[Transcriber’s Note:

This e-text includes characters that will only display in UTF-8
(Unicode) text readers, including a few words of Greek:

  Τακτικὴ [Taktikê]
  ã ẽ õ ũ [overline or tilde to show following -n or -m]
  ❧ ☞ [leaf symbol; pointing-finger symbol]
  ‡ [double-ended dagger, used in size notations (below)]

If any of these characters do not display properly-- in particular, if
the diacritic does not appear directly above the letter-- or if the
quotation marks in this paragraph appear as garbage, make sure your text
reader’s “character set” or “file encoding” is set to Unicode (UTF-8).
You may also need to change the default font. As a last resort, use the
ASCII version of this file instead.

Some aspects of the original book had to be modified for all versions
of this plain-text file.

Superscript letters are shown with ^: y^e, y^t.

Marginal quotation marks are shown inline as “ and ”, approximating the
beginning and end of the marked passage. In the original text, no
quotation marks were printed inline.

Paragraphs are broken up for sidenotes, with blank lines before and
after. Original paragraph breaks are shown as two blank lines. Brackets
within the body text are in the original.

All sidenotes except the one beginning “This noble Earle” were printed
in italics; markup has been omitted to reduce visual clutter.

At least four sizes of text were used, often in combination with
_italics_. The variants are shown here as:

  +‡very large‡+
  +larger+
  =smaller=

Further errors and anomalies are listed at the end of the text, along
with those Euclid citations identified by number.]

       *       *       *       *       *
           *       *       *       *
       *       *       *       *       *


  THE ELEMENTS
  OF GEOMETRIE

  of the most auncient
  Philosopher
  _EVCLIDE_
  of Megara.

  _Faithfully (now first) translated
  into the Englishe toung, by
  _H. Billingsley_, Citizen of London_.

  _Whereunto are annexed certaine
  Scholies, Annotations, and Inuentions,
  of the best Mathematiciens,
  both of time past, and
  in this our age._


  _With a very fruitfull Præface made by _M. I. Dee_,
  specifying the chiefe Mathematicall Sciẽces, what they are,
  and wherunto commodious: where, also, are disclosed
  certaine new Secrets Mathematicall and Mechanicall,
  vntill these our daies, greatly missed._


  Imprinted at London by _Iohn Daye_.




❧ The Translator to the Reader.


_There is (gentle Reader) nothing (the word of God onely set
apart) which so much beautifieth and adorneth the soule and
minde of mã, as doth the knowledge of good artes and sciences:
as the knowledge of naturall and morall Philosophie. The one
setteth before our eyes, the creatures of God, both in the
heauens aboue, and in the earth beneath: in which as in a
glasse, we beholde the exceding maiestie and wisedome of God,
in adorning and beautifying them as we see: in geuing vnto them
such wonderfull and manifolde proprieties, and naturall
workinges, and that so diuersly and in such varietie: farther in
maintaining and conseruing them continually, whereby to praise
and adore him, as by S. Paule we are taught. The other teacheth
vs rules and preceptes of vertue, how, in common life amongest
men, we ought to walke vprightly: what dueties pertaine to our
selues, what pertaine to the gouernment or good order both of an
housholde, and also of a citie or common wealth. The reading
likewise of histories, conduceth not a litle, to the adorning of
the soule & minde of man, a studie of all men cõmended: by it
are seene and knowen the artes and doinges of infinite wise men
gone before vs. In histories are contained infinite examples of
heroicall vertues to be of vs followed, and horrible examples of
vices to be of vs eschewed. Many other artes also there are
which beautifie the minde of man: but of all other none do more
garnishe & beautifie it, then those artes which are called
Mathematicall. Unto the knowledge of which no man can attaine,
without the perfecte knowledge and instruction of the
principles, groundes, and Elementes of Geometrie. But perfectly
to be instructed in them, requireth diligent studie and reading
of olde auncient authors. Amongest which, none for a beginner is
to be preferred before the most auncient Philosopher _Euclide_
of _Megara_. For of all others he hath in a true methode and
iuste order, gathered together whatsoeuer any before him had of
these Elementes written: inuenting also and adding many thinges
of his owne: wherby he hath in due forme accomplished the arte:
first geuing definitions, principles, & groundes, wherof he
deduceth his Propositions or conclusions, in such wonderfull
wise, that that which goeth before, is of necessitie required to
the proufe of that which followeth. So that without the diligent
studie of _Euclides_ Elementes, it is impossible to attaine vnto
the perfecte knowledge of Geometrie, and consequently of any of
the other Mathematicall sciences. Wherefore considering the want
& lacke of such good authors hitherto in our Englishe tounge,
lamenting also the negligence, and lacke of zeale to their
countrey in those of our nation, to whom God hath geuen both
knowledge, & also abilitie to translate into our tounge, and
to publishe abroad such good authors, and bookes (the chiefe
instrumentes of all learninges): seing moreouer that many good
wittes both of gentlemen and of others of all degrees, much
desirous and studious of these artes, and seeking for them as
much as they can, sparing no paines, and yet frustrate of their
intent, by no meanes attaining to that which they seeke: I haue
for their sakes, with some charge & great trauaile, faithfully
translated into our vulgare toũge, & set abroad in Print, this
booke of _Euclide_. Whereunto I haue added easie and plaine
declarations and examples by figures, of the definitions. In
which booke also ye shall in due place finde manifolde
additions, Scholies, Annotations, and Inuentions: which I haue
gathered out of many of the most famous & chiefe Mathematiciẽs,
both of old time, and in our age: as by diligent reading it in
course, ye shall well perceaue. The fruite and gaine which I
require for these my paines and trauaile, shall be nothing els,
but onely that thou gentle reader, will gratefully accept the
same: and that thou mayest thereby receaue some profite: and
moreouer to excite and stirre vp others learned, to do the like,
& to take paines in that behalfe. By meanes wherof, our Englishe
tounge shall no lesse be enriched with good Authors, then are
other straunge tounges: as the Dutch, French, Italian, and
Spanishe: in which are red all good authors in a maner, found
amongest the Grekes or Latines. Which is the chiefest cause,
that amongest thẽ do florishe so many cunning and skilfull men,
in the inuentions of straunge and wonderfull thinges, as in
these our daies we see there do. Which fruite and gaine if I
attaine vnto, it shall encourage me hereafter, in such like sort
to translate, and set abroad some other good authors, both
pertaining to religion (as partly I haue already done) and also
pertaining to the Mathematicall Artes. Thus gentle
reader farewell._ (?¿)


    [Decoration]




  ❧ TO THE VNFAINED LOVERS
  of truthe, and constant Studentes of Noble
  _Sciences, _IOHN DEE_ of London, hartily_
  wisheth grace from heauen, and most prosperous
  _successe in all their honest attemptes and_
  exercises.


