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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

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*** START OF THE PROJECT GUTENBERG EBOOK THE
MATHEMATICALL PRAEFACE TO ELEMENTS OF GEOMETRIE OF
EUCLID OF MEGARA ***

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Title Page Text

❧ The Translator to the Reader.
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. 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.
(?¿)

❧ 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.
Iuine 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.
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 The int
plesaunt, and frutefull Mathematicall Tree, with his chief armes and this Pre
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.
Of Mathematicall thinges, are two principall kindes: namely, Number,
and Magnitude. Number, we define, to be, a certayne Mathematicall Numbe
Sũme, of Vnits. And, an Vnit, is that thing Mathematicall, Indiuisible, by
participation of some likenes of whose property, any thing, which is in Note the w
Vnit, to ex
deede, or is counted One, may resonably be called One. We account an the Greke
Vnit, a thing Mathematicall, though it be no Number, and also & not Vni
indiuisible: because, of it, materially, Number doth consist: which, haue all,
commonly
principally, is a thing Mathematicall. Magnitude is a thing vsed.
Mathematicall, by participation of some likenes of whose nature, any
Magnit
thing is iudged long, broade, or thicke. A thicke Magnitude we call a
Solide, or a Body. What Magnitude so euer, is Solide or Thicke, is also
broade, & long. A broade magnitude, we call a Superficies or a Plaine.
Euery playne magnitude, hath also length. A long magnitude, we terme a
Line. A Line is neither thicke nor broade, but onely long: Euery certayne
Line, hath two endes: The endes of a line, are Pointes called. A point
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 is called A Line.
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. And Magnit
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
ue
Philosopher Anitius Boetius, sayd: Omnia quæcunq 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, *set vp 900. Conclusions, Ano. 14
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: which is, the Science that demonstrateth the properties, of Arithme
Numbers, and all operatiõs, in numbers to be performed: How often,
therfore, these fiue sundry sortes of Operations, do, for the most part, of Note.
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: the Consideration, doctrine, and 1.
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. In lyke sorte, the necessary, wonderfull and 2.
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. Moreouer, the Astronomers, for spede and 3.
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. Practise hath led Numbers farder, 4.
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
2 3
characters, (as √. √. and so of other. A 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)
2
Radicall nũbers, diuersly knit, by signes, of More & Lesse: as thus √12
3 4 3 2
+ √15. Or thus √19 + √12 - √2. &c. And some tyme with whole
2 3
numbers, or fractions of whole Number, amõg them: as 20 + √24. √16
2 4 3
+ 33 - √10. √44 + 12¼ + √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, of * Anno
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: and the principal 5.
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 (aboue 200. yeares ago) inuented. And forasmuch R. B.
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. B

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 * Take
Contrary Qualities, or of one kinde (but of diuerse intentions and part of
degrees) or a Temperate, and a Contrary, The forme resulting of their counsay
Mixture, is in the Middle betwene the degrees of the formes mixt. As for his boo
example, let A, be Moist in the first degree: and B, Dry in the third Q. Esse
degree. Adde 1. and 3. that maketh 4: the halfe or middle of 4. is 2. This
2. is the middle, equally distant from A and B (for the *Temperament is * Note.
counted none. And for it, you must put a Ciphre, if at any time, it be in
mixture). 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.
But if the Quantities of two thinges Commixt, be diuerse, and the The Sec
Intensions (of their Formes Miscible) be in diuerse degrees, and Rule.
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.
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 1X: 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 1X: to the
difference betwene 1X and 3: (the degree of the thing, in lesse quãtitie.
And with all, 1X, being alwayes in a certaine middell, betwene the two
heigthes or degrees). For the first difference, I set 4-1X: and for the
second, I set 1X-3. And, now againe, I say, as 1. is to 2. so is 4-1X to 1X-
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 1X-3. and of
the second & the third, 8-2X. Wherfore, our Æquation is betwene 1X-3:
and 8-2X. Which may be reduced, according to the Arte of Algiebar: as,
here, adding 3. to eche part, geueth the Æquation, thus, 1X=11-2X. And
yet againe, contracting, or Reducing it: Adde to eche part, 2X: Then haue
you 3X æquall to 11: thus represented 3X=11. Wherefore, diuiding 11.
by 3: the Quotient is 3⅔: the Valew of our 1X, 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 2X, it would
stand, 3X-3=8. And now adding to eche part 3: you haue (as afore)
3X=11.

