nbsp;hundred Markes of lawfull money of this realme: which,
+in dede, would seme requisit, here, to be had (though by other wayes
+prouided for) as well, as, the famous Vniuersitie of Paris, hath two
+Mathematicall Readers: and eche, two hundreth French Crownes yearly, of
+the French Kinges magnificent liberalitie onely. Now, againe, to our
+purpose returning: Moreouer, of the former knowledge Geometricall, are
+growen the Skills of <i>Geographie</i>, <i>Chorographie</i>,
+<i>Hydrographie</i>, and <i>Stratarithmetrie</i>.</p>
+
+<p><span class="sidequote">“</span>
+<span class="largest">Geographie</span> teacheth wayes, by which, in
+sũdry formes, (as <i>Sphærike</i>, <i>Plaine</i> or other), the
+Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines,
+Woods, Hauens, Riuers, Crekes, &amp; such other things, vpõ the outface
+of the earthly Globe (either in the whole, or in some principall mẽber
+and portion therof cõtayned) may be described and designed, in
+cõmensurations Analogicall to Nature and veritie: and most aptly to our
+vew, may be represented.<span class="sidequote">”</span> Of this Arte
+how great pleasure, and how manifolde commodities do come vnto vs, daily
+and hourely: of most men, is perceaued. While, some, to beautifie their
+Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other
+some, for thinges past, as battels fought, earthquakes, heauenly
+fyringes, &amp; 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&nbsp;say, in respecte of the rest. &amp;c. Some, either
+for their owne iorneyes directing into farre landes: or to vnderstand of
+other mens trauailes. To conclude, some, for one purpose: and some, for
+an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, &amp;
+Geographicall Globes. Of whose vse, to speake sufficiently, would
+require a booke peculier.</p>
+
+<p><span class="largest">Chorographie</span> seemeth to be an
+vnderling, and a twig, of <i>Geographie</i>: and yet neuerthelesse, is
+in practise manifolde, and in vse very ample. <span class="sidequote">“</span>This teacheth Analogically to describe a small
+portion or circuite of ground, with the contentes: not regarding what
+commensuration it hath to the whole, or any parcell, without it,
+contained. But in the territory or parcell of ground which it taketh in
+hand to make description of, it leaueth out (or vndescribed) no notable,
+or odde thing, aboue the ground visible. Yea and sometimes, of thinges
+vnder ground, geueth some peculier marke: or warning: as of Mettall
+mines, Cole pittes, Stone quarries. &amp;c.<span class="sidequote">”</span> Thus, a&nbsp;Dukedome, a&nbsp;Shiere,
+a&nbsp;Lordship, or lesse, may be described distinctly. But marueilous
+pleasant, and profitable it is, in the exhibiting to our eye, and
+commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true
+Symmetry: not approching to any of them: and out of Gunne shot. &amp;c.
+Hereby, the <i>Architect</i> may furnishe him selfe, with store of what
+patterns he liketh: to his great instruction: euen in those thinges
+which outwardly are proportioned: either simply in them selues: or
+respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also,
+terme this particular description of places, <i>Topographie</i>.</p>
+
+<p><span class="sidequote">“</span>
+<span class="largest">Hydrographie</span>, 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:<span class="sidequote">”</span> with the
+Iles and chiefe
+<span class="pagenum verso">||</span>
+<!-- png 24 -->
+particular places of daungers, conteyned within the boundes, and Sea
+coastes described: as, of Quicksandes, Bankes, Pittes, Rockes, Races,
+Countertides, Whorlepooles. &amp;c. This, dealeth with the Element of
+the water chiefly: as <i>Geographie</i> did principally take the Element
+of the Earthes description (with his appertenances) to taske. And
+besides thys, <i>Hydrographie</i>, requireth a particular Register of
+certaine Landmarkes (where markes may be had) from the sea, well hable
+to be skried, in what point of the Seacumpase they appeare, and what
+apparent forme, Situation, and bignes they haue, in respecte of any
+daungerous place in the sea, or nere vnto it, assigned: And in all
+Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes,
+come and go, the <i>Hydrographer</i> ought to recorde. The Soundinges
+likewise: and the Chanels wayes: their number, and depthes ordinarily,
+at ebbe and flud, ought the <i>Hydrographer</i>, by obseruation and
+diligence of <i>Measuring</i>, to haue certainly knowen. And many other
+pointes, are belonging to perfecte <i>Hydrographie</i>, and for to make
+a <i>Rutter</i>, by: of which, I&nbsp;nede not here speake: as of the
+describing, in any place, vpon Globe or Plaine, the 32. pointes of the
+Compase, truely: (wherof, scarsly foure, in England, haue right
+knowledge: bycause, the lines therof, are no straight lines, nor
+Circles.) Of making due proiection of a Sphere in plaine. Of the
+Variacion of the Compas, from true Northe: And such like matters (of
+great importance, all) I&nbsp;leaue to speake of, in this place:
+bycause, I&nbsp;may seame (al ready) to haue enlarged the boundes, and
+duety of an <ins class="correction" title="text reads ‘Hydographer’">Hydrographer</ins>, 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.</p>
+
+<p><span class="largest">Stratarithmetrie</span>, is the Skill,
+(appertainyng to the warre,) by which a man can set in figure,
+analogicall to any <i>Geometricall</i> figure appointed, any certaine
+number or summe of men: of such a figure capable: (by reason of the
+vsuall spaces betwene Souldiers allowed: and for that, of men, can be
+made no Fractions. Yet, neuertheles, he can order the giuen summe of
+men, for the greatest such figure, that of them, cã be ordred) and
+certifie, of the ouerplus: (if any be) and of the next certaine summe,
+which, with the ouerplus, will admit a figure exactly proportionall to
+the figure assigned. By which Skill, also, of any army or company of
+men: (the figure &amp; sides of whose orderly standing, or array, is
+knowen) he is able to expresse the iust number of men, within that
+figure conteined: or (orderly) able to be conteined.
+<span class="sidenote">
+* <i>Note.</i></span>
+*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 <i>Stratarithmetrie</i>: sufficient for this tyme and
+place.
+<span class="sidenote">
+The difference betwene Stratarithmetrie and Tacticie.</span>
+<span class="sidequote">“</span>It differreth from the Feate
+<i>Tacticall</i>, <i>De aciebus instruendis.</i> bycause, there, is
+necessary the wisedome and foresight, to what purpose he so ordreth the
+men: and Skillfull hability, also, for any occasion, or purpose, to
+deuise and vse the aptest and most necessary order, array and figure of
+his Company and Summe of men.<span class="sidequote">”</span> By
+figure, I&nbsp;meane: as, either of a <i>Perfect Square</i>,
+<i>Triangle</i>, <i>Circle</i>, <i>Ouale</i>, <i>long square</i>, (of
+the Grekes it is called <ins class="correction" title="text unchanged: correct form is ‘Heteromekes’"><i>Eteromekes</i></ins>)
+<i>Rhombe</i>, <i>Rhomboïd</i>, <i>Lunular</i>, <i>Ryng</i>,
+<i>Serpentine</i>, and such other Geometricall figures: Which, in
+warres, haue ben, and are to be vsed: for commodiousnes, necessity, and
+auauntage &amp;c. And no small skill ought he to haue, that should make
+true report, or nere the truth, of the numbers and Summes, of footemen
+or horsemen, in the Enemyes ordring. A&nbsp;farre of, to make an
+estimate, betwene nere termes of More and Lesse, is not a thyng very
+rife, among those that gladly would
+<span class="pagenum">b.j.</span>
+<!-- png 25 -->
+do it.
+<span class="sidenote smaller">
+I.&nbsp;D.<br />
+Frende,<br />
+you will finde it hard, to performe my description of this Feate. But by
+Chorographie, you may helpe your selfe some what: where the Figures
+knowne (in Sides and Angles) are not Regular: And where, Resolution into
+Triangles can serue. &amp;c. And yet you will finde it strange to deale
+thus generally with Arithmeticall figures: and, that for Battayle ray.
+Their contentes, differ so much from like Geometricall Figures.</span>
+Great pollicy may be vsed of the Capitaines, (at tymes fete, and in
+places conuenient) as to vse Figures, which make greatest shew, of so
+many as he hath: and vsing the aduauntage of the three kindes of vsuall
+spaces: (betwene footemen or horsemen) to take the largest: or when he
+would seme to haue few, (beyng many:) contrarywise, in Figure, and
+space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer
+is carefull to come nere the truth herein, besides the Iudgement of his
+expert eye, his skill of Ordering <i>Tacticall</i>, the helpe of his
+Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously
+framed for cariage and vse) He may wonderfully helpe him selfe, by
+perspectiue Glasses. In which, (I&nbsp;trust) our posterity will proue
+more skillfull and expert, and to greater purposes, then in these dayes,
+can (almost) be credited to be possible.</p>
+
+<p>Thus haue I lightly passed ouer the Artificiall Feates, chiefly
+dependyng vpon vulgar <i>Geometrie</i>: &amp; commonly and generally
+reckened vnder the name of <i>Geometrie</i>. But there are other (very
+many) <i>Methodicall Artes</i>, which, declyning from the purity,
+simplicitie, and Immateriality, of our Principall Science of
+<i>Magnitudes</i>: 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 <i>Geometrie</i>, (from which they
+are deriued) and one from the other. As <span class="larger">Perspectiue, Astronomie, Musike, Cosmographie, Astrologie,
+Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie,
+Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture,
+Nauigation, Thaumaturgike</span> and <span class="larger">Archemastrie</span>. I&nbsp;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
+<i>Geometrie</i>, had but serued for buildyng of an house, or a curious
+bridge, or the roufe of Westminster hall, or some witty pretty deuise,
+or engyn, appropriate to a Carpenter, or a Ioyner &amp;c. That the thing
+is farre otherwise, then the world, (commonly) to this day, hath demed,
+by worde and worke, good profe wilbe made.</p>
+
+<p>Among these Artes, by good reason, <span class="largest">Perspectiue</span> ought to be had, ere of <i>Astronomicall
+Apparences</i>, perfect knowledge can be atteyned. And bycause of the
+prerogatiue of <i>Light</i>, beyng the first of <i>Gods Creatures</i>:
+and the eye, the light of our body, and his Sense most mighty, and his
+organ most Artificiall and <i>Geometricall</i>: At <i>Perspectiue</i>,
+we will begyn therfore. <span class="larger">Perspectiue, is an Art
+Mathematicall, which demonstrateth the maner, and properties, of all
+Radiations Direct, Broken, and Reflected.</span> This Description, or
+Notation, is brief: but it reacheth so farre, as the world is wyde. It
+concerneth all Creatures, all Actions, and passions, by Emanation of
+beames perfourmed. Beames, or naturall lines, (here) I&nbsp;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
+<i>Formes</i>, both <i>Substantiall</i>, and <i>Accidentall</i>, the
+certaine and determined actiue Radiall emanations. By this Art (omitting
+to speake of the highest pointes) we may vse our eyes, and the light,
+with greater pleasure: and perfecter Iudgement: both of things, in light
+seen, &amp; of other: which by like order of Lightes Radiations, worke
+and produce their effectes. We may be ashamed to be ignorant of the
+cause, why so sundry wayes our eye is deceiued, and abused: as, while
+the eye weeneth a roũd Globe or Sphere (beyng farre of) to be a flat and
+plaine Circle, and so likewise iudgeth
+<span class="pagenum verso">||</span>
+<!-- png 26 -->
+a plaine Square, to be roũd: supposeth walles parallels, to approche,
+a&nbsp;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, &amp; 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, (&amp;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, &amp; occasion Naturall? Yea, rather, is it not, greatly,
+against the Souerainty of Mans nature, to be so ouershot and abused,
+with thinges (at hand) before his eyes? as with a Pecockes tayle, and a
+Doues necke: or a whole ore, in water, holden, to seme broken. Thynges,
+farre of, to seeme nere: and nere, to seme farre of. Small thinges, to
+seme great: and great, to seme small. One man, to seme an Army. Or a man
+to be curstly affrayed of his owne shaddow. Yea, so much, to feare,
+that, if you, being (alone) nere a certaine glasse, and proffer, with
+dagger or sword, to foyne at the glasse, you shall suddenly be moued to
+giue backe (in maner) by reason of an
+<span class="sidenote">
+A marueilous Glasse.</span>
+<span class="sidefinger">&nbsp;</span>
+Image, appearing in the ayre, betwene you &amp; the glasse, with like
+hand, sword or dagger, &amp; with like quicknes, foyning at your very
+eye, likewise as you do at the Glasse. Straunge, this is, to heare of:
+but more meruailous to behold, then these my wordes can signifie. And
+neuerthelesse by demonstration Opticall, the order and cause therof, is
+certified: euen so, as the effect is consequent. Yea, thus much more,
+dare I take vpon me, toward the satisfying of the noble courrage, that
+longeth ardently for the wisedome of Causes Naturall: as to let him
+vnderstand, that, in London, he may with his owne eyes, haue profe of
+that, which I haue sayd herein. A&nbsp;Gentleman, (which, for his good
+seruice, done to his Countrey, is famous and honorable:
+<span class="sidenote">
+S.&nbsp;W.&nbsp;P.</span>
+and for skill in the Mathematicall Sciences, and Languages, is the Od
+man of this land. &amp;c.) euen he, is hable: and (I&nbsp;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
+<i>Perspectiue</i>, 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&nbsp;speake nothing of
+<i>Naturall Philosophie</i>, which, without <i>Perspectiue</i>, can not
+be fully vnderstanded, nor perfectly atteined vnto. Nor, of
+<i>Astronomie</i>: which, without <i>Perspectiue</i>, can not well be
+grounded: Nor <i>Astrologie</i>, 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&nbsp;Beame
+reboundeth) is called <i>Catoptrike</i>: and hath so many vses, both
+merueilous, and proffitable: that, both, it would hold me to long, to
+note therin the principall conclusions, all ready knowne: And also
+(perchaunce) some thinges, might lacke due credite with you: And I,
+therby, to leese my labor: and you, to slip into light Iudgement*,
+<span class="sidefinger">
+&nbsp;</span>
+Before you haue learned sufficiently the powre of Nature and Arte.</p>
+
+<p>Now, to procede: <span class="largest">Astronomie</span>, <span
+class="larger">is an Arte Mathematicall, which demonstrateth the
+distance, magnitudes, and all naturall motions, apparences, and passions
+propre to the Planets and fixed Sterres: for</span>
+<span class="pagenum">b.ij </span>
+<!-- png 27 -->
+<span class="larger">any time past, present and to come: in respect of
+a certaine Horizon, or without respect of any Horizon.</span> By this
+Arte we are certified of the distance of the Starry Skye, and of eche
+<i>Planete</i> from the Centre of the Earth: and of the greatnes of any
+Fixed starre sene, or <i>Planete</i>, in respect of the Earthes
+greatnes. As, we are sure (by this Arte) that the Solidity, Massines and
+Body of the <i>Sonne</i>, conteineth the quantitie of the whole Earth
+and Sea, a&nbsp;hundred thre score and two times, lesse by ⅛ one eight
+parte of the earth. But the Body of the whole earthly globe and Sea, is
+bigger then the body of the Mone, three and forty times lesse by ⅛ of
+the Mone. Wherfore the <i>Sonne</i> is bigger then the <i>Mone</i>, 7000
+times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then the
+<i>Mone</i>. 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 <i>Sonne</i>, 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 <i>Mone</i> when she is fardest from the
+earth, is 68 Semidiameters of the earth and ⅓ The nerest, that the
+<i>Mone</i> commeth to the earth, is Semidiameters 52¼ The distance of
+the Starry Skye is, frõ vs, in Semidiameters of the earth 20081½ Twenty
+thousand fourescore, one, and almost a halfe. Subtract from this, the
+<i>Mones</i> nerest distance, from the Earth: and therof remaineth
+Semidiameters of the earth 20029¼ Twenty thousand nine and twenty and a
+quarter.
+<span class="sidenote">
+Note.</span>
+So thicke is the heauenly Palace, that the <i>Planetes</i> haue all
+their exercise in, and most meruailously perfourme the Commaũdement and
+Charge to them giuen by the omnipotent Maiestie of the king of kings.
+This is that, which in <i>Genesis</i> is called <i>Ha Rakia</i>.
+Consider it well. The Semidiameter of the earth, cõteineth of our common
+miles 3436 4/11 three thousand, foure hundred thirty six and foure
+eleuenth partes of one myle: Such as the whole earth and Sea, round
+about, is 21600. One and twenty thousand six hundred of our myles.
+Allowyng for euery degree of the greatest circle, thre score myles. Now
+if you way well with your selfe but this litle parcell of frute
+<i>Astronomicall</i>, as concerning the bignesse, Distances of
+<i>Sonne</i>, <i>Mone</i>, <i>Sterry Sky</i>, and the huge massines of
+<i>Ha Rakia</i>, will you not finde your Consciences moued, with the
+kingly Prophet, to sing the confession of Gods Glory, and say, <span
+class="larger"><i>The Heauens declare the glory of God, and the
+Firmament</i></span> <span class="smaller">[<i>Ha Rakia</i>]</span>
+<span class="larger"><i>sheweth forth the workes of his
+handes</i></span>. And so forth, for those fiue first staues, of that
+kingly Psalme. Well, well, It is time for some to lay hold on wisedome,
+and to Iudge truly of thinges: and notso to expound the Holy word, all
+by Allegories: as to Neglect the wisedome, powre and Goodnes of God, in,
+and by his Creatures, and Creation to be seen and learned. By parables
+and Analogies of whose natures and properties, the course of the Holy
+Scripture, also, declareth to vs very many Mysteries. The whole Frame of
+Gods Creatures, (which is the whole world,) is to vs, a&nbsp;bright
+glasse: from which, by reflexion, reboundeth to our knowledge and
+perceiuerance, Beames, and Radiations: representing the Image of his
+Infinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taught
+and persuaded to Glorifie our Creator, as God: and be thankefull
+therfore. Could the Heathenistes finde these vses, of these most pure,
+beawtifull, and Mighty Corporall Creatures: and shall we, after that the
+true <i>Sonne</i> of rightwisenesse is risen aboue the <i>Horizon</i>,
+of our temporall <i>Hemisphærie</i>, and hath so abundantly streamed
+into our hartes, the direct beames of his goodnes, mercy, and grace:
+Whose heat All Creatures feele: Spirituall and Corporall: Visible and
+<span class="pagenum verso">||</span>
+<!-- png 28 -->
+Inuisible. Shall we (I&nbsp;say) looke vpon the <i>Heauen</i>,
+<i>Sterres</i>, and <i>Planets</i>, as an Oxe and an Asse doth: no
+furder carefull or inquisitiue, what they are: why were they Created,
+How do they execute that they were Created for? Seing, All Creatures,
+were for our sake created: and both we, and they, Created, chiefly to
+glorifie the Almighty Creator: and that, by all meanes, to vs possible.
+<i>Nolite ignorare</i> (saith <i>Plato in Epinomis</i>) <i>Astronomiam,
+Sapientissimũ quiddam esse.</i> <span class="larger"><i>Be ye not
+ignorant, Astronomie to be a thyng of excellent wisedome.</i></span>
+<i>Astronomie</i>, was to vs, from the beginning commended, and in maner
+commaunded by God him selfe. In asmuch as he made the <i>Sonne</i>,
+<i>Mone</i>, and <i>Sterres</i>, to be to vs, for <i>Signes</i>, and
+knowledge of Seasons, and for Distinctions of Dayes, and yeares. Many
+wordes nede not. But I wish, euery man should way this word,
+<i>Signes</i>. And besides that, conferre it also with the tenth Chapter
+of <i>Hieremie</i>. And though Some thinke, that there, they haue found
+a rod: Yet Modest Reason, will be indifferent Iudge, who ought to be
+beaten therwith, in respect of our purpose. Leauing that: I&nbsp;pray
+you vnderstand this: that without great diligence of Obseruation,
+examination and Calculation, their periods and courses (wherby
+<i>Distinction</i> of Seasons, yeares, and New Mones might precisely be
+knowne) could not exactely be certified. Which thing to performe, is
+that <i>Art</i>, which we here haue Defined to be <i>Astronomie</i>.
+Wherby, we may haue the distinct Course of Times, dayes, yeares, and
+Ages: aswell for Consideratiõ of Sacred Prophesies, accomplished in due
+time, foretold: as for high Mysticall Solemnities holding: And for all
+other humaine affaires, Conditions, and couenantes, vpon certaine time,
+betwene man and man: with many other great vses: Wherin, (verely), would
+be great incertainty, Confusion, vntruth, and brutish Barbarousnes:
+without the wonderfull diligence and skill of this Arte: continually
+learning, and determining Times, and periodes of Time, by the Record of
+the heauenly booke, wherin all times are written: and to be read with an
+<i>Astronomicall staffe</i>, in stede of a festue.</p>
+
+<p><span class="largest">Musike</span>, 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 <i>Astronomers
+Consideration</i>: now I will Speake of an other kinde of <i>Motion</i>,
+producing sound, audible, and of Man numerable. <i>Musike</i> I call
+here that <i>Science</i>, which of the Grekes is called
+<i>Harmonice</i>. Not medling with the Controuersie betwene the auncient
+<i>Harmonistes</i>, and <i>Canonistes</i>. <span class="larger">Musike
+is a Mathematicall Science, which teacheth, by sense and reason,
+perfectly to iudge, and order the diuersities of soundes, hye and
+low.</span> <i>Astronomie</i> and <i>Musike</i> are Sisters, saith
+<i>Plato</i>. As, for <i>Astronomie</i>, the eyes: So, for <i>Harmonious
+Motion</i>, the eares were made. But as <i>Astronomie</i> hath a more
+diuine Contemplation, and cõmodity, then mortall eye can perceiue: So,
+is <i>Musike</i> to be considered, that the
+<span class="sidenote">
+1.</span>
+*Minde may be preferred, before the eare. And from audible sound, we
+ought to ascende, to the examination: which numbers are
+<i>Harmonious</i>, and which not. And why, either, the one are: or the
+other are not. I&nbsp;could at large, in the heauenly
+<span class="sidenote">
+2.</span>
+*motions and distances, describe a meruailous Harmonie, of
+<i>Pythagoras</i> Harpe
+<span class="sidenote">
+3.</span>
+with eight stringes. Also, somwhat might be sayd of <i>Mercurius</i>*
+<span class="sidenote">
+4.</span>
+two Harpes, eche of foure Stringes Elementall. And very straunge matter,
+might be alledged of the <i>Harmonie</i>, to our
+<span class="sidenote">
+5.</span>
+*Spirituall part appropriate. As in <i>Ptolomaus</i> third boke, in the
+fourth and sixth Chapters may appeare.*
+<span class="sidenote">
+6.</span>
+And what is the cause of the apt bonde, and frendly felowship, of the
+Intellectuall and Mentall part of vs, with our grosse &amp; corruptible
+body: but a certaine Meane, and <i>Harmonious Spiritualitie, with
+<span class="pagenum">b.iii </span>
+<!-- png 29 -->
+both participatyng, &amp; of both (in a maner) resultynge In the
+<span class="sidenote">
+7.</span>
+*Tune of Mans voyce, and also
+<span class="sidenote">
+8.</span>
+* the sound of Instrument</i>, what might be sayd, of <i>Harmonie</i>:
+No common Musicien would lightly beleue.
+<span class="sidenote smaller">
+I.&nbsp;D.<br />
+Read in Aristotle his 8.&nbsp;booke of Politikes: the 5, 6, and 7.
+chapters. Where you shall haue some occasion farder to thinke of Musike,
+than commonly is thought.</span>
+But of the sundry Mixture (as I may terme it) and concurse, diuerse
+collation, and Application of these <i>Harmonies</i>: 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. <i>Democritus</i> and
+<i>Theophrastus</i> affirmed, that, by <i>Musike</i>, griefes and
+diseases of the Minde, and body might be cured, or inferred. And we
+finde in Recorde, that <i>Terpander</i>, <i>Arion</i>, <i>Ismenias</i>,
+<i>Orpheus</i>, <i>Amphion</i>, <i>Dauid</i>, <i>Pythagoras</i>,
+<i>Empedocles</i>, <i>Asclepiades</i> and <i>Timotheus</i>, by
+<i>Harmonicall</i> Consonãcy, haue done, and brought to pas, thinges,
+more then meruailous, to here of. Of them then, making no farder
+discourse, in this place: Sure I am, that Common <i>Musike</i>, commonly
+vsed, is found to the <i>Musiciens</i> 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&nbsp;should finde more
+repreeuers, then I could finde priuy, or skilfull of my meaning. In
+thinges therfore euident, and better knowen, then I can expresse: and so
+allowed and liked of, (as I would wish, some other thinges, had the like
+hap) I&nbsp;will spare to enlarge my lines any farder, but consequently
+follow my purpose.</p>
+
+<p><span class="largest">Of Cosmographie</span>, I appointed briefly
+in this place, to geue you some intelligence. <span class="larger">Cosmographie, is the whole and perfect description of the
+heauenly, and also elementall parte of the world, and their homologall
+application, and mutuall collation necessarie.</span> This Art,
+requireth <i>Astronomie</i>, <i>Geographie</i>, <i>Hydrographie</i> and
+<i>Musike</i>. Therfore, it is no small Arte, nor so simple, as in
+common practise, it is (slightly) considered. This matcheth Heauen, and
+the Earth, in one frame, and aptly applieth parts Correspõdent: So, as,
+the Heauenly Globe, may (in practise) be duely described vpon the
+Geographicall, and Hydrographicall Globe. And there, for vs to consider
+an <i>Æquonoctiall Circle</i>, <i>an Ecliptike line</i>, <i>Colures</i>,
+<i>Poles</i>, <i>Sterres</i> in their true Longitudes, Latitudes,
+Declinations, and Verticalitie: also Climes, and Parallels: and by an
+<i>Horizon</i> annexed, and reuolution of the earthly Globe (as the
+Heauen, is, by the <i>Primouant</i>, caried about in 24. æquall Houres)
+to learne the Risinges and Settinges of Sterres (of <i>Virgill</i> in
+his <i>Georgikes</i>: of <i>Hesiod</i>: of <i>Hippocrates</i> in his
+<i>Medicinall Sphære</i>, to Perdicca King of the Macedonians: of
+<i>Diocles</i>, to King <i>Antigonus</i>, and of other famous
+<i>Philosophers</i> prescribed) a&nbsp;thing necessary, for due manuring
+of the earth, for <i>Nauigation</i>, for the Alteration of mans body:
+being, whole, Sicke, wounded, or brused. By the Reuolution, also, or
+mouing of the Globe Cosmographicall, the Rising and Setting of the
+Sonne: the Lengthes, of dayes and nightes: the Houres and times (both
+night and day) are knowne: with very many other pleasant and necessary
+vses: Wherof, some are knowne: but better remaine, for such to know and
+vse:
+<span class="sidefinger">
+&nbsp;</span>
+who of a sparke of true fire, can make a wonderfull bonfire, by applying
+of due matter, duely.</p>
+
+<p><span class="largest">Of Astrologie</span>, here I make an Arte,
+seuerall from <i>Astronomie</i>: not by new deuise, but by good reason
+and authoritie: for, <span class="larger">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:</span>
+<span class="pagenum verso">||</span>
+<!-- png 30 -->
+<span class="larger">at all times, in any Horizon assigned.</span>
+This Arte is furnished with many other great Artes and experiences: As
+with perfecte <i>Perspectiue</i>, <i>Astronomie</i>,
+<i>Cosmographie</i>, <i>Naturall Philosophie</i> of the 4. Elementes,
+the Arte of Graduation, and some good vnderstãding in <i>Musike</i>: and
+yet moreouer, with an other great Arte, hereafter following,
+though&nbsp;I, here, set this before, for some considerations me mouing.
+Sufficient (you see) is the stuffe, to make this rare and secrete Arte,
+of: and hard enough to frame to the Conclusion Syllogisticall. Yet both
+the manifolde and continuall trauailes of the most auncient and wise
+Philosophers, for the atteyning of this Arte: and by examples of
+effectes, to confirme the same: hath left vnto vs sufficient proufe and
+witnesse: and we, also, daily may perceaue, That mans body, and all
+other Elementall bodies, are altered, disposed, ordred, pleasured, and
+displeasured, by the Influentiall working of the <i>Sunne</i>,
+<i>Mone</i>, and the other Starres and Planets. And therfore, sayth
+<i>Aristotle</i>, in the first of his <i>Meteorologicall</i> bookes, in
+the second Chapter: <i>Est autem necessariò Mundus iste, supernis
+lationibus ferè continuus. Vt, inde, vis eius vniuersa regatur. Ea
+siquidem Causà prima putanda omnibus est, vnde motus principium
+existit.</i> That is: <span class="larger"><i>This</i></span> <span
+class="smaller">[<i>Elementall</i>]</span> <span class="larger"><i>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.</i></span> And againe, in the tenth
+Chapter. <i>Oportet igitur &amp; horum principia sumamus, &amp; causas
+omnium similiter. Principium igitur vt mouens, præcipuumq<sup>ue</sup>
+&amp; omnium primum, Circulus ille est, in quo manifeste Solis latio,
+&amp;c.</i> And so forth. His <i>Meteorologicall</i> bookes, are full of
+argumentes, and effectuall demonstrations, of the vertue, operation, and
+power of the heauenly bodies, in and vpon the fower Elementes, and other
+bodies, of them (either perfectly, or vnperfectly) composed. And in his
+second booke, <i>De Generatione &amp; Corruptione</i>, in the tenth
+Chapter. <i>Quocirca &amp; prima latio, Ortus &amp; Interitus causa non
+est: Sed obliqui Circuli latio: ea namq<sup>ue</sup> &amp; continua est,
+&amp; duobus motibus fit:</i> In Englishe, thus. <span class="larger"><i>Wherefore the vppermost motion, is not the cause of
+Generation and Corruption, but the motion of the Zodiake: for, that,
+both, is continuall, and is caused of two mouinges.</i></span> And in
+his second booke, and second Chapter of hys <i>Physikes</i>. <i>Homo
+namq<sup>ue</sup> generat hominem, atq<sup>ue</sup> Sol.</i> <span class="larger"><i>For Man (sayth he) and the Sonne, are cause of mans
+generation.</i></span> Authorities may be brought, very many: both of
+1000. 2000. yea and 3000. yeares Antiquitie: of great
+<i>Philosophers</i>, <i>Expert</i>, <i>Wise</i>, and godly men, for that
+Conclusion: which, daily and hourely, we men, may discerne and perceaue
+by sense and reason: All beastes do feele, and simply shew, by their
+actions and passions, outward and inward: All Plants, Herbes, Trees,
+Flowers, and Fruites. And finally, the Elementes, and all thinges of the
+Elementes composed, do geue Testimonie (as <i>Aristotle</i> sayd) that
+theyr <span class="larger"><i>Whole Dispositions, vertues, and
+naturall motions, depend of the Actiuitie of the heauenly motions and
+Influences. Whereby, beside the specificall order and forme, due to
+euery seede: and beside the Nature, propre to the Indiuiduall Matrix, of
+the thing produced: What shall be the heauenly Impression, the perfect
+and circumspecte Astrologien hath to Conclude.</i></span> Not onely (by
+<i>Apotelesmes</i>) <ins class="correction" title="accent unchanged"><span class="greek" title="to hoti">τὸ ὁτὶ</span></ins>.
+but by Naturall and Mathematicall demonstration <span class="greek"
+title="to dioti">τὸ διότι</span>. Whereunto, what Sciences are
+requisite (without exception) I&nbsp;partly haue here warned: And in my
+<i>Propædeumes</i> (besides other matter there disclosed) I&nbsp;haue
+Mathematically furnished vp the whole Method: To this our age, not so
+carefully handled by any, that
+<span class="pagenum">b.iiij </span>
+<!-- png 31 -->
+euer I saw, or heard of. I&nbsp;was,
+<span class="sidenote">
+* Anno. 1548 and 1549. in Louayn.</span>
+(for *21. yeares ago) by certaine earnest disputations, of the Learned
+<i>Gerardus Mercator</i>, and <i>Antonius Gogaua</i>, (and other,)
+therto so prouoked: and (by my constant and inuincible zeale to the
+veritie) in obseruations of Heauenly Influencies (to the Minute of
+time,) than, so diligent: And chiefly by the Supernaturall influence,
+from the Starre of Iacob, so directed: That any Modest and Sober
+Student, carefully and diligently seking for the Truth, will both finde
+&amp; cõfesse, therin, to be the Veritie, of these my wordes: And also
+become a Reasonable Reformer, of three Sortes of people: about these
+Influentiall Operations, greatly erring from the truth.
+<span class="sidenote">
+Note.</span>
+Wherof, the one, is <span class="larger">Light Beleuers</span>, the
+other, <span class="larger">Light Despisers</span>, and the third
+<span class="larger">Light Practisers</span>. The first, &amp; most
+cõmon Sort, thinke the Heauen and Sterres, to be answerable to any their
+doutes or desires:
+<span class="sidenote">
+1.</span>
+which is not so: and, in dede, they, to much, ouer reache. The Second
+sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare
+any Sway in Generation
+<span class="sidenote">
+2.</span>
+and Corruption, in this Elementall world. And to the <i>Sunne</i>,
+<i>Mone</i> and <i>Sterres</i> (being so many, so pure, so bright, so
+wonderfull bigge, so farre in distance, so manifold in their motions, so
+constant in their periodes. &amp;c.) they assigne a sleight, simple
+office or two, and so allow vnto thẽ (according to their capacities) as
+much vertue, and power Influentiall, as to the Signe of the
+<i>Sunne</i>, <i>Mone</i>, and seuen Sterres, hanged vp (for Signes) in
+London, for distinction of houses, &amp; such grosse helpes, in our <ins
+class="correction" title="text reads ‘wordly’">worldly</ins>
+affaires: And they vnderstand not (or will not vnderstand) of the other
+workinges, and vertues of the Heauenly <i>Sunne</i>, <i>Mone</i>, and
+<i>Sterres</i>: not so much, as the Mariner, or Husband man: no, not so
+much, as the <i>Elephant</i> doth, as the <i>Cynocephalus</i>, as the
+Porpentine doth: nor will allow these perfect, and incorruptible mighty
+bodies, so much vertuall Radiation, &amp; Force, as they see in a litle
+peece of a <i>Magnes stone</i>: which, at great distance, sheweth his
+operation. And perchaunce they thinke, the Sea &amp; Riuers (as the
+Thames) to be some quicke thing, and so to ebbe, and flow, run in and
+out, of them selues, at their owne fantasies. God helpe, God helpe.