Diuine _Plato_, the great Master of many worthy Philosophers, and the
constant auoucher, and pithy perswader of _Vnum_, _Bonum_, and _Ens_: in
his Schole and Academie, sundry times (besides his ordinary Scholers)
was visited of a certaine kinde of men, allured by the noble fame of
_Plato_, and the great commendation of hys profound and profitable
doctrine. But when such Hearers, after long harkening to him, perceaued,
that the drift of his discourses issued out, to conclude, this _Vnum_,
_Bonum_, and _Ens_, to be Spirituall, Infinite, Æternall, Omnipotent,
&c. Nothyng beyng alledged or expressed, How, worldly goods: how,
worldly dignitie: how, health, Strẽgth or lustines of body: nor yet the
meanes, how a merueilous sensible and bodyly blysse and felicitie
hereafter, might be atteyned: Straightway, the fantasies of those
hearers, were dampt: their opinion of _Plato_, was clene chaunged: yea
his doctrine was by them despised: and his schole, no more of them
visited. Which thing, his Scholer, _Aristotle_, narrowly cõsidering,
founde the cause therof, to be, “For that they had no forwarnyng and
information, in generall,” whereto his doctrine tended. For, so, might
they haue had occasion, either to haue forborne his schole hauntyng: (if
they, then, had misliked his Scope and purpose) or constantly to haue
continued therin: to their full satisfaction: if such his finall scope &
intent, had ben to their desire. Wherfore, _Aristotle_, euer, after
that, vsed in brief, to forewarne his owne Scholers and hearers, “both
of what matter, and also to what ende, he tooke in hand to speake, or
teach.” While I consider the diuerse trades of these two excellent
Philosophers (and am most sure, both, that _Plato_ right well, otherwise
could teach: and that _Aristotle_ mought boldely, with his hearers, haue
dealt in like sorte as _Plato_ did) I am in no little pang of
perplexitie: Bycause, that, which I mislike, is most easy for me to
performe (and to haue _Plato_ for my exãple.) And that, which I know to
be most commendable: and (in this first bringyng, into common handling,
the _Artes Mathematicall_) to be most necessary: is full of great
difficultie and sundry daungers. Yet, neither do I think it mete, for so
straunge matter (as now is ment to be published) and to so straunge an
audience, to be bluntly, at first, put forth, without a peculiar
Preface: Nor (Imitatyng _Aristotle_) well can I hope, that accordyng to
the amplenes and dignitie of the _State Mathematicall_, I am able,
either playnly to prescribe the materiall boundes: or precisely to
expresse the chief purposes, and most wonderfull applications therof.
And though I am sure, that such as did shrinke from _Plato_ his schole,
after they had perceiued his finall conclusion, would in these thinges
haue ben his most diligent hearers (so infinitely mought their desires,
in fine and at length, by our _Artes Mathematicall_ be satisfied) yet,
by this my Præface & forewarnyng, Aswell all such, may (to their great
behofe) the soner, hither be allured: as also the _Pythagoricall_, and
_Platonicall_ perfect scholer, and the constant profound Philosopher,
with more ease and spede, may (like the Bee,) gather, hereby, both wax
and hony.


    [The intent of this Preface.]

Wherfore, seyng I finde great occasion (for the causes alleged, and
farder, in respect of my _Art Mathematike generall_) to vse “a certaine
forewarnyng and Præface, whose content shalbe, that mighty, most
plesaunt, and frutefull _Mathematicall Tree_, with his chief armes and
second (grifted) braunches: Both, what euery one is, and also, what
commodity, in generall, is to be looked for, aswell of griff as stocke:
And forasmuch as this enterprise is so great, that, to this our tyme, it
neuer was (to my knowledge) by any achieued: And also it is most hard,
in these our drery dayes, to such rare and straunge Artes, to wyn due
and common credit:” Neuertheles, if, for my sincere endeuour to satisfie
your honest expectation, you will but lend me your thãkefull mynde a
while: and, to such matter as, for this time, my penne (with spede) is
hable to deliuer, apply your eye or eare attentifely: perchaunce, at
once, and for the first salutyng, this Preface you will finde a lesson
long enough. And either you will, for a second (by this) be made much
the apter: or shortly become, well hable your selues, of the lyons claw,
to coniecture his royall symmetrie, and farder propertie. Now then,
gentle, my frendes, and countrey men, Turne your eyes, and bend your
myndes to that doctrine, which for our present purpose, my simple talent
is hable to yeld you.


All thinges which are, & haue beyng, are found vnder a triple diuersitie
generall. For, either, they are demed Supernaturall, Naturall, or, of a
third being. Thinges Supernaturall, are immateriall, simple,
indiuisible, incorruptible, & vnchangeable. Things Naturall, are
materiall, compounded, diuisible, corruptible, and chaungeable. Thinges
Supernaturall, are, of the minde onely, comprehended: Things Naturall,
of the sense exterior, ar hable to be perceiued. In thinges Naturall,
probabilitie and coniecture hath place: But in things Supernaturall,
chief demõstration, & most sure Science is to be had. By which
properties & comparasons of these two, more easily may be described, the
state, condition, nature and property of those thinges, which, we before
termed of a third being: which, by a peculier name also, are called
_Thynges Mathematicall_. For, these, beyng (in a maner) middle, betwene
thinges supernaturall and naturall: are not so absolute and excellent,
as thinges supernatural: Nor yet so base and grosse, as things naturall:
But are thinges immateriall: and neuerthelesse, by materiall things
hable somewhat to be signified. And though their particular Images, by
Art, are aggregable and diuisible: yet the generall _Formes_,
notwithstandyng, are constant, vnchaungeable, vntrãsformable, and
incorruptible. Neither of the sense, can they, at any tyme, be perceiued
or iudged. Nor yet, for all that, in the royall mynde of man, first
conceiued. But, surmountyng the imperfectiõ of coniecture, weenyng and
opinion: and commyng short of high intellectuall cõceptiõ, are the
Mercurial fruite of _Dianœticall_ discourse, in perfect imagination
subsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_,
and also a straunge participatiõ betwene thinges supernaturall,
immortall, intellectual, simple and indiuisible: and thynges naturall,
mortall, sensible, compounded and diuisible. Probabilitie and sensible
prose, may well serue in thinges naturall: and is commendable: In
Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor
yet the testimony of sense, any whit credited: But onely a perfect
demonstration, of truthes certaine, necessary, and inuincible:
vniuersally and necessaryly concluded: is allowed as sufficient for “an
Argument exactly and purely Mathematical.”


    [Note the worde, Vnit, to expresse the Greke Monas,
    & not Vnitie: as we haue all, commonly, till now, vsed.]

Of _Mathematicall_ thinges, are two principall kindes: namely, _Number_,
and _Magnitude_.

    [Number.]