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. In procedyng first, with any two: and then, with the Forme Note.
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. That is, the Skill of Τακτικὴ
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: And of all these, liuely This nobl
dyed Ann
designementes (most curiously) to be in velame parchement described: skarse of
with Notes & peculier markes, as the Arte requireth: and all these Rules, yeares of
and descriptions Arithmeticall, inclosed in a riche Case of Gold, he vsed hauing no
his wife: D
to weare about his necke: as his Iuell most precious, and Counsaylour to the Duk
most trusty. Thus, Arithmetike, of him, was shryned in gold: Of Numbers Somerset
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, to Iustice.
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, is Numbe
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? This Science of Geome
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
*Geometria, with them, Studium quod circa planum versatur. But, well Rep.
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
ue
hoc non negabunt, Quicunq 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

ue
illius gratia, quod semper est, non & eius quod oritur quandoq &
interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur
ue
(ô Generose vir) ad Veritatem, animum: atq 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.
Megethologicall Contemplations: wherby, to trayne our Imaginations * Herei
and Myndes, by litle and litle, to forsake and abandon, the grosse and I would
corruptible Obiectes, of our vtward senses: and to apprehend, by sure shake o

doctrine demonstratiue, Things Mathematicall. And by them, readily to earthly
be holpen and conducted to conceiue, discourse, and conclude of things of Geom
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, to An Arte
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, An Art Art
Mathematicall deriuatiue, which by Mathematicall Mathem
demonstratiue Method, in Nũbers, or Magnitudes, ordreth and Deriua
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 Mechanicien, otherwise, then (hitherto) it hath ben vsed, I thinke it A
good, (for distinction sake) to giue you also a brief description, what I Mechan
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.
First, then, from the puritie, absolutenes, and Immaterialitie of Principall 1.
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. This, teacheth to measure, either at hand: Geome
and the practiser, to be by the thing Measured: and so, by due applying vulgar.
of Cumpase, Rule, Squire, Yarde, Ell, Perch, Pole, Line, Gaging rod, (or
such like instrument) to the Length, Plaine, or Solide measured, *to be 1.
certified, either of the length, perimetry, or distance lineall: and this is
called, Mecometrie. Or *to be certified of the content of any plaine 2.
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. Or els, this vulgar Geometrie, hath consideration to teach 2.
the practiser, how to measure things, with good distance betwene him
and the thing measured: and to vnderstand thereby, either *how Farre, 1.
a thing seene (on land or water) is from the measurer: and this may be
called Apomecometrie: Or, how High or depe, aboue or vnder the leuel 2.
of the measurers stãding, any thing is, which is sene on land or water,

called Hypsometrie. *Or, it informeth the measurer, how Broad any thing 3.
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: yet, know for certaine, that Note.
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.

O FLand
these Feates (farther applied) is Sprong the Feate of Geodesie, or
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) how great wrong Note.
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 Hydrographer, 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. *And this figure, and