+Surely, these men, come to short: and either are to dull: or willfully
+blind: or, perhaps, to malicious. The third man, is the common and
+vulgare <i>Astrologien</i>, or Practiser: who, being not duely,
+artificially, and perfectly
+<span class="sidenote">
+3.</span>
+furnished: yet, either for vaine glory, or gayne: or like a simple dolt,
+&amp; blinde Bayard, both in matter and maner, erreth: to the discredit
+of the <i>Wary</i>, and modest <i>Astrologien</i>: and to the robbing of
+those most noble corporall Creatures, of their Naturall Vertue: being
+most mighty: most beneficiall to all elementall Generation, Corruption
+and the appartenances: and most Harmonious in their Monarchie: For which
+thinges, being knowen, and modestly vsed: we might highly, and
+continually glorifie God, with the princely Prophet, saying. <span class="larger"><i>The Heauens declare the Glorie of God: who made the Heauẽs
+in his wisedome: who made the Sonne, for to haue dominion of the day:
+the Mone and Sterres to haue dominion of the nyght: whereby, Day to day
+vttereth talke: and night, to night declareth knowledge. Prayse him, all
+ye Sterres, and Light. Amen.</i></span></p>
+
+<p>In order, now foloweth, of <span class="largest">Statike</span>,
+somewhat to say, what we meane by that name: and what commodity, doth,
+on such Art, depend. <span class="larger">Statike, is an Arte
+Mathematicall, which demonstrateth the causes of heauynes, and lightnes
+of all thynges: and of motions and properties, to heauynes and lightnes,
+belonging.</span> And for asmuch as, by the Bilanx, or Balance (as the
+chief sensible Instrument,) Experience of these demonstrations may
+<span class="pagenum verso">||</span>
+<!-- png 32 -->
+be had: we call this Art, <i>Statike:</i> that is, <i>the Experimentes
+of the Balance</i>. Oh, that men wist, what proffit, (all maner of
+wayes) by this Arte might grow, to the hable examiner, and diligent
+practiser. <span class="sidequote">“</span>Thou onely, knowest all
+thinges precisely (O&nbsp;God) who hast made weight and Balance, thy
+Iudgement: who hast created all thinges in <i>Number, Waight, and
+Measure</i>: and hast wayed the mountaines and hils in a Balance: who
+hast peysed in thy hand, both Heauen and earth. We therfore warned by
+the Sacred word, to Consider thy Creatures: and by that consideration,
+to wynne a glyms (as it were,) or shaddow of perceiuerance, that thy
+wisedome, might, and goodnes is infinite, and vnspeakable, in thy
+Creatures declared: And being farder aduertised, by thy mercifull
+goodnes, that, three principall wayes, were, of the, vsed in Creation of
+all thy Creatures, namely, <i>Number</i>, <i>Waight</i> and
+<i>Measure</i>, And for as much as, of <i>Number</i> and <i>Measure</i>,
+the two Artes (auncient, famous, and to humaine vses most necessary,)
+are, all ready, sufficiently knowen and extant: This third key, we
+beseche thee (through thy accustomed goodnes,) that it may come to the
+nedefull and sufficient knowledge, of such thy Seruauntes, as in thy
+workemanship, would gladly finde, thy true occasions (purposely of the
+vsed) whereby we should glorifie thy name, and shew forth (to the
+weaklinges in faith) thy wondrous wisedome and Goodnes. Amen.<span class="sidequote">”</span></p>
+
+<p>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&nbsp;will giue you some ground, and withall
+some shew, of certaine commodities, by this Arte arising. And bycause
+this Arte is rare, my wordes and practises might be to darke: vnleast
+you had some light, holden before the matter: and that, best will be, in
+giuing you, out of <i>Archimedes</i> demonstrations, a&nbsp;few
+principal Conclusions, as foloweth.</p>
+
+<h5>1.</h5>
+
+<p class="larger">
+The Superficies of euery Liquor, by it selfe consistyng, and in quyet,
+is Sphæricall: the centre whereof, is the same, which is the centre of
+the Earth.</p>
+
+<h5>2.</h5>
+
+<p class="larger">
+If Solide Magnitudes, being of the same bignes, or quãtitie, that any
+Liquor is, and hauyng also the same Waight: be let downe into the same
+Liquor, they will settle downeward, so, that no parte of them, shall be
+aboue the Superficies of the Liquor: and yet neuertheles, they will not
+sinke vtterly downe, or drowne.</p>
+
+<h5>3.</h5>
+
+<p class="larger">
+If any Solide Magnitude beyng Lighter then a Liquor, be let downe into
+the same Liquor, it will settle downe, so farre into the same Liquor,
+that so great a quantitie of that Liquor, as is the parte of the Solid
+Magnitude, settled downe into the same Liquor: is in Waight, æquall, to
+the waight of the whole Solid Magnitude.</p>
+
+<h5>4.</h5>
+
+<span class="pagenum">c.j.</span>
+<p class="larger">
+Any Solide Magnitude, Lighter then a Liquor, forced downe
+<!-- png 33 -->
+into the same Liquor, will moue vpward, with so great a power, by how
+much, the Liquor hauyng æquall quantitie to the whole Magnitude, is
+heauyer then the same Magnitude.</p>
+
+<h5>5.</h5>
+
+<p class="larger">
+Any Solid Magnitude, heauyer then a Liquor, beyng let downe into the
+same Liquor, will sinke downe vtterly: And wilbe in that Liquor, Lighter
+by so much, as is the waight or heauynes of the Liquor, hauing bygnes or
+quantitie, æquall to the Solid Magnitude.</p>
+
+<h5>6.</h5>
+
+<span class="sidenote smaller">I. D.<br />
+The Cutting of a Sphære according to any proportion assigned may by this
+proposition be done Mechanically by tempering Liquor to a certayne
+waight in respect of the waight of the Sphære therein Swymming.</span>
+
+<p class="larger">
+If any Solide Magnitude, Lighter then a Liquor, be let downe into the
+same Liquor, the waight of the same Magnitude, will be, to the Waight of
+the Liquor. (Which is æquall in quantitie to the whole Magnitude,) in
+that proportion, that the parte, of the Magnitude settled downe, is to
+the whole Magnitude.</p>
+
+<p><span class="dropcap">B</span>Y these verities, great Errors may be
+reformed, in Opinion of the Naturall Motion of thinges, Light and Heauy.
+Which errors, are in Naturall Philosophie (almost) of all mẽ allowed: to
+much trusting to Authority: and false Suppositions. As, <span class="larger">Of any two bodyes, the heauyer, to moue downward faster then
+the lighter.</span>
+<span class="sidenote">
+A common error, noted.</span>
+This error, is not first by me, Noted: but by one <i>Iohn Baptist de
+Benedictis</i>. The chief of his propositions, is this: which seemeth a
+Paradox.</p>
+
+<span class="sidenote">
+A paradox.</span>
+<p><span class="larger">If there be two bodyes of one forme, and of
+one kynde, æquall in quantitie or vnæquall,
+they will moue by æquall space, in æquall tyme: So that both theyr
+mouynges be in ayre, or both in water: or in any one Middle.</span></p>
+
+<p>Hereupon, in the feate of
+<span class="sidenote">
+N. T.</span>
+<span class="larger">Gunnyng</span>, certaine good discourses
+(otherwise) may receiue great amendement, and furderance.
+<span class="sidenote">
+The wonderfull vse of these Propositions.</span>
+In the entended purpose, also, allowing somwhat to the imperfection of
+Nature: not aunswerable to the precisenes of demonstration. Moreouer, by
+the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth,
+the Fire, may be nerely, knowen, how light or heauy they are (Naturally)
+in their assigned 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 Humours in Man: their waightes: and the waight of his bones, and
+flesh. &amp;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 &amp;c. And (to lift vp your head a loft:) by waight, you may, as
+precisely, as by any instrument els, measure the Diameters of
+<i>Sonne</i> and <i>Mone. &amp;c.</i> Frende, I&nbsp;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,
+vndemonstrated: or not aunswering to Natures Law, and your experience.
+Leauing these
+<span class="pagenum verso">||</span>
+<!-- png 34 -->
+thinges, thus: I&nbsp;will giue you two or three, light practises, to
+great purpose: and so finish my Annotation <i>Staticall</i>. In
+Mathematicall matters, by the Mechaniciens ayde, we will behold, here,
+the Commodity of waight.
+<span class="sidenote">
+The practise Staticall, to know the proportion, betwene the Cube, and
+the Sphære.</span>
+Make a Cube, of any one Vniforme: and through like heauy stuffe: of the
+same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall to
+the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne,
+Lead, Siluer. &amp;c. (being, as I sayd, of like nature, 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,
+company &amp; 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, somewhat nere precisenes: by halfing
+euer the Sand: they shall, at length, come to a least common waight.
+Therein, I&nbsp;leaue the farder matter, to their discretion, whom nede
+shall pinche.] The <i>Venetians</i> consideration of waight, may seme
+precise enough: by eight descentes progressionall,*
+<span class="sidenote">
+* I. D.<br />
+For, so, haue you .256. partes of a Graine.</span>
+halfing, from a grayne. Your Cube, Sphære, apt Balance, and conuenient
+waightes, being ready: fall to worke.❉. First, way your Cube. Note the
+Number of the waight. Way, after that, your Sphære. Note likewise, the
+Nũber of the waight. If you now find the waight of your Cube, to be to
+the waight of the Sphære, as 21. is to 11: Then you see, how the
+Mechanicien and <i>Experimenter</i>, 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 <i>Euclide</i>. Often, try with the same Cube and Sphære. Then,
+chaunge, your Sphære 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.</p>
+
+<p>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 demonstrated, that a Cylinder, whose heith, and
+Diameter of his base, is æquall to the Diameter of the Sphære, is
+Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of
+the waight of the Sphære, adde halfe so much, as it is: and so haue you
+the number of the waight of that Cylinder. Which is also Comprehended of
+our former Cube: So, that the base of that Cylinder, is a Circle
+described in the Square, which is the base of our Cube. But the Cube and
+the Cylinder, 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
+<span class="sidenote smaller">
+* The proportion of the Square to the Circle inscribed.</span>
+*the proportion of the Square, to the Circle, inscribed in the same
+Square. And so are we fallen into the knowledge sensible, and
+Experimentall of <i>Archimedes</i> great Secret: of him, by great
+trauaile of minde, sought and found. Wherfore, to any Circle giuen, you
+can giue a Square æquall:
+<span class="sidenote smaller">
+* The Squaring of the Circle, Mechanically.</span>
+*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
+æquall:
+<span class="sidenote smaller">
+* To any Square geuen, to geue a Circle, equall.</span>
+*If 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 proposition of the twelfth boke of
+<i>Euclide</i>, in my third Probleme there. Your diligence 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 Sphære, &amp; the
+Cube, what
+<span class="pagenum">c.ij </span>
+<!-- png 35 -->
+their proportion is: as now, you came from the Sphære to the Circle.
+For, of Siluer, or Gold, or Latton Lamyns or plates (thorough one hole
+drawẽ, as the maner is) if you make a Square figure &amp; way it: and
+then, describing theron, the Circle inscribed: &amp; cut of, &amp; 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
+<i>Euclides</i> twelfth boke. &amp;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, by the way.
+<span class="sidenote">
+Note Squaring of the Circle without knowledge of the proportion betwene
+Circumference and Diameter.</span>
+That we may Square the Circle, without hauing knowledge of the
+proportion, of the Circumference to the Diameter: as you haue here
+perceiued. And otherwayes also, I&nbsp;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 intricate. And
+easily, you may, (and that diuersly) come to the knowledge of the
+Circumference: the Circles Quantitie, being first knowen. Which thing,
+I&nbsp;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 (before this day,) had it for you,
+that I can tell of. And that is, <i>A Mechanicall Dubblyng of the Cube:
+&amp;c.</i> Which may, thus, be done:
+<span class="sidenote">
+To Dubble the Cube redily: by Art Mechanicall: depending vppon
+Demonstration Mathematicall.</span>
+<span class="larger">Make of Copper plates, or Tyn plates,
+a&nbsp;foursquare vpright Pyramis, or a Cone: perfectly fashioned in the
+holow, within. Wherin, let great diligence be vsed, to approche (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: &amp; to
+the sides of the base of the Pyramis:</span>
+<span class="sidenote smaller">I. D.<br />
+The 4. sides of this Pyramis must be 4. Isosceles Triangles alike and
+æquall.</span>
+<span class="larger">Let 4. straight lines be drawen, in the inside of
+the Cone and Pyramis: 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 Pyramis: and as many, in the Cone)
+diuide: one, in 12. æquall 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, as experience shall teach you. Then,*</span>
+<span class="sidenote smaller">I. D.<br />
+* In all workinges with this Pyramis or Cone, Let their Situations be in
+all Pointes and Conditions, alike, or all one: while you are about one
+Worke. Els you will erre.</span>
+<span class="larger">set your Cone or Pyramis, with the vertex
+downward, perpendicularly, in respect of the Base. (Though it be
+otherwayes, it hindreth nothyng.) So let thẽ most stedily be
+stayed.</span> 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 Cubik coffen, of Copper, Siluer,
+Tynne, or Wood &amp;c. These, you may so proportiõ in respect of your
+Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the
+waight of them, in water, 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: take the waight therof exactly 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 before: put that* also,
+<span class="sidenote smaller">I. D.<br />
+* Consider well whan you must put your waters togyther: and whan, you
+must empty your first water, out of your Pyramis or Cone. Els you will
+erre.</span>
+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
+<span class="pagenum verso">||</span>
+<!-- png 36 -->
+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 æquall partes, as it is capable of, conueniently, with your
+ease, and precisenes 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 nũber belonging to the
+Radicall side, of the Cube, dubble to your Fundamentall Cube: Which
+being multiplied Cubik wise, will sone shew it selfe, whether 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 Fundamẽtall
+Cube, be to a fourth proportionall line, by the 12. proposition, of the
+sixth boke of <i>Euclide</i>. 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.
+<span class="sidefinger">
+&nbsp;</span>
+<span class="sidenote">
+God be thanked for this Inuention, &amp; the fruite ensuing.</span>
+For this, may I (with ioy) say, <span class="greek" title="EURÊKA, EURÊKA, EURÊKA">ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ</span>: thanking the holy and
+glorious Trinity: hauing greater cause therto, then
+<span class="sidenote">
+* Vitruuius. Lib. 9. Cap. 3.</span>
+*<i>Archimedes</i> 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 Cubik Coffen: the like vse, is of it: and without waight. Thus.
+Fill it with water, precisely 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.
+<span class="sidenote">
+* Note.</span>
+*And if the Cube, which you should Double, be neuer so great: you haue,
+thus, the proportion (in small) betwene 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
+Euclide.</p>
+
+<p>
+<span class="sidenote">
+Note, as concerning the Sphæricall Superficies of the Water.</span>
+Note, that all this while, I&nbsp;forget not my first Proposition
+Staticall, here rehearsed: that, the Superficies of the water, is
+Sphæricall. 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. <i>For finding the swelling of the water aboue
+leuell.</i>
+<span class="sidefinger">
+&nbsp;</span>
+<span class="sidequote">“</span>Square the Semidiameter, 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 Rote,
+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.<span class="sidequote">”</span> Which, you
+will finde, to be a thing insensible. And though it were greatly
+sensible,*
+<span class="sidenote">
+* Note.</span>
+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 moyst Sponge, before,
+made moyst your hollow Pyramis or Cone, will preuent an accidentall
+cause of Swelling, &amp;c. Experience will teach you abundantly: with
+great ease, pleasure, and cõmoditie.</p>
+
+<p>Thus, may you Double the Cube Mechanically, Treble it, and so forth,
+in any proportion.
+<span class="sidenote">
+Note this Abridgement of Dubbling the Cube. &amp;c.</span>
+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 demõstration,
+then the shortest way:
+<span class="pagenum">c.iij </span>
+<!-- png 37 -->
+and all, vpon one Mathematicall Demonstration depending. <span class="sidequote">“</span>Take water (as much as conueniently will serue your
+turne: as I warned before of your Fundamentall 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 marking of
+the lines, how the Water doth cut them, shall geue you the proportion
+betwen the Radicall sides, of any two Cubes, wherof the one is Double to
+the other: working as before I haue taught you:
+<span class="sidenote">
+* Note.</span>
+<span class="sidefinger">
+&nbsp;</span>
+*sauing that for you Fundamentall Cube his Radicall side: here, you may
+take a right line, at pleasure.<span class="sidequote">”</span></p>
+
+<p>Yet farther proceding with our droppe of Naturall truth:
+<span class="sidenote">
+To giue Cubes one to the other in any proportion, Rationall or
+Irrationall.</span>
+<span class="larger">you may (now) geue Cubes, one to the other, in
+any proportiõ geuẽ: Rationall or Irrationall</span>: on this maner. Make
+a hollow Parallelipipedon of Copper or Tinne: with one Base wãting, or
+open: as in our Cubike Coffen. Frõ 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*
+<span class="sidenote">
+* <ins class="correction" title="text reads ‘Emptyting’">Emptying</ins> the first.</span>
+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 second: 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 assigned: were it Rationall
+or Irrationall.</p>
+
+<p>Thus, in sundry waies you may furnishe your selfe with such straunge
+and profitable matter: which, long hath bene wished for. And though it
+be Naturally done and Mechanically: yet hath it a good Demonstration
+Mathematicall.
+<span class="sidenote smaller">
+The demonstrations of this Dubbling of the Cube, and of the rest.</span>
+Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones,
+in the proportions 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, or Cone to Cone, be double, then shall*
+<span class="sidenote smaller">
+I. D.<br />
+* Hereby, helpe your self to become a præcise practiser. And so
+consider, how, nothing at all, you are hindred (sensibly) by the
+Conuexitie of the water.</span>
+Line to Line, be also double, &amp;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 doubled: and therefore,
+to those twaine also, a&nbsp;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 treble, &amp;c. as the Pyramids or Cones were, one
+to an other (as we haue proued) therfore, 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 proportion, of
+the fower proportionall lines: therfore*
+<span class="sidenote smaller">
+* By the 33. of the eleuenth booke of Euclide.</span>
+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 or Cone: Wherefore the Cube is
+<span class="pagenum verso">||</span>
+<!-- png 38 -->
+to the Cube, as Pyramis is to Pyramis, or Cone is to Cone. But we*
+<span class="sidenote smaller">
+I. D.<br />
+* And your diligence in practise, can so (in waight of water) performe
+it: Therefore, now, you are able to geue good reason of your whole
+doing.</span>
+Suppose Pyramis to Pyramis, or Cone to Cone, to be double or treble.
+&amp;c. Therfore Cube, is to Cube, double, or treble, &amp;c. Which was
+to be demonstrated. And of the Parallelipipedõ, it is euidẽt, 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 proportiõ of the Pyramidal or Conik
+<i>homologall</i> 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.
+<span class="sidenote">
+* Note this Corollary.</span>
+*This, my Demonstratiõ is more generall, then onely in Square Pyramis or
+Cone: Consider well. Thus, haue&nbsp;I, both Mathematically and
+Mechanically, ben very long in wordes: yet (I&nbsp;trust) nothing
+tedious to them, who, to these thinges, are well affected. And verily I
+am forced (auoiding prolixitie) to omit sundry such things, easie to be
+practised: which to the Mathematicien, would be a great Threasure: and
+to the Mechanicien, no small gaine.
+<span class="sidenote">
+* The great Commodities following of these new Inuentions.</span>
+*Now may you, <span class="larger">Betwene two lines giuen, finde two
+middle proportionals, in Continuall proportion: by the hollow
+Parallelipipedon, and the hollow Pyramis, or Cone.</span> Now, any
+Parallelipipedon rectangle being giuen: thre right lines may be found,
+proportionall in any proportion assigned, of which, shal be produced a
+Parallelipipedon, æquall to the Parallelipipedon giuen. Hereof,
+I&nbsp;noted somwhat, vpon the 36. proposition, of the 11. boke of
+<i>Euclide</i>. Now, all those thinges, which <i>Vitruuius</i> in his
+Architecture, specified hable to be done, by dubbling of the Cube: Or,
+by finding of two middle proportionall lines, betwene two lines giuen,
+may easely be performed. Now, that Probleme, which I noted vnto you, in
+the end of my Addition, vpon the 34. of the 11. boke of <i>Euclide</i>,
+is proued possible. Now, may any regular body, be Transformed into an
+other, &amp;c. Now, any regular body: any Sphere, yea any Mixt Solid:
+and (that more is) Irregular Solides, may be made (in any proportiõ
+assigned) like vnto the body, first giuen. Thus, of a <i>Manneken</i>,
+(as the <i>Dutch</i> Painters terme it) in the same <i>Symmetrie</i>,
+may a Giant be made: and that, with any gesture, by the Manneken vsed:
+and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make
+one, of the same Mould (in any assigned proportion) bigger or lesser.
+Now, may you, of any
+<span class="sidefinger">
+* </span>
+*Gunne, or little peece of ordinaũce, make an other, with the same
+<i>Symmetrie</i> (in all pointes) as great, and as little, as you will.
+Marke that: and thinke on it. Infinitely, <span class="larger">may you
+apply this, so long sought for, and now so easily concluded: and
+withall, so willingly and frankly communicated to such, as faithfully
+deale with vertuous studies.</span>
+<span class="sidenote">
+Such is the Fruite of the Mathematicall Sciences and Artes.</span>
+Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte:
+and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he
+can, by order, Descend, to frame Naturall thinges, to wonderfull vses:
+and when he list, retire home into his owne Centre: and there, prepare
+more Meanes, to Ascend or Descend by: and, all, to the glory of God, and
+our honest delectation in earth.</p>
+
+<p>Although, the Printer, hath looked for this Præface, a&nbsp;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 <i>Statike</i>, hable to be reaped: In the rest, I&nbsp;will
+therfore, be as brief, as it is possible: and with all, describing them,
+somwhat accordingly. And that, you shall perceiue, by this, which in
+order commeth
+<span class="pagenum">c.iiij </span>
+<!-- png 39 -->
+next. For, wheras, it is so ample and wonderfull, that, an whole yeare
+long, one might finde fruitfull matter therin, to speake of: and also in
+practise, is a Threasure endeles: yet will I glanse ouer it, with wordes
+very few.</p>
+
+<p><span class="dropcap">T</span>His do I call <span class="largest">Anthropographie</span>. Which is an Art restored, and of my
+preferment to your Seruice. I&nbsp;pray you, thinke of it, as of one of
+the chief pointes, of Humane knowledge. Although it be, but now, first
+Cõfirmed, with this new name: yet the matter, hath from the beginning,
+ben in consideration of all perfect Philosophers. <span class="larger">Anthropographie, is the description of the Number, Measure,
+Waight, figure, Situation, and colour of euery diuerse thing, conteyned
+in the perfect body of MAN: with certain knowledge of the Symmetrie,
+figure, waight, Characterization, and due locall motion, of any parcell
+of the sayd body, assigned: and of Nũbers, to the sayd parcell
+appertainyng.</span> This, is the one part of the Definition, mete for
+this place: Sufficient to notifie, the particularitie, and excellency of
+the Arte: and why it is, here, ascribed to the Mathematicals. Yf the
+description of the heauenly part of the world, had a peculier Art,
+called <i>Astronomie:</i> If the description of the earthly Globe, hath
+his peculier arte, called <i>Geographie</i>. If the Matching of both,
+hath his peculier Arte, called <i>Cosmographie:</i> Which is the
+Descriptiõ of the whole, and vniuersall frame of the world: Why should
+not the description of
+<span class="sidenote">
+MAN is the Lesse World.</span>
+him, who is the Lesse world: and, frõ the beginning, called
+<i>Microcosmus</i> (that is. <i>The Lesse World.</i>) And for whose
+sake, and seruice, all bodily creatures els, were created: Who, also,
+participateth with Spirites, and Angels: and is made to the Image and
+similitude of <i>God</i>: haue his peculier Art? and be called the
+<i>Arte of Artes</i>: 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 Chyromantistes
+some. The Metaposcopistes, some: The excellent, <i>Albert Durer</i>,
+a&nbsp;good part: the Arte of Perspectiue, will somwhat, for the Eye,
+helpe forward: <i>Pythagoras</i>, <i>Hipocrates</i>, <i>Plato</i>,
+<i>Galenus</i>, <i>Meletius</i>, &amp; many other (in certaine thinges)
+will be Contributaries. And farder, the Heauen, the Earth, and all other
+Creatures, will eche shew, and offer their Harmonious seruice, to fill
+vp, that, which wanteth hereof: and with your own Experience,
+concluding: you may Methodically register the whole, for the posteritie:
+Whereby, good profe will be had, of our Harmonious, and
+<span class="sidenote">
+Micro Cosmus.</span>
+Microcosmicall constitution.
+<span class="sidefinger">
+* </span>
+The outward Image, and vew hereof: to the Art of <i>Zographie</i> 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
+<span class="sidenote">
+* Lib. 3. Cap. 1.</span>
+*<i>Vitruuius</i>, whether I deale sincerely for your behoufe, or no.
+Looke in <i>Albertus Durerus</i>, <i>De Symmetria humani Corporis</i>.
+Looke in the 27. and 28. Chapters, of the second booke, <i>De occulta
+Philosophia</i>. Consider the <i>Arke</i> of <i>Noe</i>. And by that,
+wade farther. Remember the <i>Delphicall Oracle NOSCE TEIPSVM</i> <span
+class="larger"><i>(Knowe thy selfe)</i></span> so long agoe
+pronounced: of so many a Philosopher repeated: and of the <i>Wisest</i>
+attempted: And then, you will perceaue, how long agoe, you haue bene
+called to the Schole, where this Arte might be learned. Well. I&nbsp;am
+nothing affrayde, of the disdayne of some such, as thinke Sciences and
+Artes, to be but Seuen. Perhaps, those Such, may, with ignorance, and
+shame enough, come short of them Seuen also: and yet neuerthelesse
+<span class="pagenum verso">||</span>
+<!-- png 40 -->
+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, that,
+<span class="sidefinger">
+&nbsp;</span>
+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: <i>Et bonis Omnia Cooperabuntur
+in bonum.</i></p>
+
+<p><span class="largest">Trochilike,</span> <span class="larger">is
+that Art Mathematicall, which demonstrateth the properties of all
+Circular motions, Simple and Compounde.</span> And bycause the frute
+hereof, vulgarly receiued, is in Wheles, it hath the name of
+<i>Trochilike:</i> as a man would say, <i>Whele Art</i>. By this art,
+a&nbsp;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&nbsp;Spirall line in plaine, Conicall Section lines, and
+other Irregular lines, at pleasure, may be drawen. These, and such like,
+are principall Conclusions of this Arte: and helpe forward many pleasant
+and profitable Mechanicall workes:
+<span class="sidenote">
+Saw Milles.</span>
+As Milles, to Saw great and very long Deale bordes, no man being by.
+Such haue I seene in Germany: and in the Citie of Prage: in the kingdome
+of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all
+maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight,
+Spring, Man or Beast, moued. Take in your hand, <i>Agricola De re
+Metallica:</i> 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&nbsp;wonderfull
+example of farther possibilitie, and present commoditie, was sene in my
+time, in a certaine Instrument: which by the Inuenter and Artificer
+(before) was solde for xx. Talentes of Golde: and then had (by
+misfortune) receaued some iniurie and hurt: And one <i>Ianellus</i> of
+<i>Cremona</i> did mend the same, and presented it vnto the Emperour
+<i>Charles</i> the fifth. <i>Hieronymus Cardanus</i>, 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&nbsp;thing almost incredible: But how farre, I&nbsp;keepe me within my
+boundes: very many men (yet aliue) can tell.</p>
+
+<p><span class="largest">Helicosophie</span>, is nere Sister to
+<i>Trochilike:</i> and is, <span class="larger">An Arte Mathematicall,
+which demonstrateth the designing of all Spirall lines in Plaine, on
+Cylinder, Cone, Sphære, Conoid, and Sphæroid, and their properties
+appertayning.</span> The vse hereof, in <i>Architecture</i>, and diuerse
+Instrumentes and Engines, is most necessary. For, in many thinges, the
+Skrue worketh the feate, which, els, could not be performed. By helpe
+hereof, it is
+<span class="sidenote">
+* Atheneus Lib. 5. cap. 8.</span>
+*recorded, that, where all the power of the Citie of Syracusa, was not
+hable to moue a certaine Ship (being on ground) mightie
+<i>Archimedes</i>, setting to, his Skruish Engine, caused <i>Hiero</i>
+the king, by him self, at ease, to remoue her, as he would.
+<span class="sidenote">
+Proclus. Pag. 18.</span>
+Wherat, the King wondring: <span class="greek" title="Apo tautês tês hêmêras [hêmeras], peri pantos, Archimêdei legonti pisteuteom [pisteuteon]">Απὸ τάυτης τῆς <ins class="correction"
+title="error for ἡμέρας [hêmeras]">ἡμήρας</ins>, περὶ παντὸς,
+Αρχιμήδει λέγοντι <ins class="correction" title="error for πιϛευτεόν [pisteuteon]">πιϛευτέομ</ins></span>. <i>From this
+day, forward</i> (said the King) <i>Credit ought to be giuen to
+Archimedes, what soeuer he sayth.</i></p>
+
+<p><span class="largest">Pneumatithmie</span> <span class="larger">demonstrateth by close hollow Geometricall Figures, (regular
+and irregular) the straunge properties (in motion or stay) of the Water,
+Ayre, Smoke, and Fire, in theyr cõtinuitie,</span>
+<span class="pagenum">d.j </span>
+<!-- png 41 -->
+<span class="larger">and as they are ioyned to the Elementes next
+them.</span> This Arte, to the Naturall Philosopher, is very
+proffitable: to proue, that <i>Vacuum</i>, or <i>Emptines</i> is not in
+the world. And that, all Nature, abhorreth it so much: that, contrary to
+ordinary law, the Elementes will moue or stand. As, Water to ascend:
+rather then betwene him and Ayre, Space or place should be left, more
+then (naturally) that quãtitie 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 Cõtinuitie 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.
+<span class="sidenote">
+To go to the bottom of the Sea without daunger.</span>
+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 &amp;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. &amp;c.
+Pumps and all maner of Bellowes, haue their ground of this Art: and many
+other straunge deuises. As, <i>Hydraulica</i>, Organes goyng by water.
+&amp;c. Of this Feat, (called commonly <i>Pneumatica</i>,) goodly workes
+are extant, both in Greke, and Latin. With old and learned Schole men,
+it is called <i>Scientia de pleno &amp; vacuo.</i></p>
+
+<p><span class="largest">Menadrie</span><span class="larger">, is an
+Arte Mathematicall, which demonstrateth, how, aboue Natures vertue and
+power simple: Vertue and force may be multiplied: and so, to direct, to
+lift, to pull to, and to put or cast fro, any multiplied or simple,
+determined Vertue, Waight or Force: naturally, not, so, directible or
+moueable.</span> Very much is this Art furdred by other Artes: as, in
+some pointes, by <i>Perspectiue</i>: in some, by <i>Statike</i>: in
+some, by <i>Trochilike</i>: and in other, by <i>Helicosophie</i>: and
+<i>Pneumatithmie</i>. By this Art, all Cranes, Gybbettes, &amp; 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&nbsp;mighty waghen laden, being ouerthrowne. The force of the
+Crossebow Racke, is certainly, here, demonstrated. The reason, why one
+mã, doth with a leauer, lift that, which Sixe men, with their handes
+onely, could not, so easily do. By this Arte, in our common Cranes in
+London, where powre is to Crane vp, the waight of 2000. pound: by two
+Wheles more (by good order added) Arte concludeth, that there may be
+Craned vp 200000. pound waight &amp;c. So well knew <i>Archimedes</i>
+this Arte: that he alone, with his deuises and engynes, (twise or
+thrise) spoyled and discomfited the whole Army and Hoste of the
+Romaines, besieging <i>Syracusa</i>,
+<span class="sidenote smaller">
+Plutarchus in Marco Marcello.</span>
+<i>Marcus Marcellus the Consul</i>, being their Generall Capitaine.
+<span class="sidenote smaller">
+Synesius in Epistolis.</span>
+Such huge Stones, so many, with such force, and so farre, did he with
+his engynes hayle among them, out of the Citie.
+<span class="sidenote smaller">
+Polybius.</span>
+<span class="sidenote smaller">
+Plinius.</span>
+<span class="sidenote smaller">
+Quintilianus.</span>
+<span class="sidenote smaller">
+T. Liuius.</span>
+And by Sea likewise: though their Ships might come to the walls of
+<i>Syracusa</i>, yet hee vtterly confounded the Romaine Nauye. What with
+his mighty Stones hurlyng: what with Pikes of*
+<span class="sidenote smaller">
+* Athenæus.</span>
+18 fote long, made like shaftes: which he forced almost a quarter of a
+myle. What, with his catchyng hold of their Shyps, and hoysing them vp
+aboue the water, and suddenly letting them fall into the Sea againe:
+what with his*
+<span class="sidenote smaller">
+* Galenus.</span>
+<span class="sidenote smaller">
+Anthemius.</span>
+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)
+<span class="pagenum verso">||</span>
+<!-- png 42 -->
+could nothing preuaile, for the winning of Syracusa. Wherupon, the
+Romanes named <i>Archimedes</i>, <i>Briareus</i>, and <i>Centimanus</i>.
+<i>Zonaras</i> maketh mention of one <i>Proclus</i>, who so well had
+perceiued <i>Archimedes</i> Arte of <i>Menadrie</i>, and had so well
+inuented of his owne, that with his Burning Glasses,
+<span class="sidenote">
+Burning Glasses.</span>
+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 lightening) that he burned and
+destroyed both man and ship. And <i>Dion</i>specifieth of
+<i>Priscus</i>, a&nbsp;<i>Geometricien</i> in Bysance, who inuented and
+vsed sondry Engins, of Force multiplied: Which was cause, that the
+<i>Emperour Seuerus</i> pardoned him, his life, after he had wonne
+Bysance: Bycause he honored the Arte, wytt, and rare industrie of
+<i>Priscus</i>. But nothing inferior to the inuention of these engines
+of Force, was the inuention of Gunnes.
+<span class="sidenote">
+Gunnes.</span>
+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,
+<span class="sidequote">“</span>small thinges, slight, and cõmon:
+comming to wise mens consideration, and industrious mens handling, may
+grow to be of force incredible.<span class="sidequote">”</span></p>
+
+<p><span class="largest">Hypogeiodie</span><span class="larger">, is
+an Arte Mathematicall, demonstratyng, how, vnder the Sphæricall
+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 entrance, is knowen) certaine
+way may be præscribed and gone: And how, any way aboue the Superficies
+of the earth designed, may vnder earth, at any depth limited, be kept:
+goyng alwayes, perpendicularly, 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
+Sphæricall Superficies of the earth, shall precisely fall in the
+Correspondent pointes of those two wayes. This, with all other Cases and
+circumstances herein, and appertenances, this Arte demonstrateth.</span>
+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 (before this) was of me published,
+<i>De Itinere Subterraneo</i>: The rest, be at Gods will. For Pioners,
+Miners, Diggers for Mettalls, Stone, Cole, and for secrete passages
+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.</p>
+
+<p><span class="largest">Hydragogie</span><span class="larger">,
+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.</span> <span class="pagenum">d.ij
+</span>
+<!-- png 43 -->
+Long, hath this Arte bene in vse: and much thereof written: and very
+marueilous workes therein, performed: as may yet 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 Water 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 waters labour be farre, and great) I&nbsp;neede not: Seing, at
+hand (about vs) many expert men can sufficiently testifie, in effecte,
+the order: though the Demonstration of the Necessitie thereof, they know
+not: Nor yet, if they should be led, vp and downe, and about Mountaines,
+from the head of the Spring: and then, a&nbsp;place being 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. <i>Geometrie</i> therefore, is necessary to
+<i>Hydragogie</i>. Of the sundry wayes to force water to ascend, eyther
+by <i>Tympane</i>, <i>Kettell mills</i>, <i>Skrue</i>, <i>Ctesibike</i>,
+or such like: in <i>Vitruuius</i>, <i>Agricola</i>, (and other,) fully,
+the maner may appeare. And so, thereby, also be most euident, how the
+Artes, of <i>Pneumatithmie</i>, <i>Helicosophie</i>, <i>Statike</i>,
+<i>Trochilike</i>, and <i>Menadrie</i>, come to the furniture of this,
+in Speculation, and to the Commoditie of the Common Wealth, in
+practise.</p>
+
+<p><span class="largest">Horometrie</span><span class="larger">, is
+an Arte Mathematicall, which demõstrateth, how, at all times appointed,
+the precise vsuall denominatiõ of time, may be knowen, for any place
+assigned.</span> These wordes, are smoth and plaine easie Englishe, but
+the reach of their meaning, is farther, then you woulde lightly imagine.
+Some part of this Arte, was called in olde time, <i>Gnomonice</i>: and
+of late, <i>Horologiographia</i>: and in Englishe, may be termed,
+<i>Dialling</i>. 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*
+<span class="sidenote">
+4. Reg. 20.</span>
+King <i>Achaz</i> 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 <i>Astronomie</i>, and <i>Geometrie</i> Elementall, Sphæricall,
+Phænomenall, 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) theron, to describe (all maner of wayes) how, vsuall
+howers, may be (by the <i>Sunnes</i> shadow) 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&nbsp;Inuented a way, <span class="larger">How in any Horizontall,
+Murall, or Æquinoctiall Diall, &amp;c. At all howers (the Sunne shining)
+the Signe and Degree ascendent, may be knowen.</span> Which is a thing
+very necessary for the Rising of those fixed Sterres: whose Operation in
+the Ayre, is of great might, euidently. I&nbsp;speake no further, of the
+vse hereof. Bur forasmuch as, Mans affaires require knowledge of Times
+&amp; Momentes, when, neither Sunne, Mone, or Sterre, can be sene:
+Therefore, by Industrie Mechanicall, was inuented, first, how, by Water,
+running orderly, the Time and howers might be knowen: whereof, the
+famous <i>Ctesibius</i>, was Inuentor: a&nbsp;man, of <i>Vitruuius</i>,
+to the Skie (iustly) extolled. Then, after that, by Sand running, were
+howers measured: Then, by <i>Trochilike</i> with waight: And of late
+time, by <i>Trochilike</i> with Spring: without waight. All these,
+<span class="pagenum verso">||</span>
+<!-- png 44 -->
+by Sunne or Sterres direction (in certaine time) require ouersight and
+reformation, according to the heauenly Æquinoctiall Motion: besides the
+inæqualitie of their owne Operation. There remayneth (without
+parabolicall meaning herein) among the Philosophers,
+<span class="sidenote">
+A perpetuall Motion.</span>
+a&nbsp;more excellent, more commodious, and more marueilous way, then
+all these: of hauing the motion of the Primouant (or first æquinoctiall
+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.</p>
+
+<p><span class="largest">Zographie</span><span class="larger">, is
+an Arte Mathematicall, which teacheth and demonstrateth, how, the
+Intersection of all visuall Pyramides, made by any playne assigned, (the
+Centre, distance, and lightes, beyng determined) may be, by lynes, and
+due propre colours, represented.</span> A notable Arte, is this: and
+would require a whole Volume, to declare the property thereof: and the
+Commodities ensuyng. Great skill of <i>Geometrie</i>,
+<i>Arithmetike</i>, <i>Perspectiue</i>, and <i>Anthropographie</i>, with
+many other particular Artes, hath the <i>Zographer</i>, nede of, for his
+perfection. For, the most excellent Painter, (who is but the propre
+Mechanicien, &amp; 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 (commonly 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 foule, 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&nbsp;great
+pleasure and Commoditie? Which of them all, will refuse the Direction
+and ayde of Picture? The Architect, the Goldsmith, 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 commodious and meruailous: what shall be thought of
+<i>Zographie</i>, the Scholemaster of Picture, and chief gouernor?