_Number_, we define, to be, a certayne Mathematicall Sũme, of _Vnits_.
And, an _Vnit_, is that thing Mathematicall, Indiuisible, by
participation of some likenes of whose property, any thing, which is in
deede, or is counted One, may resonably be called One. We account an
_Vnit_, a thing _Mathematicall_, though it be no Number, and also
indiuisible: because, of it, materially, Number doth consist: which,
principally, is a thing _Mathematicall_.

    [Magnitude.]

_Magnitude_ is a thing _Mathematicall_, by participation of some likenes
of whose nature, any thing is iudged long, broade, or thicke. “A thicke
_Magnitude_ we call a _Solide_, or a _Body_. What _Magnitude_ so euer,
is Solide or Thicke, is also broade, & long. A broade magnitude, we call
a _Superficies_ or a Plaine. Euery playne magnitude, hath also length.
A long magnitude, we terme a _Line_. A _Line_ is neither thicke nor
broade, but onely long: Euery certayne Line, hath two endes:

    [A point.]

The endes of a line, are _Pointes_ called. A _Point_, is a thing
_Mathematicall_, indiuisible, which may haue a certayne determined
situation.” If a Poynt moue from a determined situation, the way wherein
it moued, is also a _Line_: mathematically produced, whereupon, of the
auncient Mathematiciens,

    [A Line.]

a _Line_ is called the race or course of a _Point_. A Poynt we define,
by the name of a thing Mathematicall: though it be no Magnitude, and
indiuisible: because it is the propre ende, and bound of a Line: which
is a true _Magnitude_.

    [Magnitude.]

And _Magnitude_ we may define to be that thing _Mathematicall_, which is
diuisible for euer, in partes diuisible, long, broade or thicke.
Therefore though a Poynt be no _Magnitude_, yet _Terminatiuely_, we
recken it a thing _Mathematicall_ (as I sayd) by reason it is properly
the end, and bound of a line. Neither _Number_, nor _Magnitude_, haue
any Materialitie. First, we will consider of _Number_, and of the
Science _Mathematicall_, to it appropriate, called _Arithmetike_: and
afterward of _Magnitude_, and his Science, called _Geometrie_. But that
name contenteth me not: whereof a word or two hereafter shall be sayd.
How Immateriall and free from all matter, _Number_ is, who doth not
perceaue? yea, who doth not wonderfully wõder at it? For, neither pure
_Element_, nor _Aristoteles, Quinta Essentia_, is hable to serue for
Number, as his propre matter. Nor yet the puritie and simplenes of
Substance Spirituall or Angelicall, will be found propre enough thereto.
And therefore the great & godly Philosopher _Anitius Boetius_, sayd:
_Omnia quæcun[que] a primæua rerum natura constructa sunt, Numerorum
videntur ratione formata. Hoc enim fuit principale in animo Conditoris
Exemplar_. That is: +_All thinges (which from the very first originall
being of thinges, haue bene framed and made) do appeare to be Formed by
the reason of Numbers. For this was the principall example or patterne
in the minde of the Creator_.+ O comfortable allurement, O rauishing
perswasion, to deale with a Science, whose Subiect, is so Auncient, so
pure, so excellent, so surmounting all creatures, so vsed of the
Almighty and incomprehensible wisdome of the Creator, in the distinct
creation of all creatures: in all their distinct partes, properties,
natures, and vertues, by order, and most absolute number, brought, from
_Nothing_, to the _Formalitie_ of their being and state. By _Numbers_
propertie therefore, of vs, by all possible meanes, (to the perfection
of the Science) learned, we may both winde and draw our selues into the
inward and deepe search and vew, of all creatures distinct vertues,
natures, properties, and _Formes_: And also, farder, arise, clime,
ascend, and mount vp (with Speculatiue winges) in spirit, to behold in
the Glas of Creation, the _Forme of Formes_, the _Exemplar Number_ of
all thinges _Numerable_: both visible and inuisible, mortall and
immortall, Corporall and Spirituall. Part of this profound and diuine
Science, had _Ioachim_ the Prophesier atteyned vnto: by _Numbers
Formall, Naturall_, and _Rationall_, forseyng, concludyng, and
forshewyng great particular euents, long before their comming. His
bookes yet remainyng, hereof, are good profe: And the noble Earle of
_Mirandula_, (besides that,) a sufficient witnesse: that _Ioachim, in
his prophesies, proceded by no other way, then by Numbers Formall_. And
this Earle hym selfe, in Rome,

    [Ano. 1488.]

* set vp 900. Conclusions, in all kinde of Sciences, openly to be
disputed of: and among the rest, in his Conclusions _Mathematicall_, (in
the eleuenth Conclusion) hath in Latin, this English sentence. _By
Numbers, a way is had, to the searchyng out, and vnderstandyng of euery
thyng, hable to be knowen. For the verifying of which Conclusion,
I promise to aunswere to the 74. Questions, vnder written, by the way of
Numbers_. Which Cõclusions, I omit here to rehearse: aswell auoidyng
superfluous prolixitie: as, bycause _Ioannes Picus, workes_, are
commonly had. But, in any case, I would wish that those Conclusions were
red diligently, and perceiued of such, as are earnest Obseruers and
Considerers of the constant law of nũbers: which is planted in thyngs
Naturall and Supernaturall: and is prescribed to all Creatures,
inuiolably to be kept. For, so, besides many other thinges, in those
Conclusions to be marked, it would apeare, how sincerely, & within my
boundes, I disclose the wonderfull mysteries, by numbers, to be atteyned
vnto.


Of my former wordes, easy it is to be gathered, that _Number_ hath a
treble state: One, in the Creator: an other in euery Creature (in
respect of his complete constitution:) and the third, in Spirituall and
Angelicall Myndes, and in the Soule of mã. In the first and third state,
_Number_, is termed _Number Numbryng_. But in all Creatures, otherwise,
_Number_, is termed _Nũber Numbred_. And in our Soule, Nũber beareth
such a swaye, and hath such an affinitie therwith: that some of the old
_Philosophers_ taught, _Mans Soule, to be a Number mouyng it selfe_. And
in dede, in vs, though it be a very Accident: yet such an Accident it
is, that before all Creatures it had perfect beyng, in the Creator,
Sempiternally. _Number Numbryng_ therfore, is the discretion discerning,
and distincting of thinges. But in God the Creator, This discretion, in
the beginnyng, produced orderly and distinctly all thinges. For his
_Numbryng_, then, was his Creatyng of all thinges. And his Continuall
_Numbryng_, of all thinges, is the Conseruation of them in being: And,
where and when he will lacke an _Vnit_: there and then, that particular
thyng shalbe _Discreated_. Here I stay. But our Seuerallyng,
distinctyng, and _Numbryng_, createth nothyng: but of Multitude
considered, maketh certaine and distinct determination. And albeit these
thynges be waighty and truthes of great importance, yet (by the infinite
goodnes of the Almighty _Ternarie_,) Artificiall Methods and easy wayes
are made, by which the zelous Philosopher, may wyn nere this Riuerish
_Ida_, this Mountayne of Contemplation: and more then Contemplation. And
also, though _Number_, be a thyng so Immateriall, so diuine, and
æternall: yet by degrees, by litle and litle, stretchyng forth, and
applying some likenes of it, as first, to thinges Spirituall: and then,
bryngyng it lower, to thynges sensibly perceiued: as of a momentanye
sounde iterated: then to the least thynges that may be seen, numerable:
And at length, (most grossely,) to a multitude of any corporall thynges
seen, or felt: and so, of these grosse and sensible thynges, we are
trayned to learne a certaine Image or likenes of numbers: and to vse
Arte in them to our pleasure and proffit. So grosse is our conuersation,
and dull is our apprehension: while mortall Sense, in vs, ruleth the
common wealth of our litle world. Hereby we say, Three Lyons, are three:
or a _Ternarie_. Three Egles, are three, or a _Ternarie_.