sides therof, he is hable to know: either beyng by, and at hand: or a farre * Note.
of. Thus farre, stretcheth the description and property of
Stratarithmetrie: sufficient for this tyme and place. It differreth from the The diff
Feate Tacticall, De aciebus instruendis. bycause, there, is necessary the betwen
wisedome and foresight, to what purpose he so ordreth the men: and Stratar
Skillfull hability, also, for any occasion, or purpose, to deuise and vse ie and
the aptest and most necessary order, array and figure of his Company Tacticie
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,
Great pollicy may be vsed of the Capitaines, (at tymes fete, and in places you will f
conuenient) as to vse Figures, which make greatest shew, of so many as hard, to p
he hath: and vsing the aduauntage of the three kindes of vsuall spaces: my descri
this Feate
(betwene footemen or horsemen) to take the largest: or when he would Chorogra
seme to haue few, (beyng many:) contrarywise, in Figure, and space. The may helpe
selfe some
Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer is carefull where the
to come nere the truth herein, besides the Iudgement of his expert eye, knowne (i
his skill of Ordering Tacticall, the helpe of his Geometricall instrument: and Angle
not Regul
Ring, or Staffe Astronomicall: (commodiously framed for cariage and where, Re
vse) He may wonderfully helpe him selfe, by perspectiue Glasses. In into Trian
serue. &c
which, (I trust) our posterity will proue more skillfull and expert, and to you will f
greater purposes, then in these dayes, can (almost) be credited to be strange to
thus gene
possible. Arithmeti
figures: a
Thus haue I lightly passed ouer the Artificiall Feates, chiefly dependyng for Battay
vpon vulgar Geometrie: & commonly and generally reckened vnder the Their con
name of Geometrie. But there are other (very many) Methodicall Artes, differ so m
from like
which, declyning from the purity, simplicitie, and Immateriality, of our Geometri
Principall Science of Magnitudes: do yet neuertheles vse the great ayde, Figures.
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 maru
appearing in the ayre, betwene you & the glasse, with like hand, sword Glasse.
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: and for skill in the Mathematicall Sciences, and S. W. P.
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. So thicke is the heauenly Palace, that the Planetes haue all Note.
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, that the *Minde may be 1.
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, in the
heauenly *motions and distances, describe a meruailous Harmonie, of

Pythagoras Harpe with eight stringes. Also, somwhat might be sayd of 2.
Mercurius* two Harpes, eche of foure Stringes Elementall. And very
3.
straunge matter, might be alledged of the Harmonie, to our *Spirituall
part appropriate. As in Ptolomaus third boke, in the fourth and sixth 4.
Chapters may appeare.* And what is the cause of the apt bonde, and 5.
frendly felowship, of the Intellectuall and Mentall part of vs, with our
grosse & corruptible body: but a certaine Meane, and Harmonious 6.
Spiritualitie, with both participatyng, & of both (in a maner) resultynge
In the *Tune of Mans voyce, and also * the sound of Instrument, what 7.
might be sayd, of Harmonie: No common Musicien would lightly
8.
beleue. But of the sundry Mixture (as I may terme it) and concurse,
diuerse collation, and Application of these Harmonies: as of thre, foure, I. D.
Read in A
fiue, or mo: Maruailous haue the effectes ben: and yet may be founde, his 8. boo
and produced the like: with some proportionall consideration for our Politikes:
time, and being: in respect of the State, of the thinges then: in which, and and 7. cha
Where yo
by which, the wondrous effectes were wrought. Democritus and haue som
Theophrastus affirmed, that, by Musike, griefes and diseases of the occasion
thinke of M
Minde, and body might be cured, or inferred. And we finde in Recorde, than comm
that Terpander, Arion, Ismenias, Orpheus, Amphion, Dauid, Pythagoras, thought.
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.
ue
Principium igitur vt mouens, præcipuumq & 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
ue
namq & 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
ue
booke, and second Chapter of hys Physikes. Homo namq generat
ue
hominem, atq 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, (for *21. * Anno
yeares ago) by certaine earnest disputations, of the Learned Gerardus and 154
Mercator, and Antonius Gogaua, (and other,) therto so prouoked: and Louayn
(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. Wherof, the one, is Light Beleuers, the other, Light Note.
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: which is not so: and, in dede, they, to much, ouer reache. The 1.
Second sorte thinke no Influentiall vertue (frõ the heauenly bodies) to
beare any Sway in Generation and Corruption, in this Elementall world. 2.
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 furnished: yet, either for 3.
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.
If any Solide Magnitude, Lighter then a Liquor, be let downe I. D.
The Cutti
into the same Liquor, the waight of the same Magnitude, will be, Sphære ac
to the Waight of the Liquor. (Which is æquall in quantitie to the to any pro
assigned
whole Magnitude,) in that proportion, that the parte, of the this propo
Magnitude settled downe, is to the whole Magnitude. done Mec
by temper