+Though I mencion not <i>Sculpture</i>, in my Table of Artes
+Mathematicall: yet may all men perceiue, How, that <i>Picture</i> and
+<i>Sculpture</i>, are Sisters germaine: and both, right profitable, in a
+Commõ wealth. and of <i>Sculpture</i>, aswell as of Picture, excellent
+Artificers haue written great bokes in commendation. Witnesse I take, of
+<i>Georgio Vasari</i>, <i>Pittore Aretino</i>: of <i>Pomponius
+Gauricus</i>: and other. To these two Artes, (with other,) is a certaine
+od Arte, called <i>Althalmasat</i>, much beholdyng: more, then the
+common <i>Sculptor</i>, <i>Entayler</i>, <i>Keruer</i>, <i>Cutter</i>,
+<i>Grauer</i>, <i>Founder</i>,
+<span class="pagenum">d.iij </span>
+<!-- png 45 -->
+or <i>Paynter (&amp;c)</i> know their Arte, to be commodious.</p>
+
+<p><span class="sidenote">
+An objection.</span>
+<span class="largest">Architecture</span>, to many may seme not
+worthy, or not mete, to be reckned among the <i>Artes Mathematicall</i>.
+To whom, I&nbsp;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 <i>Artes Mathematicall</i>, to be such, as dealed with no
+Materiall or corruptible thing: and also did demonstratiuely procede in
+their faculty, by Number or Magnitude. First,
+<span class="sidenote">
+The Answer.</span>
+you see, that I count, here, <i>Architecture</i>, among those <i>Artes
+Mathematicall</i>, which are Deriued from the Principals: and you know,
+that such, may deale with Naturall thinges, and sensible matter. Of
+which, <span class="sidequote">“</span>some draw nerer, to the Simple
+and absolute Mathematicall Speculation, then other do.
+<span class="sidefinger">
+&nbsp;</span>
+And though, the <i>Architect</i> procureth, enformeth, &amp; directeth,
+the <i>Mechanicien</i>, to handworke, &amp; the building actuall, of
+house, Castell, or Pallace, and is chief Iudge of the same: yet, with
+him selfe (as chief <i>Master</i> and <i>Architect</i>,) remaineth the
+Demonstratiue reason and cause, of the Mechaniciens worke: in Lyne,
+plaine, and Solid: by <i>Geometricall</i>, <i>Arithmeticall</i>,
+<i>Opticall</i>, <i>Musicall</i>, <i>Astronomicall</i>,
+<i>Cosmographicall</i><span class="sidequote">”</span> (&amp;&nbsp;to
+be brief) by all the former Deriued <i>Artes Mathematicall</i>, and
+other Naturall Artes, hable to be confirmed and stablished. If this be
+so: then, may you thinke, that <i>Architecture</i>, hath good and due
+allowance, in this honest Company of <i>Artes Mathematicall</i>
+Deriuatiue. I&nbsp;will, herein, craue Iudgement of two most perfect
+<i>Architectes</i>: the one, being <i>Vitruuius</i>, the Romaine: who
+did write ten bookes thereof, to the Emperour <i>Augustus</i> (in whose
+daies our Heauenly Archemaster, was borne): and the other, <i>Leo
+Baptista Albertus</i>, a&nbsp;Florentine: who also published ten bookes
+therof. <i>Architectura</i> (sayth <i>Vitruuius</i>) <i>est Scientia
+pluribus disciplinis &amp; varijs eruditionibus ornata: cuius Iudicio
+probantur omnia, quæ ab cæteris Artificibus perficiuntur opera.</i> That
+is. <span class="larger">Architecture, is a Science garnished with
+many doctrines &amp; diuerse instructions: by whose Iudgement, all
+workes, by other workmen finished, are Iudged.</span> It followeth.
+<i>Ea nascitur ex Fabrica, &amp; Ratiocinatione. &amp;c. Ratiocinatio
+autem est, quæ, res fabricatas, Solertia ac ratione proportionis,
+demonstrare atq<sup>ue</sup> explicare potest. <span class="larger">Architecture, groweth of Framing, and Reasoning. &amp;c.
+Reasoning, is that, which of thinges framed, with forecast, and
+proportion: can make demonstration, and manifest declaration.</span></i>
+Againe. <i>Cùm, in omnibus enim rebus, tùm maximè etiam in Architectura,
+hæc duo insunt: quod significatur, &amp; quod significat. Significatur
+proposita res, de qua dicitur: hanc autem Significat Demonstratio,
+rationibus doctrinarum explicata. <span class="larger">Forasmuch as,
+in all thinges: therefore chiefly in Architecture, these two thinges
+are: the thing signified: and that which signifieth. The thing
+propounded, whereof we speake, is the thing Signified. But
+Demonstration, expressed with the reasons of diuerse doctrines, doth
+signifie the same thing.</span></i> After that. <i>Vt literatus sit,
+peritus Graphidos, eruditus Geometriæ, &amp; Optices non ignarus:
+instructus Arithmetica: historias complures nouerit, Philosophos
+diligenter audiuerit: Musicam sciuerit: Medicinæ non sit ignarus,
+responsa Iurisperitorũ nouerit: Astrologiam, Cæliq<sup>ue</sup> rationes
+cognitas habeat. <span class="larger">An Architect</span></i> (sayth
+he) <span class="larger"><i>ought to vnderstand Languages, to be
+skilfull of Painting, well instructed in Geometrie, not ignorant of
+Perspectiue, furnished with Arithmetike, haue knowledge of many
+histories, and diligently haue heard Philosophers, haue skill of Musike,
+not ignorant of Physike, know the aunsweres of Lawyers, and haue
+Astronomie,</i></span>
+<span class="pagenum verso">||</span>
+<!-- png 46 -->
+<span class="larger"><i>and the courses Cælestiall, in good
+knowledge.</i></span> He geueth reason, orderly, wherefore all these
+Artes, Doctrines, and Instructions, are requisite in an excellent
+<i>Architect</i>. And (for breuitie) omitting the Latin text, thus he
+hath. <span class="larger"><i>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 heauen, 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. &amp;c. Besides this, of the Nature of thinges (which in
+Greke is called <span class="greek" title="phusiologia">φυσιολογία</span>) Philosophie doth make declaration.
+Which, it is necessary, for an Architect, with diligence to haue
+learned: because it hath many and diuers naturall questions: as
+specially, in Aqueductes. For in their courses, leadinges about, in the
+leuell ground, and in the mountinges, the naturall 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 Ctesibius, or
+Archimedes 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 Balistes, Catapultes, and Scorpions. &amp;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<sup>e</sup> Grecians call <span class="greek"
+title="êcheia">ηχεῖα</span>) are ordred according to Musicall
+Symphonies &amp; Harmonies: being distributed in y<sup>e</sup>
+Circuites, by Diatessaron, Diapente, and Diapason. That the conuenient
+voyce, of the players sound, whẽ it came to these preparations, made in
+order, there being increased: with y<sup>t</sup> increasing, might come
+more cleare &amp; pleasant, to y<sup>e</sup> eares of the lokers on.
+&amp;c. And of Astronomie, is knowẽ y<sup>e</sup> East, West, South, and
+North. The fashion of the heauen, the Æquinox, the Solsticie, and the
+course of the sterres. Which thinges, vnleast one know: he can not
+perceiue, any thyng at all, the reason of Horologies. Seyng therfore
+this ample Science, is garnished, beautified and stored, with so many
+and sundry skils and knowledges: I&nbsp;thinke, that none can iustly
+account them selues Architectes, of the <ins class="correction" title="text reads ‘snddeyne’">suddeyne</ins>. 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 Tabernacle of Architecture. &amp;c. And to whom Nature hath giuen
+such quicke Circumspection, sharpnes of witt, and Memorie, that they may
+be very absolutely skillfull in Geometrie, Astronomie, Musike, and the
+rest of the Artes Mathematicall:</i></span>
+<span class="pagenum">d.iiij </span>
+<!-- png 47 -->
+<span class="larger"><i>Such, surmount and passe the callyng, and
+state, of Architectes:</i></span>
+<span class="sidenote">
+A Mathematicien.</span>
+<span class="larger"><i>and are become Mathematiciens. &amp;c. And
+they are found, seldome. As, in tymes past, was Aristarchus Samius:
+Philolaus, and Archytas, Tarentynes: Apollonius Pergęus: Eratosthenes
+Cyreneus: Archimedes, and Scopas, Syracusians. Who also, left to theyr
+posteritie, many Engines and Gnomonicall workes: by numbers and naturall
+meanes, inuented and declared.</i></span></p>
+
+<p>Thus much, and the same wordes (in sense) in one onely Chapter of
+this Incõparable <i>Architect Vitruuius</i>, shall you finde. And if you
+should, but take his boke in your hand, and slightly loke thorough it,
+you would say straight way:
+<span class="sidenote">
+Vitruuius.</span>
+This is <i>Geometrie</i>, <i>Arithmetike</i>, <i>Astronomie</i>,
+<i>Musike</i>, <i>Anthropographie</i>, <i>Hydragogie</i>,
+<i>Horometrie</i>. <i>&amp;c</i>. and (to cõclude) the Storehouse of all
+workmãship. Now, let vs listen to our other Iudge, our Florentine,
+<i>Leo Baptista</i>: and narrowly consider, how he doth determine of
+<i>Architecture</i>. <i>Sed anteq<sup>ue</sup> vltra progrediar. &amp;c.
+<span class="larger">But before I procede any further
+</span></i>(sayth he) <span class="larger"><i>I thinke, that I ought
+to expresse, what man I would haue to bee allowed an Architect. For,
+I&nbsp;will not bryng in place a Carpenter: as though you might Compare
+him to the Chief Masters of other Artes. For the hand of the Carpenter,
+is the Architectes Instrument.</i></span>
+<span class="sidenote">
+VVho is an Architect.</span>
+<span class="larger"><i>But I will appoint the Architect to be <span
+class="sidequote">“</span>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 bodyes, may most aptly be
+Commodious for the worthiest Vses of Man.<span class="sidequote">”</span> And that he may be able to performe these thinges,
+he hath nede of atteynyng and knowledge of the best, and most worthy
+thynges. &amp;c. The whole Feate of Architecture in buildyng, consisteth
+in Lineamentes, and in Framyng. And the whole power and skill of
+Lineamentes, tendeth to this: that the right and absolute way may <ins
+class="correction" title="text reads ‘he’">be</ins> had, of Coaptyng
+and ioyning Lines and angles: by which, the face of the buildyng 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, &amp; certaine nũber: a&nbsp;worthy maner, and a semely
+order: that, so, y<sup>e</sup> whole forme and figure of the buildyng,
+may rest in the very Lineamentes. &amp;c. And we may prescribe in mynde
+and imagination the whole formes,* </i></span><span class="sidenote">
+* The Immaterialitie of perfect Architecture.</span>
+<span class="larger"><i>all material stuffe beyng secluded. Which
+point we shall atteyne, by Notyng and forepointyng the angles, and
+lines, by a sure and certaine direction and connexion. Seyng then, these
+thinges, are thus:</i></span>
+<span class="sidenote">
+What, Lineament is.</span>
+<span class="larger"><i>Lineamente, shalbe the certaine and constant
+prescribyng, conceiued in mynde: made in lines and angles: and finished
+with a learned minde and wyt.</i></span> <span class="sidequote">“</span>We thanke you Master <i>Baptist</i>, that you haue
+so aptly brought your Arte, and phrase therof, to haue some
+Mathematicall perfection:
+<span class="sidenote">
+Note.</span>
+by certaine order, nũber, forme, figure, and <i>Symmetrie</i>
+mentall:<span class="sidequote">”</span> all naturall &amp; sensible
+stuffe set a part. Now, then, it is euident, (Gentle reader) how aptely
+and worthely, I&nbsp;haue preferred <i>Architecture</i>, to be bred and
+fostered vp in the Dominion of the pereles <i>Princesse</i>,
+<i>Mathematica</i>: and to be a naturall Subiect of hers. And the name
+of <i>Architecture</i>, is of the principalitie, which this Science
+hath, aboue all other Artes. And <i>Plato</i> affirmeth, the
+<i>Architect</i> to be <i>Master</i> ouer all, that make any worke.
+Wherupon, he is neither Smith, nor Builder: nor, separately, any
+Artificer: but the
+<span class="pagenum verso">||</span>
+<!-- png 48 -->
+Hed, the Prouost, the Directer, and Iudge of all Artificiall workes, and
+all Artificers. For, the true <i>Architect</i>, is hable to teach,
+Demonstrate, distribute, describe, and Iudge all workes wrought. And he,
+onely, searcheth out the causes and reasons of all Artificiall thynges.
+Thus excellent, is <i>Architecture</i>: though few (in our dayes)
+atteyne thereto: 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 Definitions in our dayes: for scarsitie of Artificers: No
+more, than we may pynche in, the Definitions of <i>Wisedome</i>, or
+<i>Honestie</i>, or of <i>Frendeshyp</i> or of <i>Iustice</i>. No more
+will I consent, to Diminish any whit, of the perfection and dignitie,
+(by iust cause) allowed to absolute <i>Architecture</i>. Vnder the
+Direction of this Arte, are thre principall, necessary <i>Mechanicall
+Artes</i>. Namely, <i>Howsing</i>, <i>Fortification</i>, and
+<i>Naupegie</i>. <i>Howsing</i>, I&nbsp;vnderstand, both for Diuine
+Seruice, and Mans common vsage: publike, and priuate. Of
+<i>Fortification</i> and <i>Naupegie</i>, straunge matter might be told
+you: But perchaunce, some will be tyred, with this Bederoll, all ready
+rehearsed: and other some, will 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 <i>Power
+Mathematicall</i>. Lyfe is short, and vncertaine: Tymes are perilouse:
+&amp;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.</p>
+
+<p><span class="dropcap">T</span>He <span class="larger">Arte of
+</span><span class="largest">Nauigation</span><span class="larger">,
+demonstrateth how, by the shortest good way, by the aptest Directiõ,
+&amp; in the shortest time, a&nbsp;sufficient Ship, betwene any two
+places (in passage Nauigable,) assigned: may be cõducted: and in all
+stormes, &amp; naturall disturbances chauncyng, how, to vse the best
+possible meanes, whereby to recouer the place first assigned.</span>
+What nede, the <i>Master Pilote</i>, hath of other Artes, here before
+recited, it is easie to know: as, of <i>Hydrographie</i>,
+<i>Astronomie</i>, <i>Astrologie</i>, and <i>Horometrie</i>.
+Presupposing continually, the common Base, and foundacion of all: namely
+<i>Arithmetike</i> and <i>Geometrie</i>. 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 Astronomers Ryng, The
+Astronomers staffe, The Astrolabe vniuersall. An Hydrographicall Globe.
+Charts Hydrographicall, true, (not with parallell Meridians). The Common
+Sea Compas: The Compas of variacion: The Proportionall, and Paradoxall
+Compasses
+<span class="sidenote">
+Anno. 1559.</span>
+(of me Inuented, for our two Moscouy Master Pilotes, at the request of
+the Company) Clockes with spryng: houre, halfe houre, and three houre
+Sandglasses: &amp; sundry other Instrumẽtes: 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.</p>
+
+<p>Moreouer, with Sonne Mone or Sterre (or without) be hable to define
+the Longitude &amp; Latitude of the place, which he is in: So that, the
+Longitude &amp; Latitude of the place, from which he sayled, be giuen:
+or by him, be knowne. whereto, appertayneth expert meanes, to be
+certified euer, of the Ships way. &amp;c. And by foreseing the Rising,
+Settyng, Nonestedyng, or Midnightyng of certaine tempestuous fixed
+Sterres: or their Coniunctions, and Anglynges with the Planetes, &amp;c.
+he ought to haue expert coniecture of Stormes, Tempestes, and Spoutes:
+and such lyke Meteorologicall effectes, daungerous on Sea. For (as
+<i>Plato</i> sayth,) <i>Mutationes,
+<span class="pagenum">A.j </span>
+<!-- png 49 -->
+opportunitatesq<sup>ue</sup> temporum presentire, non minus rei
+militari, quàm Agriculturæ, Nauigationiq<sup>ue</sup> conuenit. <span
+class="larger">To foresee the alterations and opportunities of tymes,
+is conuenient, no lesse to the Art of Warre, then to Husbandry and
+Nauigation.</span></i> And besides such cunnyng meanes, more euident
+tokens in Sonne and Mone, ought of hym to be knowen: such as (the
+Philosophicall Poëte) <i>Virgilius</i> teacheth, in hys
+<i>Georgikes</i>. Where he sayth,<a class="tag" name="endtag3" id="endtag3" href="#endnote3">C</a></p>
+
+<div class="poem">
+<p>Sol quoq<sup>ue</sup> &amp; exoriens &amp; quum se condet in
+vndas,</p>
+<p>Signa dabit, Solem certissima signa sequuntur. &amp;c.</p>
+<span class="sidenote">
+Georgic. 1.</span>
+<p>&nbsp; &nbsp; &mdash;&mdash;&mdash;&mdash; Nam sæpe videmus,</p>
+<p>Ipsius in vultu varios errare colores.</p>
+<p>Cæruleus, pluuiam denunciat, igneus Euros.</p>
+<p>Sin maculæ incipient rutilo immiscerier igni,</p>
+<p>Omnia tum pariter vento, nimbisq<sup>ue</sup> videbis</p>
+<p>Feruere: non illa quisquam me nocte per altum</p>
+<p>Ire, neq<sup>ue</sup> a terra moueat conuellere funem. &amp;c.</p>
+<p>Sol tibi signa dabit. Solem quis dicere falsum</p>
+<p>Audeat? &mdash;&mdash;&mdash;&mdash; &amp;c.</p>
+</div>
+
+<p>And so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and
+Beastes, and of many thynges els, a&nbsp;certaine Sympathicall
+forewarnyng may be had: sometymes to great pleasure and proffit, both on
+Sea and Land. Sufficiently, for my present purpose, it doth appeare, by
+the premisses, how <i>Mathematicall</i>, the <i>Arte</i> of
+<i>Nauigation</i>, is: and how it nedeth and also vseth other
+<i>Mathematicall Artes</i>: And now, if I would go about to speake of
+the manifold Commodities, commyng to this Land, and others, by Shypps
+and <i>Nauigation</i>, you might thinke, that I catch at occasions, to
+vse many wordes, where no nede is.</p>
+
+<p>Yet, this one thyng may I, (iustly) say. In <i>Nauigation</i>, none
+ought to haue greater care, to be skillfull, then our English Pylotes.
+And perchaunce, Some, would more attempt: And other Some, more willingly
+would be aydyng, it they wist certainely, What Priuiledge, God had
+endued this Iland with, by reason of Situation, most commodious for
+<i>Nauigation</i>, to Places most Famous &amp; Riche. And though,
+<span class="sidenote">
+* Anno. 1567 S.&nbsp;H.&nbsp;G.</span>
+(of* Late) a&nbsp;young Gentleman, a&nbsp;Courragious Capitaine, was in
+a great readynes, with good hope, and great causes of persuasion, to
+haue ventured, for a Discouerye, (either <i>Westerly</i>, by <i>Cape de
+Paramantia</i>: or <i>Esterly</i>, aboue <i>Noua Zemla</i>, and the
+<i>Cyremisses</i>) and was, at the very nere tyme of Attemptyng, called
+and employed otherwise (both then, and since,) in great good seruice to
+his Countrey, as the Irish Rebels haue *tasted:
+<span class="sidenote">
+* Anno. 1569</span>
+Yet, I&nbsp;say, (though the same Gentleman, doo not hereafter, deale
+therewith) Some one, or other, should listen to the Matter: and by good
+aduise, and discrete Circumspection, by little, and little, wynne to the
+sufficient knowledge of that <span class="larger">Trade</span> and
+<span class="larger">Voyage</span>: Which, now, I&nbsp;would be sory,
+(through Carelesnesse, want of Skill, and Courrage,) should remayne
+Vnknowne and vnheard of. Seyng, also, we are herein, halfe Challenged,
+by the learned, by halfe request, published. Therof, verely, might grow
+Commoditye, to this Land chiefly, and to the rest of the Christen Common
+wealth, farre passing all riches and worldly Threasure.</p>
+
+<p><span class="largest">Thaumaturgike</span><span class="larger">,
+is that Art Mathematicall, which giueth certaine order to make straunge
+workes, of the sense to be perceiued, and of men greatly to be wondred
+at.</span> By sundry meanes, this <i>Wonder-worke</i> is wrought. Some,
+by <i>Pneumatithmie</i>. As the workes of <i>Ctesibius</i> and
+<i>Hero</i>,
+<span class="pagenum verso">||</span>
+<!-- png 50 -->
+Some by waight. wherof <i>Timæus</i> speaketh. Some, by Stringes
+strayned, or Springs, therwith Imitating liuely Motions. Some, by other
+meanes, as the Images of Mercurie: and the brasen hed, made by
+<i>Albertus Magnus</i>, which dyd seme to speake. <i>Boethius</i> was
+excellent in these feates. To whom, <i>Cassiodorus</i> writyng, sayth.
+<span class="larger"><i>Your purpose 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&nbsp;brasen Serpent
+hisseth: byrdes made, sing swetely. Small thynges we rehearse of you,
+who can Imitate the heauen. &amp;c.</i></span> Of the straunge
+Selfmouyng, which, at Saint Denys, by Paris,
+<span class="sidenote">
+* Anno. 1551</span>
+*I saw, ones or twise (<i>Orontius</i> beyng then with me, in Company)
+it were to straunge to tell. But some haue written it. And yet,
+(I&nbsp;hope) it is there, of other to be sene. And by
+<i>Perspectiue</i> also straunge thinges, are done. As partly (before)
+I&nbsp;gaue you to vnderstand in <i>Perspectiue</i>. As, to see in the
+Ayre, a&nbsp;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 thẽ selues:
+misdeaming of the meanes. Therfore sayd <i>Claudius Cælestinus</i>.
+<span class="sidenote">
+De his quæ Mundo mirabiliter eueniunt. cap. 8.</span>
+<i>Hodie magnæ literaturæ viros &amp; magna reputationis videmus, opera
+quedam quasi miranda, supra Naturã putare: de quibus in Perspectiua
+doctus causam faciliter reddidisset.</i> That is. <span class="larger"><i>Now a dayes, we see some men, yea of great learnyng and
+reputation, to Iudge certain workes as meruaylous, aboue the power of
+Nature: Of which workes, one that were skillfull in Perspectiue might
+easely haue giuen the Cause.</i></span> Of <i>Archimedes Sphære</i>,
+<i>Cicero</i> witnesseth.
+<span class="sidenote">
+Tusc. 1.</span>
+Which is very straunge to thinke on. <span class="larger"><i>For when
+Archimedes</i></span> (sayth he) <span class="larger"><i>did fasten in
+a Sphære, the mouynges of the Sonne, Mone, and of the fiue other
+Planets, he did, as the God, which (in Timæus of Plato) did make the
+world. That, one turnyng, should rule motions most vnlike in slownes,
+and swiftnes.</i></span> But a greater cause of meruayling we haue by
+<i>Claudianus</i> report hereof. Who affirmeth this <i>Archimedes
+worke</i>, to haue ben of Glasse. And discourseth of it more at large:
+which I omit. The Doue of wood, which the <i>Mathematicien Archytas</i>
+did make to flye, is by <i>Agellius</i> spoken of. Of <i>Dædalus</i>
+straunge Images, <i>Plato</i> reporteth. <i>Homere</i> of <i>Vulcans
+Selfmouers</i>, (by secret wheles) leaueth in writyng. <i>Aristotle</i>,
+in hys <i>Politikes</i>, of both, maketh mention. Meruaylous was the
+workemanshyp, of late dayes, performed by good skill of <i>Trochilike.
+&amp;c.</i> For in Noremberge, A&nbsp;flye of Iern, beyng let out of the
+Artificers hand, did (as it were) fly about by the gestes, at the table,
+and at length, as though it were weary, retourne to his masters hand
+agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of the
+same Towne, a&nbsp;mighty way, and that a loft in the Ayre, toward the
+Emperour comming thether: and followed hym, beyng come to the gate of
+the towne.*
+<span class="sidefinger">
+*&nbsp;</span>
+Thus, you see, what, Arte Mathematicall can performe, when Skill, will,
+Industry, and Hability, are duely applyed to profe.</p>
+
+<p>
+<span class="sidenote">
+A Digression.</span>
+And for these, and such like marueilous Actes and Feates, Naturally,
+Mathematically, and Mechanically, wrought and contriued:
+<span class="sidenote">
+Apologeticall.</span>
+ought any honest Student, and Modest Christian Philosopher, be counted,
+&amp; called a <span class="larger">Coniurer</span>? Shall the folly
+of Idiotes, 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, &amp; knowledge of pure veritie:
+Shall he (I&nbsp;say) in the meane
+<span class="pagenum">A.ij </span>
+<!-- png 51 -->
+space, be robbed and spoiled of his honest name and fame? He that seketh
+(by S.&nbsp;Paules aduertisement) in the Creatures Properties, and
+wonderfull vertues, to finde iuste cause, to glorifie the Æternall, 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
+mã 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
+peruerter 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, æternally? Surely (for my part,
+somewhat to say herein) I&nbsp;haue not learned to make so brutish, and
+so wicked a Bargaine. Should&nbsp;I, for my xx. or xxv. yeares Studie:
+for two or three thousand Markes spending: seuen or eight thousand Miles
+going and trauailing, onely for good learninges sake: And that, in all
+maner of wethers: in all maner 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 bereuing life:
+(as God knoweth): with lodginges, oft times, to small ease: and somtime
+to lesse securitie. And for much more (then all this) done &amp;
+suffred, for Learning and attaining of Wisedome: Should I (I&nbsp;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&nbsp;Nette, so long time in drawing (and that with the helpe and
+aduise of Lady Philosophie, &amp; Queene Theologie): but at length, to
+haue catched, and drawen vp,*
+<span class="sidenote">
+* A&nbsp;prouerb. Fayre fisht, and caught a Frog.</span>
+a&nbsp;Frog? Nay, a&nbsp;Deuill? For, so, doth the Common peuish Pratler
+Imagine and Iangle: And, so, doth the Malicious skorner, secretly wishe,
+&amp; brauely and boldly face down, behinde my backe. Ah, what a
+miserable thing, is this kinde of Men? How great is the blindnes &amp;
+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, them selues: 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 giltlesse: 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 &amp; 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&nbsp;(you such) my vnkinde
+Countrey men. O&nbsp;vnnaturall Countrey men. O&nbsp;vnthankfull
+Countrey men. O&nbsp;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&nbsp;I, to this hower, neither by worde, deede, or thought, haue
+bene, any way, hurtfull, damageable, or iniurious to you, or yours?
+Haue&nbsp;I, so long, so dearly, so farre, so carefully, so painfully,
+so daungerously sought &amp; trauailed for the learning of Wisedome,
+&amp; atteyning of Vertue: And in the end (in your iudgemẽt) am I
+become, worse, then when I begã? Worse, thẽ a Mad man? A&nbsp;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 Wisedome? To forsake the straight heauenly way: and to wallow in the
+broad way of
+<span class="pagenum verso">||</span>
+<!-- png 52 -->
+damnation? To forsake the light of heauenly Wisedome: and to lurke in
+the dungeon of the Prince of darkenesse? To forsake the Veritie of God,
+&amp; 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 &amp; Blisse Æternall: and to cleaue vnto the
+Author of Death euerlasting? that Murderous Tyrant, most gredily
+awaiting the Pray of Mans Soule? Well: I&nbsp;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&nbsp;am worthy to be compared: and yet, they sustained
+the very like Iniuries, that I do: or rather, greater. Pacient
+<i>Socrates</i>, his <i>Apologie</i> will testifie: <i>Apuleius</i> his
+<i>Apologies</i>, will declare the Brutishnesse of the Multitude.
+<i>Ioannes Picus</i>, Earle of Mirandula, his <i>Apologie</i> will teach
+you, of the Raging slaunder of the Malicious Ignorant against him.
+<i>Ioannes Trithemius</i>, his <i>Apologie</i> will specifie, how he had
+occasion to make publike Protestation: as well by reason of the Rude
+Simple: as also, in respect of such, as were counted to be of the wisest
+sort of men. <span class="sidequote">“</span>Many could I recite: But
+I deferre the precise and determined handling of this matter: being loth
+to detect the Folly &amp; Mallice of my Natiue Countrey men.*
+<span class="sidefinger">
+*&nbsp;</span>
+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.<span class="sidequote">”</span> These men, are of fower sortes, chiefly. The first,
+I&nbsp;may name, <i>Vaine pratling busie bodies</i>: The second, <i>Fond
+Frendes</i>: The third, <i>Imperfectly zelous</i>: and the fourth,
+<i>Malicious Ignorant</i>. To eche of these (briefly, and in charitie)
+I&nbsp;will say a word or two, and so returne to my Præface.
+<span class="sidenote">
+1.</span>
+<i>Vaine pratling busie bodies</i>, 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.
+<span class="sidenote">
+2.</span>
+<i>Fonde Frendes</i>, 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 <i>Coniuring</i>. 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, <i>Vntrue</i>, both to God, and his Soueraigne. Such
+<i>Frendes</i> and <i>Fondlinges</i>, I&nbsp;shake of, and renounce you:
+Shake you of, your Folly.
+<span class="sidenote">
+3.</span>
+<i>Imperfectly zelous</i>, to you, 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
+ædified. Nor your faire pretense, by such rashe ragged Rhetorike, 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, &amp;
+let Charitie be your guide.
+<span class="sidenote">
+4.</span>
+<i>Malicious Ignorant</i>, what shall I say to thee? <i>Prohibe linguam
+tuam a malo. A&nbsp;detractione parcite linguæ. <span class="larger">Cause thy toung to refraine frõ euill. Refraine your toung from
+slaunder.</span></i> Though your tounges be sharpned, Serpent like,
+&amp; Adders poyson lye in your lippes:
+<span class="sidenote">
+Psal. 140.</span>
+yet take heede, and thinke, betimes, with your selfe, <i>Vir linguosus
+non stabilietur in terra. Virum violentum venabitur malum, donec
+præcipitetur.</i> For, sure I am, <i>Quia faciet Dominus Iudicium
+afflicti: &amp;&nbsp;vindictam pauperum.</i></p>
+
+<p>Thus, I require you, my assured frendes, and Countrey men (you
+Mathematiciens, Mechaniciens, and Philosophers, Charitable and discrete)
+to deale in my
+<span class="pagenum">A.iij </span>
+<!-- png 53 -->
+behalf, with the light &amp; vntrue tounged, my enuious Aduersaries, or
+Fond frends. And farther, I&nbsp;would wishe, that at leysor, you would
+consider, how <i>Basilius Magnus</i>, layeth <i>Moses</i> and
+<i>Daniel</i>, before the eyes of those, which count all such Studies
+Philosophicall (as mine hath bene) to be vngodly, or vnprofitable. Waye
+well <i>S.&nbsp;Stephen</i> his witnesse of <i>Moses</i>.
+<span class="sidenote">
+Act. 7. C.</span>
+<i>Eruditus est Moses omni Sapientia Ægyptiorũ: &amp; erat potens in
+verbis &amp; operibus suis. <span class="larger">Moses was instructed
+in all maner of wisedome of the Ægyptians: and he was of power both in
+his wordes, and workes.</span></i> You see this Philosophicall Power
+&amp; Wisedome, which <i>Moses</i> had, to be nothing misliked of the
+Holy Ghost. Yet <i>Plinius</i> hath recorded, <i>Moses</i> to be a
+wicked <i>Magicien</i>. And that (of force) must be, either for this
+Philosophicall wisedome, learned, before his calling to the leading of
+the Children of <i>Israel</i>: or for those his wonders, wrought before
+King <i>Pharao</i>, after he had the conducting of the
+<i>Israelites</i>. As concerning the first, you perceaue, how
+<i>S.&nbsp;Stephen</i>, at his Martyrdome (being full of the Holy Ghost)
+in his Recapitulation of the olde Testament, hath made mention of
+<i>Moses</i> Philosophie: with good liking of it: And <i>Basilius
+Magnus</i> also, auoucheth it, to haue bene to <i>Moses</i> profitable
+(and therefore, I&nbsp;say, to the Church of God, necessary). But as
+cõcerning <i>Moses</i> wonders, done before King <i>Pharao</i>: God, him
+selfe, sayd: <i>Vide vt omnia ostenta, quæ posui in manu tua, facias
+coram Pharaone. <span class="larger">See that thou do all those
+wonders before Pharao, which I haue put in thy hand.</span></i> Thus,
+you euidently perceaue, how rashly, <i>Plinius</i> hath slaundered
+<i>Moses</i>,
+<span class="sidenote">
+Lib. 30. Cap. 1.</span>
+of vayne fraudulent <i>Magike</i>, saying: <i>Est &amp; alia Magices
+Factio, a&nbsp;Mose, Iamne, &amp; Iotape, Iudæis pendens: sed multis
+millibus annorum post Zoroastrem. &amp;c.</i>
+<span class="sidenote">
+1.</span>
+Let all such, therefore, who, in Iudgement and Skill of Philosophie, are
+farre Inferior to <i>Plinie</i>, <span class="sidequote">“</span>take
+good heede, least they ouershoote them selues rashly,<span class="sidequote">”</span> in
+<span class="sidefinger">
+&nbsp;</span>
+Iudging of <i>Philosophers straunge Actes</i>: and the Meanes, how they
+are done.
+<span class="sidenote">
+2.</span>
+But, much more, ought they 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.
+<span class="sidenote">
+3.</span>
+And (to conclude) most of all, let them be ashamed of Man, and afraide
+of the dreadfull and Iuste Iudge: both Folishly or Maliciously to
+deuise: and then, deuilishly to father their new fond Monsters 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&nbsp;hope, will be more manifest.</p>
+
+<p><span class="dropcap">N</span>Ow end I, with <span class="largest">Archemastrie</span>. Which name, is not so new, as this Arte
+is rare. For an other Arte, vnder this, a&nbsp;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. <span class="larger">This Arte, teacheth to bryng to actuall experience sensible,
+all worthy conclusions by all the Artes Mathematicall purposed, &amp; by
+true Naturall Philosophie concluded: &amp; both addeth to them a farder
+scope, in the termes of the same Artes, &amp; also by hys propre Method,
+and in peculier termes, procedeth, with helpe of the foresayd Artes, to
+the performance of complet Experiẽces, which of no particular Art, are
+hable (Formally) to be challenged.</span> If you remember, how we
+considered <i>Architecture</i>, in respect of all common handworkes:
+some light may you haue, therby, to vnderstand the Souerainty and
+propertie of this Science. <i>Science</i> I may call it, rather, then an
+Arte: for the excellency and Mastershyp it hath, ouer so many, and so
+mighty Artes and
+<span class="pagenum verso">||</span>
+<!-- png 54 -->
+Sciences. And bycause it procedeth by <i>Experiences</i>, and searcheth
+forth the causes of Conclusions, by <i>Experiences</i>: and also putteth
+the Conclusions them selues, in <i>Experience</i>, it is named of some,
+<i>Scientia Experimentalis</i>. The <span class="larger"><i>Experimentall Science</i></span>. <i>Nicolaus Cusanus</i>
+termeth it so, in hys <i>Experimentes Statikall</i>, And an other
+<i>Philosopher</i>,
+<span class="sidenote">
+R.&nbsp;B.</span>
+of this land Natiue (the floure of whose worthy fame, can neuer dye nor
+wither) did write therof largely, at the request of <i>Clement the
+sixt</i>. The Arte carrieth with it, a&nbsp;wonderfull Credit: By
+reason, it certefieth, sensibly, fully, and completely to the vtmost
+power of Nature, and Arte. This Arte, certifieth by <i>Experience</i>
+complete and absolute: and other Artes, with their Argumentes, and
+Demonstrations, persuade: and in wordes, proue very well their
+Conclusions.* But
+<span class="sidefinger">
+&nbsp;</span>
+wordes, and Argumentes, are no sensible certifying: nor the full and
+finall frute of Sciences practisable. And though some Artes, haue in
+them, <i>Experiences</i>, 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 Opticall Mechanicien, put some
+thynges in <i>Experience</i>: but neither, all, that they may: nor yet
+sufficiently, and to the vtmost, those, which they do, There, then, the
+<i>Archemaster</i> steppeth in, and leadeth forth on, the
+<i>Experiences</i>, by order of his doctrine <i>Experimentall</i>, to
+the chief and finall power of Naturall and Mathematicall Artes. Of two
+or three men, in whom, this Description of <i>Archemastry</i> was
+<i>Experimentally</i>, verified, I&nbsp;haue read and hard: and good
+record, is of their such perfection. So that, this Art, is no
+fantasticall Imagination: as some Sophister, might, <i>Cum suis
+Insolubilibus</i>, make a florish: and dassell your Imagination: and
+dash your honest desire and Courage, from beleuing these thinges, so
+vnheard of, so meruaylous, &amp; of such Importance. Well: as you will.
+I&nbsp;haue forewarned you. I&nbsp;haue done the part of a frende:
+I&nbsp;haue discharged my Duety toward God: for my small Talent, at hys
+most mercyfull handes receiued. To this Science, doth the <i>Science
+Alnirangiat</i>, great Seruice. Muse nothyng of this name.