    [☞]

Which * _Ternaries_, are eche, the _Vnion_, _knot_, and _Vniformitie_,
of three discrete and distinct _Vnits_. That is, we may in eche
_Ternarie_, thrise, seuerally pointe, and shew a part, _One_, _One_, and
_One_. Where, in Numbryng, we say One, two, Three. But how farre, these
visible Ones, do differre from our Indiuisible Vnits (in pure
_Arithmetike_, principally considered) no man is ignorant. Yet from
these grosse and materiall thynges, may we be led vpward, by degrees,
so, informyng our rude Imagination, toward the cõceiuyng of _Numbers_,
absolutely (:Not supposing, nor admixtyng any thyng created, Corporall
or Spirituall, to support, conteyne, or represent those _Numbers_
imagined:) that at length, we may be hable, to finde the number of our
owne name, gloriously exemplified and registred in the booke of the
_Trinitie_ most blessed and æternall.


But farder vnderstand, that vulgar Practisers, haue Numbers, otherwise,
in sundry Considerations: and extend their name farder, then to Numbers,
whose least part is an _Vnit_. For the common Logist, Reckenmaster, or
Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse
partes: and calleth them _Fractions_. As of an _Vnit_, he maketh an
halfe, and thus noteth it, ½. and so of other, (infinitely diuerse)
partes of an _Vnit_. Yea and farder, hath, _Fractions of Fractions. &c_.
And, forasmuch, as, _Addition_, _Substraction_, _Multiplication_,
_Diuision_ and _Extraction of Rotes_, are the chief, and sufficient
partes of _Arithmetike_:

    [Arithmetike.]

which is, the _Science that demonstrateth the properties, of Numbers,
and all operatiõs, in numbers to be performed_:

    [Note.]

“How often, therfore, these fiue sundry sortes of Operations, do, for
the most part, of their execution, differre from the fiue operations of
like generall property and name, in our Whole numbers practisable, So
often, (for a more distinct doctrine) we, vulgarly account and name it,
an other kynde of _Arithmetike_.” And by this reason:

    [1.]

the Consideration, doctrine, and working, in whole numbers onely: where,
of an _Vnit_, is no lesse part to be allowed: is named (as it were) an
_Arithmetike_ by it selfe. And so of the _Arithmetike of Fractions_.

    [2.]

In lyke sorte, the necessary, wonderfull and Secret doctrine of
Proportion, and proportionalytie hath purchased vnto it selfe a peculier
maner of handlyng and workyng: and so may seme an other forme of
_Arithmetike_.

    [3.]

Moreouer, the _Astronomers_, for spede and more commodious calculation,
haue deuised a peculier maner of orderyng nũbers, about theyr circular
motions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes
&c. which commonly is called the _Arithmetike_ of _Astronomical_ or
_Phisicall Fractions_. That, haue I briefly noted, by the name of
_Arithmetike Circular_. Bycause it is also vsed in circles, not
_Astronomicall. &c._

    [4.]

Practise hath led _Numbers_ farder, and hath framed them, to take vpon
them, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_
and _Irrationalitie_. (For in pure _Arithmetike_, an _Vnit_, is the
common Measure of all Numbers.) And, here, Nũbers are become, as Lynes,
Playnes and Solides: some tymes _Rationall_, some tymes _Irrationall_.
And haue propre and peculier characters, (as ²√. ³√. and so of other.
Which is to signifie _Rote Square, Rote Cubik: and so forth_:) & propre
and peculier fashions in the fiue principall partes: Wherfore the
practiser, estemeth this, a diuerse _Arithmetike_ from the other.
Practise bryngeth in, here, diuerse compoundyng of Numbers: as some
tyme, two, three, foure (or more) _Radicall_ nũbers, diuersly knit, by
signes, of More & Lesse: as thus ²√12 + ³√15. Or thus ⁴√19 + ³√12 - ²√2.
&c. And some tyme with whole numbers, or fractions of whole Number, amõg
them: as 20 + ²√24. ³√16 + 33 - ²√10. ⁴√44 + 12¼ + ³√9. And so,
infinitely, may hap the varietie. After this: Both the one and the other
hath fractions incident: and so is this _Arithmetike_ greately enlarged,
by diuerse exhibityng and vse of Compositions and mixtynges. Consider
how, I (beyng desirous to deliuer the student from error and
Cauillation) do giue to this _Practise_, the name of the _Arithmetike of
Radicall numbers_: Not, of _Irrationall_ or _Surd Numbers_: which other
while, are Rationall: though they haue the Signe of a Rote before them,
which, _Arithmetike_ of whole Numbers most vsuall, would say they had no
such Roote: and so account them _Surd Numbers_: which, generally spokẽ,
is vntrue: as _Euclides_ tenth booke may teach you. Therfore to call
them, generally, _Radicall Numbers_, (by reason of the signe √.
prefixed,) is a sure way: and a sufficient generall distinction from all
other ordryng and vsing of Numbers: And yet (beside all this) Consider:
the infinite desire of knowledge, and incredible power of mans Search
and Capacitye: how, they, ioyntly haue waded farder (by mixtyng of
speculation and practise) and haue found out, and atteyned to the very
chief perfection (almost) of _Numbers_ Practicall vse. Which thing, is
well to be perceiued in that great Arithmeticall Arte of _Æquation_:
commonly called the _Rule of Coss._ or _Algebra_. The Latines termed it,
_Regulam Rei & Census_, that is, the +_Rule of the thyng and his
value_+. With an apt name: comprehendyng the first and last pointes of
the worke. And the vulgar names, both in Italian, Frenche and Spanish,
depend (in namyng it,) vpon the signification of the Latin word, _Res_:
+_A thing_+: vnleast they vse the name of _Algebra_. And therin
(commonly) is a dubble error. The one, of them, which thinke it to be of
_Geber_ his inuentyng: the other of such as call it _Algebra_. For,
first, though _Geber_ for his great skill in Numbers, Geometry,
Astronomy, and other maruailous Artes, mought haue semed hable to haue
first deuised the sayd Rule: and also the name carryeth with it a very
nere likenes of _Geber_ his name: yet true it is, that a _Greke_
Philosopher and Mathematicien, named _Diophantus_, before _Geber_ his
tyme, wrote 13. bookes therof (of which, six are yet extant: and I had
them to *vse,