B YNaturall
these verities, great Errors may be reformed, in Opinion of the
Motion of thinges, Light and Heauy. Which errors, are in
Liquor to
certayne w
respect of
Naturall Philosophie (almost) of all mẽ allowed: to much trusting to waight of
Sphære th
Authority: and false Suppositions. As, Of any two bodyes, the Swymmin
heauyer, to moue downward faster then the lighter. This error, is
not first by me, Noted: but by one Iohn Baptist de Benedictis. The chief A comm
of his propositions, is this: which seemeth a Paradox. error, n
If there be two bodyes of one forme, and of one kynde, æquall in A parad
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.
Hereupon, in the feate of Gunnyng, certaine good discourses N. T.
(otherwise) may receiue great amendement, and furderance. In the
entended purpose, also, allowing somwhat to the imperfection of Nature: The
not aunswerable to the precisenes of demonstration. Moreouer, by the wonder
foresaid propositions (wisely vsed.) The Ayre, the water, the Earth, the vse of t
Fire, may be nerely, knowen, how light or heauy they are (Naturally) in Propos

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. Make a Cube, of any one Vniforme: and The pra
through like heauy stuffe: of the same Stuffe, make a Sphære or Globe, Statical
precisely, of a Diameter æquall to the Radicall side of the Cube. Your know th
stuffe, may be wood, Copper, Tinne, Lead, Siluer. &c. (being, as I sayd, proport
of like nature, condition, and like waight throughout.) And you may, by betwen
Say Balance, haue prepared a great number of the smallest waightes: Cube, a
which, by those Balance can be discerned or tryed: and so, haue Sphære
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. Your Cube, Sphære, * I. D.
apt Balance, and conuenient waightes, being ready: fall to worke.❉. For, so,
First, way your Cube. Note the Number of the waight. Way, after that, you .25
your Sphære. Note likewise, the Nũber of the waight. If you now find partes o
the waight of your Cube, to be to the waight of the Sphære, as 21. is to Graine
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, we haue *the proportion of the Square, to the Circle, * The pro
of the Squ
inscribed in the same Square. And so are we fallen into the knowledge Circle ins
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: *as I haue taught, in my Annotation, vpon the * The Squ
the Circle
first proposition of the twelfth boke, And likewise, to any Square giuen, Mechanic
you may giue a Circle æquall: *If you describe a Circle, which shall be
in that proportion, to your Circle inscribed, as the Square is to the same * To any S
geuen, to
Circle: This, you may do, by my Annotations, vpon the second Circle, eq
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. That we may Square Note Sq
the Circle, without hauing knowledge of the proportion, of the of the C
Circumference to the Diameter: as you haue here perceiued. And without
otherwayes also, I can demonstrate it. So that, many haue cumberd them knowle
selues superfluously, by trauailing in that point first, which was not of the prop
necessitie, first: and also very intricate. And easily, you may, (and that betwen
diuersly) come to the knowledge of the Circumference: the Circles Circum
Quantitie, being first knowen. Which thing, I leaue to your e and
consideration: making hast to despatch an other Magistrall Probleme: Diamet
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 Dub
Make of Copper plates, or Tyn plates, a foursquare vpright Cube re
Pyramis, or a Cone: perfectly fashioned in the holow, within. by Art
Wherin, let great diligence be vsed, to approche (as nere as may Mechan
be) to the Mathematicall perfection of those figures. At their depend
bases, let them be all open: euery where, els, most close, and iust vppon
Demon
to. From the vertex, to the Circumference of the base of the n
Cone: & to the sides of the base of the Pyramis: Let 4. straight Mathem
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 I. D.
sides them selues, with the sayd perimeters. These 4. lines (in The 4. sid
Pyramis m
the Pyramis: and as many, in the Cone) diuide: one, in 12. Isosceles
æquall partes: and an other, in 24. an other, in 60, and an other, alike and