+I&nbsp;chaunge not the name, so vsed, and in Print published by other:
+beyng a name, propre to the Science. Vnder this, commeth <i>Ars
+Sintrillia</i>, by <i>Artephius</i>, 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.</p>
+
+<p>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) requested me, (an old forworne Mathematicien) to take pen
+in hand: (through the confidence they reposed in my long experience: and
+tryed sincerity) for the declaryng and reportyng somewhat, of the frute
+and commodity, by the <span class="larger">Artes Mathematicall, to be
+atteyned vnto</span>: euen they, Sore agaynst their willes, are forced,
+for sundry causes, to satisfie the workemans request, in endyng
+forthwith: He, so feareth this, so new an attempt, &amp; so costly: And
+in matter so slenderly (hetherto) among the common Sorte of Studentes,
+considered or estemed.</p>
+
+<p>And where I was willed, somewhat to alledge, why, in our vulgare
+Speche, this part of the Principall Science of <i>Geometrie</i>, called
+<i>Euclides Geometricall Elementes</i>, is published, to your handlyng:
+being vnlatined people, and not Vniuersitie Scholers: Verily,
+I&nbsp;thinke it nedelesse.</p>
+
+<p><span class="sidenote">
+1.</span>
+For, the Honour, and Estimation of the <span class="larger">Vniuersities, and Graduates</span>, is, hereby, nothing
+diminished. Seing, from, and by their Nurse Children, you receaue all
+this Benefite: how great soeuer it be.</p>
+
+<span class="pagenum">A.iiij </span>
+<!-- png 55 -->
+<p><span class="sidenote">
+2.</span>
+Neither are their Studies, hereby, any whit hindred. No more, then the
+Italian <i>Vniuersities</i>, as <i>Academia Bononiensis</i>,
+<i>Ferrariensis</i>, <i>Florentina</i>, <i>Mediolanensis</i>,
+<i>Patauina</i>, <i>Papiensis</i>, <i>Perusina</i>, <i>Pisana</i>,
+<i>Romana</i>, <i>Senensis</i>, or any one of them, finde them selues,
+any deale, disgraced, or their Studies any thing hindred, by <i>Frater
+Lucas de Burgo</i>, or by <i>Nicolaus Tartalea</i>, who in vulgar
+Italian language, haue published, not onely <i>Euclides Geometrie</i>,
+but of <i>Archimedes</i> somewhat: and in Arithmetike and Practicall
+Geometrie, very large volumes, all in their vulgar speche. Nor in
+Germany haue the famous <i>Vniuersities</i>, any thing bene discontent
+with <i>Albertus Durerus</i>, his Geometricall Institutions in Dutch: or
+with <i>Gulielmus Xylander</i>, his learned translation of the first
+sixe bookes of <i>Euclide</i>, out of the Greke into the high Dutch. Nor
+with <i>Gualterus H. Riffius</i>, his Geometricall Volume: very
+diligently translated into the high Dutch tounge, and published. Nor yet
+the <i>Vniuersities</i> of Spaine, or Portugall, thinke their reputation
+to be decayed: or suppose any their Studies to be hindred by the
+Excellent <i>P.&nbsp;Nonnius</i>, his Mathematicall workes, in vulgare
+speche by him put forth. Haue you not, likewise, in the French tounge,
+the whole Mathematicall Quadriuie? and yet neither Paris, Orleance, or
+any of the other Vniuersities of Fraunce, at any time, with the
+Translaters, or Publishers offended: or any mans Studie thereby
+hindred?</p>
+
+<p><span class="sidenote">
+3.</span>
+And surely, the Common and Vulgar Scholer (much more, the Gramarian)
+before his comming to the <i>Vniuersitie</i>, shall (or may) be, now
+(according to <i>Plato</i> his Counsell) sufficiently instructed in
+<i>Arithmetike</i> and <i>Geometrie</i>, for the better and easier
+learning of all maner of <i>Philosophie</i>, <i>Academicall</i>, or
+<i>Peripateticall</i>. And by that meanes, goe more cherefully, more
+skilfully, and spedily forwarde, in his Studies, there to be learned.
+And, so, in lesse time, profite more, then (otherwise) he should, or
+could do.</p>
+
+<p><span class="sidenote">
+4.</span>
+Also many good and pregnant Englishe wittes, of young Gentlemen, and of
+other, who neuer intend to meddle with the profound search and Studie of
+Philosophie (in the <i>Vniuersities</i> to be learned) may
+neuerthelesse, now, with more ease and libertie, haue good occasion,
+vertuously to occupie the sharpnesse of their wittes: where, els
+(perchance) otherwise, they would in fond exercises, spend (or rather
+leese) their time: neither seruing God: nor furdering the Weale, common
+or priuate.</p>
+
+<p><span class="sidenote">
+5.</span>
+And great Comfort, with good hope, may the <i>Vniuersities</i> haue, by
+reason of this <i>Englishe</i> <span class="larger">Geometrie, and
+Mathematicall Præface</span>, that they (hereafter) shall be the more
+regarded, esteemed, and resorted vnto. For, when it shall be knowen and
+reported, that of the <i>Mathematicall Sciences</i> onely, such great
+Commodities are ensuing (as I haue 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: or els,
+not so fully credited: <span class="sidequote">“</span>Well, may all
+men coniecture, that farre greater ayde, and better furniture, to winne
+to the Perfection of all Philosophie,
+<span class="sidenote">
+Vniuersities.</span>
+may in the Vniuersities be had: being the Storehouses &amp; Threasory of
+all Sciences,
+<span class="sidefinger">
+&nbsp;</span>
+and all Artes, necessary for the best, and most noble State of Common
+Wealthes.<span class="sidequote">”</span></p>
+
+<p><span class="sidenote">
+6.</span>
+Besides this, how many a Common Artificer, is there, in these Realmes of
+England and Ireland, that dealeth with Numbers, Rule, &amp; Cumpasse:
+Who, with their owne Skill and experience, already had, will be hable
+(by these good helpes and informations) to finde out, and deuise, new
+workes, straunge Engines, and Instrumentes: for sundry purposes in the
+Common Wealth? or for priuate pleasure? and for the better maintayning
+of their owne estate? I&nbsp;will not (therefore)
+<span class="pagenum verso">||</span>
+<!-- png 56 -->
+fight against myne owne shadowe. For, no man (I&nbsp;am sure) will open
+his mouth against this Enterprise. No mã (I&nbsp;say) who either hath
+Charitie toward his brother (and would be glad of his furtherance in
+vertuous knowledge): or that hath any care &amp; zeale for the bettering
+of the Cõmon state of this Realme. Neither any, that make accompt, what
+the wiser sort of men (Sage and Stayed) do thinke of them. To none
+(therefore) will I make any <i>Apologie,</i> for a vertuous acte doing:
+and for cõmending, or setting forth, Profitable Artes to English men, in
+the English toung. <span class="sidequote">“</span>But, vnto God our
+Creator, let vs all be thankefull: for that, <span class="larger"><i>As he, of his Goodnes, by his Powre, and in his wisedome,
+<span class="sidefinger">
+&nbsp;</span>
+hath Created all thynges, in Number, Waight, and Measure</i></span>: So,
+to vs, of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne
+the sufficient and necessary knowledge of the foresayd hys three
+principall Instrumentes: Which Meanes, I&nbsp;haue abundantly proued
+vnto you, to be the <i>Sciences</i> and <i>Artes Mathematicall</i>.<span
+class="sidequote">”</span></p>
+
+<p>And though I haue ben pinched with straightnes of tyme: that, no way,
+I&nbsp;could so pen downe the matter (in my Mynde) as I determined:
+hopyng of conuenient laysure: Yet. if vertuous zeale, and honest Intent
+prouoke and bryng you to the readyng and examinyng of this Compendious
+treatise, I&nbsp;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 pleasure you.
+And that you may the easier perceiue, and better remember, the
+principall pointes, whereof my Preface treateth,
+<span class="sidenote">
+The Ground platt of this Præface in a Table.</span>
+I&nbsp;will giue you the <span class="larger">Groundplatt</span> of my
+whole discourse, in a Table annexed: from the first to the last,
+somewhat Methodically contriued.</p>
+
+<p class="center">
+If Hast, hath caused my poore pen, any where, to stumble: You will,
+(I&nbsp;am<br />
+sure) in part of recompence, (for my earnest and sincere good will to
+pleasure<br />
+you), Consider the rockish huge mountaines, and the perilous<br />
+vnbeaten wayes, which (both night and day, for the while) it<br />
+hath toyled and labored through, to bryng you this good<br />
+Newes, and Comfortable profe, of Vertues frute.</p>
+
+<p class="center">
+So, I Commit you vnto Gods Mercyfull direction, for the rest:
+hartely<br />
+besechyng hym, to prosper your Studyes, and honest Intentes:<br />
+to his Glory, &amp; the Commodity of our Countrey. <i>Amen</i>.</p>
+
+<p class="right">
+<i>Written at my poore House<br />
+At Mortlake.</i></p>
+
+<p class="right">
+<i>Anno. 1570. February. 9.</i></p>
+
+<p class="illustration">
+<img src="images/dec_end.png" width="119" height="121"
+alt="decoration" />
+</p>
+
+</div> <!-- end div main-->
+
+<hr />
+
+<p>
+<a name="groundplat" id="groundplat">&nbsp;</a>
+</p>
+
+<p class="center smaller"><i>J. DEE</i></p>
+
+<p class="center larger">Here haue you (according to my promisse) the
+Groundplat of</p>
+
+<p class="center">my MATHEMATICALL Præface: annexed to <i>Euclide</i>
+(now first)</p>
+
+<p class="center smaller">published in our Englishe tounge. An. 1570.
+Febr. 3.</p>
+
+<table class="groundplat" summary="impossible to summarize">
+
+<!-- principal, 6 rows -->
+<!-- row 1 of 6 -->
+
+<tr>
+<td class="bracket righttop" rowspan="3">
+&nbsp;
+</td>
+<td class="bracket righttop" rowspan="2">
+&nbsp;
+</td>
+<td class="bracket righttop">
+&nbsp;
+</td>
+<td class="lower" colspan="4">
+<p><i><span class="larger">Simple,</span> <span class="small">Which
+dealeth with Numbers onely: and demon&shy;strateth all their properties
+and apper&shy;tenances: where, an Vnit, is
+Indiui&shy;sible.</span></i></p>
+</td>
+<td class="bracket lefttop">
+&nbsp;
+</td>
+<td class="bracket righttop">
+&nbsp;
+</td>
+<td rowspan="2">
+<p class="small"><i>In thinges Super&shy;naturall, æternall, &amp;
+Diuine: By Appli&shy;cation</i>, Ascending.</p></td>
+<td class="bracket lefttop">
+&nbsp;
+</td>
+</tr>
+
+<!-- row 2 of 6 -->
+<tr>
+<!-- <td></td> <td></td> -->
+<td class="bracket rightline">
+<p class="larger">Arithmetike.</p>
+</td>
+<td colspan="4" rowspan="2">
+<p><i><span class="larger">Mixt</span>, <span class="small">Which
+with aide of Geometrie principall, demon&shy;strateth some
+Arith&shy;meticall Conclusion, or Purpose.</span></i></p>
+</td>
+<td class="bracket leftline" rowspan="4">
+&nbsp;
+</td>
+<td class="bracket rightline" rowspan="4">
+<p><i><span class="larger">The vse</span><br />
+whereof, is either,</i></p>
+</td>
+<!-- <td></td> -->
+<td class="bracket leftline" rowspan="4">
+<p><i><span class="small">The like Vses and Appli&shy;cations are,
+(though in a degree lower) in the</span></i> Artes Mathe&shy;maticall
+Deriuatiue.</p>
+</td>
+</tr>
+
+<!-- row 3 of 6 -->
+<tr>
+<!-- <td></td> -->
+<td class="bracket rightline" rowspan="2">
+<p><i><span class="larger">Principall,</span> which are two,
+onely,</i></p>
+</td>
+<td class="bracket rightbottom">
+<p>&nbsp;<br />&nbsp;</p>
+</td>
+<!-- <td></td> <td></td> <td></td> <td></td> <td></td> -->
+<td rowspan="2">
+<p class="small"><i>In thinges Mathe&shy;maticall: without farther
+Appli&shy;cation.</i></p>
+</td>
+<!-- <td></td> -->
+</tr>
+
+<!-- row 4 of 6 -->
+<tr>
+<td class="bracket rightline" rowspan="6">
+<p><span class="largest">Sciences, and Artes
+Mathe&shy;maticall,</span> <span class="larger">are, either</span></p>
+</td>
+<!-- <td></td> -->
+<td class="bracket righttop">
+<p>&nbsp;<br />&nbsp;</p>
+</td>
+<td colspan="4" rowspan="2">
+<p><i><span class="larger">Simple</span>, <span class="small">Which
+dealeth with Magni&shy;tudes, onely: and demon&shy;strateth all their
+properties, passions, and apper&shy;tenances: whose Point, is
+Indiui&shy;sible.</span></i></p>
+</td>
+<!-- <td></td> <td></td> <td></td> <td></td> -->
+</tr>
+
+<!-- row 5 of 6 -->
+<tr>
+<!-- <td></td> -->
+<td class="bracket rightbottom">
+<p>&nbsp;<br />&nbsp;</p>
+</td>
+<td class="bracket rightline">
+<p class="larger">Geometrie.</p>
+</td>
+<!-- <td></td> <td></td> <td></td> <td></td> <td></td> -->
+<td rowspan="2">
+<p class="small"><i>In thinges Naturall: both Substã&shy;tiall, &amp;
+Accidentall, Visible, &amp; Inuisible. &amp;c. By Appli&shy;cation:</i>
+Descending.</p>
+</td>
+<!-- <td></td> -->
+</tr>
+
+<!-- row 6 of 6 -->
+<tr>
+<!-- <td></td> -->
+<td>
+&nbsp;
+</td>
+<td class="bracket rightbottom">
+&nbsp;
+</td>
+<td class="upper" colspan="4">
+<p><i><span class="larger">Mixt</span>, <span class="small">Which
+with aide of Arith&shy;metike principall, demon&shy;strateth some
+Geometricall purpose, as</span></i> EVCLIDES ELEMENTES.</p>
+</td>
+<td class="bracket leftbottom">
+&nbsp;
+</td>
+<td class="bracket rightbottom">
+&nbsp;
+</td>
+<!-- <td></td> -->
+<td class="bracket leftbottom">
+&nbsp;
+</td>
+</tr>
+
+<!-- derivative, 5 rows -->
+<!-- row 1 of 5 -->
+
+<tr>
+<!-- <td></td> -->
+<td class="bracket righttop" rowspan="2">
+&nbsp;
+</td>
+<td class="bracket righttop">
+&nbsp;
+</td>
+<td class="lower bracket single">
+<p><span class="larger">Arith&shy;metike,</span> <span class="small"><i>vulgar: which consi&shy;dereth</i></span></p>
+</td>
+<td colspan="7">
+<p>Arith&shy;metike of most vsuall whole numbers: And of Fractions to
+them apper&shy;taining.</p>
+<p>Arith&shy;metike of Propor&shy;tions.</p>
+<p>Arith&shy;metike Circular.</p>
+<p>Arith&shy;metike of Radicall Nũbers: Simple, Compound, Mixt: And of
+their Fractions.</p>
+<p>Arith&shy;metike of Cossike Nũbers: with their Fractions: And the
+great Arte of Algiebar.</p>
+</td>
+</tr>
+
+<!-- row 2 of 5 -->
+<tr>
+<!-- <td></td> <td></td> -->
+<td class="lower bracket rightline">
+<p><i><span class="larger">The names of</span> the Princi&shy;palls:
+as,</i></p>
+</td>
+<td class="bracket righttop">
+&nbsp;
+</td>
+<td class="bracket single" colspan="2">
+<p class="larger"><i>At hand</i></p>
+</td>
+<td class="bracket single" colspan="3">
+<p>All Lengthes.&mdash;</p>
+<p>All Plaines: As, Land, Borde, Glasse, &amp;c.</p>
+<p>All Solids: As, Timber, Stone, Vessels, &amp;c.</p>
+</td>
+<td colspan="2">
+<p class="larger">Mecometrie.</p>
+<p class="larger">Embadometrie.</p>
+<p class="larger">Stereometrie.</p>
+</td>
+</tr>
+
+<!-- row 3 of 5 -->
+<tr>
+<!-- <td></td> -->
+<td class="bracket rightline" rowspan="3">
+<p><i><span class="larger">Deriuatiue</span> frõ the Princi&shy;palls:
+of which, some haue</i></p>
+</td>
+<td class="bracket rightbottom">
+&nbsp;
+</td>
+<td class="upper bracket rightline">
+<p><span class="larger">Geometrie,</span> <span class="small"><i>vulgar: which teacheth Measuring</i></span></p>
+</td>
+<td class="bracket righttop" colspan="2">
+&nbsp;
+</td>
+<td class="lower">
+<p><i><span class="larger">How farre</span>, <span class="small">from the Measurer, any thing is: of him sene, on Land or Water:
+called</span></i> <span class="larger">Apomecometrie</span>.</p>
+</td>
+<td class="bracket lefttop">
+&nbsp;
+</td>
+<td class="bracket righttop">
+&nbsp;
+</td>
+<td class="lower" colspan="2">
+<p><span class="larger">Geodesie:</span> <span class="small"><i>more
+cunningly to Measure and Suruey Landes, Woods, Waters.
+&amp;c.</i></span></p>
+</td>
+</tr>
+
+<!-- row 4 of 5 -->
+<tr>
+<td class="bracket rightbottom" rowspan="2">
+&nbsp;
+</td>
+<!-- <td></td> -->
+<td rowspan="2">
+&nbsp;
+</td>
+<td class="bracket rightbottom">
+&nbsp;
+</td>
+<td class="upper bracket rightline" colspan="2">
+<p><i><span class="larger">With distãce</span> <span class="small">from the thing Measured, as,</span></i></p>
+</td>
+<td>
+<p><i><span class="larger">How high or deepe</span>, <span class="small">from the leuell of the Measurers standing, any thing is: Seene
+of hym, on Land or Water: called</span></i> <span class="larger">Hypso&shy;metrie</span>.</p>
+</td>
+<td class="bracket leftline">
+&nbsp;
+</td>
+<td class="bracket rightline">
+<p><i><span class="larger">Of which</span> are growen the Feates &amp;
+Artes of</i></p>
+</td>
+<td colspan="2">
+<p><span class="larger">Geographie.</span></p>
+<p>&nbsp;</p>
+<p><span class="larger">Choro&shy;graphie.</span></p>
+<p>&nbsp;</p>
+<p><span class="larger">Hydro&shy;graphie.</span></p>
+</td>
+</tr>
+
+<!-- row 5 of 5 -->
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td>
+&nbsp;
+</td>
+<td class="bracket rightbottom" colspan="2">
+&nbsp;
+</td>
+<td>
+<p><i><span class="larger">How broad</span>, <span class="small">a&nbsp;thing is, which is in the Measurers view: so it be
+situated on Land or Water: called</span></i> <span class="larger">Plato&shy;metrie</span>.</p>
+</td>
+<td class="bracket leftbottom">
+&nbsp;
+</td>
+<td class="bracket rightbottom">
+&nbsp;
+</td>
+<td class="upper" colspan="2">
+<p><span class="larger">Strat&shy;arith&shy;metrie.</span></p>
+</td>
+</tr>
+
+<!-- proper names, 19 rows -->
+
+<tr>
+<td rowspan="19">&nbsp;</td>
+<td class="bracket rightline" rowspan="8">
+&nbsp;
+</td>
+<td class="bracket righttop">
+<p>&nbsp;<br />&nbsp;</p>
+</td>
+<td class="upper" colspan="2">
+<p class="larger">Perspectiue,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth the maners and properties
+of all Radia&shy;tions: Directe, Broken, and Reflected.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> -->
+<td class="bracket rightline" rowspan="6">
+&nbsp;
+</td>
+<td class="upper" colspan="2">
+<p class="larger">Astro&shy;nomie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth the Distances,
+Magni&shy;tudes, and all Naturall motions, Apparences, and Passions,
+proper to the Planets and fixed Starres: for any time, past, present,
+and to come: in respecte of a certaine Horizon, or without respecte of
+any Horizon.</i></p>
+</td>
+</tr>
+
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Musike,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth by reason, and teacheth
+by sense, perfectly to iudge and order the diuer&shy;sitie of Soundes,
+hie or low.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Cosmo&shy;graphie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which, wholy and perfectly maketh description of
+the Heauenlym and also Elementall part of the World: and of these
+partes, maketh homologall appli&shy;cation, and mutuall collation
+necessary.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Astro&shy;logie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which reasonably demon&shy;strateth the
+opera&shy;tions and effectes of the naturall beames of light, and
+secrete Influence of the Planets, and fixed Starres, in euery Element
+and Elementall body: at all times, in any Horizon assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Statike,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth the causes of heauines
+and lightnes of all thinges: and of the motions and properties to
+heauines and lightnes belonging.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Anthropographie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which describeth the Nũber, Measure, Waight,
+Figure, Situation, and colour of euery diuers thing contained in the
+perfecte body of MAN: and geueth certaine knowledge of the Figure,
+Symmetrie, Waight, Charac&shy;terization, &amp; due Locall motion of any
+percell of the said body assigned: and of numbers to the said percell
+apper&shy;taining.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> -->
+<td class="bracket rightline" rowspan="3">
+<p><i><span class="larger">Propre names</span><br />
+as,</i></p>
+</td>
+<td class="upper" colspan="2">
+<p class="larger">Trochilike,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth the properties of all
+Circular motions: Simple and Compound.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> -->
+<td class="bracket rightbottom" rowspan="2">
+&nbsp;
+</td>
+<!-- <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Helico&shy;sophie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth the designing of all
+Spirall lines: in Plaine, on Cylinder, Cone, Sphære, Conoïd, and
+Sphæroid: and their properties.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Pneuma&shy;tithmie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth by close hollow
+Geometricall figures (Regular and Irregular) the straunge properties (in
+motion or stay) of the Water, Ayre, Smoke, and Fire, in their
+Conti&shy;nuitie, and as they are ioyned to the Elementes next
+them.</i></p>
+</td>
+</tr>
+
+<tr>
+<!-- <td></td> -->
+<td rowspan="9">
+&nbsp;
+</td>
+<td class="bracket rightline" rowspan="8">
+&nbsp;
+</td>
+<td class="upper" colspan="2">
+<p class="larger">Menadrie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth, how, aboue Natures
+Vertue, and power simple: Vertue and force, may be multi&shy;plied: and
+so to directe, to lift, to pull to, and to put or cast fro, any
+multi&shy;plied, or simple deter&shy;mined Vertue, Waight, or Force:
+naturally, not, so, direc&shy;tible, or moueable.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Hypogeiodie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth, how, vnder the
+Sphæricall Super&shy;ficies of the Earth, at any depth, to any
+perpen&shy;dicular line assigned (whose distance from the
+perpen&shy;dicular of the entrance: and the Azimuth likewise, in
+respecte of the sayd entrance, is knowen) certaine way, may be
+prescribed and gone, &amp;c.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Hydra&shy;gogie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth the possible leading of
+water by Natures law, and by artificiall helpe, from any head (being
+Spring, standing, or running water) to any other place assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Horometrie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth, how, at all times
+appointed, the precise, vsuall denomi&shy;nation of time, may be knowen,
+for any place assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Zographie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth and teacheth, how, the
+Inter&shy;section of all visuall Pyramids, made by any plaine assigned
+(the Center, distance, and lightes being deter&shy;mined) may be, by
+lines, and proper colours repre&shy;sented.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Archi&shy;tecture,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which is a Science garnished with many doctrines,
+and diuers Instructions: by whose iudgement, all workes by other workmen
+finished, are iudged.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Nauigation,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which demon&shy;strateth, how, by the Shortest
+good way, by the aptest direction, and in the shortest time:
+a&nbsp;suffi&shy;cient Shippe, betwene any two places (in passage
+nauigable) assigned, may be conducted: and in all stormes and naturall
+distur&shy;bances chauncing, how to vse the best possible meanes, to
+recouer the place first assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class="upper" colspan="2">
+<p class="larger">Thaumaturgike,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which geueth certaine order to make straunge
+workes, of the sense to be perceiued: and of men greatly to be wondred
+at.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> -->
+<td class="bracket rightbottom">
+&nbsp;
+</td>
+<td class="upper" colspan="2">
+<p class="larger">Arche&shy;mastrie,</p>
+</td>
+<td colspan="6">
+<p class="small"><i>Which teacheth to bring to actuall experience
+sensible, all worthy conclu&shy;sions, by all the Artes
+Mathe&shy;maticall purposed: and by true Naturall philo&shy;sophie,
+concluded: And both addeth to them a farder Scope, in the termes of the
+same Artes: and also, by his proper Method, and in peculiar termes,
+procedeth, with helpe of the forsayd Artes, to the perfor&shy;mance of
+complete Experi&shy;ences: which, of no parti&shy;cular Arte, are hable
+(Formally) to be challenged.</i></p>
+</td>
+</tr>
+</table>
+
+
+<hr class="spacer" />
+
+<h4><a name="title_text" id="title_text">Title Page Text</a></h4>
+
+<p class="illustration">
+<img src="images/titletext.png" width="394" height="543"
+alt="full text below" />
+</p>
+
+<h5>THE ELEMENTS<br />
+<span class="smaller">OF GEOMETRIE</span><br />
+of the most auncient<br />
+Philosopher<br />
+<i>EVCLIDE</i><br />
+of Megara.</h5>
+
+<h5><i>Faithfully (now first) translated<br />
+into the Englishe toung, by<br />
+</i>H. Billingsley<i>, Citizen of London</i>.<br />
+<i>Whereunto are annexed certaine<br />
+<span class="smaller">Scholies, Annotations, and Inuentions,<br />
+of the best Mathematiciens,<br />
+both of time past, and<br />
+in this our age.</span></i></h5>
+
+<hr class="tiny" />
+
+<h5><i>With a very fruitfull Præface made by </i>M. I. Dee<i>,<br />
+specifying the chiefe Mathematicall Sciẽces, what<br />
+<span class="smaller">they are, and wherunto commodious: where, also,
+are disclosed certaine new Secrets Mathematicall and Mechanicall, vntill
+these our daies, greatly missed.</span></i></h5>
+
+<hr class="tiny" />
+
+<h5>Imprinted at London by <i>Iohn Daye</i>.</h5>
+
+
+<div class="mynote end">
+<h5>Transcriber’s Footnotes</h5>
+
+<p><a name="endnote1" id="endnote1" href="#endtag1">A.</a>
+<b>Mathematical Notation.</b> John Dee used the “root” sign √ in
+combination with some less familiar symbols:</p>
+
+<table class="symbols" summary="list of mathematical symbols">
+<tr>
+<td>
+<img src="images/power_1.gif" width="38" height="35"
+alt="‘power of 1’ symbol" />
+</td>
+<td><p>“First power”, here used to express an unknown. <i>Shown in this
+e-text as</i> <b>X</b> <i>(capitalized).</i></p>
+</td>
+</tr>
+<tr>
+<td>
+<img src="images/root_square.gif" width="54" height="33"
+alt="square root" />
+</td>
+<td><p>Root sign combined with “second power” symbol = square root.
+<i>Shown in this e-text as</i> <sup>2</sup>√.</p>
+</td>
+</tr>
+<tr>
+<td>
+<img src="images/root_cubic.gif" width="60" height="33"
+alt="cube root" />
+</td>
+<td><p>Root sign combined with “third power” symbol = cube root.
+<i>Shown in this e-text as</i> <sup>3</sup>√.</p>
+</td>
+</tr>
+<tr>
+<td>
+<img src="images/root_fourth.gif" width="84" height="34"
+alt="cube root" />
+</td>
+<td><p>Doubled “second power” symbol = 4th power; with root sign =
+fourth root. <i>Shown in this e-text as</i> <sup>4</sup>√.</p>
+</td>
+</tr>
+</table>
+
+<p><a name="endnote2" id="endnote2" href="#endtag2">B.</a>
+<b>Diagrams:</b> The symbol drawn as <i>P</i> (Pounds) is shown here as
+<span class="pmath">P</span>. See above for <b>X</b> symbol.</p>
+
+<pre>
+                            HOTE
+                              +C
+                              |
+                              |
+                              +
+                              |
+                              |
+                              +
+                              |
+                              |
+                              +E
+                              |
+MOIST                  A  TEMPERATE                B     DRYE
+  +------+------+------+------+------+------+------+------+
+                              |D
+                              |
+                              +
+                              |
+                              |
+                              +
+                              |
+                              |
+                              +
+                              |
+                              |
+                              +
+                            COLD
+</pre>
+
+<pre>
+ _____________________
+|         |           |
+| {P}. 2. |  Hote. 4. |
+|         |           |
+| {P}. 1. |  Hote. 3. |
+|_________|___________|
+</pre>
+
+<pre>
+ _____________________
+|         |           | _
+| {P}. 2. |  Hote. 4. |   ⅓ _   The forme_
+|         |           |     _   3⅔ resulting.
+| {P}. 1. |  Hote. 3. | _ ⅔
+|_________|___________|
+</pre>
+
+<p><a name="endnote3" id="endnote3" href="#endtag3">C.</a>
+<b>“Vergilius teaches in his Georgikes.”</b> The quoted lines, with
+breaks at each “&amp;c.”, are 438-439; 451-457; 463-464.</p>
+
+<hr class="tiny" />
+
+<h5><a name="euclid" id="euclid">
+Euclid citations</a></h5>
+
+<p>The following Propositions were identified by number.</p>
+
+<p><b>6.12</b>: (How) to find a fourth (line) proportional to three
+given straight lines.</p>
+
+<p><b>11.34</b>: In equal parallelepipedal solids the bases are
+reciprocally proportional to the heights; and those parallelepipedal
+solids in which the bases are reciprocally proportional to the heights
+are equal.</p>
+
+<p><b>11.36</b>: If three straight lines are proportional, then the
+parallelepipedal solid formed out of the three equals the
+parallelepipedal solid on the mean which is equilateral, but equiangular
+with the aforesaid solid.</p>
+
+<p><b>12.1</b>: Similar polygons inscribed in circles are to one another
+as the squares on their diameters.</p>
+
+<p><b>12.2</b>: Circles are to one another as the squares on their
+diameters.</p>
+
+<p><b>12.18</b> (“last”): Spheres are to one another in triplicate ratio
+of their respective diameters.</p>
+
+<hr class="tiny" />
+
+<h5><a name="textnotes" id="textnotes">
+Notes on the text</a></h5>
+
+<p>The Greek letter η (eta) was consistently printed as if it were the
+ou-ligature&nbsp;ȣ.</p>
+
+<p>The Latin <i>-que</i> was written as an abbreviation resembling
+<b>-q´;</b>. It is shown here as q<sup>ue</sup>.</p>
+
+<p>Less common words include “fatch” (probably used as a variant of
+“fetch”) and the mathematical terms “sexagene” and “sexagesme”.</p>
+
+</div>
+
+<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE ***</div>
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@@ -0,0 +1,4324 @@
+The Project Gutenberg EBook of The Mathematicall Praeface to Elements of
+Geometrie of Euclid of Megara, by John Dee
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara
+
+Author: John Dee
+
+Release Date: July 13, 2007 [EBook #22062]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE ***
+
+
+
+
+Produced by Louise Hope, David Starner, Suzanne Lybarger
+and the Online Distributed Proofreading Team at
+http://www.pgdp.net
+
+
+
+
+
+[Transcriber's Note:
+
+This e-text is for readers who cannot use the "real" (unicode, utf-8)
+version of the file. Characters that could not be fully displayed
+have been shown indirectly:
+
+  [leaf] "leaf" or "floral" symbol
+  -> pointing-finger symbol
+  The letters a, e, o, u (never i) were sometimes written with an
+    overline instead of a following m or n. All have been silently
+    "unpacked" without further notation.
+  The "oe" and "ae" ligature have also been unpacked.
+  % replaces double-ended dagger, used in size notations (below)
+  Mathematical "root" symbols are shown as [2rt] [3rt] [4rt] (see end of
+    text for more detail).
+  Greek has been transliterated and shown between #marks#. Eta is
+    written E: or e:. (Omega does not occur.)
+
+Some aspects of the original book had to be modified for all versions
+of this plain-text file.
+
+Superscript letters are shown with ^: y^e, y^t.
+
+Marginal quotation marks are shown inline as "...", approximating the
+beginning and end of the marked passage. In the original text, no
+quotation marks were printed inline.
+
+Paragraphs are broken up for sidenotes, with blank lines before and
+after. Original paragraph breaks are shown as two blank lines. Brackets
+within the body text are in the original.
+
+All sidenotes except the one beginning "This noble Earle" were printed
+in italics; markup has been omitted to reduce visual clutter.
+
+At least four sizes of text were used, often in combination with
+_italics_. The variants are shown here as:
+
+  +%very large%+
+  +larger+
+  =smaller=
+
+Further errors and anomalies are listed at the end of the text, along
+with those Euclid citations identified by number.]
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+
+  THE ELEMENTS
+  OF GEOMETRIE
+
+  of the most auncient
+  Philosopher
+  _EVCLIDE_
+  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 Praeface made by _M. I. Dee_,
+  specifying the chiefe Mathematicall Sciences, what they are,
+  and wherunto commodious: where, also, are disclosed
+  certaine new Secrets Mathematicall and Mechanicall,
+  vntill these our daies, greatly missed._
+
+
+  Imprinted at London by _Iohn Daye_.
+
+
+
+
+[leaf] The Translator to the Reader.
+
+
+_There is (gentle Reader) nothing (the word of God onely set
+apart) which so much beautifieth and adorneth the soule and
+minde of man, as doth the knowledge of good artes and sciences:
+as the knowledge of naturall and morall Philosophie. The one
+setteth before our eyes, the creatures of God, both in the
+heauens aboue, and in the earth beneath: in which as in a
+glasse, we beholde the exceding maiestie and wisedome of God,
+in adorning and beautifying them as we see: in geuing vnto them
+such wonderfull and manifolde proprieties, and naturall
+workinges, and that so diuersly and in such varietie: farther in
+maintaining and conseruing them continually, whereby to praise
+and adore him, as by S. Paule we are taught. The other teacheth
+vs rules and preceptes of vertue, how, in common life amongest
+men, we ought to walke vprightly: what dueties pertaine to our
+selues, what pertaine to the gouernment or good order both of an
+housholde, and also of a citie or common wealth. The reading
+likewise of histories, conduceth not a litle, to the adorning of
+the soule & minde of man, a studie of all men commended: by it
+are seene and knowen the artes and doinges of infinite wise men
+gone before vs. In histories are contained infinite examples of
+heroicall vertues to be of vs followed, and horrible examples of
+vices to be of vs eschewed. Many other artes also there are
+which beautifie the minde of man: but of all other none do more
+garnishe & beautifie it, then those artes which are called
+Mathematicall. Unto the knowledge of which no man can attaine,
+without the perfecte knowledge and instruction of the
+principles, groundes, and Elementes of Geometrie. But perfectly
+to be instructed in them, requireth diligent studie and reading
+of olde auncient authors. Amongest which, none for a beginner is
+to be preferred before the most auncient Philosopher _Euclide_
+of _Megara_. For of all others he hath in a true methode and
+iuste order, gathered together whatsoeuer any before him had of
+these Elementes written: inuenting also and adding many thinges
+of his owne: wherby he hath in due forme accomplished the arte:
+first geuing definitions, principles, & groundes, wherof he
+deduceth his Propositions or conclusions, in such wonderfull
+wise, that that which goeth before, is of necessitie required to
+the proufe of that which followeth. So that without the diligent
+studie of _Euclides_ Elementes, it is impossible to attaine vnto
+the perfecte knowledge of Geometrie, and consequently of any of
+the other Mathematicall sciences. Wherefore considering the want
+& lacke of such good authors hitherto in our Englishe tounge,
+lamenting also the negligence, and lacke of zeale to their
+countrey in those of our nation, to whom God hath geuen both
+knowledge, & also abilitie to translate into our tounge, and
+to publishe abroad such good authors, and bookes (the chiefe
+instrumentes of all learninges): seing moreouer that many good
+wittes both of gentlemen and of others of all degrees, much
+desirous and studious of these artes, and seeking for them as
+much as they can, sparing no paines, and yet frustrate of their
+intent, by no meanes attaining to that which they seeke: I haue
+for their sakes, with some charge & great trauaile, faithfully
+translated into our vulgare tounge, & set abroad in Print, this
+booke of _Euclide_. Whereunto I haue added easie and plaine
+declarations and examples by figures, of the definitions. In
+which booke also ye shall in due place finde manifolde
+additions, Scholies, Annotations, and Inuentions: which I haue
+gathered out of many of the most famous & chiefe Mathematiciens,
+both of old time, and in our age: as by diligent reading it in
+course, ye shall well perceaue. The fruite and gaine which I
+require for these my paines and trauaile, shall be nothing els,
+but onely that thou gentle reader, will gratefully accept the
+same: and that thou mayest thereby receaue some profite: and
+moreouer to excite and stirre vp others learned, to do the like,
+& to take paines in that behalfe. By meanes wherof, our Englishe
+tounge shall no lesse be enriched with good Authors, then are
+other straunge tounges: as the Dutch, French, Italian, and
+Spanishe: in which are red all good authors in a maner, found
+amongest the Grekes or Latines. Which is the chiefest cause,
+that amongest them do florishe so many cunning and skilfull men,
+in the inuentions of straunge and wonderfull thinges, as in
+these our daies we see there do. Which fruite and gaine if I
+attaine vnto, it shall encourage me hereafter, in such like sort
+to translate, and set abroad some other good authors, both
+pertaining to religion (as partly I haue already done) and also
+pertaining to the Mathematicall Artes. Thus gentle
+reader farewell._ (?[?])