    [* Anno. 1550.]

of the famous Mathematicien, and my great frende, _Petrus Montaureus_:)
And secondly, the very name, is _Algiebar_, and not _Algebra_: as by the
Arabien _Auicen_, may be proued: who hath these precise wordes in
Latine, by _Andreas Alpagus_ (most perfect in the Arabik tung) so
translated. _Scientia faciendi Algiebar & Almachabel. i. Scientia
inueniendi numerum ignotum, per additionem Numeri, & diuisionem &
æquationem_. Which is to say: +_The Science of workyng Algiebar and
Almachabel_+, that is, the +_Science of findyng an vnknowen number, by
Addyng of a Number, & Diuision & æquation_+. Here haue you the name: and
also the principall partes of the Rule, touched. To name it, _The rule,
or Art of Æquation_, doth signifie the middle part and the State of the
Rule. This Rule, hath his peculier Characters:

    [5.]

and the principal partes of _Arithmetike_, to it appertayning, do
differre from the other _Arithmeticall operations_. This _Arithmetike,
hath Nũbers_ Simple, Cõpound, Mixt: and Fractions, accordingly. This
Rule, and _Arithmetike of Algiebar_, is so profound, so generall and so
(in maner) conteyneth the whole power of Numbers Application practicall:
that mans witt, can deale with nothyng, more proffitable about numbers:
nor match, with a thyng, more mete for the diuine force of the Soule,
(in humane Studies, affaires, or exercises) to be tryed in. Perchaunce
you looked for, (long ere now,) to haue had some particular profe, or
euident testimony of the vse, proffit and Commodity of Arithmetike
vulgar, in the Common lyfe and trade of men. Therto, then, I will now
frame my selfe: But herein great care I haue, least length of sundry
profes, might make you deme, that either I did misdoute your zelous
mynde to vertues schole: or els mistrust your hable witts, by some, to
gesse much more. A profe then, foure, fiue, or six, such, will I bryng,
as any reasonable man, therwith may be persuaded, to loue & honor, yea
learne and exercise the excellent Science of _Arithmetike_.


And first: who, nerer at hand, can be a better witnesse of the frute
receiued by _Arithmetike_, then all kynde of Marchants? Though not all,
alike, either nede it, or vse it. How could they forbeare the vse and
helpe of the Rule, called the Golden Rule? Simple and Compounde: both
forward and backward? How might they misse _Arithmeticall_ helpe in the
Rules of Felowshyp: either without tyme, or with tyme? and betwene the
Marchant & his Factor? The Rules of Bartering in wares onely: or part in
wares, and part in money, would they gladly want? Our Marchant
venturers, and Trauaylers ouer Sea, how could they order their doynges
iustly and without losse, vnleast certaine and generall Rules for
Exchaũge of money, and Rechaunge, were, for their vse, deuised? The Rule
of Alligation, in how sundry cases, doth it conclude for them, such
precise verities, as neither by naturall witt, nor other experience,
they, were hable, els, to know? And (with the Marchant then to make an
end) how ample & wonderfull is the Rule of False positions? especially
as it is now, by two excellent Mathematiciens (of my familier
acquayntance in their life time) enlarged? I meane _Gemma Frisius_, and
_Simon Iacob_. Who can either in brief conclude, the generall and
Capitall Rules? or who can Imagine the Myriades of sundry Cases, and
particular examples, in Act and earnest, continually wrought, tried and
concluded by the forenamed Rules, onely? How sundry other _Arithmeticall
practises_, are commonly in Marchantes handes, and knowledge: They them
selues, can, at large, testifie.


The Mintmaster, and Goldsmith, in their Mixture of Metals, either of
diuerse kindes, or diuerse values: how are they, or may they, exactly be
directed, and meruailously pleasured, if _Arithmetike_ be their guide?
And the honorable Phisiciãs, will gladly confesse them selues, much
beholding to the Science of _Arithmetike_, and that sundry wayes: But
chiefly in their Art of Graduation, and compounde Medicines. And though
_Galenus_, _Auerrois_, _Arnoldus_, _Lullus_, and other haue published
their positions, aswell in the quantities of the Degrees aboue
Temperament, as in the Rules, concluding the new _Forme_ resulting: yet
a more precise, commodious, and easy _Method_, is extant: by a
Countreyman of ours

    [R. B.]

(aboue 200. yeares ago) inuented. And forasmuch as I am vncertaine, who
hath the same: or when that litle Latin treatise, (as the Author writ
it,) shall come to be Printed: (Both to declare the desire I haue to
pleasure my Countrey, wherin I may: and also, for very good profe of
Numbers vse, in this most subtile and frutefull, Philosophicall
Conclusion,) I entend in the meane while, most briefly, and with my
farder helpe, to communicate the pith therof vnto you.


First describe a circle: whose diameter let be an inch. Diuide the
Circumference into foure equall partes. Frõ the Center, by those 4.
sections, extend 4. right lines: eche of 4. inches and a halfe long: or
of as many as you liste, aboue 4. without the circumference of the
circle: So that they shall be of 4. inches long (at the least) without
the Circle. Make good euident markes, at euery inches end. If you list,
you may subdiuide the inches againe into 10. or 12. smaller partes,
equall. At the endes of the lines, write the names of the 4. principall
elementall Qualities. _Hote_ and _Colde_, one against the other. And
likewise _Moyst_ and _Dry_, one against the other. And in the Circle
write _Temperate_. Which _Temperature_ hath a good Latitude: as
appeareth by the Complexion of man. And therefore we haue allowed vnto
it, the foresayd Circle: and not a point Mathematicall or Physicall.


    [* Take some part of Lullus counsayle in his booke
    de Q. Essentia.]

Now, when you haue two thinges Miscible, whose degrees are * truely
knowen: Of necessitie, either they are of one Quantitie and waight, or
of diuerse. If they be of one Quantitie and waight: whether their
formes, be Contrary Qualities, or of one kinde (but of diuerse
intentions and degrees) or a _Temperate_, and a Contrary, _The forme
resulting of their Mixture, is in the Middle betwene the degrees of the
formes mixt_. As for example, let _A_, be _Moist_ in the first degree:
and _B_, _Dry_ in the third degree. Adde 1. and 3. that maketh 4: the
halfe or middle of 4. is 2. This 2. is the middle, equally distant from
_A_ and _B_

    [* Note.]