in 100. (reckenyng vp from the vertex.) Or vse other numbers of
diuision, as experience shall teach you. Then,* set your Cone or I. D.
* In all w
Pyramis, with the vertex downward, perpendicularly, in respect with this P
of the Base. (Though it be otherwayes, it hindreth nothyng.) So or Cone, L
Situations
let thẽ most stedily be stayed. Now, if there be a Cube, which you Pointes a
wold haue Dubbled. Make you a prety Cube of Copper, Siluer, Lead, Condition
or all one
Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or Cubik you are a
coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so Worke. El
erre.
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, into the Pyramis or Cone, where you did put the first. I. D.
* Conside
Marke now againe, in what number or place of the lines, the water whan you
Cutteth them. Two wayes you may conclude your purpose: it is to wete, your wate
either by numbers or lines. By numbers: as, if you diuide the side of your togyther:
whan, you
Fundamentall Cube into so many æquall partes, as it is capable of, empty you
conueniently, with your ease, and precisenes of the diuision. For, as the water, out
Pyramis o
number of your first and lesse line (in your hollow Pyramis or Cone,) is Els you w
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. For this, may I (with ioy) say, God be
ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking the holy and glorious Trinity: thanked
hauing greater cause therto, then *Archimedes had (for finding the this Inu
fraude vsed in the Kinges Crowne, of Gold): as all men may easily & the fr
Iudge: by the diuersitie of the frute following of the one, and the other. ensuing
Where I spake before, of a hollow Cubik Coffen: the like vse, is of it: * Vitruu
and without waight. Thus. Fill it with water, precisely full, and poure Lib. 9.
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. *And if the * Note.
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, that all this while, I forget not my first Proposition Staticall, here Note, a
rehearsed: that, the Superficies of the water, is Sphæricall. Wherein, vse concern
your discretion: to the first line, adding a small heare breadth, more: and the Sph
to the second, halfe a heare breadth more, to his length. For, you will Superfi
easily perceaue, that the difference can be no greater, in any Pyramis or the Wat
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,* yet, by helpe of my sixt Theoreme vpon the last * Note.
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. Now will I Abridge your paine, cost, and Care herein. Note th
Without all preparing of your Fundamentall Cubes: you may (alike) Abridge
worke this Conclusion. For, that, was rather a kinde of Experimentall of Dubb
demõstration, then the shortest way: and all, vpon one Mathematicall the Cub
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: *sauing * Note.
that for you Fundamentall Cube his Radicall side: here, you may take a
right line, at pleasure.

Yet farther proceding with our droppe of Naturall truth: you may To giue
(now) geue Cubes, one to the other, in any proportiõ geuẽ: one to t
Rationall or Irrationall: on this maner. Make a hollow other in
Parallelipipedon of Copper or Tinne: with one Base wãting, or open: as proport
in our Cubike Coffen. Frõ the bottome of that Parallelipipedon, raise vp, Rationa
many perpendiculars, in euery of his fower sides. Now if any proportion Irration
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* duely into the hollow Pyramis or * Empty
Cone: Marke now againe, where the water cutteth the same line which the firs
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. Which is this: Alwaies, you haue two Like Pyramids: or The
demonstra
two Like Cones, in the proportions assigned: and like Pyramids or this Dubb
Cones, are in proportion, one to the other, in the proportion of their the Cube,
Homologall sides (or lines) tripled. Wherefore, if to the first, and second the rest.
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* I. D.
* Hereby,
Line to Line, be also double, &c. But, as our first line, is to the second, your self t
so is the Radicall side of our Fundamentall Cube, to the Radicall side of a præcise
the Cube to be made, or to be doubled: and therefore, to those twaine practiser.
consider,
also, a third and a fourth line, in continuall proportion, ioyned: will geue nothing a
the fourth line in that proportion to the first, as our fourth Pyramidall, or are hindre
(sensibly)
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, Conuexiti
water.
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* as the * By the 3
eleuenth b
fourth line, is to the first, so is that Cube, to the first Cube: and we haue Euclide.
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. But we* Suppose Pyramis to Pyramis, or I. D.
* And you
Cone to Cone, to be double or treble. &c. Therfore Cube, is to Cube, diligence
double, or treble, &c. Which was to be demonstrated. And of the practise, c
Parallelipipedõ, it is euidẽt, that the water Solide Parallelipipedons, are waight of
performe
one to the other, as their heithes are, seing they haue one base. Wherfore Therefore
the Pyramids or Cones, made of those water Parallelipipedons, are one you are a
geue good
to the other, as the lines are (one to the other) betwene which, our of your w
proportion was assigned. But the Cubes made of lines, after the doing.
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, my Demonstratiõ is more generall, then onely in Square Pyramis Corolla
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. *Now may you, Betwene two lines * The g
giuen, finde two middle proportionals, in Continuall proportion: Commo
by the hollow Parallelipipedon, and the hollow Pyramis, or followin
Cone. Now, any Parallelipipedon rectangle being giuen: thre right lines these n
may be found, proportionall in any proportion assigned, of which, shal Inuenti
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
Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte: Fruite o
and by good meanes, Mount aboue the cloudes and sterres: And thirdly, Mathem
he can, by order, Descend, to frame Naturall thinges, to wonderfull vses: Science
and when he list, retire home into his owne Centre: and there, prepare Artes.
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.