+
+
+    [Decoration]
+
+
+
+
+  [leaf] TO THE VNFAINED LOVERS
+  of truthe, and constant Studentes of Noble
+  _Sciences, _IOHN DEE_ of London, hartily_
+  wisheth grace from heauen, and most prosperous
+  _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_, _Bonum_, 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_,
+_Bonum_, and _Ens_, to be Spirituall, Infinite, Aeternall, Omnipotent,
+&c. Nothyng beyng alledged or expressed, How, worldly goods: how,
+worldly dignitie: how, health, Strength or lustines of body: nor yet the
+meanes, how a merueilous sensible and bodyly blysse and felicitie
+hereafter, might be atteyned: Straightway, the fantasies of those
+hearers, were dampt: their opinion of _Plato_, was clene chaunged: yea
+his doctrine was by them despised: and his schole, no more of them
+visited. Which thing, his Scholer, _Aristotle_, narrowly considering,
+founde the cause therof, 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 constantly to haue
+continued therin: to their full satisfaction: if such his 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, otherwise
+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 example.) 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 mete, for so
+straunge matter (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 schole,
+after they had perceiued his finall conclusion, would in these thinges
+haue ben his most diligent hearers (so infinitely mought their desires,
+in fine and at length, by our _Artes Mathematicall_ be satisfied) 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.
+
+
+    [The intent of this Preface.]
+
+Wherfore, seyng I finde great occasion (for the causes alleged, and
+farder, in respect 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: 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 thankefull 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 either 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 chaungeable. Thinges
+Supernaturall, are, of the minde onely, comprehended: Things Naturall,
+of the sense exterior, ar hable to be perceiued. In thinges Naturall,
+probabilitie and coniecture hath place: But in things Supernaturall,
+chief demonstration, & 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 supernaturall 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 neuerthelesse, by materiall things
+hable somewhat to be signified. And though their particular Images, by
+Art, are aggregable and diuisible: yet the generall _Formes_,
+notwithstandyng, are constant, vnchaungeable, vntransformable, 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 imperfection of coniecture, weenyng and
+opinion: and commyng short of high intellectuall conception, are the
+Mercurial fruite of _Dianoeticall_ discourse, in perfect imagination
+subsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_,
+and also a straunge participation betwene thinges supernaturall,
+immortall, intellectual, simple and indiuisible: and thynges naturall,
+mortall, sensible, compounded and diuisible. Probabilitie and sensible
+prose, may well serue in thinges naturall: and is commendable: In
+Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor
+yet the testimony of sense, any whit credited: But onely a perfect
+demonstration, of truthes certaine, necessary, and inuincible:
+vniuersally and necessaryly concluded: is allowed as sufficient for "an
+Argument exactly and purely Mathematical."
+
+
+    [Note the worde, Vnit, to expresse the Greke Monas,
+    & not Vnitie: as we haue all, commonly, till now, vsed.]
+
+Of _Mathematicall_ thinges, are two principall kindes: namely, _Number_,
+and _Magnitude_.
+
+    [Number.]
+
+_Number_, we define, to be, a certayne Mathematicall Summe, of _Vnits_.
+And, an _Vnit_, is that thing Mathematicall, Indiuisible, by
+participation of some likenes of whose property, any thing, which is in
+deede, 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,
+principally, is a thing _Mathematicall_.
+
+    [Magnitude.]
+
+_Magnitude_ is a thing _Mathematicall_, 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:
+
+    [A point.]
+
+The endes of a line, are _Pointes_ called. A _Point_, is a thing
+_Mathematicall_, indiuisible, 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.]
+
+a _Line_ is called the race or course of a _Point_. 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_.
+
+    [Magnitude.]
+
+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
+recken 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 _Mathematicall_, to it appropriate, called _Arithmetike_: and
+afterward of _Magnitude_, and his Science, called _Geometrie_. But that
+name contenteth me not: whereof a word or two hereafter shall be sayd.
+How Immateriall and free from all matter, _Number_ is, who doth not
+perceaue? yea, who doth not wonderfully wonder 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 simplenes of
+Substance Spirituall or Angelicall, will be found propre enough thereto.
+And therefore the great & godly Philosopher _Anitius Boetius_, sayd:
+_Omnia quaecun[que] a primaeua rerum natura constructa sunt, Numerorum
+videntur ratione formata. Hoc enim fuit principale in animo Conditoris
+Exemplar_. That is: +_All thinges (which from the very first originall
+being 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 comfortable allurement, O rauishing
+perswasion, to deale with a Science, whose Subiect, is so Auncient, so
+pure, so excellent, so surmounting all creatures, so vsed of the
+Almighty 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 perfection
+of the Science) learned, we may both winde and draw our selues into the
+inward and deepe search and vew, of all creatures distinct vertues,
+natures, properties, and _Formes_: And also, farder, arise, clime,
+ascend, and mount vp (with Speculatiue 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, proceded by no other way, then by Numbers Formall_. And
+this Earle hym selfe, in Rome,
+
+    [Ano. 1488.]
+
+* 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, and vnderstandyng of euery
+thyng, hable to be knowen. For the verifying of which Conclusion,
+I promise to aunswere to the 74. Questions, vnder written, by the way of
+Numbers_. Which Conclusions, I omit here to rehearse: aswell auoidyng
+superfluous prolixitie: as, bycause _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 Obseruers and
+Considerers of the constant law of numbers: 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 mysteries, 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 constitution:) and the third, in Spirituall and
+Angelicall Myndes, and in the Soule of man. In the first and third
+state, _Number_, is termed _Number Numbryng_. But in all Creatures,
+otherwise, _Number_, is termed _Number Numbred_. And in our Soule,
+Number beareth such a swaye, and hath such an affinitie therwith: that
+some of the old _Philosophers_ 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 perfect beyng,
+in the Creator, Sempiternally. _Number Numbryng_ therfore, is the
+discretion discerning, and distincting of thinges. But in God the
+Creator, This discretion, in the beginnyng, produced orderly and
+distinctly all thinges. For his _Numbryng_, 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 considered, maketh certaine and distinct
+determination. And albeit these thynges be waighty and truthes of great
+importance, yet (by the infinite goodnes of the Almighty _Ternarie_,)
+Artificiall Methods and easy wayes are made, by which the zelous
+Philosopher, may wyn nere this Riuerish _Ida_, this Mountayne of
+Contemplation: and more then Contemplation. And also, though _Number_,
+be a thyng so Immateriall, 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 * _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 conceiuyng 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 sundry Considerations: and extend their name farder, then to Numbers,
+whose least part is an _Vnit_. For the common Logist, Reckenmaster, or
+Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse
+partes: and calleth them _Fractions_. As of an _Vnit_, he maketh an
+halfe, and thus noteth it, 1/2. and so of other, (infinitely diuerse)
+partes of an _Vnit_. Yea and farder, hath, _Fractions of Fractions. &c_.
+And, forasmuch, as, _Addition_, _Substraction_, _Multiplication_,
+_Diuision_ and _Extraction of Rotes_, are the chief, and sufficient
+partes of _Arithmetike_:
+
+    [Arithmetike.]
+
+which is, the _Science that demonstrateth the properties, of Numbers,
+and all operations, in numbers to be performed_:
+
+    [Note.]
+
+"How often, therfore, these fiue sundry sortes of Operations, do, for
+the most part, of their execution, 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:
+
+    [1.]
+
+the Consideration, doctrine, and working, in whole numbers onely: where,
+of an _Vnit_, is no lesse part to be allowed: is named (as it were) an
+_Arithmetike_ by it selfe. And so of the _Arithmetike of Fractions_.
+
+    [2.]
+
+In lyke sorte, the necessary, wonderfull and Secret doctrine of
+Proportion, and proportionalytie hath purchased vnto it selfe a peculier
+maner of handlyng and workyng: and so may seme an other forme of
+_Arithmetike_.
+
+    [3.]
+
+Moreouer, the _Astronomers_, for spede and more commodious calculation,
+haue deuised a peculier maner of orderyng numbers, about theyr circular
+motions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes
+&c. which commonly is called the _Arithmetike_ of _Astronomical_ or
+_Phisicall Fractions_. That, haue I briefly noted, by the name of
+_Arithmetike Circular_. Bycause it is also vsed in circles, not
+_Astronomicall. &c._
+
+    [4.]
+
+Practise hath led _Numbers_ farder, and hath framed them, to take vpon
+them, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_
+and _Irrationalitie_. (For in pure _Arithmetike_, an _Vnit_, is the
+common Measure of all Numbers.) And, here, Numbers are become, as Lynes,
+Playnes and Solides: some tymes _Rationall_, some tymes _Irrationall_.
+And haue propre and peculier characters, (as [2rt]. [3rt]. and so of
+other. Which is to signifie _Rote 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 compoundyng of Numbers: as some
+tyme, two, three, foure (or more) _Radicall_ numbers, diuersly knit, by
+signes, of More & Lesse: as thus [2rt]12 + [3rt]15. Or thus [4rt]19 +
+[3rt]12 - [2rt]2. &c. And some tyme with whole numbers, or fractions of
+whole Number, among them: as 20 + [2rt]24. [3rt]16 + 33 - [2rt]10.
+[4rt]44 + 12-1/4 + [3rt]9. And so, infinitely, may hap the varietie.
+After this: Both the one and the other hath fractions incident: and so
+is this _Arithmetike_ greately enlarged, by diuerse exhibityng and vse
+of Compositions and mixtynges. Consider how, I (beyng desirous 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 spoken, is vntrue: as
+_Euclides_ tenth booke may teach you. Therfore to call them, generally,
+_Radicall Numbers_, (by reason of the signe [rt]. 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 speculation and
+practise) and haue found out, and atteyned to the very chief perfection
+(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 _Algebra_. 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 *vse,
+
+    [* Anno. 1550.]
+
+of the famous Mathematicien, and my great frende, _Petrus Montaureus_:)
+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 Alpagus_ (most perfect in the Arabik tung) so
+translated. _Scientia faciendi Algiebar & Almachabel. i. Scientia
+inueniendi numerum ignotum, per additionem Numeri, & diuisionem &
+aequationem_. Which is to say: +_The Science of workyng Algiebar and
+Almachabel_+, that is, the +_Science of findyng an vnknowen number, by
+Addyng of a Number, & Diuision & aequation_+. Here haue you the name:
+and also the principall partes of the Rule, touched. To name it, _The
+rule, or Art of Aequation_, doth signifie the middle part and the State
+of the Rule. This Rule, hath his peculier Characters:
+
+    [5.]
+
+and the principal partes of _Arithmetike_, to it appertayning, do
+differre from the other _Arithmeticall operations_. This _Arithmetike,
+hath Numbers_ Simple, Compound, Mixt: and Fractions, accordingly. This
+Rule, and _Arithmetike of Algiebar_, is so profound, so generall and so
+(in maner) conteyneth the whole power of Numbers Application practicall:
+that mans witt, can deale with nothyng, more proffitable about 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 testimony 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 misdoute 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
+forward and backward? How might they misse _Arithmeticall_ helpe in the
+Rules of Felowshyp: either without tyme, or with tyme? and betwene the
+Marchant & his Factor? The Rules of Bartering in wares onely: or part in
+wares, and part in money, would they gladly want? Our Marchant
+venturers, and Trauaylers ouer Sea, how could they order their doynges
+iustly and without losse, vnleast certaine and generall Rules for
+Exchaunge of money, 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 Marchant then to make an
+end) how ample & wonderfull is the Rule of False positions? especially
+as it is now, by two excellent Mathematiciens (of my familier
+acquayntance 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 Imagine 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 meruailously pleasured, if _Arithmetike_ be their guide?
+And the honorable Phisicians, 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_, _Auerrois_, _Arnoldus_, _Lullus_, and other haue published
+their positions, aswell in the quantities of the Degrees aboue
+Temperament, as in the Rules, concluding the new _Forme_ resulting: yet
+a more precise, commodious, and easy _Method_, is extant: by a
+Countreyman of ours
+
+    [R. B.]
+
+(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 also, for very good profe of
+Numbers vse, in this most subtile and frutefull, Philosophicall
+Conclusion,) I entend in the meane while, most briefly, and with my
+farder helpe, to communicate the pith therof vnto you.
+
+
+First describe a circle: whose diameter let be an inch. Diuide the
+Circumference into foure equall partes. From 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. without the circumference of the
+circle: So that they shall be of 4. inches long (at the least) 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 _Colde_, one against the other. And
+likewise _Moyst_ and _Dry_, one against the other. And in the Circle
+write _Temperate_. Which _Temperature_ hath a good Latitude: as
+appeareth by the Complexion of man. And therefore we haue allowed vnto
+it, the foresayd Circle: and not a point Mathematicall or Physicall.
+
+
+    [* Take some part of Lullus counsayle in his booke
+    de Q. Essentia.]
+
+Now, when you haue two thinges Miscible, whose degrees are * 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 Qualities, or of one kinde (but of diuerse
+intentions and degrees) or a _Temperate_, and a Contrary, _The forme
+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_
+
+    [* Note.]
+
+(for the * _Temperament_ is counted none. And for it, you must put a
+Ciphre, if at any time, it be in mixture).
+
+                              HOTE
+                                +C
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +E
+                                |
+  MOIST                  A  TEMPERATE                B     DRYE
+    +------+------+------+------+------+------+------+------+
+                                |D
+                                |
+                                +
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +
+                                |
+                                |
+                                +
+                              COLD
+
+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, accounted from _C_, toward _D_, ende
+iuste in the 2. degree of heate. Of the third maner, I will geue also an
+example: which let be this:
+
+    [Note.]
+
+I haue a liquid Medicine whose Qualitie of heate is in the 4. degree
+exalted: 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 from 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
+1-1/2: Adde to 1. this 1-1/2: you haue 2-1/2. Or subtract from 4. this
+1-1/2: you haue likewise 2-1/2 remayning. Which declareth, the _Forme
+resulting_, to be _Heate_, in the middle of the third degree.
+
+
+    [The Second Rule.]
+
+"But if the Quantities of two thinges Commixt, be diuerse, and the
+Intensions (of their Formes Miscible) be in diuerse degrees, and
+heigthes. (Whether those Formes be of one kinde, or of Contrary kindes,
+or of a Temperate and a Contrary, _What proportion is of the lesse
+quantitie 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
+quantitie_. As for example. Let two pound of Liquor be geuen, hote in
+the 4. degree: & one pound of Liquor be geuen, hote in the third
+degree." I would gladly know the Forme resulting, in the Mixture of
+these two Liquors. Set downe your numbers in order, thus.
+   ___________________________
+  |            |              |
+  |  {P}. _2._ |  _Hote. 4._  |
+  |            |              |
+  |  {P}. _1._ |  _Hote. 3._  |
+  |____________|______________|
+
+Now by the rule of Algiebar, haue I deuised a very easie, briefe, and
+generall maner of working in this case. Let vs first, suppose that
+_Middle Forme resulting_, to be 1{x}: as 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{x}: to
+the difference betwene 1{x} and 3: (the degree of the thing, in lesse
+quantitie. And with all, 1{x}, being alwayes in a certaine middell,
+betwene the two heigthes or degrees). For the first difference, I set
+4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1.
+is to 2. so is 4-1{x} to 1{x}-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 the other.
+Let these Multiplications be made accordingly. And of the first and the
+fourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore,
+our Aequation is betwene 1{x}-3: and 8-2{x}. Which may be reduced,
+according to the Arte of Algiebar: as, here, adding 3. to eche part,
+geueth the Aequation, thus, 1{x}=11-2{x}. And yet againe, contracting,
+or Reducing it: Adde to eche part, 2{x}: Then haue you 3{x} aequall to
+11: thus represented 3{x}=11. Wherefore, diuiding 11. by 3: the Quotient
+is 3-2/3: the _Valew_ of our 1{x}, _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 3-2/3, resteth 2/3. Subtracte the same heigth of the
+Forme resulting, (which is 3-2/3) from 4: then resteth 1/3: You see,
+that 2/3 is double to 1/3: as 2.{P}. is double to 1.{P}. So should it
+be: by the rule here geuen. Note. As you added to eche part of the
+Aequation, 3: so if ye first added to eche part 2{x}, it would stand,
+3{x}-3=8. And now adding to eche part 3: you haue (as afore) 3{x}=11.
+    _________________________
+   |           |             | _
+   | {P}. _2._ |  _Hote. 4._ |   1/3 _   _The forme_
+   |           |             |       _   _3-2/3 resulting._
+   | {P}. _1._ |  _Hote. 3._ | _ 2/3
+   |___________|_____________|
+
+
+And though I, here, speake onely of two thyngs Miscible: and most
+commonly 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 Rules are sufficient: duely repeated and
+iterated.
+
+    [Note.]
+
+In procedyng first, with any two: and then, with the Forme Resulting,
+and an other: & so forth: For, the last worke, concludeth 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 definition, 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 varieties, by the
+foresayd two generall 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 particular: would haue larger
+discoursing, then is mete in this place to be dealt withall: And other
+(perchaunce) with a proude snuffe will disdaine this litle: 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 Qualities Naturall_, is so Matcht and maryed with
+the most simple, easie, and short way of the noble Rule of _Algiebar_.
+Who can remaine, therfore vnpersuaded, to loue, alow, 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
+hunderd thousand mens wittes, of the middle sorte, are hable to
+perfourme, or truely to conclude, with out helpe thereof.
+
+
+Now will we farder, 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
+
+    [#Taktike:#.]
+
+#Taktike:#. "That is, the Skill of Ordring 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, to 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_: _Cyneas_ of
+_Thessaly_: _Pyrrhus Epirota_: and _Alexander_ his sonne: _Clearchus_:
+_Pausanias_: _Euangelus_: _Polybius_, familier frende to _Scipio_:
+_Eupolemus_: _Iphicrates_, _Possidonius_: and very many other worthy
+Capitaines, Philosophers and Princes of Immortall fame and memory: Whose
+fayrest floure of their garland (in this feat) was _Arithmetike_: and a
+litle perceiuerance, in _Geometricall_ Figures. But in many other cases
+doth _Arithmetike_ 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 company, to make the
+victuals, longer to serue the remanent, & for a certaine determined
+tyme: if nede so require. And so in sundry his other accountes,
+Reckeninges, Measurynges, and proportionynges, the wise, expert, and
+Circumspect Capitaine 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, the loyall, and
+Curteous
+
+    [->]
+
+_Iohn_, late Earle of Warwicke. Who was a yong Gentleman, throughly
+knowne to very few. Albeit his lusty valiantnes, force, and Skill in
+Chiualrous feates and exercises: his humblenes, and frendelynes 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 achieue: 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, forecastinges and determinations, no twayne,
+(I thinke) beside my selfe, can so perfectly, and truely report. And
+therfore, in Conscience, I count it my part, for the honor, preferment,
+& procuring of vertue (thus, briefly) to haue put his Name, in the
+Register of _Fame Immortall_.
+
+
+To our purpose. This _Iohn_, by one of his actes (besides many other:
+both in England and Fraunce, by me, in him noted.) did disclose his
+harty loue to vertuous Sciences: and his noble intent, to excell in
+Martiall prowesse: When he, with humble 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 deuised) for
+ordring of all Companies, summes and Numbers of men, (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 seem few, being many. To marche in battaile or Iornay: with many such
+feates, to Foughten field, Skarmoush, or Ambushe appartaining:
+
+    [This noble Earle, dyed Anno. 1554. skarse of 24. yeares
+    of age: hauing no issue by his wife: Daughter to the Duke
+    of Somerset.]
+
+And of all these, liuely designementes (most curiously) to be in velame
+parchement described: with Notes & peculier markes, as the Arte
+requireth: and all these Rules, and descriptions Arithmeticall, inclosed
+in a riche Case of Gold, he vsed to weare about his necke: as his Iuell
+most precious, and Counsaylour most trusty. Thus, _Arithmetike_, of him,
+was shryned in gold: Of _Numbers_ frute, he had good hope. Now, Numbers
+therfore innumerable, in _Numbers_ prayse, his shryne shall finde.
+
+
+What nede I, (for farder profe 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 meruaylously declare: how, neither the Auncient Romaine
+lawes, without good knowledge of _Numbers art_, can be perceiued: Nor
+(Iustice in infinite Cases) without due proportion, (narrowly
+considered,) is hable to be executed. How Iustly, & with great knowledge
+of Arte, did _Papinianus_ institute a law of partition, and allowance,
+betwene man and wife after a diuorce? But how _Accursius_, _Baldus_,
+_Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (being
+otherwise, notably well learned) do iumble, gesse, and erre, from the
+aequity, art and Intent 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: and with _Accursius_
+Glosse, much cumbred: burst out, and sayd: _Nulla est in toto libro, hac
+glossa difficilior: Cuius computationem nec Scholastici nec Doctores
+intelligunt. &c._ That is: +_In the whole booke, there is no Glosse
+harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor
+the Doctours vnderstand. &c._+ What can they say of _Iulianus_ law, _Si
+ita Scriptum. &c._ Of the Testators will iustly performing, betwene the
+wife, Sonne and daughter? How can they perceiue the aequitie of
+_Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _Lex
+Falcidia_? How can they deliuer him, from his Reprouers: and their
+maintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? How
+Iustly and artificially, was _Africanus_ 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
+contributed by the legataries to the heire. For, what proportion, 100
+hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that
+is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law,
+require an expert _Arithmeticien_, to vnderstand the deepe Iudgement, &
+Iust determination 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 equity might be
+greately preferred, and skilfully executed, through due skill of
+Arithmetike, and proportions appertainyng. The worthy Philosophers, and
+prudent lawmakers (who haue written many bookes _De Republica:_ How the
+best state of Common wealthes might be procured and mainteined,) haue
+very well determined 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:) defining it,
+
+    [Iustice.]
+
+"to be that vertue, by which, to euery one, is rendred, that to him
+appertaineth." God challengeth this at our handes, to be honored as God:
+to be loued, as a father: to be feared as a Lord & master. Our
+neighbours proportion, is also prescribed 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 same) 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,
+briefly, of other Sciences discoursing, findeth them, not hable to bring
+it to passe: But of the Science of Numbers, he sayth. _Illa, quae
+numerum mortalium generi dedit, id profecto efficiet. Deum autem
+aliquem, magis quam fortunam, ad salutem nostram, hoc munus nobis
+arbitror contulisse. &c. Nam ipsum bonorum omnium Authorem, cur non
+maximi boni, Prudentiae dico, causam arbitramur? +That Science, verely,
+which hath taught mankynde 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 the greatest good
+thyng, namely, Wisedome?+_ There, at length, he proueth _Wisedome_ to be
+atteyned, by good Skill of _Numbers_. 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, 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 entend, 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 vnspeakable behofe) our heauenly father, hath for vs prepared, and
+reuealed, by sundry _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 Originall causes: But the diuersitie neuerthelesse, is
+great. We defined an _Vnit_, to be a thing Mathematicall Indiuisible:
+A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And
+farder, that a Point may haue a certaine determined Situation: that is,
+that we may assigne, and prescribe a Point, to be here, there, yonder.
+&c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to
+be tyed to any place, or seat: diuisible or indiuisible. Agayne, by
+reason, a Point may haue a Situation limited to him: a certaine motion,
+therfore (to a place, and from a place) is to a Point incident and
+appertainyng. But an _Vnit_, can not be imagined to haue any motion.
+A Point, by his motion, produceth, Mathematically, a line: (as we sayd
+before) which is the first kinde of Magnitudes, and most simple: An
+_Vnit_, can not produce any number. A Line, though it be produced of a
+Point moued, yet, it doth not consist of pointes: Number, though it be
+not produced of an _Vnit_, yet doth it Consist of vnits, as a materiall
+cause. But formally,
+
+    [Number.]
+
+Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting,
+is the workemanship of our minde: which, of distinct and discrete Vnits,
+maketh a Number: by vniformitie, resulting of a certaine multitude of
+Vnits. And so, euery number, may haue his least part, giuen: namely, an
+Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be
+giuen: by cause, infinitly, diuision therof, may be conceiued. All
+Magnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine,
+or Solid, of no Sense, can be perceiued, nor exactly by hand (any way)
+represented: nor of Nature produced: But, as (by degrees) Number did
+come to our perceiuerance: So, by visible formes, we are holpen to
+imagine, what our Line Mathematicall, is. What our Point, is. So
+precise, are our Magnitudes, that one Line is no broader then an other:
+for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our
+Bodies, any weight: be they neuer so large of dimension. Our Bodyes, we
+can haue Smaller, then either Arte or Nature can produce any: and
+Greater also, then all the world can comprehend. Our least Magnitudes,
+can be diuided into so many partes, as the greatest. As, a Line of an
+inch long, (with vs) may be diuided into as many partes, as may the
+diameter of the whole world, from East to West: or any way extended:
+What priuiledges, aboue all manual Arte, and Natures might, haue our two
+Sciences Mathematicall? to exhibite, and to deale with thinges of such
+power, liberty, simplicity, puritie, and perfection? And in them, so
+certainly, so orderly, so precisely to procede: as, excellent is that
+workeman Mechanicall Iudged, who nerest can approche to the representing
+of workes, Mathematically demonstrated?
+
+    [->]
+
+And our two Sciences, remaining pure, and absolute, in their proper
+termes, and in their owne Matter: to haue, and allowe, onely such
+Demonstrations, as are plaine, certaine, vniuersall, and of an aeternall
+veritye?
+
+    [Geometrie.]
+
+This Science of _Magnitude_, his properties, conditions, and
+appertenances: commonly, now is, and from the beginnyng, hath of all
+Philosophers, ben called _Geometrie_. But, veryly, with a name to base
+and scant, for a Science of such dignitie and amplenes. And, perchaunce,
+that name, by common and secret consent, of all wisemen, hitherto hath
+ben suffred to remayne: that it might carry with it a perpetuall
+memorye, of the first and notablest benefite, by that Science, to common
+people shewed: Which was, when Boundes and meres of land and ground were
+lost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of
+_Nilus_, the greatest and longest riuer in the world) or, that ground
+bequeathed, were to be assigned: or, ground sold, were to be layd out:
+or (when disorder preuailed) that Commons were distributed into
+seueralties. For, where, vpon these & such like occasions, Some by
+ignorance, some by negligence, Some by fraude, and some by violence, did
+wrongfully limite, measure, encroach, or challenge (by pretence of iust
+content, and measure) those landes and groundes: great losse,
+disquietnes, murder, and warre did (full oft) ensue: Till, by Gods
+mercy, and mans Industrie, The perfect Science of Lines, Plaines, and
+Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. The
+people then, by this art pleasured, and greatly relieued, in their
+landes iust measuring: & other Philosophers, writing Rules for land
+measuring: betwene them both, thus, confirmed the name of _Geometria_,
+that is, (according to the very etimologie of the word) Land measuring.
+Wherin, the people knew no farder, of Magnitudes vse, but in Plaines:
+and the Philosophers, of them, had no feet hearers, or Scholers: farder
+to disclose vnto, then of flat, plaine _Geometrie_. And though, these
+Philosophers, knew of farder vse, and best vnderstode the etymologye of
+the worde, yet this name _Geometria_, was of them applyed generally to
+all sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and
+_Pythagoras_: When they would precisely declare their owne doctrine.
+Then, was
+
+    [* Plato. 7. de Rep.]
+
+* _Geometria_, with them, _Studium quod circa planum versatur_. But,
+well you may perceiue by _Euclides Elementes_, that more ample is our
+Science, then to measure Plaines: and nothyng lesse therin is tought (of
+purpose) then how to measure Land. An other name, therfore, must nedes
+be had, for our Mathematicall Science of Magnitudes: which regardeth
+neither clod, nor turff: neither hill, nor dale: neither earth nor
+heauen: but is absolute _Megethologia_: not creping on ground, and
+dasseling the eye, with pole perche, rod or lyne: but "liftyng the hart
+aboue the heauens, by inuisible lines, and
+
+    [->]
+
+immortall beames meteth with the reflexions, of the light
+incomprehensible: and so procureth Ioye, and perfection vnspeakable." Of
+which true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_
+seemed to haue good taste, and iudgement: and (by the name of
+_Geometrie_) so noted it: and warned his Scholers therof: as, in hys
+seuenth _Dialog_, of the Common wealth, may euidently be sene. Where (in
+Latin) thus it is: right well translated: _Profecto, nobis hoc non
+negabunt, Quicun[que] vel paululum quid Geometriae gustarunt, quin haec
+Scientia, contra, omnino se habeat, quam de ea loquuntur, qui in ipsa
+versantur._ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuer
+haue, (but euen very litle) tasted of Geometrie, will not denye vnto vs,
+this: but that this Science, is of an other condicion, quite contrary to
+that, which they that are exercised in it, do speake of it._+ And there
+it followeth, of our _Geometrie_, _Quod quaeritur cognoscendi illius
+gratia, quod semper est, non & eius quod oritur quando[que] & interit.
+Geometria, eius quod est semper, Cognitio est. Attollet igitur
+(o Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum
+preparabit cogitationem, vt ad supera conuertamus: quae, nunc, contra
+quam decet, ad inferiora deijcimus. &c. Quam maxime igitur praecipiendum
+est, vt qui praeclarissimam hanc habitant Civitatem, nullo modo,
+Geometriam spernant. Nam & quae praeter ipsius propositum, quodam modo
+esse videntur, haud exigua sunt. &c._ It must nedes be confessed (saith
+_Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, which
+is euer: and not of that, which, in tyme, both is bred and is brought to
+an ende. &c. Geometrie is the knowledge of that which is euerlastyng. It
+will lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by
+that meanes, it will prepare the Thought, to the Philosophicall loue of
+wisdome: that we may turne or conuert, toward heauenly thinges =[both
+mynde and thought]= which now, otherwise then becommeth vs, we cast down
+on base or inferior things. &c. Chiefly, therfore, Commaundement must be
+giuen, that such as do inhabit this most honorable Citie, by no meanes,
+despise Geometrie. For euen those thinges =[done by it]= which, in
+manner, seame to be, beside the purpose of Geometrie: are of no small
+importance. &c._+ And besides the manifold vses of _Geometrie_, in
+matters appertainyng to warre, he addeth more, of second vnpurposed
+frute, and commoditye, arrising by _Geometrie_: saying: _Scimus quin
+etiam, ad Disciplinas omnes facilius per discendas, interesse omnino,
+attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam,
+secundo loco discendam Iuuenibus statuamus._ That is. +_But, also, we
+know, that for the more easy learnyng of all Artes, it importeth much,
+whether one haue any knowledge in Geometrie, or no. &c. Let vs therfore
+make an ordinance or decree, that this Science, of young men shall be
+learned in the second place._+ This was _Diuine Plato_ his Iudgement,
+both of the purposed, chief, and perfect vse of _Geometrie_: and of his
+second, dependyng, deriuatiue commodities. And for vs, Christen men,
+a thousand thousand mo occasions are, to haue nede of the helpe of *
+
+    [I. D.
+    * Herein, I would gladly shake of, the earthly name,
+    of Geometrie.]
+
+_Megethologicall_ Contemplations: wherby, to trayne our Imaginations and
+Myndes, by litle and litle, to forsake and abandon, the grosse and
+corruptible Obiectes, of our vtward senses: and to apprehend, by sure
+doctrine demonstratiue, Things Mathematicall. And by them, readily to be
+holpen and conducted to conceiue, discourse, and conclude of things
+Intellectual, Spirituall, aeternall, and such as concerne our Blisse
+euerlasting: which, otherwise (without Speciall priuiledge of
+Illumination, or Reuelation from heauen) No mortall mans wyt (naturally)
+is hable to reach vnto, or to Compasse. And, veryly, by my small Talent
+(from aboue) I am hable to proue and testifie, that the litterall Text,
+and order of our diuine Law, Oracles, and Mysteries, require more skill
+in Numbers, and Magnitudes: then (commonly) the expositors haue vttered:
+but rather onely (at the most) so warned: & shewed their own want
+therin. (To name any, is nedeles: and to note the places, is, here, no
+place: But if I be duely asked, my answere is ready.) And without the
+litterall, Grammaticall, Mathematicall or Naturall verities of such
+places, by good and certaine Arte, perceiued, no Spirituall sense
+(propre to those places, by Absolute _Theologie_) will thereon depend.
+
+    [->]
+
+"No man, therfore, can doute, but toward the atteyning of knowledge
+incomparable, and Heauenly Wisedome: Mathematicall Speculations, both of
+Numbers and Magnitudes: are meanes, aydes, and guides: ready, certaine,
+and necessary." From henceforth, in this my Preface, will I frame my
+talke, to _Plato_ his fugitiue Scholers: or, rather, to such, who well
+can, (and also wil,) vse their vtward senses, to the glory of God, the
+benefite of their Countrey, and their owne secret contentation, or
+honest preferment, on this earthly Scaffold. To them, I will orderly
+recite, describe & declare a great Number of Artes, from our two
+Mathematicall fountaines, deriued into the fieldes of _Nature_. Wherby,
+such Sedes, and Rotes, as lye depe hyd in the ground of _Nature_, are
+refreshed, quickened, and prouoked to grow, shote vp, floure, and giue
+frute, infinite, and incredible. And these Artes, shalbe such, as vpon
+Magnitudes properties do depende, more, then vpon Number. And by good
+reason we may call them Artes, and Artes Mathematicall Deriuatiue: for
+(at this tyme) I Define
+
+    [An Arte.]
+
++An Arte, to be a Methodicall complete Doctrine, hauing abundancy of
+sufficient, and peculier matter to deale with, by the allowance of the
+Metaphisicall Philosopher: the knowledge whereof, to humaine state is
+necessarye.+ And that I account,
+
+    [Art Mathematicall Deriuatiue.]
+
++An Art Mathematicall deriuatiue, which by Mathematicall demonstratiue
+Method, in Numbers, or Magnitudes, ordreth and confirmeth his doctrine,
+as much & as perfectly, as the matter subiect will admit.+ And for that,
+I entend to vse the name and propertie of a
+
+    [A Mechanitien.]
+
+_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke it
+good, (for distinction sake) to giue you also a brief description, what
+I meane therby. +A Mechanicien, or a Mechanicall workman is he, whose
+skill is, without knowledge of Mathematicall demonstration, perfectly to
+worke and finishe any sensible worke, by the Mathematicien principall or
+deriuatiue, demonstrated or demonstrable.+ Full well I know, that he
+which inuenteth, or maketh these demonstrations, is generally called _A
+speculatiue Mechanicien_: which differreth nothyng from a _Mechanicall
+Mathematicien_. So, in respect of diuerse actions, one man may haue the
+name of sundry artes: as, some tyme, of a Logicien, some tymes (in the
+same matter otherwise handled) of a Rethoricien. Of these trifles,
+I make, (as now, in respect of my Preface,) small account: to fyle them
+for the fine handlyng of subtile curious disputers. In other places,
+they may commaunde me, to giue good reason: and yet, here, I will not be
+vnreasonable.
+
+
+    [+1.+]
+
+First, then, from the puritie, absolutenes, and Immaterialitie of
+Principall _Geometrie_, is that kinde of _Geometrie_ deriued, which
+vulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible
+magnitudes, their iust quantities and contentes.+
+
+    [Geometrie vulgar.]
+
+This, teacheth to measure, either at hand: and the practiser, to be by
+the thing Measured: and so, by due applying of Cumpase, Rule, Squire,
+Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to
+the Length, Plaine, or Solide measured,
+
+    [1.]
+
+* to be certified, either of the length, perimetry, or distance lineall:
+and this is called, _Mecometrie_. Or
+
+    [2.]
+
+* to be certified of the content of any plaine Superficies: whether it
+be in ground Surueyed, Borde, or Glasse measured, or such like thing:
+which measuring, is named _Embadometrie_.
+
+    [3.]
+
+* Or els to vnderstand the Soliditie, and content of any bodily thing:
+as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels,
+small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale
+vessells, &c, the Measuring, commonly, hath a peculier name: and is
+called _Gaging_. And the generall name of these Solide measures, is
+_Stereometrie_.
+
+    [+2.+]
+
+Or els, this _vulgar Geometrie_, hath consideration to teach the
+practiser, how to measure things, with good distance betwene him and the
+thing measured: and to vnderstand thereby, either
+
+    [1.]
+
+* how Farre, a thing seene (on land or water) is from the measurer: and
+this may be called _Apomecometrie_:
+
+    [2.]
+
+Or, how High or depe, aboue or vnder the leuel of the measurers
+standing, any thing is, which is sene on land or water, called
+_Hypsometrie_.
+
+    [3.]
+
+* Or, it informeth the measurer, how Broad any thing is, which is in the
+measurers vew: so it be on Land or Water, situated: and may be called
+_Platometrie_. Though I vse here to condition, the thing measured, to be
+on Land, or Water Situated:
+
+    [Note.]
+
+yet, know for certaine, that the sundry heigthe of Cloudes, blasing
+Starres, and of the Mone, may (by these meanes) haue their distances
+from the earth: and, of the blasing Starres and Mone, the Soliditie
+(aswell as distances) to be measured: But because, neither these things
+are vulgarly taught: nor of a common practiser so ready to be executed:
+I, rather, let such measures be reckened incident to some of our other
+Artes, dealing with thinges on high, more purposely, then this vulgar
+Land measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c._
+
+
+Of these Feates (farther applied) is Sprong the Feate of _Geodesie_, or
+Land Measuring: more cunningly to measure & Suruey Land, Woods, and
+Waters, a farre of. More cunningly, I say: But God knoweth (hitherto) in
+these Realmes of England and Ireland (whether through ignorance or
+fraude, I can not tell, in euery particular)
+
+    [Note.]