(for the * _Temperament_ is counted none. And for it, you must put a
Ciphre, if at any time, it be in mixture).

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

Counting then from _B_, 2. degrees, toward _A_: you finde it to be _Dry_
in the first degree: So is the _Forme resulting_ of the Mixture of _A_,
and _B_, in our example. I will geue you an other example. Suppose, you
haue two thinges, as _C_, and _D_: and of _C_, the Heate to be in the 4.
degree: and of _D_, the Colde, to be remisse, euen vnto the
_Temperament_. Now, for _C_, you take 4: and for _D_, you take a Ciphre:
which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is
2. Wherefore the _Forme resulting_ of _C_, and _D_, is Hote in the
second degree: for, 2. degrees, accounted from _C_, toward _D_, ende
iuste in the 2. degree of heate. Of the third maner, I will geue also an
example: which let be this:

    [Note.]

I haue a liquid Medicine whose Qualitie of heate is in the 4. degree
exalted: as was _C_, in the example foregoing: and an other liquid
Medicine I haue: whose Qualitie, is heate, in the first degree. Of eche
of these, I mixt a like quantitie: Subtract here, the lesse frõ the
more: and the residue diuide into two equall partes: whereof, the one
part, either added to the lesse, or subtracted from the higher degree,
doth produce the degree of the Forme resulting, by this mixture of _C_,
and _E_. As, if from 4. ye abate 1. there resteth 3. the halfe of 3. is
1½: Adde to 1. this 1½: you haue 2½. Or subtract from 4. this 1½: you
haue likewise 2½ remayning. Which declareth, the _Forme resulting_, to
be _Heate_, in the middle of the third degree.


    [The Second Rule.]

“But if the Quantities of two thinges Commixt, be diuerse, and the
Intensions (of their Formes Miscible) be in diuerse degrees, and
heigthes. (Whether those Formes be of one kinde, or of Contrary kindes,
or of a Temperate and a Contrary, _What proportion is of the lesse
quantitie to the greater, the same shall be of the difference, which is
betwene the degree of the Forme resulting, and the degree of the greater
quantitie of the thing miscible, to the difference, which is betwene the
same degree of the Forme resulting, and the degree of the lesse
quantitie_. As for example. Let two pound of Liquor be geuen, hote in
the 4. degree: & one pound of Liquor be geuen, hote in the third
degree.” I would gladly know the Forme resulting, in the Mixture of
these two Liquors. Set downe your nũbers in order, thus.
   ___________________________
  |            |              |
  |  {P}. _2._ |  _Hote. 4._  |
  |            |              |
  |  {P}. _1._ |  _Hote. 3._  |
  |____________|______________|

Now by the rule of Algiebar, haue I deuised a very easie, briefe, and
generall maner of working in this case. Let vs first, suppose that
_Middle Forme resulting_, to be 1{x}: as that Rule teacheth. And because
(by our Rule, here geuen) as the waight of 1. is to 2: So is the
difference betwene 4. (the degree of the greater quantitie) and 1{x}: to
the difference betwene 1{x} and 3: (the degree of the thing, in lesse
quãtitie. And with all, 1{x}, being alwayes in a certaine middell,
betwene the two heigthes or degrees). For the first difference, I set
4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1.
is to 2. so is 4-1{x} to 1{x}-3. Wherfore, of these foure proportionall
numbers, the first and the fourth Multiplied, one by the other, do make
as much, as the second and the third Multiplied the one by the other.
Let these Multiplications be made accordingly. And of the first and the
fourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore,
our Æquation is betwene 1{x}-3: and 8-2{x}. Which may be reduced,
according to the Arte of Algiebar: as, here, adding 3. to eche part,
geueth the Æquation, thus, 1{x}=11-2{x}. And yet againe, contracting, or
Reducing it: Adde to eche part, 2{x}: Then haue you 3{x} æquall to 11:
thus represented 3{x}=11. Wherefore, diuiding 11. by 3: the Quotient is
3⅔: the _Valew_ of our 1{x}, _Coss_, or _Thing_, first supposed. And
that is the heigth, or Intension of the _Forme resulting:_ which is,
_Heate_, in two thirdes of the fourth degree: And here I set the shew of
the worke in conclusion, thus. The proufe hereof is easie: by
subtracting 3. from 3⅔, resteth ⅔. Subtracte the same heigth of the
Forme resulting, (which is 3⅔) frõ 4: then resteth ⅓: You see, that ⅔ is
double to ⅓: as 2.{P}. is double to 1.{P}. So should it be: by the rule
here geuen. Note. As you added to eche part of the Æquation, 3: so if ye
first added to eche part 2{x}, it would stand, 3{x}-3=8. And now adding
to eche part 3: you haue (as afore) 3{x}=11.
    _________________________
   |           |             | _
   | {P}. _2._ |  _Hote. 4._ |   ⅓ _   _The forme_
   |           |             |     _   _3⅔ resulting._
   | {P}. _1._ |  _Hote. 3._ | _ ⅔
   |___________|_____________|


And though I, here, speake onely of two thyngs Miscible: and most
commonly mo then three, foure, fiue or six, (&c.) are to be Mixed: (and
in one Compound to be reduced: & the Forme resultyng of the same, to
serue the turne) yet these Rules are sufficient: duely repeated and
iterated.

    [Note.]

In procedyng first, with any two: and then, with the Forme Resulting,
and an other: & so forth: For, the last worke, concludeth the Forme
resultyng of them all: I nede nothing to speake, of the Mixture (here
supposed) what it is. Common Philosophie hath defined it, saying,
_Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio_.
Euery word in the definition, is of great importance. I nede not also
spend any time, to shew, how, the other manner of distributing of
degrees, doth agree to these Rules. Neither nede I of the farder vse
belonging to the Crosse of Graduation (before described) in this place
declare, vnto such as are capable of that, which I haue all ready sayd.
Neither yet with examples specifie the Manifold varieties, by the
foresayd two generall Rules, to be ordered. The witty and Studious,
here, haue sufficient: And they which are not hable to atteine to this,
without liuely teaching, and more in particular: would haue larger
discoursing, then is mete in this place to be dealt withall: And other
(perchaunce) with a proude snuffe will disdaine this litle: and would be
vnthankefull for much more. I, therfore conclude: and wish such as haue
modest and earnest Philosophicall mindes, to laude God highly for this:
and to Meruayle, that the profoundest and subtilest point, concerning
_Mixture of Formes and Qualities Naturall_, is so Matcht and maryed with
the most simple, easie, and short way of the noble Rule of _Algiebar_.
Who can remaine, therfore vnpersuaded, to loue, alow, and honor the
excellent Science of _Arithmetike_? For, here, you may perceiue that the
litle finger of _Arithmetike_, is of more might and contriuing, then a
hunderd thousand mens wittes, of the middle sorte, are hable to
perfourme, or truely to conclude, with out helpe thereof.