T His 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 him, who is the Lesse world: and, frõ the beginning, MAN is
called Microcosmus (that is. The Lesse World.) And for whose sake, and Lesse W
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 Microcosmicall constitution. The outward Image, and vew hereof: to Micro
the Art of Zographie and Painting, to Sculpture, and Architecture: (for Cosmus
Church, House, Fort, or Ship) is most necessary and profitable: for that,
it is the chiefe base and foundation of them. Looke in *Vitruuius, * Lib. 3
whether I deale sincerely for your behoufe, or no. Looke in Albertus 1.
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: As Saw Mi
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, it is *recorded, that, where all the power * Athen
of the Citie of Syracusa, was not hable to moue a certaine Ship (being on Lib. 5.
ground) mightie Archimedes, setting to, his Skruish Engine, caused
Hiero the king, by him self, at ease, to remoue her, as he would. Wherat, Proclus
the King wondring: Απὸ τάυτης τῆς ἡμήρας, περὶ παντὸς, Αρχιμήδει 18.
λέγοντι πιϛευτέομ. 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. Hereupon, two or three men together, by keping Ayre vnder a To go to
great Cauldron, and forcyng the same downe, orderly, may without bottom
harme descend to the Sea bottome: and continue there a tyme &c. Sea wit
Where, Note, how the thicker Element (as the Water) giueth place to the daunge
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, Plutarchu
Marco Ma
Marcus Marcellus the Consul, being their Generall Capitaine. Such huge
Stones, so many, with such force, and so farre, did he with his engynes Synesius i
Epistolis.
hayle among them, out of the Citie. And by Sea likewise: though their
Ships might come to the walls of Syracusa, yet hee vtterly confounded Polybius.
the Romaine Nauye. What with his mighty Stones hurlyng: what with Plinius.
Pikes of* 18 fote long, made like shaftes: which he forced almost a Quintilian
quarter of a myle. What, with his catchyng hold of their Shyps, and T. Liuius.
hoysing them vp aboue the water, and suddenly letting them fall into the * Athenæ
Sea againe: what with his* Burning Glasses: by which he fired their
other Shippes a far-of: what, with his other pollicies, deuises, and * Galenus
engines, he so manfully acquit him selfe: that all the Force, courage, and Anthemiu
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, being placed Burning
vpon the walles of Bysance, he multiplied so the heate of the Sunne, and Glasses
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 Dionspecifieth 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. Which, from an English Gunnes
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: in* King 4. Reg.
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 more excellent, more commodious, and more A perpe
marueilous way, then all these: of hauing the motion of the Primouant Motion
(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.

Architecture, to many may seme not worthy, or not mete, to be An obje
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 An
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,
ue
res fabricatas, Solertia ac ratione proportionis, demonstrare atq
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
ue
Iurisperitorũ nouerit: Astrologiam, Cæliq 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
e
diuersities of the soundes (which y Grecians call ηχεῖα) are
ordred according to Musicall Symphonies & Harmonies: being
e
distributed in y 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
t
increased: with y increasing, might come more cleare &
e
pleasant, to y eares of the lokers on. &c. And of Astronomie, is
e
knowẽ y 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: and are become A
Mathematiciens. &c. And they are found, seldome. As, in tymes Mathem
past, was Aristarchus Samius: Philolaus, and Archytas, n.
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: This is Geometrie, Arithmetike, Astronomie, Musike, Vitruuiu
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
ue
of Architecture. Sed anteq 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. But I will appoint the Architect to be VVho is
that man, who hath the skill, (by a certaine and meruailous Archite
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,
e
y 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,* all material stuffe beyng * The
secluded. Which point we shall atteyne, by Notyng and Immate
forepointyng the angles, and lines, by a sure and certaine of perfe
direction and connexion. Seyng then, these thinges, are thus: Archite
Lineamente, shalbe the certaine and constant prescribyng, What,
conceiued in mynde: made in lines and angles: and finished with Lineam
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: by certaine order, nũber, forme, figure, and Note.
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.