+
+how great wrong and iniurie hath (in my time) bene committed by vntrue
+measuring and surueying of Land or Woods, any way. And, this I am sure:
+that the Value of the difference, betwene the truth and such Surueyes,
+would haue bene hable to haue found (for euer) in eche of our two
+Vniuersities, an excellent Mathematicall Reader: to eche, allowing
+(yearly) a hundred Markes of lawfull money of this realme: which, in
+dede, would seme requisit, here, to be had (though by other wayes
+prouided for) as well, as, the famous Vniuersitie of Paris, hath two
+Mathematicall Readers: and eche, two hundreth French Crownes yearly, of
+the French Kinges magnificent liberalitie onely. Now, againe, to our
+purpose returning: Moreouer, of the former knowledge Geometricall, are
+growen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and
+_Stratarithmetrie_.
+
+
+"+%Geographie%+ teacheth wayes, by which, in sundry formes, (as
+_Sphaerike_, _Plaine_ or other), the Situation of Cities, Townes,
+Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, &
+such other things, vpon the outface of the earthly Globe (either in the
+whole, or in some principall member and portion therof contayned) may be
+described and designed, in commensurations Analogicall to Nature and
+veritie: and most aptly to our vew, may be represented." Of this Arte
+how great pleasure, and how manifolde commodities do come vnto vs, daily
+and hourely: of most men, is perceaued. While, some, to beautifie their
+Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other
+some, for thinges past, as battels fought, earthquakes, heauenly
+fyringes, & such occurentes, in histories mentioned: therby liuely, as
+it were, to vewe the place, the region adioyning, the distance from vs:
+and such other circumstances. Some other, presently to vewe the large
+dominion of the Turke: the wide Empire of the Moschouite: and the litle
+morsell of ground, where Christendome (by profession) is certainly
+knowen. Litle, I say, in respecte of the rest. &c. Some, either for
+their owne iorneyes directing into farre landes: or to vnderstand of
+other mens trauailes. To conclude, some, for one purpose: and some, for
+an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, &
+Geographicall Globes. Of whose vse, to speake sufficiently, would
+require a booke peculier.
+
+
++%Chorographie%+ seemeth to be an vnderling, and a twig, of
+_Geographie_: and yet neuerthelesse, is in practise manifolde, and in
+vse very ample. "This teacheth Analogically to describe a small portion
+or circuite of ground, with the contentes: not regarding what
+commensuration it hath to the whole, or any parcell, without it,
+contained. But in the territory or parcell of ground which it taketh in
+hand to make description of, it leaueth out (or vndescribed) no notable,
+or odde thing, aboue the ground visible. Yea and sometimes, of thinges
+vnder ground, geueth some peculier marke: or warning: as of Mettall
+mines, Cole pittes, Stone quarries. &c." Thus, a Dukedome, a Shiere,
+a Lordship, or lesse, may be described distinctly. But marueilous
+pleasant, and profitable it is, in the exhibiting to our eye, and
+commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true
+Symmetry: not approching to any of them: and out of Gunne shot. &c.
+Hereby, the _Architect_ may furnishe him selfe, with store of what
+patterns he liketh: to his great instruction: euen in those thinges
+which outwardly are proportioned: either simply in them selues: or
+respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also,
+terme this particular description of places, _Topographie_.
+
+
+"+%Hydrographie%+, deliuereth to our knowledge, on Globe or in Plaine,
+the perfect Analogicall description of the Ocean Sea coastes, through
+the whole world: or in the chiefe and principall partes thereof:" with
+the Iles and chiefe particular places of daungers, conteyned within the
+boundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes,
+Rockes, Races, Countertides, Whorlepooles. &c. This, dealeth with the
+Element of the water chiefly: as _Geographie_ did principally take the
+Element of the Earthes description (with his appertenances) to taske.
+And besides thys, _Hydrographie_, requireth a particular Register of
+certaine Landmarkes (where markes may be had) from the sea, well hable
+to be skried, in what point of the Seacumpase they appeare, and what
+apparent forme, Situation, and bignes they haue, in respecte of any
+daungerous place in the sea, or nere vnto it, assigned: And in all
+Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes,
+come and go, the _Hydrographer_ ought to recorde. The Soundinges
+likewise: and the Chanels wayes: their number, and depthes ordinarily,
+at ebbe and flud, ought the _Hydrographer_, by obseruation and diligence
+of _Measuring_, to haue certainly knowen. And many other pointes, are
+belonging to perfecte _Hydrographie_, and for to make a _Rutter_, by: of
+which, I nede not here speake: as of the describing, in any place, vpon
+Globe or Plaine, the 32. pointes of the Compase, truely: (wherof,
+scarsly foure, in England, haue right knowledge: bycause, the lines
+therof, are no straight lines, nor Circles.) Of making due proiection of
+a Sphere in plaine. Of the Variacion of the Compas, from true Northe:
+And such like matters (of great importance, all) I leaue to speake of,
+in this place: bycause, I may seame (al ready) to haue enlarged the
+boundes, and duety of an Hydographer, much more, then any man (to this
+day) hath noted, or prescribed. Yet am I well hable to proue, all these
+thinges, to appertaine, and also to be proper to the Hydrographer. The
+chief vse and ende of this Art, is the Art of Nauigation: but it hath
+other diuerse vses: euen by them to be enioyed, that neuer lacke sight
+of land.
+
+
++%Stratarithmetrie%+, is the Skill, (appertainyng to the warre,) by
+which a man can set in figure, analogicall to any _Geometricall_ figure
+appointed, any certaine number or summe of men: of such a figure
+capable: (by reason of the vsuall spaces betwene Souldiers allowed: and
+for that, of men, can be made no Fractions. Yet, neuertheles, he can
+order the giuen summe of men, for the greatest such figure, that of
+them, can be ordred) and certifie, of the ouerplus: (if any be) and of
+the next certaine summe, which, with the ouerplus, will admit a figure
+exactly proportionall to the figure assigned. By which Skill, also, of
+any army or company of men: (the figure & sides of whose orderly
+standing, or array, is knowen) he is able to expresse the iust number of
+men, within that figure conteined: or (orderly) able to be conteined.
+
+    [* Note.]
+
+* And this figure, and sides therof, he is hable to know: either beyng
+by, and at hand: or a farre of. Thus farre, stretcheth the description
+and property of _Stratarithmetrie_: sufficient for this tyme and place.
+
+    [The difference betwene Stratarithmetrie and Tacticie.]
+
+"It differreth from the Feate _Tacticall_, _De aciebus instruendis._
+bycause, there, is necessary the wisedome and foresight, to what purpose
+he so ordreth the men: and Skillfull hability, also, for any occasion,
+or purpose, to deuise and vse the aptest and most necessary order, array
+and figure of his Company and Summe of men." By figure, I meane: as,
+either of a _Perfect Square_, _Triangle_, _Circle_, _Ouale_, _long
+square_, (of the Grekes it is called _Eteromekes_) _Rhombe_, _Rhomboid_,
+_Lunular_, _Ryng_, _Serpentine_, and such other Geometricall figures:
+Which, in warres, haue ben, and are to be vsed: for commodiousnes,
+necessity, and auauntage &c. And no small skill ought he to haue, that
+should make true report, or nere the truth, of the numbers and Summes,
+of footemen or horsemen, in the Enemyes ordring. A farre of, to make an
+estimate, betwene nere termes of More and Lesse, is not a thyng very
+rife, among those that gladly would do it.
+
+    [I. D.
+    Frende, you will finde it hard, to performe my description
+    of this Feate. But by Chorographie, you may helpe your selfe
+    some what: where the Figures knowne (in Sides and Angles)
+    are not Regular: And where, Resolution into Triangles can
+    serue. &c. And yet you will finde it strange to deale thus
+    generally with Arithmeticall figures: and, that for Battayle
+    ray. Their contentes, differ so much from like Geometricall
+    Figures.]
+
+Great pollicy may be vsed of the Capitaines, (at tymes fete, and in
+places conuenient) as to vse Figures, which make greatest shew, of so
+many as he hath: and vsing the aduauntage of the three kindes of vsuall
+spaces: (betwene footemen or horsemen) to take the largest: or when he
+would seme to haue few, (beyng many:) contrarywise, in Figure, and
+space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer
+is carefull to come nere the truth herein, besides the Iudgement of his
+expert eye, his skill of Ordering _Tacticall_, the helpe of his
+Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously
+framed for cariage and vse) He may wonderfully helpe him selfe, by
+perspectiue Glasses. In which, (I trust) our posterity will proue more
+skillfull and expert, and to greater purposes, then in these dayes, can
+(almost) be credited to be possible.
+
+
+Thus haue I lightly passed ouer the Artificiall Feates, chiefly
+dependyng vpon vulgar _Geometrie_: & commonly and generally reckened
+vnder the name of _Geometrie_. But there are other (very many)
+_Methodicall Artes_, which, declyning from the purity, simplicitie, and
+Immateriality, of our Principall Science of _Magnitudes_: do yet
+neuertheles vse the great ayde, direction, and Method of the sayd
+principall Science, and haue propre names, and distinct: both from the
+Science of _Geometrie_, (from which they are deriued) and one from the
+other. As +Perspectiue, Astronomie, Musike, Cosmographie, Astrologie,
+Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie,
+Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture,
+Nauigation, Thaumaturgike+ and +Archemastrie+. I thinke it necessary,
+orderly, of these to giue some peculier descriptions: and withall, to
+touch some of their commodious vses, and so to make this Preface, to be
+a little swete, pleasant Nosegaye for you: to comfort your Spirites,
+beyng almost out of courage, and in despayre, (through brutish brute)
+Weenyng that _Geometrie_, had but serued for buildyng of an house, or a
+curious bridge, or the roufe of Westminster hall, or some witty pretty
+deuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That the
+thing is farre otherwise, then the world, (commonly) to this day, hath
+demed, by worde and worke, good profe wilbe made.
+
+
+Among these Artes, by good reason, +%Perspectiue%+ ought to be had, ere
+of _Astronomicall Apparences_, perfect knowledge can be atteyned. And
+bycause of the prerogatiue of _Light_, beyng the first of _Gods
+Creatures_: and the eye, the light of our body, and his Sense most
+mighty, and his organ most Artificiall and _Geometricall_: At
+_Perspectiue_, we will begyn therfore. +Perspectiue, is an Art
+Mathematicall, which demonstrateth the maner, and properties, of all
+Radiations Direct, Broken, and Reflected.+ This Description, or
+Notation, is brief: but it reacheth so farre, as the world is wyde. It
+concerneth all Creatures, all Actions, and passions, by Emanation of
+beames perfourmed. Beames, or naturall lines, (here) I meane, not of
+light onely, or of colour (though they, to eye, giue shew, witnes, and
+profe, wherby to ground the Arte vpon) but also of other _Formes_, both
+_Substantiall_, and _Accidentall_, the certaine and determined actiue
+Radiall emanations. By this Art (omitting to speake of the highest
+pointes) we may vse our eyes, and the light, with greater pleasure: and
+perfecter Iudgement: both of things, in light seen, & of other: which by
+like order of Lightes Radiations, worke and produce their effectes. We
+may be ashamed to be ignorant of the cause, why so sundry wayes our eye
+is deceiued, and abused: as, while the eye weeneth a round Globe or
+Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise
+iudgeth a plaine Square, to be round: supposeth walles parallels, to
+approche, a farre of: rofe and floure parallels, the one to bend
+downward, the other to rise vpward, at a little distance from you.
+Againe, of thinges being in like swiftnes of mouing, to thinke the
+nerer, to moue faster: and the farder, much slower. Nay, of two thinges,
+wherof the one (incomparably) doth moue swifter then the other, to deme
+the slower to moue very swift, & the other to stand: what an error is
+this, of our eye? Of the Raynbow, both of his Colours, of the order of
+the colours, of the bignes of it, the place and heith of it, (&c) to
+know the causes demonstratiue, is it not pleasant, is it not necessary?
+of two or three Sonnes appearing: of Blasing Sterres: and such like
+thinges: by naturall causes, brought to passe, (and yet neuertheles, of
+farder matter, Significatiue) is it not commodious for man to know the
+very true cause, & occasion Naturall? Yea, rather, is it not, greatly,
+against the Souerainty of Mans nature, to be so ouershot and abused,
+with thinges (at hand) before his eyes? as with a Pecockes tayle, and a
+Doues necke: or a whole ore, in water, holden, to seme broken. Thynges,
+farre of, to seeme nere: and nere, to seme farre of. Small thinges, to
+seme great: and great, to seme small. One man, to seme an Army. Or a man
+to be curstly affrayed of his owne shaddow. Yea, so much, to feare,
+that, if you, being (alone) nere a certaine glasse, and proffer, with
+dagger or sword, to foyne at the glasse, you shall suddenly be moued to
+giue backe (in maner) by reason of an Image,
+
+    [-> A marueilous Glasse.]
+
+appearing in the ayre, betwene you & the glasse, with like hand, sword
+or dagger, & with like quicknes, foyning at your very eye, likewise as
+you do at the Glasse. Straunge, this is, to heare of: but more
+meruailous to behold, then these my wordes can signifie. And
+neuerthelesse by demonstration Opticall, the order and cause therof, is
+certified: euen so, as the effect is consequent. Yea, thus much more,
+dare I take vpon me, toward the satisfying of the noble courrage, that
+longeth ardently for the wisedome of Causes Naturall: as to let him
+vnderstand, that, in London, he may with his owne eyes, haue profe of
+that, which I haue sayd herein. A Gentleman, (which, for his good
+seruice, done to his Countrey, is famous and honorable:
+
+    [S. W. P.]
+
+and for skill in the Mathematicall Sciences, and Languages, is the Od
+man of this land. &c.) euen he, is hable: and (I am sure) will, very
+willingly, let the Glasse, and profe be sene: and so I (here) request
+him: for the encrease of wisedome, in the honorable: and for the
+stopping of the mouthes malicious: and repressing the arrogancy of the
+ignorant. Ye may easily gesse, what I meane. This Art of _Perspectiue_,
+is of that excellency, and may be led, to the certifying, and executing
+of such thinges, as no man would easily beleue: without Actuall profe
+perceiued. I speake nothing of _Naturall Philosophie_, which, without
+_Perspectiue_, can not be fully vnderstanded, nor perfectly atteined
+vnto. Nor, of _Astronomie_: which, without _Perspectiue_, can not well
+be grounded: Nor _Astrologie_, naturally Verified, and auouched. That
+part hereof, which dealeth with Glasses (which name, Glasse, is a
+generall name, in this Arte, for any thing, from which, a Beame
+reboundeth) is called _Catoptrike_: and hath so many vses, both
+merueilous, and proffitable: that, both, it would hold me to long, to
+note therin the principall conclusions, all ready knowne: And also
+(perchaunce) some thinges, might lacke due credite with you: And I,
+therby, to leese my labor: and you, to slip into light Iudgement,
+
+    [* ->]
+
+Before you haue learned sufficiently the powre of Nature and Arte.
+
+
+Now, to procede: +%Astronomie%, is an Arte Mathematicall, which
+demonstrateth the distance, magnitudes, and all naturall motions,
+apparences, and passions propre to the Planets and fixed Sterres: for
+any time past, present and to come: in respect of a certaine Horizon, or
+without respect of any Horizon.+ By this Arte we are certified of the
+distance of the Starry Skye, and of eche _Planete_ from the Centre of
+the Earth: and of the greatnes of any Fixed starre sene, or _Planete_,
+in respect of the Earthes greatnes. As, we are sure (by this Arte) that
+the Solidity, Massines and Body of the _Sonne_, conteineth the quantitie
+of the whole Earth and Sea, a hundred thre score and two times, lesse by
+1/8 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 1/8 of the Mone. Wherfore the _Sonne_ is bigger then the
+_Mone_, 7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64
+bigger then the _Mone_. And yet the vnskillfull man, would iudge them a
+like bigge. Wherfore, of Necessity, the one is much farder from vs, then
+the other. The _Sonne_, when he is fardest from the earth (which, now,
+in our age, is, when he is in the 8. degree, of Cancer) is, 1179
+Semidiameters of the Earth, distante. And the _Mone_ when she is fardest
+from the earth, is 68 Semidiameters of the earth and 1/3 The nerest,
+that the _Mone_ commeth to the earth, is Semidiameters 52-1/4 The
+distance of the Starry Skye is, from vs, in Semidiameters of the earth
+20081-1/2 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 20029-1/4 Twenty thousand nine and
+twenty and a quarter.
+
+    [Note.]
+
+So thicke is the heauenly Palace, that the _Planetes_ haue all their
+exercise in, and most meruailously perfourme the Commaundement and
+Charge to them giuen by the omnipotent Maiestie of the king of kings.
+This is that, which in _Genesis_ is called _Ha Rakia_. Consider it well.
+The Semidiameter of the earth, conteineth of our common miles 3436 4/11
+three thousand, foure hundred thirty six and foure eleuenth partes of
+one myle: Such as the whole earth and Sea, round about, is 21600. One
+and twenty thousand six hundred of our myles. Allowyng for euery degree
+of the greatest circle, thre score myles. Now if you way well with your
+selfe but this litle parcell of frute _Astronomicall_, as concerning the
+bignesse, Distances of _Sonne_, _Mone_, _Sterry Sky_, and the huge
+massines of _Ha Rakia_, will you not finde your Consciences moued, with
+the kingly Prophet, to sing the confession of Gods Glory, and say, +_The
+Heauens declare the glory of God, and the Firmament =[Ha Rakia]= sheweth
+forth the workes of his handes_+. And so forth, for those fiue first
+staues, of that kingly Psalme. Well, well, It is time for some to lay
+hold on wisedome, and to Iudge truly of thinges: and notso to expound
+the Holy word, all by Allegories: as to Neglect the wisedome, powre and
+Goodnes of God, in, and by his Creatures, and Creation to be seen and
+learned. By parables and Analogies of whose natures and properties, the
+course of the Holy Scripture, also, declareth to vs very many Mysteries.
+The whole Frame of Gods Creatures, (which is the whole world,) is to vs,
+a bright glasse: from which, by reflexion, reboundeth to our knowledge
+and perceiuerance, Beames, and Radiations: representing the Image of his
+Infinite goodnes, Omnipotency, and wisedome. And we therby, are taught
+and persuaded to Glorifie our Creator, as God: and be thankefull
+therfore. Could the Heathenistes finde these vses, of these most pure,
+beawtifull, and Mighty Corporall Creatures: and shall we, after that the
+true _Sonne_ of rightwisenesse is risen aboue the _Horizon_, of our
+temporall _Hemisphaerie_, and hath so abundantly streamed into our
+hartes, the direct beames of his goodnes, mercy, and grace: Whose heat
+All Creatures feele: Spirituall and Corporall: Visible and Inuisible.
+Shall we (I say) looke vpon the _Heauen_, _Sterres_, and _Planets_, as
+an Oxe and an Asse doth: no furder carefull or inquisitiue, what they
+are: why were they Created, How do they execute that they were Created
+for? Seing, All Creatures, were for our sake created: and both we, and
+they, Created, chiefly to glorifie the Almighty Creator: and that, by
+all meanes, to vs possible. _Nolite ignorare_ (saith _Plato in
+Epinomis_) _Astronomiam, Sapientissimum quiddam esse._ +_Be ye not
+ignorant, Astronomie to be a thyng of excellent wisedome._+
+_Astronomie_, was to vs, from the beginning commended, and in maner
+commaunded by God him selfe. In asmuch as he made the _Sonne_, _Mone_,
+and _Sterres_, to be to vs, for _Signes_, and knowledge of Seasons, and
+for Distinctions of Dayes, and yeares. Many wordes nede not. But I wish,
+euery man should way this word, _Signes_. And besides that, conferre it
+also with the tenth Chapter of _Hieremie_. And though Some thinke, that
+there, they haue found a rod: Yet Modest Reason, will be indifferent
+Iudge, who ought to be beaten therwith, in respect of our purpose.
+Leauing that: I pray you vnderstand this: that without great diligence
+of Obseruation, examination and Calculation, their periods and courses
+(wherby _Distinction_ of Seasons, yeares, and New Mones might precisely
+be knowne) could not exactely be certified. Which thing to performe, is
+that _Art_, which we here haue Defined to be _Astronomie_. Wherby, we
+may haue the distinct Course of Times, dayes, yeares, and Ages: aswell
+for Consideration of Sacred Prophesies, accomplished in due time,
+foretold: as for high Mysticall Solemnities holding: And for all other
+humaine affaires, Conditions, and couenantes, vpon certaine time,
+betwene man and man: with many other great vses: Wherin, (verely), would
+be great incertainty, Confusion, vntruth, and brutish Barbarousnes:
+without the wonderfull diligence and skill of this Arte: continually
+learning, and determining Times, and periodes of Time, by the Record of
+the heauenly booke, wherin all times are written: and to be read with an
+_Astronomicall staffe_, in stede of a festue.
+
+
++%Musike%+, of Motion, hath his Originall cause: Therfore, after the
+motions most swift, and most Slow, which are in the Firmament, of Nature
+performed: and vnder the _Astronomers Consideration_: now I will Speake
+of an other kinde of _Motion_, producing sound, audible, and of Man
+numerable. _Musike_ I call here that _Science_, which of the Grekes is
+called _Harmonice_. Not medling with the Controuersie betwene the
+auncient _Harmonistes_, and _Canonistes_. +Musike is a Mathematicall
+Science, which teacheth, by sense and reason, perfectly to iudge, and
+order the diuersities of soundes, hye and low.+ _Astronomie_ and
+_Musike_ are Sisters, saith _Plato_. As, for _Astronomie_, the eyes: So,
+for _Harmonious Motion_, the eares were made. But as _Astronomie_ hath a
+more diuine Contemplation, and commodity, then mortall eye can perceiue:
+So, is _Musike_ to be considered,
+
+    [1.]
+
+that the * Minde may be preferred, before the eare. And from audible
+sound, we ought to ascende, to the examination: which numbers are
+_Harmonious_, and which not. And why, either, the one are: or the other
+are not. I could at large,
+
+    [2.]
+
+in the heauenly * motions and distances, describe a meruailous Harmonie,
+of _Pythagoras_ Harpe
+
+    [3.]
+
+with eight stringes. Also, somwhat might be sayd of _Mercurius_ * two
+Harpes,
+
+    [4.]
+
+eche of foure Stringes Elementall. And very straunge matter, might be
+alledged of the _Harmonie_,
+
+    [5.]
+
+to our * Spirituall part appropriate. As in _Ptolomaus_ third boke, in
+the fourth and sixth Chapters may appeare. *
+
+    [6.]
+
+And what is the cause of the apt bonde, and frendly felowship, of the
+Intellectuall and Mentall part of vs, with our grosse & corruptible
+body: but a certaine Meane, and _Harmonious Spiritualitie, with both
+participatyng, & of both (in a maner) resultynge In
+
+    [7.]
+
+the * Tune of Mans voyce, and also
+
+    [8.]
+
+* the sound of Instrument_, what might be sayd, of _Harmonie_: No common
+Musicien would lightly beleue.
+
+    [I. D.
+    Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7.
+    chapters. Where you shall haue some occasion farder to thinke
+    of Musike, than commonly is thought.]
+
+But of the sundry Mixture (as I may terme it) and concurse, diuerse
+collation, and Application of these _Harmonies_: as of thre, foure,
+fiue, or mo: Maruailous haue the effectes ben: and yet may be founde,
+and produced the like: with some proportionall consideration for our
+time, and being: in respect of the State, of the thinges then: in which,
+and by which, the wondrous effectes were wrought. _Democritus_ and
+_Theophrastus_ affirmed, that, by _Musike_, griefes and diseases of the
+Minde, and body might be cured, or inferred. And we finde in Recorde,
+that _Terpander_, _Arion_, _Ismenias_, _Orpheus_, _Amphion_, _Dauid_,
+_Pythagoras_, _Empedocles_, _Asclepiades_ and _Timotheus_, by
+_Harmonicall_ Consonancy, haue done, and brought to pas, thinges, more
+then meruailous, to here of. Of them then, making no farder discourse,
+in this place: Sure I am, that Common _Musike_, commonly vsed, is found
+to the _Musiciens_ and Hearers, to be so Commodious and pleasant, That
+if I would say and dispute, but thus much: That it were to be otherwise
+vsed, then it is, I should finde more repreeuers, then I could finde
+priuy, or skilfull of my meaning. In thinges therfore euident, and
+better knowen, then I can expresse: and so allowed and liked of, (as I
+would wish, some other thinges, had the like hap) I will spare to
+enlarge my lines any farder, but consequently follow my purpose.
+
+
++%Of Cosmographie%+, I appointed briefly in this place, to geue you some
+intelligence. +Cosmographie, is the whole and perfect description of the
+heauenly, and also elementall parte of the world, and their homologall
+application, and mutuall collation necessarie.+ This Art, requireth
+_Astronomie_, _Geographie_, _Hydrographie_ and _Musike_. Therfore, it is
+no small Arte, nor so simple, as in common practise, it is (slightly)
+considered. This matcheth Heauen, and the Earth, in one frame, and aptly
+applieth parts Correspondent: So, as, the Heauenly Globe, may (in
+practise) be duely described vpon the Geographicall, and Hydrographicall
+Globe. And there, for vs to consider an _Aequonoctiall Circle_, _an
+Ecliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes,
+Latitudes, Declinations, and Verticalitie: also Climes, and Parallels:
+and by an _Horizon_ annexed, and reuolution of the earthly Globe (as the
+Heauen, is, by the _Primouant_, caried about in 24. aequall Houres) to
+learne the Risinges and Settinges of Sterres (of _Virgill_ in his
+_Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sphaere_,
+to Perdicca King of the Macedonians: of _Diocles_, to King _Antigonus_,
+and of other famous _Philosophers_ prescribed) a thing necessary, for
+due manuring of the earth, for _Nauigation_, for the Alteration of mans
+body: being, whole, Sicke, wounded, or brused. By the Reuolution, also,
+or mouing of the Globe Cosmographicall, the Rising and Setting of the
+Sonne: the Lengthes, of dayes and nightes: the Houres and times (both
+night and day) are knowne: with very many other pleasant and necessary
+vses: Wherof, some are knowne: but better remaine, for such to know and
+vse:
+
+    [->]
+
+who of a sparke of true fire, can make a wonderfull bonfire, by applying
+of due matter, duely.
+
+
++%Of Astrologie%+, here I make an Arte, seuerall from _Astronomie_: not
+by new deuise, but by good reason and authoritie: for, +Astrologie, is
+an Arte Mathematicall, which reasonably demonstrateth the operations and
+effectes, of the naturall beames, of light, and secrete influence: of
+the Sterres and Planets: in euery element and elementall body: at all
+times, in any Horizon assigned.+ This Arte is furnished with many other
+great Artes and experiences: As with perfecte _Perspectiue_,
+_Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4.
+Elementes, the Arte of Graduation, and some good vnderstanding in
+_Musike_: and yet moreouer, with an other great Arte, hereafter
+following, though I, here, set this before, for some considerations me
+mouing. Sufficient (you see) is the stuffe, to make this rare and
+secrete Arte, of: and hard enough to frame to the Conclusion
+Syllogisticall. Yet both the manifolde and continuall trauailes of the
+most auncient and wise Philosophers, for the atteyning of this Arte: and
+by examples of effectes, to confirme the same: hath left vnto vs
+sufficient proufe and witnesse: and we, also, daily may perceaue, That
+mans body, and all other Elementall bodies, are altered, disposed,
+ordred, pleasured, and displeasured, by the Influentiall working of the
+_Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth
+_Aristotle_, in the first of his _Meteorologicall_ bookes, in the second
+Chapter: _Est autem necessario Mundus iste, supernis lationibus fere
+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.
+_Oportet igitur & horum principia sumamus, & causas omnium similiter.
+Principium igitur vt mouens, praecipuum[que] & omnium primum, Circulus
+ille est, in quo manifeste Solis latio, &c._ And so forth. His
+_Meteorologicall_ bookes, are full of argumentes, and effectuall
+demonstrations, of the vertue, operation, and power of the heauenly
+bodies, in and vpon the fower Elementes, and other bodies, of them
+(either perfectly, or vnperfectly) composed. And in his second booke,
+_De Generatione & Corruptione_, in the tenth Chapter. _Quocirca & prima
+latio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea
+nam[que] & continua est, & duobus motibus fit:_ In Englishe, thus.
++_Wherefore the vppermost motion, is not the cause of Generation and
+Corruption, but the motion of the Zodiake: for, that, both, is
+continuall, and is caused of two mouinges._+ And in his second booke,
+and second Chapter of hys _Physikes_. _Homo nam[que] generat hominem,
+at[que] Sol._ +_For Man (sayth he) and the Sonne, are cause of mans
+generation._+ Authorities may be brought, very many: both of 1000. 2000.
+yea and 3000. yeares Antiquitie: of great _Philosophers_, _Expert_,
+_Wise_, and godly men, for that Conclusion: which, daily and hourely, we
+men, may discerne and perceaue by sense and reason: All beastes do
+feele, and simply shew, by their actions and passions, outward and
+inward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally,
+the Elementes, and all thinges of the Elementes composed, do geue
+Testimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions,
+vertues, and naturall motions, depend of the Actiuitie of the heauenly
+motions and Influences. Whereby, beside the specificall order and forme,
+due to euery seede: and beside the Nature, propre to the Indiuiduall
+Matrix, of the thing produced: What shall be the heauenly Impression,
+the perfect and circumspecte Astrologien hath to Conclude._+ Not onely
+(by _Apotelesmes_) #to hoti#. but by Naturall and Mathematicall
+demonstration #to dioti#. Whereunto, what Sciences are requisite
+(without exception) I partly haue here warned: And in my _Propaedeumes_
+(besides other matter there disclosed) I haue Mathematically furnished
+vp the whole Method: To this our age, not so carefully handled by any,
+that euer I saw, or heard of. I was,
+
+    [* Anno. 1548 and 1549. in Louayn.]
+
+(for * 21. yeares ago) by certaine earnest disputations, of the Learned
+_Gerardus Mercator_, and _Antonius Gogaua_, (and other,) therto so
+prouoked: and (by my constant and inuincible zeale to the veritie) in
+obseruations of Heauenly Influencies (to the Minute of time,) than, so
+diligent: And chiefly by the Supernaturall influence, from the Starre of
+Iacob, so directed: That any Modest and Sober Student, carefully and
+diligently seking for the Truth, will both finde & confesse, therin, to
+be the Veritie, of these my wordes: And also become a Reasonable
+Reformer, of three Sortes of people: about these Influentiall
+Operations, greatly erring from the truth.
+
+    [Note.]
+
+Wherof, the one, is +Light Beleuers+, the other, +Light Despisers+, and
+the third +Light Practisers+. The first, & most common Sort, thinke the
+Heauen and Sterres, to be answerable to any their doutes or desires:
+
+    [1.]
+
+which is not so: and, in dede, they, to much, ouer reache. The Second
+sorte thinke no Influentiall vertue (from the heauenly bodies) to beare
+any Sway in Generation
+
+    [2.]
+
+and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and
+_Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so
+farre in distance, so manifold in their motions, so constant in their
+periodes. &c.) they assigne a sleight, simple office or two, and so
+allow vnto them (according to their capacities) as much vertue, and
+power Influentiall, as to the Signe of the _Sunne_, _Mone_, and seuen
+Sterres, hanged vp (for Signes) in London, for distinction of houses, &
+such grosse helpes, in our worldly affaires: And they vnderstand not (or
+will not vnderstand) of the other workinges, and vertues of the Heauenly
+_Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husband
+man: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, as
+the Porpentine doth: nor will allow these perfect, and incorruptible
+mighty bodies, so much vertuall Radiation, & Force, as they see in a
+litle peece of a _Magnes stone_: which, at great distance, sheweth his
+operation. And perchaunce they thinke, the Sea & Riuers (as the Thames)
+to be some quicke thing, and so to ebbe, and flow, run in and out, of
+them selues, at their owne fantasies. God helpe, God helpe. Surely,
+these men, come to short: and either are to dull: or willfully blind:
+or, perhaps, to malicious. The third man, is the common and vulgare
+_Astrologien_, or Practiser: who, being not duely, artificially, and
+perfectly
+
+    [3.]
+
+furnished: yet, either for vaine glory, or gayne: or like a simple dolt,
+& blinde Bayard, both in matter and maner, erreth: to the discredit of
+the _Wary_, and modest _Astrologien_: and to the robbing of those most
+noble corporall Creatures, of their Naturall Vertue: being most mighty:
+most beneficiall to all elementall Generation, Corruption and the
+appartenances: and most Harmonious in their Monarchie: For which
+thinges, being knowen, and modestly vsed: we might highly, and
+continually glorifie God, with the princely Prophet, saying. +_The
+Heauens declare the Glorie of God: who made the Heauens in his wisedome:
+who made the Sonne, for to haue dominion of the day: the Mone and
+Sterres to haue dominion of the nyght: whereby, Day to day vttereth
+talke: and night, to night declareth knowledge. Prayse him, all ye
+Sterres, and Light. Amen._+
+
+
+In order, now foloweth, of +%Statike%+, somewhat to say, what we meane
+by that name: and what commodity, doth, on such Art, depend. +Statike,
+is an Arte Mathematicall, which demonstrateth the causes of heauynes,
+and lightnes of all thynges: and of motions and properties, to heauynes
+and lightnes, belonging.+ And for asmuch as, by the Bilanx, or Balance
+(as the chief sensible Instrument,) Experience of these demonstrations
+may be had: we call this Art, _Statike:_ that is, _the Experimentes of
+the Balance_. Oh, that men wist, what proffit, (all maner of wayes) by
+this Arte might grow, to the hable examiner, and diligent practiser.
+"Thou onely, knowest all thinges precisely (O God) who hast made weight
+and Balance, thy Iudgement: who hast created all thinges in _Number,
+Waight, and Measure_: and hast wayed the mountaines and hils in a
+Balance: who hast peysed in thy hand, both Heauen and earth. We therfore
+warned by the Sacred word, to Consider thy Creatures: and by that
+consideration, to wynne a glyms (as it were,) or shaddow of
+perceiuerance, that thy wisedome, might, and goodnes is infinite, and
+vnspeakable, in thy Creatures declared: And being farder aduertised, by
+thy mercifull goodnes, that, three principall wayes, were, of the, vsed
+in Creation of all thy Creatures, namely, _Number_, _Waight_ and
+_Measure_, And for as much as, of _Number_ and _Measure_, the two Artes
+(auncient, famous, and to humaine vses most necessary,) are, all ready,
+sufficiently knowen and extant: This third key, we beseche thee (through
+thy accustomed goodnes,) that it may come to the nedefull and sufficient
+knowledge, of such thy Seruauntes, as in thy workemanship, would gladly
+finde, thy true occasions (purposely of the vsed) whereby we should
+glorifie thy name, and shew forth (to the weaklinges in faith) thy
+wondrous wisedome and Goodnes. Amen."
+
+
+Meruaile nothing at this pang (godly frend, you Gentle and zelous
+Student.) An other day, perchaunce, you will perceiue, what occasion
+moued me. Here, as now, I will giue you some ground, and withall some
+shew, of certaine commodities, by this Arte arising. And bycause this
+Arte is rare, my wordes and practises might be to darke: vnleast you had
+some light, holden before the matter: and that, best will be, in giuing
+you, out of _Archimedes_ demonstrations, a few principal Conclusions, as
+foloweth.
+
+  +1.+
+
+  +The Superficies of euery Liquor, by it selfe consistyng, and in
+  quyet, is Sphaericall: the centre whereof, is the same, which is the
+  centre of the Earth.+
+
+  +2.+
+
+  +If Solide Magnitudes, being of the same bignes, or quantitie, that
+  any Liquor is, and hauyng also the same Waight: be let downe into
+  the same Liquor, they will settle downeward, so, that no parte of
+  them, shall be aboue the Superficies of the Liquor: and yet
+  neuertheles, they will not sinke vtterly downe, or drowne.+
+
+  +3.+
+
+  +If any Solide Magnitude beyng Lighter then a Liquor, be let downe
+  into the same Liquor, it will settle downe, so farre into the same
+  Liquor, that so great a quantitie of that Liquor, as is the parte of
+  the Solid Magnitude, settled downe into the same Liquor: is in
+  Waight, aequall, to the waight of the whole Solid Magnitude.+
+
+  +4.+
+
+  +Any Solide Magnitude, Lighter then a Liquor, forced downe into the
+  same Liquor, will moue vpward, with so great a power, by how much,
+  the Liquor hauyng aequall quantitie to the whole Magnitude, is
+  heauyer then the same Magnitude.+
+
+  +5.+
+
+  +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into
+  the same Liquor, will sinke downe vtterly: And wilbe in that Liquor,
+  Lighter by so much, as is the waight or heauynes of the Liquor,
+  hauing bygnes or quantitie, aequall to the Solid Magnitude.+
+
+  +6.+
+
+      [I. D.
+      The Cutting of a Sphaere according to any proportion assigned
+      may by this proposition be done Mechanically by tempering
+      Liquor to a certayne waight in respect of the waight of the
+      Sphaere therein Swymming.]