Now will we farder, by the wise and valiant Capitaine, be certified,
what helpe he hath, by the Rules of _Arithmetike_: in one of the Artes
to him appertaining: And of the Grekes named

    [Τακτικὴ.]

Τακτικὴ. “That is, the Skill of Ordring Souldiers in Battell ray after
the best maner to all purposes.” This Art so much dependeth vppon
Numbers vse, and the Mathematicals, that _Ælianus_ (the best writer
therof,) in his worke, to the _Emperour Hadrianus_, by his perfection,
in the Mathematicals, (beyng greater, then other before him had,)
thinketh his booke to passe all other the excellent workes, written of
that Art, vnto his dayes. For, of it, had written _Æneas_: _Cyneas_ of
_Thessaly_: _Pyrrhus Epirota_: and _Alexander_ his sonne: _Clearchus_:
_Pausanias_: _Euangelus_: _Polybius_, familier frende to _Scipio_:
_Eupolemus_: _Iphicrates_, _Possidonius_: and very many other worthy
Capitaines, Philosophers and Princes of Immortall fame and memory: Whose
fayrest floure of their garland (in this feat) was _Arithmetike_: and a
litle perceiuerance, in _Geometricall_ Figures. But in many other cases
doth _Arithmetike_ stand the Capitaine in great stede. As in
proportionyng of vittayles, for the Army, either remaining at a stay: or
suddenly to be encreased with a certaine number of Souldiers: and for a
certain tyme. Or by good Art to diminish his company, to make the
victuals, longer to serue the remanent, & for a certaine determined
tyme: if nede so require. And so in sundry his other accountes,
Reckeninges, Measurynges, and proportionynges, the wise, expert, and
Circumspect Capitaine will affirme the Science of _Arithmetike_, to be
one of his chief Counsaylors, directers and aiders. Which thing (by good
meanes) was euident to the Noble, the Couragious, the loyall, and
Curteous

    [☞]

_Iohn_, late Earle of Warwicke. Who was a yong Gentleman, throughly
knowne to very few. Albeit his lusty valiantnes, force, and Skill in
Chiualrous feates and exercises: his humblenes, and frendelynes to all
men, were thinges, openly, of the world perceiued. But what rotes
(otherwise,) vertue had fastened in his brest, what Rules of godly and
honorable life he had framed to him selfe: what vices, (in some then
liuing) notable, he tooke great care to eschew: what manly vertues, in
other noble men, (florishing before his eyes,) he Sythingly aspired
after: what prowesses he purposed and ment to achieue: with what feats
and Artes, he began to furnish and fraught him selfe, for the better
seruice of his Kyng and Countrey, both in peace & warre. These (I say)
his Heroicall Meditations, forecastinges and determinations, no twayne,
(I thinke) beside my selfe, can so perfectly, and truely report. And
therfore, in Conscience, I count it my part, for the honor, preferment,
& procuring of vertue (thus, briefly) to haue put his Name, in the
Register of _Fame Immortall_.


To our purpose. This _Iohn_, by one of his actes (besides many other:
both in England and Fraunce, by me, in him noted.) did disclose his
harty loue to vertuous Sciences: and his noble intent, to excell in
Martiall prowesse: When he, with humble request, and instant
Solliciting: got the best Rules (either in time past by Greke or
Romaine, or in our time vsed: and new Stratagemes therin deuised) for
ordring of all Companies, summes and Numbers of mẽ, (Many, or few) with
one kinde of weapon, or mo, appointed: with Artillery, or without: on
horsebacke, or on fote: to giue, or take onset: to seem many, being few:
to seem few, being many. To marche in battaile or Iornay: with many such
feates, to Foughten field, Skarmoush, or Ambushe appartaining:

    [This noble Earle, dyed Anno. 1554. skarse of 24. yeares
    of age: hauing no issue by his wife: Daughter to the Duke
    of Somerset.]

And of all these, liuely designementes (most curiously) to be in velame
parchement described: with Notes & peculier markes, as the Arte
requireth: and all these Rules, and descriptions Arithmeticall, inclosed
in a riche Case of Gold, he vsed to weare about his necke: as his Iuell
most precious, and Counsaylour most trusty. Thus, _Arithmetike_, of him,
was shryned in gold: Of _Numbers_ frute, he had good hope. Now, Numbers
therfore innumerable, in _Numbers_ prayse, his shryne shall finde.


What nede I, (for farder profe to you) of the Scholemasters of Iustice,
to require testimony: how nedefull, how frutefull, how skillfull a thing
_Arithmetike_ is? I meane, the Lawyers of all sortes. Vndoubtedly, the
Ciuilians, can meruaylously declare: how, neither the Auncient Romaine
lawes, without good knowledge of _Numbers art_, can be perceiued: Nor
(Iustice in infinite Cases) without due proportion, (narrowly
considered,) is hable to be executed. How Iustly, & with great knowledge
of Arte, did _Papinianus_ institute a law of partition, and allowance,
betwene man and wife after a diuorce? But how _Accursius_, _Baldus_,
_Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (being
otherwise, notably well learned) do iumble, gesse, and erre, from the
æquity, art and Intent of the lawmaker: _Arithmetike_ can detect, and
conuince: and clerely, make the truth to shine. Good _Bartolus_, tyred
in the examining & proportioning of the matter: and with _Accursius_
Glosse, much cumbred: burst out, and sayd: _Nulla est in toto libro, hac
glossa difficilior: Cuius computationem nec Scholastici nec Doctores
intelligunt. &c._ That is: +_In the whole booke, there is no Glosse
harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor
the Doctours vnderstand. &c._+ What can they say of _Iulianus_ law, _Si
ita Scriptum. &c._ Of the Testators will iustly performing, betwene the
wife, Sonne and daughter? How can they perceiue the æquitie of
_Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _Lex
Falcidia_? How can they deliuer him, from his Reprouers: and their
maintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? How
Iustly and artificially, was _Africanus_ reckening made? Proportionating
to the Sommes bequeathed, the Contributions of eche part? Namely, for
the hundred presently receiued, 17-1/7. And for the hundred, receiued
after ten monethes, 12-6/7: which make the 30: which were to be
cõtributed by the legataries to the heire. For, what proportion, 100
hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that
is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law,
require an expert _Arithmeticien_, to vnderstand the deepe Iudgemẽt, &
Iust determinatiõ of the Auncient Romaine Lawmakers. But much more
expert ought he to be, who should be hable, to decide with æquitie, the
infinite varietie of Cases, which do, or may happen, vnder euery one of
those lawes and ordinances Ciuile. Hereby, easely, ye may now
coniecture: that in the Canon law: and in the lawes of the Realme (which
with vs, beare the chief Authoritie), Iustice and equity might be
greately preferred, and skilfully executed, through due skill of
Arithmetike, and proportions appertainyng. The worthy Philosophers, and
prudent lawmakers (who haue written many bookes _De Republica:_ How the
best state of Common wealthes might be procured and mainteined,) haue
very well determined of Iustice: (which, not onely, is the Base and
foundacion of Common weales: but also the totall perfection of all our
workes, words, and thoughtes:) defining it,

    [Iustice.]