T He 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 (of me Inuented, for our two Moscouy Master Anno. 1
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,)
ue
Mutationes, opportunitatesq temporum presentire, non minus rei
ue
militari, quàm Agriculturæ, Nauigationiq 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, C
ue
Sol quoq & exoriens & quum se condet in vndas,
Signa dabit, Solem certissima signa sequuntur. &c.
———— Nam sæpe videmus, Georgic
Ipsius in vultu varios errare colores.
Cæruleus, pluuiam denunciat, igneus Euros.
Sin maculæ incipient rutilo immiscerier igni,
ue
Omnia tum pariter vento, nimbisq videbis
Feruere: non illa quisquam me nocte per altum
ue
Ire, neq 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, (of* Late) a young * Anno
Gentleman, a Courragious Capitaine, was in a great readynes, with good S. H. G
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: Yet, I say, * Anno
(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, *I saw, ones or twise * Anno
(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. Hodie magnæ literaturæ De his
viros & magna reputationis videmus, opera quedam quasi miranda, Mundo
supra Naturã putare: de quibus in Perspectiua doctus causam faciliter mirabil
reddidisset. That is. Now a dayes, we see some men, yea of great eueniun
learnyng and reputation, to Iudge certain workes as meruaylous, 8.
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. Which is very straunge to thinke Tusc. 1
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.
And for these, and such like marueilous Actes and Feates, Naturally, A Digre
Mathematically, and Mechanically, wrought and contriued: ought any

honest Student, and Modest Christian Philosopher, be counted, & called Apolog
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? Nay, a Deuill? For, so, doth the Common * A pro
peuish Pratler Imagine and Iangle: And, so, doth the Malicious skorner, Fayre f
secretly wishe, & brauely and boldly face down, behinde my backe. Ah, and cau
what a miserable thing, is this kinde of Men? How great is the blindnes Frog.
& 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. Vaine pratling busie bodies, vse your idle assemblies, and 1.
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. Malicious Ignorant, what shall I say to thee? 4.
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: yet take heede, and thinke, betimes, with your selfe, Vir Psal. 14
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. Eruditus est Moses omni Act. 7.
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, of vayne fraudulent Magike, saying: Est & alia Lib. 30
Magices Factio, a Mose, Iamne, & Iotape, Iudæis pendens: sed multis 1.
millibus annorum post Zoroastrem. &c. Let all such, therefore, who, in
Iudgement and Skill of Philosophie, are farre Inferior to Plinie, take 1.
good heede, least they ouershoote them selues rashly, in Iudging of
Philosophers straunge Actes: and the Meanes, how they are done. But,
much more, ought they to beware of forging, deuising, and imagining 2.
monstrous feates, and wonderfull workes, when and where, no such
were done: no, not any sparke or likelihode, of such, as they, without all
shame, do report. And (to conclude) most of all, let them be ashamed of 3.
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.

N Ow 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, of this land Natiue R. B.
(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.
For, the Honour, and Estimation of the Vniuersities, and Graduates, 1.
is, hereby, nothing diminished. Seing, from, and by their Nurse Children,

you receaue all this Benefite: how great soeuer it be.
Neither are their Studies, hereby, any whit hindred. No more, then the 2.
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?
And surely, the Common and Vulgar Scholer (much more, the 3.
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.
Also many good and pregnant Englishe wittes, of young Gentlemen, and 4.
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.
And great Comfort, with good hope, may the Vniuersities haue, by 5.
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, may in the Vniuersities be had: being Vniuers
the Storehouses & Threasory of all Sciences, and all Artes, necessary for
the best, and most noble State of Common Wealthes.
Besides this, how many a Common Artificer, is there, in these Realmes 6.
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, I will giue you the Groundplatt of The Gr
my whole discourse, in a Table annexed: from the first to the last, platt of
somewhat Methodically contriued. Præfac
Table.
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.