+
+  +If any Solide Magnitude, Lighter then a Liquor, be let downe into
+  the same Liquor, the waight of the same Magnitude, will be, to the
+  Waight of the Liquor. (Which is aequall in quantitie to the whole
+  Magnitude,) in that proportion, that the parte, of the Magnitude
+  settled downe, is to the whole Magnitude.+
+
+
+By these verities, great Errors may be reformed, in Opinion of the
+Naturall Motion of thinges, Light and Heauy. Which errors, are in
+Naturall Philosophie (almost) of all men allowed: to much trusting to
+Authority: and false Suppositions. As, +Of any two bodyes, the heauyer,
+to moue downward faster then the lighter.+
+
+    [A common error, noted.]
+
+This error, is not first by me, Noted: but by one _Iohn Baptist de
+Benedictis_. The chief of his propositions, is this: which seemeth a
+Paradox.
+
+
++If there be two bodyes of one forme, and of one kynde, aequall in
+quantitie or vnaequall,
+
+    [A paradox.]
+
+they will moue by aequall space, 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+,
+
+    [N. T.]
+
+certaine good discourses (otherwise) may receiue great amendement, and
+furderance.
+
+    [The wonderfull vse of these Propositions.]
+
+In the entended purpose, also, allowing somwhat to the imperfection of
+Nature: not aunswerable to the precisenes of demonstration. Moreouer, by
+the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth,
+the Fire, may be nerely, knowen, how light or heauy they are (Naturally)
+in their assigned 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 Humours 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, vndemonstrated: 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.
+
+    [The practise Staticall, to know the proportion, betwene
+    the Cube, and the Sphaere.]
+
+Make a Cube, of any one Vniforme: 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 nature, 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,
+company & 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, somewhat 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: by eight descentes progressionall, * halfing, from a grayne.
+
+    [I. D.
+    * For, so, haue you .256. partes of a Graine.]
+
+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 Number 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 Mechanicien 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 demonstrated, 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 Comprehended of
+our former Cube: So, that the base of that Cylinder, is a Circle
+described in the Square, which is the base of our Cube. But the Cube and
+the Cylinder, 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,
+
+    [* =The proportion of the Square to the Circle inscribed.=]
+
+we haue * the proportion of the Square, to the Circle, inscribed in the
+same Square. And so are we fallen into the knowledge sensible, and
+Experimentall 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:
+
+    [* =The Squaring of the Circle, Mechanically.=]
+
+* 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:
+
+    [* =To any Square geuen, to geue a Circle, equall.=]
+
+* If 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 proposition of the twelfth boke of
+_Euclide_, in my third Probleme there. Your diligence 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 Sphaere to the
+Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough
+one hole drawen, as the maner is) if you make a Square figure & way it:
+and then, describing theron, the Circle inscribed: & 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, by the way.
+
+    [Note Squaring of the Circle without knowledge of the
+    proportion betwene Circumference and Diameter.]
+
+That we may Square the Circle, without hauing knowledge of the
+proportion, of the Circumference 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 intricate. 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 (before this day,) had it for you, that I can tell of. And that
+is, _A Mechanicall Dubblyng of the Cube: &c._ Which may, thus, be done:
+
+    [To Dubble the Cube redily: by Art Mechanicall: depending
+    vppon Demonstration Mathematicall.]
+
++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 approche (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:+
+
+    [=I. D.=
+    =The 4. sides of this Pyramis must be 4. Isosceles
+    Triangles alike and aequall.=]
+
++Let 4. straight lines be drawen, in the inside of the Cone and Pyramis:
+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
+Pyramis: 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, as experience
+shall teach you.+
+
+    [=I. D.=
+    =* In all workinges with this Pyramis or Cone, Let their
+    Situations be in all Pointes and Conditions, alike,
+    or all one: while you are about one Worke. Els you
+    will erre.=]
+
++Then, * set your Cone or Pyramis, with the vertex downward,
+perpendicularly, in respect of the Base. (Though it be otherwayes, it
+hindreth nothyng.) So let them 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
+Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so
+proportion in respect of your Pyramis or Cone, that the Pyramis or Cone,
+will be hable to conteine the waight of them, in water, 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: take the waight therof exactly 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 before:
+
+    [=I. D.=
+    =* Consider well whan you must put your waters togyther:
+    and whan, you must empty your first water, out of your
+    Pyramis or Cone. Els you will erre.=]
+
+put that* 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 precisenes 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
+number belonging to the Radicall side, of the Cube, dubble to your
+Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it
+selfe, whether 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 Fundamentall 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.
+
+    [-> God be thanked for this Inuention, & the fruite ensuing.]
+
+For this, may I (with ioy) say, #EURE:KA, EURE:KA, EURE:KA#: thanking
+the holy and glorious Trinity: hauing greater cause therto, then
+
+    [* Vitruuius. Lib. 9. Cap. 3.]
+
+* _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
+Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it
+with water, precisely 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.
+
+    [* Note.]
+
+* And if the Cube, which you should Double, be neuer so great: you haue,
+thus, the proportion (in small) betwene 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
+Euclide.
+
+
+    [Note, as concerning the Sphaericall Superficies of the Water.]
+
+Note, that all this while, I forget not my first Proposition Staticall,
+here rehearsed: that, the Superficies 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 finding the swelling of the water aboue leuell._
+
+    [->]
+
+"Square the Semidiameter, 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 Rote, 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, *
+
+    [* Note.]
+
+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 moyst 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 commoditie.
+
+
+Thus, may you Double the Cube Mechanically, Treble it, and so forth, in
+any proportion.
+
+    [Note this Abridgement of Dubbling the Cube. &c.]
+
+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
+demonstration, 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 Fundamentall 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 marking of
+the lines, how the Water doth cut them, shall geue you the proportion
+betwen the Radicall sides, of any two Cubes, wherof the one is Double to
+the other: working as before I haue taught you:
+
+    [* -> Note.]
+
+* 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:
+
+    [To giue Cubes one to the other in any proportion,
+    Rationall or Irrationall.]
+
++you may (now) geue Cubes, one to the other, in any proportion geuen:
+Rationall or Irrationall+: on this maner. Make a hollow Parallelipipedon
+of Copper or Tinne: with one Base wanting, or open: as in our Cubike
+Coffen. From 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.
+
+    [* Emptying the first.]
+
+Poure that water * 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 second: 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
+assigned: were it Rationall or Irrationall.
+
+
+Thus, in sundry waies you may furnishe your selfe with such straunge and
+profitable matter: which, long hath bene wished for. And though it be
+Naturally done and Mechanically: yet hath it a good Demonstration
+Mathematicall.
+
+    [=The demonstrations of this Dubbling of the Cube, and of the
+rest.=]
+
+Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones,
+in the proportions 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, or Cone to Cone, be double,
+
+    [I. D.
+    = * Hereby, helpe your self to become a praecise practiser.
+    And so consider, how, nothing at all, you are hindred
+    (sensibly) by the Conuexitie of the water.=]
+
+then shall * 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 doubled: 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 treble, &c. as the Pyramids or
+Cones were, one to an other (as we haue proued) therfore, 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
+proportion, of the fower proportionall lines:
+
+    [= * By the 33. of the eleuenth booke of Euclide.=]
+
+therfore * 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 or Cone: Wherefore the Cube is to
+the Cube, as Pyramis is to Pyramis, or Cone is to Cone.
+
+    [I. D.
+    = * And your diligence in practise, can so (in waight of
+    water) performe it: Therefore, now, you are able to geue
+    good reason of your whole doing.=]
+
+But we * Suppose Pyramis 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 Parallelipipedon, it is euident, 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 proportion 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.
+
+    [* _Note this Corollary._]
+
+* This, my Demonstration is more generall, then onely in Square Pyramis
+or Cone: Consider well. Thus, haue I, both Mathematically and
+Mechanically, ben very long in wordes: yet (I trust) nothing tedious to
+them, who, to these thinges, are well affected. And verily I am forced
+(auoiding prolixitie) to omit sundry such things, easie to be practised:
+which to the Mathematicien, would be a great Threasure: and to the
+Mechanicien, no small gaine.
+
+    [* The great Commodities following of these new Inuentions.]
+
+* Now may you, +Betwene two lines giuen, finde two middle proportionals,
+in Continuall proportion: by the hollow Parallelipipedon, and the hollow
+Pyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre
+right lines may be found, proportionall in any proportion assigned, of
+which, shal be produced a Parallelipipedon, aequall to the
+Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36.
+proposition, of the 11. boke of _Euclide_. Now, all those thinges, which
+_Vitruuius_ in his Architecture, specified hable to be done, by dubbling
+of the Cube: Or, by finding of two middle proportionall lines, betwene
+two lines giuen, may easely be performed. Now, that Probleme, which I
+noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke
+of _Euclide_, is proued possible. Now, may any regular body, be
+Transformed into an other, &c. Now, any regular body: any Sphere, yea
+any Mixt Solid: and (that more is) Irregular Solides, may be made (in
+any proportion 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 Manneken 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 * Gunne, or little peece of ordinaunce, make an
+other, with the same _Symmetrie_ (in all pointes) as great, and as
+little, as you will. Marke that: and thinke on it. Infinitely, +may you
+apply this, so long sought for, and now so easily concluded: and
+withall, so willingly and frankly communicated to such, as faithfully
+deale with vertuous studies.+
+
+    [Such is the Fruite of the Mathematicall Sciences and Artes.]
+
+Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte:
+and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he
+can, by order, Descend, to frame Naturall thinges, to wonderfull vses:
+and when he list, retire home into his owne Centre: and there, prepare
+more Meanes, to Ascend or Descend by: and, all, to the glory of 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
+commeth next. For, wheras, it is so ample and wonderfull, that, an whole
+yeare long, one might finde fruitfull matter therin, to speake of: and
+also in practise, is a Threasure endeles: yet will I glanse ouer it,
+with wordes very few.
+
+
+This do I call +%Anthropographie%+. Which is an Art restored, and of my
+preferment to your Seruice. I pray you, thinke of it, as of one of the
+chief pointes, of Humane knowledge. Although it be, but now, first
+Confirmed, with this new name: yet the matter, hath from the beginning,
+ben in consideration of all perfect Philosophers. +Anthropographie, is
+the description of the Number, Measure, Waight, figure, Situation, and
+colour of euery diuerse thing, conteyned in the perfect body of MAN:
+with certain knowledge of the Symmetrie, figure, waight,
+Characterization, and due locall motion, of any parcell of the sayd
+body, assigned: and of Numbers, to the sayd parcell appertainyng.+ This,
+is the one part of the Definition, mete for this place: Sufficient to
+notifie, the particularitie, and excellency of the Arte: and why it is,
+here, ascribed to the Mathematicals. Yf the description of the heauenly
+part of the world, had a peculier Art, called _Astronomie:_ If the
+description of the earthly Globe, hath his peculier arte, called
+_Geographie_. If the Matching of both, hath his peculier Arte, called
+_Cosmographie:_ Which is the Description of the whole, and vniuersall
+frame of the world: Why should not the description of
+
+    [MAN is the Lesse World.]
+
+him, who is the Lesse world: and, from the beginning, called
+_Microcosmus_ (that is. _The Lesse World._) And for whose sake, and
+seruice, all bodily creatures els, were created: Who, also,
+participateth with Spirites, and Angels: and is made to the Image 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 Chyromantistes some. The
+Metaposcopistes, some: The excellent, _Albert Durer_, a good part: the
+Arte of Perspectiue, will somwhat, for the Eye, helpe forward:
+_Pythagoras_, _Hipocrates_, _Plato_, _Galenus_, _Meletius_, & many other
+(in certaine thinges) will be Contributaries. And farder, the Heauen,
+the Earth, and all other Creatures, will eche shew, and offer their
+Harmonious seruice, to fill vp, that, which wanteth hereof: and with
+your own Experience, concluding: you may Methodically register the
+whole, for the posteritie: Whereby, good profe will be had, of our
+Harmonious, and
+
+    [Micro Cosmus.]
+
+Microcosmicall constitution.
+
+    [* ->]
+
+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.
+
+    [* Lib. 3. Cap. 1.]
+
+Looke in * _Vitruuius_, whether I deale sincerely for your behoufe, or
+no. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Looke
+in the 27. and 28. Chapters, of the second booke, _De occulta
+Philosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther.
+Remember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ so
+long agoe 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 ignorance, and
+shame enough, come short of them Seuen also: and yet neuerthelesse they
+can not prescribe a certaine number of Artes: and in eche, certaine
+vnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly
+rise vp: and there was no such order taken, that,
+
+    [->]
+
+All Artes, should in one age, or in one land, or of one man, be made
+knowen to the world. Let vs embrace the giftes of 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 Irregular lines, at pleasure,
+may be drawen. These, and such like, are principall Conclusions of this
+Arte: and helpe forward many pleasant and profitable Mechanicall workes:
+
+    [Saw Milles.]
+
+As Milles, to Saw great and very long Deale bordes, no man being by.
+Such haue I seene in Germany: and in the Citie of Prage: in the kingdome
+of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all
+maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight,
+Spring, Man or Beast, moued. Take in your hand, _Agricola De re
+Metallica:_ and then shall you (in all Mines) perceaue, how great nede
+is, of Whele worke. By Wheles, straunge workes and incredible, are done:
+as will, in other Artes hereafter, appeare. A wonderfull example of
+farther possibilitie, and present commoditie, was sene in my time, in a
+certaine Instrument: which by the Inuenter and Artificer (before) was
+solde for xx. Talentes of Golde: and then had (by misfortune) receaued
+some iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend the
+same, and presented it vnto the Emperour _Charles_ the fifth.
+_Hieronymus 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
+Mathematicall, which demonstrateth the designing of all Spirall lines in
+Plaine, on Cylinder, Cone, Sphaere, Conoid, and Sphaeroid, and their
+properties appertayning.+ The vse hereof, in _Architecture_, and diuerse
+Instrumentes and Engines, is most necessary. For, in many thinges, the
+Skrue worketh the feate, which, els, could not be performed. By helpe
+hereof,
+
+    [* Atheneus Lib. 5. cap. 8.]
+
+it is * recorded, that, where all the power of the Citie of Syracusa,
+was not hable to moue a certaine 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.
+
+    [Proclus. Pag. 18.]
+
+Wherat, the King wondring: #Apo taute:s te:s he:meras, peri pantos,
+Archime:dei legonti pisteuteon#. _From this day, forward_ (said the
+King) _Credit ought to be giuen to Archimedes, what soeuer he sayth._
+
+
++%Pneumatithmie% demonstrateth by close hollow Geometricall Figures,
+(regular and irregular) the straunge properties (in motion or stay) of
+the Water, Ayre, Smoke, and Fire, in theyr continuitie, and as they are
+ioyned to the Elementes next them.+ This Arte, to the Naturall
+Philosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_
+is not in the world. And that, all Nature, abhorreth it so much: that,
+contrary to ordinary law, the Elementes will moue or stand. As, Water to
+ascend: rather then betwene him and Ayre, Space or place should be left,
+more then (naturally) that quantitie 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 Continuitie 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.
+
+    [To go to the bottom of the Sea without daunger.]
+
+Hereupon, two or three men together, by keping Ayre vnder a great
+Cauldron, and forcyng the same downe, orderly, may without harme descend
+to the Sea bottome: and continue there a tyme &c. Where, Note, how the
+thicker Element (as the Water) giueth place to the thynner (as, is the
+ayre:) and receiueth violence of the thinner, in maner. &c. Pumps and
+all maner of Bellowes, haue their ground of this Art: and many other
+straunge deuises. As, _Hydraulica_, Organes goyng by water. &c. Of this
+Feat, (called commonly _Pneumatica_,) goodly workes are extant, both in
+Greke, and Latin. With old and learned Schole men, it is called
+_Scientia de pleno & vacuo._
+
+
++%Menadrie%, is an Arte Mathematicall, which demonstrateth, how, aboue
+Natures vertue and power simple: Vertue and force may be multiplied: and
+so, to direct, to lift, to pull to, and to put or cast fro, any
+multiplied or simple, determined Vertue, Waight or Force: naturally,
+not, so, directible or moueable.+ Very much is this Art furdred by other
+Artes: as, in some pointes, by _Perspectiue_: in some, by _Statike_: in
+some, by _Trochilike_: and in other, by _Helicosophie_: and
+_Pneumatithmie_. By this Art, all Cranes, Gybbettes, & Ingines to lift
+vp, or to force any thing, any maner way, are ordred: and the certaine
+cause of their force, is knowne: As, the force which one man hath with
+the Duche waghen Racke: therwith, to set vp agayne, a mighty waghen
+laden, being ouerthrowne. The force of the Crossebow Racke, is
+certainly, here, demonstrated. The reason, why one man, doth with a
+leauer, lift that, which Sixe men, with their handes onely, could not,
+so easily do. By this Arte, in our common Cranes in London, where powre
+is to Crane vp, the waight of 2000. pound: by two Wheles more (by good
+order added) Arte concludeth, that there may be Craned vp 200000. pound
+waight &c. So well knew _Archimedes_ this Arte: that he alone, with his
+deuises and engynes, (twise or thrise) spoyled and discomfited the whole
+Army and Hoste of the Romaines, besieging _Syracusa_,
+
+    [=Plutarchus in Marco Marcello.=]
+
+_Marcus Marcellus the Consul_, being their Generall Capitaine.
+
+    [=Synesius in Epistolis.=]
+
+Such huge Stones, so many, with such force, and so farre, did he with
+his engynes hayle among them, out of the Citie.
+
+    [=Polybius.=]
+
+    [=Plinius.=]
+
+    [=Quintilianus.=]
+
+    [=T. Liuius.=]
+
+And by Sea likewise: though their Ships might come to the walls of
+_Syracusa_, yet hee vtterly confounded the Romaine Nauye. What with his
+mighty Stones hurlyng:
+
+    [=* Athenaeus.=]
+
+what with Pikes of * 18 fote long, made like shaftes: which he forced
+almost a quarter of a myle. What, with his catchyng hold of their Shyps,
+and hoysing them vp aboue the water, and suddenly letting them fall into
+the Sea againe:
+
+    [= * Galenus.=]
+
+    [=Anthemius.=]
+
+what with his * 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 _Proclus_,
+who so well had perceiued _Archimedes_ Arte of _Menadrie_, and had so
+well inuented of his owne, that with his Burning Glasses,
+
+    [Burning Glasses.]
+
+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 lightening) 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 Seuerus_ pardoned
+him, his life, after 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.
+
+    [Gunnes.]
+
+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, slight, and common: 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 entrance, is knowen)
+certaine way may be praescribed and gone: And how, any way aboue the
+Superficies of the earth designed, may vnder earth, at any depth
+limited, be kept: goyng alwayes, perpendicularly, 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 circumstances 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
+(before this) was of me published, _De Itinere Subterraneo_: The rest,
+be at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole,
+and for secrete passages 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 yet 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 Water 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 waters
+labour be farre, and great) I neede not: Seing, at hand (about vs) many
+expert men can sufficiently testifie, in effecte, the order: though the
+Demonstration of the Necessitie thereof, they know not: Nor yet, if they
+should be led, vp and downe, and about Mountaines, from the head of the
+Spring: and then, a place being 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
+water to ascend, eyther by _Tympane_, _Kettell mills_, _Skrue_,
+_Ctesibike_, or such like: in _Vitruuius_, _Agricola_, (and other,)
+fully, the maner may appeare. And so, thereby, also be most euident, how
+the Artes, of _Pneumatithmie_, _Helicosophie_, _Statike_, _Trochilike_,
+and _Menadrie_, come to the furniture of this, in Speculation, and to
+the Commoditie of the Common Wealth, in practise.
+
+
++%Horometrie%, is an Arte Mathematicall, which demonstrateth, how, at
+all times appointed, the precise vsuall denomination of time, may be
+knowen, for any place assigned.+ These wordes, are smoth and plaine
+easie 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, _Horologiographia_: 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:
+
+    [4. Reg. 20.]
+
+in * King _Achaz_ 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, Phaenomenall,
+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) theron, to describe (all maner of wayes) how, vsuall howers, may
+be (by the _Sunnes_ shadow) 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 Aequinoctiall Diall, &c. At all howers
+(the Sunne shining) the Signe and Degree 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. Bur 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 Water, 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 reformation,
+according to the heauenly Aequinoctiall Motion: besides the inaequalitie
+of their owne Operation. There remayneth (without parabolicall meaning
+herein) among the Philosophers,
+
+    [A perpetuall Motion.]
+
+a more excellent, more commodious, and more marueilous way, then all
+these: of hauing the motion of the Primouant (or first aequinoctiall
+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
+demonstrateth, how, the Intersection of all visuall Pyramides, made by
+any playne assigned, (the Centre, distance, and lightes, beyng
+determined) may be, by lynes, and due propre colours, represented.+ A
+notable Arte, is this: and would require a whole Volume, to declare the
+property thereof: and the Commodities ensuyng. Great skill of
+_Geometrie_, _Arithmetike_, _Perspectiue_, and _Anthropographie_, with
+many other particular Artes, hath the _Zographer_, 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 (commonly 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 foule, 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 Goldsmith, 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 commodious and meruailous: what shall be thought of _Zographie_,
+the Scholemaster of Picture, 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 Sisters germaine: and
+both, right profitable, in a Common wealth. and of _Sculpture_, aswell
+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_, _Cutter_, _Grauer_, _Founder_,
+or _Paynter (&c)_ know their Arte, to be commodious.
+
+
+    [An objection.]
+
++%Architecture%+, to many may seme not worthy, or not mete, 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 also did
+demonstratiuely procede in their faculty, by Number or Magnitude. First,
+
+    [The Answer.]
+
+you see, that I count, here, _Architecture_, among those _Artes
+Mathematicall_, which are Deriued from the Principals: and you know,
+that such, may deale with Naturall thinges, and sensible matter. Of
+which, "some draw nerer, to the Simple and absolute Mathematicall
+Speculation, then other do.
+
+    [->]
+
+And though, the _Architect_ 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_, _Arithmeticall_, _Opticall_, _Musicall_,
+_Astronomicall_, _Cosmographicall_" (& to be brief) by all the former
+Deriued _Artes Mathematicall_, and other Naturall Artes, hable to be
+confirmed 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 _Architectes_: the one, being _Vitruuius_, the Romaine: who
+did write ten bookes thereof, to the Emperour _Augustus_ (in whose daies
+our Heauenly Archemaster, was borne): and the other, _Leo Baptista
+Albertus_, a Florentine: who also published ten bookes therof.
+_Architectura_ (sayth _Vitruuius_) _est Scientia pluribus disciplinis &
+varijs eruditionibus ornata: cuius Iudicio probantur omnia, quae ab
+caeteris 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 Fabrica, & Ratiocinatione. &c. Ratiocinatio
+autem est, quae, res fabricatas, Solertia ac ratione proportionis,
+demonstrare at[que] explicare potest. +Architecture, groweth of Framing,
+and Reasoning. &c. Reasoning, is that, which of thinges framed, with
+forecast, and proportion: can make demonstration, and manifest
+declaration.+_ Againe. _Cum, in omnibus enim rebus, tum maxime etiam in
+Architectura, haec duo insunt: quod significatur, & quod significat.
+Significatur proposita res, de qua dicitur: hanc autem Significat
+Demonstratio, rationibus doctrinarum explicata. +Forasmuch as, in all
+thinges: therefore chiefly in Architecture, these two thinges are: the
+thing signified: and that which signifieth. The thing propounded,
+whereof we speake, is the thing Signified. But Demonstration, expressed
+with the reasons of diuerse doctrines, doth signifie 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 Iurisperitorum nouerit: Astrologiam,
+Caeli[que] rationes cognitas habeat. +An Architect+_ (sayth he) +_ought
+to vnderstand Languages, to be skilfull of Painting, well instructed in
+Geometrie, not ignorant of Perspectiue, furnished with Arithmetike, haue
+knowledge of many histories, and diligently haue heard Philosophers,
+haue skill of Musike, not ignorant of Physike, know the aunsweres of
+Lawyers, and haue Astronomie, and the courses Caelestiall, in good
+knowledge._+ He geueth reason, orderly, wherefore all these Artes,
+Doctrines, and Instructions, are requisite in an excellent _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 heauen, 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 #phusiologia#) Philosophie doth make declaration.
+Which, it is necessary, for an Architect, with diligence to haue
+learned: because it hath many and diuers naturall questions: as
+specially, in Aqueductes. For in their courses, leadinges about, in the
+leuell ground, and in the mountinges, the naturall 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 Ctesibius, or
+Archimedes 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 Balistes, 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 #e:cheia#) are ordred according
+to Musicall Symphonies & Harmonies: being distributed in y^e Circuites,
+by Diatessaron, Diapente, and Diapason. That the conuenient voyce, of
+the players sound, when 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
+knowen 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
+Horologies. Seyng therfore this ample Science, is garnished, beautified
+and stored, with so many and sundry skils and knowledges: I thinke, that
+none can iustly account 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 Tabernacle of Architecture. &c. And to
+whom Nature hath giuen such quicke Circumspection, sharpnes of witt, and
+Memorie, that they may be very absolutely skillfull in Geometrie,
+Astronomie, Musike, and the rest of the Artes Mathematicall: Such,
+surmount and passe the callyng, and state, of Architectes:
+
+    [A Mathematicien.]
+
+and are become Mathematiciens. &c. And they are found, seldome. As, in
+tymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes:
+Apollonius Pergaeus: Eratosthenes Cyreneus: Archimedes, and Scopas,
+Syracusians. Who also, left to theyr posteritie, many Engines and
+Gnomonicall workes: by numbers and naturall meanes, inuented and
+declared._+
+
+
+Thus much, and the same wordes (in sense) in one onely Chapter of this
+Incomparable _Architect Vitruuius_, shall you finde. And if you should,
+but take his boke in your hand, and slightly loke thorough it, you would
+say straight way:
+
+    [Vitruuius.]
+
+This is _Geometrie_, _Arithmetike_, _Astronomie_, _Musike_,
+_Anthropographie_, _Hydragogie_, _Horometrie_. _&c_. and (to conclude)
+the Storehouse of all workmanship. 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 allowed an Architect. For,
+I will not bryng in place a Carpenter: as though you might Compare him
+to the Chief Masters of other Artes. For the hand of the Carpenter, is
+the Architectes Instrument._+
+
+    [VVho is an Architect.]
+
++_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 bodyes,
+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, consisteth in Lineamentes, and in Framyng. And
+the whole power and skill of Lineamentes, tendeth to this: that the
+right and absolute way may be had, of Coaptyng and ioyning Lines and
+angles: by which, the face of the buildyng 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 number:
+a worthy maner, and a semely order: that, so, y^e whole forme and figure
+of the buildyng, may rest in the very Lineamentes. &c. And we may
+prescribe in mynde and imagination the whole formes, *
+
+    [* The Immaterialitie of perfect Architecture.]
+
+all material stuffe beyng secluded. Which point we shall atteyne, by
+Notyng and forepointyng the angles, and lines, by a sure and certaine
+direction and connexion. Seyng then, these thinges, are thus:_+
+
+    [What, Lineament is.]
+
++_Lineamente, shalbe the certaine 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:
+
+    [Note.]
+
+by certaine order, number, forme, figure, and _Symmetrie_ mentall:" all
+naturall & sensible stuffe set a part. Now, then, it is euident, (Gentle
+reader) how aptely and worthely, I haue preferred _Architecture_, to be
+bred and fostered vp in the Dominion of the pereles _Princesse_,
+_Mathematica_: and to be a naturall Subiect of hers. And the name of
+_Architecture_, 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, nor
+Builder: nor, separately, any Artificer: but the Hed, the Prouost, the
+Directer, and Iudge of all Artificiall workes, and all Artificers. For,
+the true _Architect_, is hable to teach, Demonstrate, distribute,
+describe, 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) atteyne thereto: 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 Definitions 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) allowed to absolute
+_Architecture_. Vnder the Direction of this Arte, are thre principall,
+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, will 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 Direction, & in the shortest time, a sufficient Ship,
+betwene any two places (in passage Nauigable,) assigned: may be
+conducted: 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 _Horometrie_. 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 Astronomers Ryng, The Astronomers staffe, The
+Astrolabe vniuersall. An Hydrographicall Globe. Charts Hydrographicall,
+true, (not with parallell Meridians). The Common Sea Compas: The Compas
+of variacion: The Proportionall, and Paradoxall Compasses
+
+    [Anno. 1559.]
+
+(of me Inuented, for our two Moscouy Master Pilotes, at the request of
+the Company) Clockes with spryng: houre, halfe houre, and three houre
+Sandglasses: & sundry other Instrumentes: 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
+Longitude & 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, appertayneth expert meanes, to be certified
+euer, of the Ships way. &c. And by foreseing 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,)
+_Mutationes, opportunitates[que] temporum presentire, non minus rei
+militari, quam Agriculturae, Nauigationi[que] conuenit. +To foresee the
+alterations and opportunities of tymes, is conuenient, no lesse to the
+Art of Warre, then to Husbandry and Nauigation.+_ And besides such
+cunnyng meanes, more euident tokens in Sonne and Mone, ought of hym to
+be knowen: such as (the Philosophicall Poete) _Virgilius_ teacheth, in
+hys _Georgikes_. Where he sayth,
+
+    [Sidenote: Georgic. 1.]
+
+  _Sol quo[que] & exoriens & quum se condet in vndas,
+  Signa dabit, Solem certissima signa sequuntur. &c.
+    -------- Nam saepe videmus,
+  Ipsius in vultu varios errare colores.
+  Caeruleus, pluuiam denunciat, igneus Euros.
+  Sin maculae incipient rutilo immiscerier igni,
+  Omnia tum pariter vento, nimbis[que] videbis
+  Feruere: non illa quisquam me nocte per altum
+  Ire, ne[que] a terra moueat conuellere funem. &c.
+  Sol tibi signa dabit. Solem quis dicere falsum
+  Audeat? -------- &c._
+
+And so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and
+Beastes, and of many thynges els, a certaine Sympathicall forewarnyng
+may be had: sometymes to great pleasure and proffit, both on Sea and
+Land. Sufficiently, for my present purpose, it doth appeare, by the
+premisses, how _Mathematicall_, the _Arte_ of _Nauigation_, is: and how
+it nedeth and also vseth other _Mathematicall Artes_: And now, if I
+would go about to speake of the manifold Commodities, commyng to this
+Land, and others, by Shypps and _Nauigation_, you might thinke, that I
+catch at occasions, to vse many wordes, where no nede is.
+
+
+Yet, this one thyng may I, (iustly) say. In _Nauigation_, none ought to
+haue greater care, to be skillfull, then our English Pylotes. And
+perchaunce, Some, would more attempt: And other Some, more willingly
+would be aydyng, it they wist certainely, What Priuiledge, God had
+endued this Iland with, by reason of Situation, most commodious for
+_Nauigation_, to Places most Famous & Riche. And though,
+
+    [* Anno. 1567 S. H. G.]
+
+(of * Late) a young Gentleman, a Courragious Capitaine, was in a great
+readynes, with good hope, and great causes of persuasion, to haue
+ventured, for a Discouerye, (either _Westerly_, by _Cape de Paramantia_:
+or _Esterly_, aboue _Noua Zemla_, and the _Cyremisses_) and was, at the
+very nere tyme of Attemptyng, called and employed otherwise (both then,
+and since,) in great good seruice to his Countrey, as the Irish Rebels
+haue * tasted:
+
+    [* Anno. 1569]
+
+Yet, I say, (though the same Gentleman, doo not hereafter, deale
+therewith) Some one, or other, should listen to the Matter: and by good
+aduise, and discrete Circumspection, by little, and little, wynne to the
+sufficient knowledge of that +Trade+ and +Voyage+: Which, now, I would
+be sory, (through Carelesnesse, want of Skill, and Courrage,) should
+remayne Vnknowne and vnheard of. Seyng, also, we are herein, halfe
+Challenged, by the learned, by halfe request, published. Therof, verely,
+might grow Commoditye, to this Land chiefly, and to the rest of the
+Christen Common wealth, farre passing all riches and worldly Threasure.
+
+
++%Thaumaturgike%, is that Art Mathematicall, which giueth certaine order
+to make straunge workes, of the sense to be perceiued, and of men
+greatly to be wondred at.+ By sundry meanes, this _Wonder-worke_ is
+wrought. Some, by _Pneumatithmie_. As the workes of _Ctesibius_ and
+_Hero_, Some by waight. wherof _Timaeus_ speaketh. Some, by Stringes
+strayned, or Springs, therwith Imitating liuely Motions. Some, by other
+meanes, as the Images of Mercurie: and the brasen hed, made by _Albertus
+Magnus_, which dyd seme to speake. _Boethius_ was excellent in these
+feates. To whom, _Cassiodorus_ writyng, sayth. +_Your purpose 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,
+
+    [* Anno. 1551]
+
+* 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 them selues: misdeaming of the meanes. Therfore sayd
+_Claudius Caelestinus_.
+
+    [De his quae Mundo mirabiliter eueniunt. cap. 8.]
+
+_Hodie magnae literaturae viros & magna reputationis videmus, opera
+quedam quasi miranda, supra Naturan putare: de quibus in Perspectiua
+doctus causam faciliter reddidisset._ That is. +_Now a dayes, we see
+some men, yea of great learnyng and reputation, to Iudge certain workes
+as meruaylous, 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.
+
+    [Tusc. 1.]
+
+Which is very straunge to thinke on. +_For when Archimedes_+ (sayth he)
++_did fasten in a Sphaere, the mouynges of the Sonne, Mone, and of the
+fiue other Planets, he did, as the God, which (in Timaeus of Plato) did
+make the world. That, one turnyng, should rule motions most vnlike in
+slownes, and swiftnes._+ But a greater cause of meruayling we haue by
+_Claudianus_ report hereof. Who affirmeth this _Archimedes worke_, to
+haue ben of Glasse. And discourseth of it more at large: which I omit.
+The Doue of wood, which the _Mathematicien Archytas_ did make to flye,
+is by _Agellius_ spoken of. Of _Daedalus_ straunge Images, _Plato_
+reporteth. _Homere_ of _Vulcans Selfmouers_, (by secret wheles) leaueth
+in writyng. _Aristotle_, in hys _Politikes_, of both, maketh mention.
+Meruaylous was the workemanshyp, of late dayes, performed by good skill
+of _Trochilike. &c._ For in Noremberge, A flye of Iern, beyng let out of
+the Artificers hand, did (as it were) fly about by the gestes, at the
+table, and at length, as though it were weary, retourne to his masters
+hand agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of
+the same Towne, a mighty way, and that a loft in the Ayre, toward the
+Emperour comming thether: and followed hym, beyng come to the gate of
+the towne. *
+
+    [* ->]
+
+Thus, you see, what, Arte Mathematicall can performe, when Skill, will,
+Industry, and Hability, are duely applyed to profe.
+
+
+    [A Digression.]
+
+And for these, and such like marueilous Actes and Feates, Naturally,
+Mathematically, and Mechanically, wrought and contriued:
+
+    [Apologeticall.]
+
+ought any honest Student, and Modest Christian Philosopher, be counted,
+& called a +Coniurer+? Shall the folly of Idiotes, 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 pure 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 man 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 peruerter 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 trauailing,
+onely for good learninges sake: And that, in all maner of wethers: in
+all maner 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 bereuing life: (as God knoweth): with
+lodginges, oft times, to small ease: and somtime to lesse securitie. And
+for much more (then all this) done & suffred, for Learning 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 aduise of Lady Philosophie, & Queene Theologie): but at
+length, to haue catched, and drawen vp, * a Frog?
+
+    [* A prouerb. Fayre fisht, and caught 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, them selues:
+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 giltlesse: 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 vnnaturall 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 iudgement) am I become,
+worse, then when I began? Worse, then 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
+Wisedome? 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 dungeon 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 Aeternall: and
+to cleaue vnto the Author of Death euerlasting? that Murderous Tyrant,
+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 Iniuries, that I do: or rather, greater. Pacient
+_Socrates_, his _Apologie_ will testifie: _Apuleius_ 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 Malicious Ignorant against him. _Ioannes Trithemius_, his _Apologie_
+will specifie, how he had occasion to make publike Protestation: as well
+by reason of the Rude Simple: as also, in respect of such, as were
+counted to be of the wisest sort of men. "Many could I recite: But I
+deferre the precise and determined handling of this matter: being loth
+to detect the Folly & Mallice of my Natiue Countrey men. *
+
+    [* ->]
+
+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, chiefly. The first, I may name, _Vaine pratling busie
+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 or two, and so returne to my Praeface.
+
+    [1.]
+
+_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.
+
+    [2.]
+
+_Fonde 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.
+
+    [3.]
+
+_Imperfectly zelous_, to you, 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 Rhetorike, 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 detractione parcite linguae. +Cause thy toung to refraine from
+euill. Refraine your toung from slaunder.+_ Though your tounges be
+sharpned, Serpent like, & Adders poyson lye in your lippes:
+
+    [Psal. 140.]
+
+yet take heede, and thinke, betimes, with your selfe, _Vir linguosus 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
+Mathematiciens, 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 Magnus_, layeth _Moses_ and _Daniel_,
+before the eyes of those, which count all such Studies Philosophicall
+(as mine hath bene) to be vngodly, or vnprofitable. Waye well
+_S. Stephen_ his witnesse of _Moses_.