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.
Simple, Which dealeth with Numbers onely: and demon­strateth all their In thinges Super­‐
naturall, æternall,
properties and apper­tenances: where, an Vnit, is Indiui­sible. & Diuine: By
Appli­cation,
Arithmetike.
Mixt, Which with aide of Geometrie principall, demon­strateth some Arith­‐ Ascending.
The like Vses and
meticall Conclusion, or Purpose. In thinges Mathe­‐ Appli­cations are,
Principall,
which are two,
The vse maticall: without (though in a
whereof, is farther Appli­‐ degree lower) in
onely, cation. the Artes Mathe­‐
either,
Simple, Which dealeth with Magni­tudes, onely: and demon­strateth all In thinges maticall
their properties, passions, and apper­tenances: whose Point, is Indiui­sible. Naturall: both Deriuatiue.
Geometrie.
Substã­tiall, &
Accidentall,
Mixt, Which with aide of Arith­metike principall, demon­strateth some Visible, &
Geometricall purpose, as EVCLIDES ELEMENTES. Inuisible. &c. By
Appli­cation:
Sciences, Descending.

and Artes Arith­metike of most vsuall whole numbers: And of Fractions to them apper­taining.
Arith­metike of Propor­tions.
Mathe­‐ Arith­metike, Arith­metike Circular.
maticall, vulgar: which consi­‐ Arith­metike of Radicall Nũbers: Simple, Compound, Mixt: And of their Fractions.
dereth Arith­metike of Cossike Nũbers: with their Fractions: And the great Arte of Algiebar.
are, either
All Lengthes.—
Mecometrie.
The names of At hand All Plaines: As, Land, Borde, Glasse, &c. Embadometrie.
All Solids: As, Timber, Stone, Vessels, &c.
the Princi­palls: as, Stereometrie.
Geometrie, vulgar: How farre, from the
which teacheth Measurer, any thing is: of him Geodesie: more cunningly to
Measuring sene, on Land or Water: called
Measure and Suruey Landes, Woods,
Apomecometrie. Waters. &c.
With How high or deepe, Of Geographie.
Deriuatiue distãce from the leuell of the Measurers
frõ the Princi­‐ which
from the standing, any thing is: Seene of
are growen Choro­graphie.
palls: of which, thing hym, on Land or Water: called
some haue the Feates
Measured, Hypso­metrie. & Artes of Hydro­graphie.
as,
How broad, a thing is, Strat­arith­metrie.
which is in the Measurers view:
so it be situated on Land or
Water: called Plato­metrie.

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.
Propre Helico­sophie, Which demon­strateth the designing of all Spirall lines: in Plaine, on Cylinder, Cone, Sphære,
Conoïd, and Sphæroid: and their properties.
names
as, 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.

Title Page Text

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.

Transcriber’s Footnotes
A. Mathematical Notation. John Dee used the “root” sign √ in combination
with some less familiar symbols:

“First power”, here used to express an
unknown. Shown in this e-text as X
(capitalized).
Root sign combined with “second power”
symbol = square root. Shown in this e-text
2
as √.
Root sign combined with “third power” symbol =
3
cube root. Shown in this e-text as √.
Doubled “second power” symbol = 4th power;
with root sign = fourth root. Shown in this e-
4
text as √.

B. Diagrams: The symbol drawn as P (Pounds) is shown here as P. See
above for X symbol.

HOTE
+C
|
|
+
|
|
+
|
|
+E
|
MOIST A TEMPERATE B
DRYE
+------+------+------+------+------+------+------
+------+
|D
|
+

|
|
+
|
|
+
|
|
+
COLD

_____________________
| | |
| {P}. 2. | Hote. 4. |
| | |
| {P}. 1. | Hote. 3. |
|_________|___________|

_____________________
| | | _
| {P}. 2. | Hote. 4. | ⅓ _ The forme_
| | | _ 3⅔ resulting.
| {P}. 1. | Hote. 3. | _ ⅔
|_________|___________|

C. “Vergilius teaches in his Georgikes.” The quoted lines, with breaks at
each “&c.”, are 438-439; 451-457; 463-464.

Euclid citations
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.

Notes on the text
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
ue
here as q .
Less common words include “fatch” (probably used as a variant of “fetch”)
and the mathematical terms “sexagene” and “sexagesme”.

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