+
+    [Act. 7. C.]
+
+_Eruditus est Moses omni Sapientia Aegyptiorum: & erat potens in verbis
+& operibus suis. +Moses was instructed in all maner of wisedome 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 _Magicien_. 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 wonders, 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 mention of _Moses_ Philosophie: with good liking of it: And
+_Basilius Magnus_ also, auoucheth it, to haue bene to _Moses_ profitable
+(and therefore, I say, to the Church of God, necessary). But as
+concerning _Moses_ wonders, done before King _Pharao_: God, him selfe,
+sayd: _Vide vt omnia ostenta, quae posui in manu tua, 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_,
+
+    [Lib. 30. Cap. 1.]
+
+of vayne fraudulent _Magike_, saying: _Est & alia Magices Factio,
+a Mose, Iamne, & Iotape, Iudaeis pendens: sed multis millibus annorum
+post Zoroastrem. &c._
+
+    [1.]
+
+Let all such, therefore, who, in Iudgement and Skill of Philosophie, are
+farre Inferior to _Plinie_, "take good heede, least they ouershoote them
+selues rashly," in
+
+    [->]
+
+Iudging of _Philosophers straunge Actes_: and the Meanes, how they are
+done.
+
+    [2.]
+
+But, much more, ought they 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.
+
+    [3.]
+
+And (to conclude) most of all, let them be ashamed of Man, and afraide
+of the dreadfull and Iuste Iudge: both Folishly or Maliciously to
+deuise: and then, deuilishly to father their new fond Monsters 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 experience sensible, all worthy conclusions
+by all the Artes Mathematicall purposed, & by true Naturall Philosophie
+concluded: & both addeth to them a farder scope, in the termes of the
+same Artes, & also by hys propre Method, and in peculier termes,
+procedeth, with helpe of the foresayd Artes, to the performance of
+complet Experiences, which of no particular Art, are hable (Formally) to
+be challenged.+ If you remember, how we considered _Architecture_, in
+respect of all common handworkes: some light may you haue, therby, to
+vnderstand the Souerainty 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 Science_+. _Nicolaus Cusanus_ termeth it so, in hys
+_Experimentes Statikall_, And an other _Philosopher_,
+
+    [R. B.]
+
+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 _Experience_ complete and
+absolute: and other Artes, with their Argumentes, and Demonstrations,
+persuade: and in wordes, proue very well their Conclusions. *
+
+    [->]
+
+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 Opticall 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 fantasticall Imagination: as some Sophister, might, _Cum suis
+Insolubilibus_, make a florish: 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 discharged my
+Duety toward God: for my small Talent, at hys most mercyfull handes
+receiued. 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) requested me, (an old forworne Mathematicien) to take pen
+in hand: (through the confidence they reposed in my long experience: and
+tryed sincerity) for the declaryng and reportyng somewhat, of the frute
+and commodity, by the +Artes Mathematicall, to be atteyned vnto+: euen
+they, Sore agaynst their willes, are forced, for sundry causes, to
+satisfie the workemans request, in endyng forthwith: He, so feareth
+this, so new an attempt, & so costly: And in matter so slenderly
+(hetherto) 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.
+
+
+    [2.]
+
+Neither are their Studies, hereby, any whit hindred. No more, then the
+Italian _Vniuersities_, as _Academia Bononiensis_, _Ferrariensis_,
+_Florentina_, _Mediolanensis_, _Patauina_, _Papiensis_, _Perusina_,
+_Pisana_, _Romana_, _Senensis_, or any one of them, finde them selues,
+any deale, disgraced, or their Studies any thing hindred, by _Frater
+Lucas de Burgo_, or by _Nicolaus Tartalea_, who in vulgar Italian
+language, haue published, not onely _Euclides Geometrie_, but of
+_Archimedes_ somewhat: and in Arithmetike and Practicall Geometrie, very
+large volumes, all in their vulgar speche. Nor in Germany haue the
+famous _Vniuersities_, any thing bene discontent with _Albertus
+Durerus_, his Geometricall Institutions in Dutch: or with _Gulielmus
+Xylander_, his learned translation of the first sixe bookes of
+_Euclide_, out of the Greke into the high Dutch. Nor with _Gualterus H.
+Riffius_, his Geometricall Volume: very diligently translated into the
+high Dutch tounge, and published. Nor yet the _Vniuersities_ of Spaine,
+or Portugall, thinke their reputation to be decayed: or suppose any
+their Studies to be hindred by the Excellent _P. Nonnius_, his
+Mathematicall workes, in vulgare speche by him put forth. Haue you not,
+likewise, in the French tounge, the whole Mathematicall Quadriuie? and
+yet neither Paris, Orleance, or any of the other Vniuersities of
+Fraunce, at any time, with the Translaters, or Publishers offended: or
+any mans Studie thereby hindred?
+
+
+    [3.]
+
+And surely, the Common and Vulgar Scholer (much more, the Gramarian)
+before his comming to the _Vniuersitie_, shall (or may) be, now
+(according to _Plato_ his Counsell) sufficiently instructed in
+_Arithmetike_ and _Geometrie_, for the better and easier learning of all
+maner of _Philosophie_, _Academicall_, or _Peripateticall_. And by that
+meanes, goe more cherefully, more skilfully, and spedily forwarde, in
+his Studies, there to be learned. And, so, in lesse time, profite more,
+then (otherwise) he should, or could do.
+
+
+    [4.]
+
+Also many good and pregnant Englishe wittes, of young Gentlemen, and of
+other, who neuer intend to meddle with the profound search and Studie of
+Philosophie (in the _Vniuersities_ to be learned) may neuerthelesse,
+now, with more ease and libertie, haue good occasion, vertuously to
+occupie the sharpnesse of their wittes: where, els (perchance)
+otherwise, they would in fond exercises, spend (or rather leese) their
+time: neither seruing God: nor furdering the Weale, common or priuate.
+
+
+    [5.]
+
+And great Comfort, with good hope, may the _Vniuersities_ haue, by
+reason of 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 Commodities are ensuing (as I
+haue 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: or els, not so fully credited:
+"Well, may all men coniecture, that farre greater ayde, and better
+furniture, to winne to the Perfection of all Philosophie,
+
+    [Vniuersities.]
+
+may in the Vniuersities be had: being the Storehouses & Threasory of all
+Sciences,
+
+    [->]
+
+and all Artes, necessary for the best, and most noble State of Common
+Wealthes."
+
+    [6.]
+
+Besides this, how many a Common Artificer, is there, in these Realmes of
+England and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who,
+with their owne Skill and experience, already had, will be hable (by
+these good helpes and informations) to finde out, and deuise, new
+workes, straunge Engines, and Instrumentes: for sundry purposes in the
+Common Wealth? or for priuate pleasure? and for the better maintayning
+of their owne estate? I will not (therefore) fight against myne owne
+shadowe. For, no man (I am sure) will open his mouth against this
+Enterprise. No man (I say) who either hath Charitie toward his brother
+(and would be glad of his furtherance in vertuous knowledge): or that
+hath any care & zeale for the bettering of the Common state of this
+Realme. Neither any, that make accompt, what the wiser sort of men (Sage
+and Stayed) do thinke of them. To none (therefore) will I make any
+_Apologie,_ for a vertuous acte doing: and for commending, 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
+Goodnes, by his Powre, and in his wisedome,
+
+    [->]
+
+hath Created all thynges, in Number, Waight, and Measure_+: So, to vs,
+of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne the
+sufficient and necessary knowledge of the foresayd hys 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 matter (in my Mynde) as I determined: hopyng of
+conuenient laysure: Yet. if vertuous zeale, and honest Intent 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 pleasure you. And that you
+may the easier perceiue, and better remember, the principall pointes,
+whereof my Preface treateth,
+
+    [The Ground platt of this Praeface in a Table.]
+
+I will giue you the +Groundplatt+ of my whole discourse, in a Table
+annexed: from the first to the last, somewhat Methodically 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 pleasure you), Consider the rockish huge
+      mountaines, and the perilous vnbeaten wayes, which
+        (both night and day, for the while) it hath
+          toyled 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 House
+                     At Mortlake._
+
+                      _Anno. 1570. February. 9._
+
+
+  [Decoration]
+
+
+
+
+  [Transcriber's Note:
+
+  The "Groundplat" was printed in the form of a stemma, or tree, on an
+  oversized fold-out page. The layout was impossible to reproduce for
+  this e-text, so the information has been rearranged in nested-list
+  form. Size markings (see note at beginning of e-text) are relative
+  within each paragraph.]
+
+
+      _J. DEE_
+
+  +%Here haue you (according to my promisse) the Groundplat of%+
+    +my MATHEMATICALL Praeface: annexed to _Euclide_ (now first)+
+       published in our Englishe tounge. An. 1570. Febr. 3.
+
+
++%Sciences, and Artes Mathematicall,% are, either+
+
+  +%Principall,% which are two, onely,+
+
+    +Arithmetike.+
+
+      +%Simple%+, Which dealeth with Numbers onely: and demonstrateth
+        all their properties and appertenances: where, an Vnit, is
+        Indiuisible.
+      +%Mixt%+, Which with aide of Geometrie principall, demonstrateth
+        some Arithmeticall Conclusion, or Purpose.
+
+    +Geometrie.+
+
+      +%Simple%+, Which dealeth with Magnitudes, onely: and
+        demonstrateth all their properties, passions, and
+        appertenances: whose Point, is Indiuisible.
+      +%Mixt%+, Which with aide of Arithmetike principall,
+        demonstrateth some Geometricall purpose, as +EVCLIDES
+        ELEMENTES+.
+
+    +%The vse% whereof, is either,+
+
+      In thinges Supernaturall, aeternall, & Diuine: By Application,
+        _Ascending_.
+      In thinges Mathematicall: without farther Application.
+      In thinges Naturall: both Substantiall, & Accidentall, Visible,
+        & Inuisible. &c. By Application: _Descending_.
+
+    The like Vses and Applications are, (though in a degree lower) in
+    the +Artes Mathematicall Deriuatiue+.
+
+  +%Deriuatiue% from the Principalls: of which, some haue+
+
+    +%The names of% the Principalls: as,+
+
+      +_Arithmetike_, vulgar: which considereth+
+
+        --Arithmetike of most vsuall whole numbers: And of Fractions to
+        them appertaining.
+        --Arithmetike of Proportions.
+        --Arithmetike Circular.
+        --Arithmetike of Radicall Numbers: Simple, Compound, Mixt: And
+of
+        their Fractions.
+        --Arithmetike of Cossike Numbers: with their Fractions: And the
+        great Arte of Algiebar.
+
+      +_Geometrie_, vulgar: which teacheth Measuring+
+
+        +%At hand%+
+
+          All Lengthes.--+Mecometrie.+
+          All Plaines: As, Land, Borde, Glasse, &c.--+Embadometrie.+
+          All Solids: As, Timber, Stone, Vessels, &c.--+Stereometrie.+
+
+        +%With distance%+ from the thing Measured, as,
+
+          +%How farre%+, from the Measurer, any thing is: of him
+            sene, on Land or Water: called +Apomecometrie+.
+          +%How high or deepe%+, from the leuell of the Measurers
+            standing, any thing is: Seene of hym, on Land or Water:
+            called +Hypsometrie+.
+          +%How broad%+, a thing is, which is in the Measurers view:
+            so it be situated on Land or Water: called
+            +Platometrie+.
+
+        +%Of which% are growen the Feates & Artes of+
+
+          +Geodesie+: more cunningly to Measure and Suruey Landes,
+            Woods, Waters. &c.
+          +Geographie.+
+          +Chorographie.+
+          +Hydrographie.+
+          +Stratarithmetrie.+
+
+    +%Propre names% as+,
+
+      +Perspectiue,+--Which demonstrateth the maners and properties
+      of all Radiations: Directe, Broken, and Reflected.
+
+      +Astronomie,+--Which demonstrateth the Distances, Magnitudes,
+      and all Naturall motions, Apparences, and Passions, proper to
+      the Planets and fixed Starres: for any time, past, present, and
+      to come: in respecte of a certaine Horizon, or without respecte
+      of any Horizon.
+
+      +Musike,+--Which demonstrateth by reason, and teacheth by
+      sense, perfectly to iudge and order the diuersitie of Soundes,
+      hie or low.
+
+      +Cosmographie,+--Which, wholy and perfectly maketh description
+      of the Heauenlym and also Elementall part of the World: and of
+      these partes, maketh homologall application, and mutuall
+      collation necessary.
+
+      +Astrologie,+--Which reasonably demonstrateth the operations
+      and effectes of the naturall beames of light, and secrete
+      Influence of the Planets, and fixed Starres, in euery Element
+      and Elementall body: at all times, in any Horizon assigned.
+
+      +Statike,+--Which demonstrateth the causes of heauines and
+      lightnes of all thinges: and of the motions and properties to
+      heauines and lightnes belonging.
+
+      +Anthropographie,+ Which describeth the Number, Measure, Waight,
+      Figure, Situation, and colour of euery diuers thing contained in
+      the perfecte body of MAN: and geueth certaine knowledge of the
+      Figure, Symmetrie, Waight, Characterization, & due Locall motion
+      of any percell of the said body assigned: and of numbers to the
+      said percell appertaining.
+
+      +Trochilike,+--Which demonstrateth the properties of all
+      Circular motions: Simple and Compound.
+
+      +Helicosophie,+--Which demonstrateth the designing of all
+      Spirall lines: in Plaine, on Cylinder, Cone, Sphaere, Conoid, and
+      Sphaeroid: and their properties.
+
+      +Pneumatithmie,+--Which demonstrateth by close hollow
+      Geometricall figures (Regular and Irregular) the straunge
+      properties (in motion or stay) of the Water, Ayre, Smoke, and
+      Fire, in their Continuitie, and as they are ioyned to the
+      Elementes next them.
+
+      +Menadrie,+--Which demonstrateth, how, aboue Natures Vertue,
+      and power simple: Vertue and force, may be multiplied: and so
+      to directe, to lift, to pull to, and to put or cast fro, any
+      multiplied, or simple determined Vertue, Waight, or Force:
+      naturally, not, so, directible, or moueable.
+
+      +Hypogeiodie,+--Which demonstrateth, 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 respecte of the sayd
+      entrance, is knowen) certaine way, may be prescribed and gone,
+      &c.
+
+      +Hydragogie,+--Which demonstrateth the possible leading of
+      water by Natures law, and by artificiall helpe, from any head
+      (being Spring, standing, or running water) to any other place
+      assigned.
+
+      +Horometrie,+--Which demonstrateth, how, at all times
+      appointed, the precise, vsuall denomination of time, may be
+      knowen, for any place assigned.
+
+      +Zographie,+--Which demonstrateth and teacheth, how, the
+      Intersection of all visuall Pyramids, made by any plaine
+      assigned (the Center, distance, and lightes being determined)
+      may be, by lines, and proper colours represented.
+
+      +Architecture,+--Which is a Science garnished with many
+      doctrines, and diuers Instructions: by whose iudgement, all
+      workes by other workmen finished, are iudged.
+
+      +Nauigation,+--Which demonstrateth, how, by the Shortest good
+      way, by the aptest direction, and in the shortest time:
+      a sufficient Shippe, betwene any two places (in passage
+      nauigable) assigned, may be conducted: and in all stormes and
+      naturall disturbances chauncing, how to vse the best possible
+      meanes, to recouer the place first assigned.
+
+      +Thaumaturgike,+--Which geueth certaine order to make straunge
+      workes, of the sense to be perceiued: and of men greatly to be
+      wondred at.
+
+      +Archemastrie,+--Which teacheth to bring to actuall experience
+      sensible, all worthy conclusions, by all the Artes Mathematicall
+      purposed: and by true Naturall philosophie, concluded: And both
+      addeth to them a farder Scope, in the termes of the same Artes:
+      and also, by his proper Method, and in peculiar termes,
+      procedeth, with helpe of the forsayd Artes, to the performance
+      of complete Experiences: which, of no particular Arte, are hable
+      (Formally) to be challenged.
+
+
++Imprinted by _Iohn Day_.+
+
+An. 1570. Feb. 25.
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+Errors and Anomalies:
+
+Unless otherwise noted, spelling and punctuation are unchanged.
+Errors are listed below, with the original form, if changed, shown in
+[brackets]. Unusual words include "fatch" (probably used as a variant
+of "fetch") and the mathematical terms "sexagene" and "sexagesme".
+
+  How, worldly goods: how, worldly dignitie
+    [_"o" in second "worldly" invisible_]
+  his most diligent hearers (so infinitely mought  [hearers) so]
+  the boundes, and duety of an Hydrographer  [Hydographer]
+  of the Grekes it is called _Eteromekes_
+    [_text unchanged: correct form is "Heteromekes"_]
+  #to hoti#  [_Greek printed with incorrect accent_]
+  in our worldly affaires  [wordly]
+  fall to worke.[[*]].
+    [_In this place only, the text has an oversized asterisk symbol._]
+  _Emptying the first._  [Emptyting]
+  #Apo taute:s te:s he:meras, peri pantos, Archime:dei legonti
+    pisteuteon#  [#he:me:ras ... pisteuteom#]
+  of the suddeyne  [snddeyne]
+  that the right and absolute way may be had  [he had]
+  Georgic I: [_The quoted segments, each ending in "&c.", are
+    438-439; 451-457; 463-464._]
+
+Additional Notes:
+
+  The Greek letter #e:# (eta) was consistently printed as if it were the
+    #ou# ligature.
+  The Latin "-que" was written as an abbreviation resembling "-q';".
+    It is shown here as [que].
+
+  Mathematical symbols seen in the section accompanying the diagrams
+  could not be reproduced. The following substitutions were made:
+    --The curly "P" used for "Pounds" is shown as {P}.
+    --The "potestas" symbol, used to represent "x" (the unknown),
+    is shown as {x}.
+    --All roots were expressed as the "root" sign combined with
+    symbols for the power of 2 (doubled for power of 4, or fourth root)
+    and 3. They are shown here as [2rt] [3rt] [4rt].
+
+Euclid:
+
+The following Propositions were identified by number.
+
+6.12: (How) to find a fourth (line) proportional to three given straight
+lines.
+
+11.34: In equal parallelepipedal solids the bases are reciprocally
+proportional to the heights; and those parallelepipedal solids in which
+the bases are reciprocally proportional to the heights are equal.
+
+11.36: If three straight lines are proportional, then the
+parallelepipedal solid formed out of the three equals the
+parallelepipedal solid on the mean which is equilateral, but equiangular
+with the aforesaid solid.
+
+12.1: Similar polygons inscribed in circles are to one another as the
+squares on their diameters.
+
+12.2: Circles are to one another as the squares on their diameters.
+
+12.18 ("last"): Spheres are to one another in triplicate ratio of their
+respective diameters.
+
+
+
+
+
+
+
+End of the Project Gutenberg EBook of The Mathematicall Praeface to Elements
+of Geometrie of Euclid of Megara, by John Dee
+
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+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
+<html>
+<head>
+<title>The Groundplat</title>
+<meta http-equiv = "Content-Type" content = "text/html; charset=UTF-8">
+
+
+<style type = "text/css">
+
+body {margin-left: 10%; margin-right: 10%;}
+
+hr {width: 80%; margin-top: 1em; margin-bottom: 1em;}
+
+p {margin-top: .25em; margin-bottom: .25em; line-height: normal;}
+p.center {text-align: center;}
+
+
+/* tables */
+
+table {margin-left: auto; margin-right: auto; margin-top: 1em; margin-bottom: 1em;}
+
+table.groundplat {font-size: 87%; border-collapse: collapse;}
+
+td {vertical-align: middle; text-align: left; padding: .2em;
+/*border: 1px dotted #999;*/}
+
+td.number {text-align: right;}
+td.lower {vertical-align: bottom;}
+td.upper {vertical-align: top;}
+
+td.bracket {padding: .25em .1em;}
+
+td.bracket.leftline {padding-left: 1em; background-repeat: repeat-y; background-position: center left; background-image: url("../images/leftline.gif");}
+td.bracket.rightline {padding-right: 1em; background-repeat: repeat-y; background-position: center right; background-image: url("../images/rightline.gif");}
+
+td.bracket.lefttop {padding-left: 1em; background-repeat: no-repeat; background-position: bottom left; background-image: url("../images/topleft.gif");}
+td.bracket.righttop {padding-right: 1em; background-repeat: no-repeat; background-position: bottom right; background-image: url("../images/topright.gif");}
+
+td.bracket.leftbottom {padding-left: 1em; background-repeat: no-repeat; background-position: top left; background-image: url("../images/bottomleft.gif");}
+td.bracket.rightbottom {padding-right: 1em; background-repeat: no-repeat; background-position: top right; background-image: url("../images/bottomright.gif");}
+
+td.bracket.single {padding-right: 1em; background-repeat: no-repeat; background-position: center right;
+background-image: url("../images/bracket3_rt.gif");}
+
+
+/* conditional */
+table p {margin-top: 0em; line-height: normal;}
+
+/* sizes */
+.larger {font-size: 120%;}
+.largest {font-size: 150%;}
+
+
+/* my additions */
+
+p.mynote {background-color: #DDE; color: #000; padding: 1em;
+margin: 1em 5%; font-family: sans-serif; font-size: 90%;}
+
+</style>
+</head>
+
+<body>
+
+<p class = "mynote">
+Return to <a href = "../main.html#groundplat">John Dee’s Preface</a>.
+</p>
+
+<p class = "center"><i>J. DEE</i></p>
+
+<p class = "center largest">Here haue you (according to my promisse) the Groundplat of</p>
+
+<p class = "center larger">my MATHEMATICALL Præface: annexed to <i>Euclide</i> (now first)</p>
+
+<p class = "center">published in our Englishe tounge. An. 1570. Febr. 3.</p>
+
+
+<table class = "groundplat">
+
+<!-- principal, 6 rows -->
+<!-- row 1 of 6 -->
+
+<tr>
+<td class = "bracket righttop" rowspan = "3">
+&nbsp;
+</td>
+<td class = "bracket righttop" rowspan = "2">
+&nbsp;
+</td>
+<td class = "bracket righttop">
+&nbsp;
+</td>
+<td class = "lower" colspan = "4">
+<p><i><span class = "larger">Simple,</span> <span class = "small">Which dealeth with Numbers onely: and demon&shy;strateth all their properties and apper&shy;tenances: where, an Vnit, is Indiui&shy;sible.</span></i></p>
+</td>
+<td class = "bracket lefttop">
+&nbsp;
+</td>
+<td class = "bracket righttop">
+&nbsp;
+</td>
+<td rowspan = "2">
+<p class = "small"><i>In thinges Super&shy;naturall, æternall, &amp; Diuine: By Appli&shy;cation</i>, Ascending.</p></td>
+<td class = "bracket lefttop">
+&nbsp;
+</td>
+</tr>
+
+<!-- row 2 of 6 -->
+<tr>
+<!-- <td></td> <td></td> -->
+<td class = "bracket rightline">
+<p class = "larger">Arithmetike.</p>
+</td>
+<td colspan = "4" rowspan = "2">
+<p><i><span class = "larger">Mixt</span>, <span class = "small">Which with aide of Geometrie principall, demon&shy;strateth some Arith&shy;meticall Conclusion, or Purpose.</span></i></p>
+</td>
+<td class = "bracket leftline" rowspan = "4">
+&nbsp;
+</td>
+<td class = "bracket rightline" rowspan = "4">
+<p><i><span class = "larger">The vse</span><br>
+whereof, is either,</i></p>
+</td>
+<!-- <td></td> -->
+<td class = "bracket leftline" rowspan = "4">
+<p><i><span class = "small">The like Vses and Appli&shy;cations are, (though in a degree lower) in the</span></i> Artes Mathe&shy;maticall Deriuatiue.</p>
+</td>
+</tr>
+
+<!-- row 3 of 6 -->
+<tr>
+<!-- <td></td> -->
+<td class = "bracket rightline" rowspan = "2">
+<p><i><span class = "larger">Principall,</span> which are two, onely,</i></p>
+</td>
+<td class = "bracket rightbottom">
+<p>&nbsp;<br>&nbsp;</p>
+</td>
+<!-- <td></td> <td></td> <td></td> <td></td> <td></td> -->
+<td rowspan = "2">
+<p class = "small"><i>In thinges Mathe&shy;maticall: without farther Appli&shy;cation.</i></p>
+</td>
+<!-- <td></td> -->
+</tr>
+
+<!-- row 4 of 6 -->
+<tr>
+<td class = "bracket rightline" rowspan = "6">
+<p><span class = "largest">Sciences, and Artes Mathe&shy;maticall,</span> <span class = "larger">are, either</span></p>
+</td>
+<!-- <td></td> -->
+<td class = "bracket righttop">
+<p>&nbsp;<br>&nbsp;</p>
+</td>
+<td colspan = "4" rowspan = "2">
+<p><i><span class = "larger">Simple</span>, <span class = "small">Which dealeth with Magni&shy;tudes, onely: and demon&shy;strateth all their properties, passions, and apper&shy;tenances: whose Point, is Indiui&shy;sible.</span></i></p>
+</td>
+<!-- <td></td> <td></td> <td></td> <td></td> -->
+</tr>
+
+<!-- row 5 of 6 -->
+<tr>
+<!-- <td></td> -->
+<td class = "bracket rightbottom">
+<p>&nbsp;<br>&nbsp;</p>
+</td>
+<td class = "bracket rightline">
+<p class = "larger">Geometrie.</p>
+</td>
+<!-- <td></td> <td></td> <td></td> <td></td> <td></td> -->
+<td rowspan = "2">
+<p class = "small"><i>In thinges Naturall: both Substã&shy;tiall, &amp; Accidentall, Visible, &amp; Inuisible. &amp;c. By Appli&shy;cation:</i> Descending.</p>
+</td>
+<!-- <td></td> -->
+</tr>
+
+<!-- row 6 of 6 -->
+<tr>
+<!-- <td></td> -->
+<td>
+&nbsp;
+</td>
+<td class = "bracket rightbottom">
+&nbsp;
+</td>
+<td class = "upper" colspan = "4">
+<p><i><span class = "larger">Mixt</span>, <span class = "small">Which with aide of Arith&shy;metike principall, demon&shy;strateth some Geometricall purpose, as</span></i> EVCLIDES ELEMENTES.</p>
+</td>
+<td class = "bracket leftbottom">
+&nbsp;
+</td>
+<td class = "bracket rightbottom">
+&nbsp;
+</td>
+<!-- <td></td> -->
+<td class = "bracket leftbottom">
+&nbsp;
+</td>
+</tr>
+ 
+<!-- derivative, 5 rows -->
+<!-- row 1 of 5 -->
+
+<tr>
+<!-- <td></td> -->
+<td class = "bracket righttop" rowspan = "2">
+&nbsp;
+</td>
+<td class = "bracket righttop">
+&nbsp;
+</td>
+<td class = "lower bracket single">
+<p><span class = "larger">Arith&shy;metike,</span> <span class = "small"><i>vulgar: which consi&shy;dereth</i></span></p>
+</td>
+<td colspan = "7">
+<p>Arith&shy;metike of most vsuall whole numbers: And of Fractions to them apper&shy;taining.</p>
+<p>Arith&shy;metike of Propor&shy;tions.</p>
+<p>Arith&shy;metike Circular.</p>
+<p>Arith&shy;metike of Radicall Nũbers: Simple, Compound, Mixt: And of their Fractions.</p>
+<p>Arith&shy;metike of Cossike Nũbers: with their Fractions: And the great Arte of Algiebar.</p>
+</td>
+</tr>
+
+<!-- row 2 of 5 -->
+<tr>
+<!-- <td></td> <td></td> -->
+<td class = "lower bracket rightline">
+<p><i><span class = "larger">The names of</span> the Princi&shy;palls: as,</i></p>
+</td>
+<td class = "bracket righttop">
+&nbsp;
+</td>
+<td class = "bracket single" colspan = "2">
+<p class = "larger"><i>At hand</i></p>
+</td>
+<td class = "bracket single" colspan = "3">
+<p>All Lengthes.&mdash;</p>
+<p>All Plaines: As, Land, Borde, Glasse, &amp;c.</p>
+<p>All Solids: As, Timber, Stone, Vessels, &amp;c.</p>
+</td>
+<td colspan = "2">
+<p class = "larger">Mecometrie.</p>
+<p class = "larger">Embadometrie.</p>
+<p class = "larger">Stereometrie.</p>
+</td>
+</tr>
+
+<!-- row 3 of 5 -->
+<tr>
+<!-- <td></td> -->
+<td class = "bracket rightline" rowspan = "3">
+<p><i><span class = "larger">Deriuatiue</span> frõ the Princi&shy;palls: of which, some haue</i></p>
+</td>
+<td class = "bracket rightbottom">
+&nbsp;
+</td>
+<td class = "upper bracket rightline">
+<p><span class = "larger">Geometrie,</span> <span class = "small"><i>vulgar: which teacheth Measuring</i></span></p>
+</td>
+<td class = "bracket righttop" colspan = "2">
+&nbsp;
+</td>
+<td class = "lower">
+<p><i><span class = "larger">How farre</span>, <span class = "small">from the Measurer, any thing is: of him sene, on Land or Water: called</span></i> <span class = "larger">Apomecometrie</span>.</p>
+</td>
+<td class = "bracket lefttop">
+&nbsp;
+</td>
+<td class = "bracket righttop">
+&nbsp;
+</td>
+<td class = "lower" colspan = "2">
+<p><span class = "larger">Geodesie:</span> <span class = "small"><i>more cunningly to Measure and Suruey Landes, Woods, Waters. &amp;c.</i></span></p>
+</td>
+</tr>
+
+<!-- row 4 of 5 -->
+<tr>
+<td class = "bracket rightbottom" rowspan = "2">
+&nbsp;
+</td>
+<!-- <td></td> -->
+<td rowspan = "2">
+&nbsp;
+</td>
+<td class = "bracket rightbottom">
+&nbsp;
+</td>
+<td class = "upper bracket rightline" colspan = "2">
+<p><i><span class = "larger">With distãce</span> <span class = "small">from the thing Measured, as,</span></i></p>
+</td>
+<td>
+<p><i><span class = "larger">How high or deepe</span>, <span class = "small">from the leuell of the Measurers standing, any thing is: Seene of hym, on Land or Water: called</span></i> <span class = "larger">Hypso&shy;metrie</span>.</p>
+</td>
+<td class = "bracket leftline">
+&nbsp;
+</td>
+<td class = "bracket rightline">
+<p><i><span class = "larger">Of which</span> are growen the Feates &amp; Artes of</i></p>
+</td>
+<td colspan = "2">
+<p><span class = "larger">Geographie.</span></p>
+<p>&nbsp;</p>
+<p><span class = "larger">Choro&shy;graphie.</span></p>
+<p>&nbsp;</p>
+<p><span class = "larger">Hydro&shy;graphie.</span></p>
+</td>
+</tr>
+
+<!-- row 5 of 5 -->
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td>
+&nbsp;
+</td>
+<td class = "bracket rightbottom" colspan = "2">
+&nbsp;
+</td>
+<td>
+<p><i><span class = "larger">How broad</span>, <span class = "small">a&nbsp;thing is, which is in the Measurers view: so it be situated on Land or Water: called</span></i> <span class = "larger">Plato&shy;metrie</span>.</p>
+</td>
+<td class = "bracket leftbottom">
+&nbsp;
+</td>
+<td class = "bracket rightbottom">
+&nbsp;
+</td>
+<td class = "upper" colspan = "2">
+<p><span class = "larger">Strat&shy;arith&shy;metrie.</span></p>
+</td>
+</tr>
+
+<!-- proper names, 19 rows -->
+
+<tr>
+<td rowspan = "19">&nbsp;</td>
+<td class = "bracket rightline" rowspan = "8">
+&nbsp;
+</td>
+<td class = "bracket righttop">
+<p>&nbsp;<br>&nbsp;</p>
+</td>
+<td class = "upper" colspan = "2">
+<p class = "larger">Perspectiue,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth the maners and properties of all Radia&shy;tions: Directe, Broken, and Reflected.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> -->
+<td class = "bracket rightline" rowspan = "6">
+&nbsp;
+</td>
+<td class = "upper" colspan = "2">
+<p class = "larger">Astro&shy;nomie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth the Distances, Magni&shy;tudes, and all Naturall motions, Apparences, and Passions, proper to the Planets and fixed Starres: for any time, past, present, and to come: in respecte of a certaine Horizon, or without respecte of any Horizon.</i></p>
+</td>
+</tr>
+
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Musike,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth by reason, and teacheth by sense, perfectly to iudge and order the diuer&shy;sitie of Soundes, hie or low.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Cosmo&shy;graphie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which, wholy and perfectly maketh description of the Heauenlym and also Elementall part of the World: and of these partes, maketh homologall appli&shy;cation, and mutuall collation necessary.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Astro&shy;logie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which reasonably demon&shy;strateth the opera&shy;tions and effectes of the naturall beames of light, and secrete Influence of the Planets, and fixed Starres, in euery Element and Elementall body: at all times, in any Horizon assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Statike,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth the causes of heauines and lightnes of all thinges: and of the motions and properties to heauines and lightnes belonging.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Anthropographie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which describeth the Nũber, Measure, Waight, Figure, Situation, and colour of euery diuers thing contained in the perfecte body of MAN: and geueth certaine knowledge of the Figure, Symmetrie, Waight, Charac&shy;terization, &amp; due Locall motion of any percell of the said body assigned: and of numbers to the said percell apper&shy;taining.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> -->
+<td class = "bracket rightline" rowspan = "3">
+<p><i><span class = "larger">Propre names</span><br>
+as,</i></p>
+</td>
+<td class = "upper" colspan = "2">
+<p class = "larger">Trochilike,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth the properties of all Circular motions: Simple and Compound.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> -->
+<td class = "bracket rightbottom" rowspan = "2">
+&nbsp;
+</td>
+<!-- <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Helico&shy;sophie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth the designing of all Spirall lines: in Plaine, on Cylinder, Cone, Sphære, Conoïd, and Sphæroid: and their properties.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Pneuma&shy;tithmie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth by close hollow Geometricall figures (Regular and Irregular) the straunge properties (in motion or stay) of the Water, Ayre, Smoke, and Fire, in their Conti&shy;nuitie, and as they are ioyned to the Elementes next them.</i></p>
+</td>
+</tr>
+
+<tr>
+<!-- <td></td> -->
+<td rowspan = "9">
+&nbsp;
+</td>
+<td class = "bracket rightline" rowspan = "8">
+&nbsp;
+</td>
+<td class = "upper" colspan = "2">
+<p class = "larger">Menadrie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth, how, aboue Natures Vertue, and power simple: Vertue and force, may be multi&shy;plied: and so to directe, to lift, to pull to, and to put or cast fro, any multi&shy;plied, or simple deter&shy;mined Vertue, Waight, or Force: naturally, not, so, direc&shy;tible, or moueable.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Hypogeiodie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth, how, vnder the Sphæricall Super&shy;ficies of the Earth, at any depth, to any perpen&shy;dicular line assigned (whose distance from the perpen&shy;dicular of the entrance: and the Azimuth likewise, in respecte of the sayd entrance, is knowen) certaine way, may be prescribed and gone, &amp;c.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Hydra&shy;gogie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth the possible leading of water by Natures law, and by artificiall helpe, from any head (being Spring, standing, or running water) to any other place assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Horometrie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth, how, at all times appointed, the precise, vsuall denomi&shy;nation of time, may be knowen, for any place assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Zographie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth and teacheth, how, the Inter&shy;section of all visuall Pyramids, made by any plaine assigned (the Center, distance, and lightes being deter&shy;mined) may be, by lines, and proper colours repre&shy;sented.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Archi&shy;tecture,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which is a Science garnished with many doctrines, and diuers Instructions: by whose iudgement, all workes by other workmen finished, are iudged.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Nauigation,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which demon&shy;strateth, how, by the Shortest good way, by the aptest direction, and in the shortest time: a&nbsp;suffi&shy;cient Shippe, betwene any two places (in passage nauigable) assigned, may be conducted: and in all stormes and naturall distur&shy;bances chauncing, how to vse the best possible meanes, to recouer the place first assigned.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> <td></td> -->
+<td class = "upper" colspan = "2">
+<p class = "larger">Thaumaturgike,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which geueth certaine order to make straunge workes, of the sense to be perceiued: and of men greatly to be wondred at.</i></p>
+</td>
+</tr>
+<tr>
+<!-- <td></td> <td></td> -->
+<td class = "bracket rightbottom">
+&nbsp;
+</td>
+<td class = "upper" colspan = "2">
+<p class = "larger">Arche&shy;mastrie,</p>
+</td>
+<td colspan = "6">
+<p class = "small"><i>Which teacheth to bring to actuall experience sensible, all worthy conclu&shy;sions, by all the Artes Mathe&shy;maticall purposed: and by true Naturall philo&shy;sophie, concluded: And both addeth to them a farder Scope, in the termes of the same Artes: and also, by his proper Method, and in peculiar termes, procedeth, with helpe of the forsayd Artes, to the perfor&shy;mance of complete Experi&shy;ences: which, of no parti&shy;cular Arte, are hable (Formally) to be challenged.</i></p>
+</td>
+</tr>
+</table>
+
+<p class = "mynote">
+Return to <a href = "../main.html#groundplat">John Dee’s Preface</a>.
+</p>
+
+</body>
+</html>