“to be that vertue, by which, to euery one, is rendred, that to him
appertaineth.” God challengeth this at our handes, to be honored as God:
to be loued, as a father: to be feared as a Lord & master. Our
neighbours proportiõ, is also prescribed of the Almighty lawmaker: which
is, to do to other, euen as we would be done vnto. These proportions,
are in Iustice necessary: in duety, commendable: and of Common wealthes,
the life, strength, stay and florishing. _Aristotle_ in his _Ethikes_
(to fatch the sede of Iustice, and light of direction, to vse and
execute the same) was fayne to fly to the perfection, and power of
Numbers: for proportions Arithmeticall and Geometricall. _Plato_ in his
booke called _Epinomis_ (which boke, is the Threasury of all his
doctrine) where, his purpose is, to seke a Science, which, when a man
had it, perfectly: he might seme, and so be, in dede, _Wise_. He,
briefly, of other Sciences discoursing, findeth them, not hable to bring
it to passe: But of the Science of Numbers, he sayth. _Illa, quæ numerum
mortalium generi dedit, id profecto efficiet. Deum autem aliquem, magis
quam fortunam, ad salutem nostram, hoc munus nobis arbitror contulisse.
&c. Nam ipsum bonorum omnium Authorem, cur non maximi boni, Prudentiæ
dico, causam arbitramur? +That Science, verely, which hath taught
mankynde number, shall be able to bryng it to passe. And, I thinke,
a certaine God, rather then fortune, to haue giuen vs this gift, for our
blisse. For, why should we not Iudge him, who is the Author of all good
things, to be also the cause of the greatest good thyng, namely,
Wisedome?+_ There, at length, he proueth _Wisedome_ to be atteyned, by
good Skill of _Numbers_. With which great Testimony, and the manifold
profes, and reasons, before expressed, you may be sufficiently and fully
persuaded: of the perfect Science of _Arithmetike_, to make this
accounte: That

    [☞]

of all Sciences, next to _Theologie_, it is most diuine, most pure, most
ample and generall, most profounde, most subtile, most commodious and
most necessary. Whose next Sister, is the Absolute Science of
_Magnitudes_: of which (by the Direction and aide of him, whose
_Magnitude_ is Infinite, and of vs Incomprehensible) I now entend, so to
write, that both with the _Multitude_, and also with the _Magnitude_ of
Meruaylous and frutefull verities, you (my frendes and Countreymen) may
be stird vp, and awaked, to behold what certaine Artes and Sciences, (to
our vnspeakable behofe) our heauenly father, hath for vs prepared, and
reuealed, by sundry _Philosophers_ and _Mathematiciens_.


Both, _Number_ and _Magnitude_, haue a certaine Originall sede, (as it
were) of an incredible property: and of man, neuer hable, Fully, to be
declared. Of _Number_, an Vnit, and of _Magnitude_, a Poynte, doo seeme
to be much like Originall causes: But the diuersitie neuerthelesse, is
great. We defined an _Vnit_, to be a thing Mathematicall Indiuisible:
A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And
farder, that a Point may haue a certaine determined Situation: that is,
that we may assigne, and prescribe a Point, to be here, there, yonder.
&c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to
be tyed to any place, or seat: diuisible or indiuisible. Agayne, by
reason, a Point may haue a Situation limited to him: a certaine motion,
therfore (to a place, and from a place) is to a Point incident and
appertainyng. But an _Vnit_, can not be imagined to haue any motion.
A Point, by his motion, produceth, Mathematically, a line: (as we sayd
before) which is the first kinde of Magnitudes, and most simple: An
_Vnit_, can not produce any number. A Line, though it be produced of a
Point moued, yet, it doth not consist of pointes: Number, though it be
not produced of an _Vnit_, yet doth it Consist of vnits, as a materiall
cause. But formally,

    [Number.]

Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting,
is the workemanship of our minde: which, of distinct and discrete Vnits,
maketh a Number: by vniformitie, resulting of a certaine multitude of
Vnits. And so, euery number, may haue his least part, giuen: namely, an
Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be
giuẽ: by cause, infinitly, diuision therof, may be conceiued. All
Magnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine,
or Solid, of no Sense, can be perceiued, nor exactly by hãd (any way)
represented: nor of Nature produced: But, as (by degrees) Number did
come to our perceiuerance: So, by visible formes, we are holpen to
imagine, what our Line Mathematicall, is. What our Point, is. So
precise, are our Magnitudes, that one Line is no broader then an other:
for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our
Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we
can haue Smaller, then either Arte or Nature can produce any: and
Greater also, then all the world can comprehend. Our least Magnitudes,
can be diuided into so many partes, as the greatest. As, a Line of an
inch long, (with vs) may be diuided into as many partes, as may the
diameter of the whole world, from East to West: or any way extended:
What priuiledges, aboue all manual Arte, and Natures might, haue our two
Sciences Mathematicall? to exhibite, and to deale with thinges of such
power, liberty, simplicity, puritie, and perfection? And in them, so
certainly, so orderly, so precisely to procede: as, excellent is that
workemã Mechanicall Iudged, who nerest can approche to the representing
of workes, Mathematically demonstrated?

    [☞]

And our two Sciences, remaining pure, and absolute, in their proper
termes, and in their owne Matter: to haue, and allowe, onely such
Demonstrations, as are plaine, certaine, vniuersall, and of an æternall
veritye?

    [Geometrie.]

This Science of _Magnitude_, his properties, conditions, and
appertenances: commonly, now is, and from the beginnyng, hath of all
Philosophers, ben called _Geometrie_. But, veryly, with a name to base
and scant, for a Science of such dignitie and amplenes. And, perchaunce,
that name, by cõmon and secret consent, of all wisemen, hitherto hath
ben suffred to remayne: that it might carry with it a perpetuall
memorye, of the first and notablest benefite, by that Science, to common
people shewed: Which was, when Boundes and meres of land and ground were
lost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of
_Nilus_, the greatest and longest riuer in the world) or, that ground
bequeathed, were to be assigned: or, ground sold, were to be layd out:
or (when disorder preuailed) that Commõs were distributed into
seueralties. For, where, vpon these & such like occasiõs, Some by
ignorãce, some by negligẽce, Some by fraude, and some by violence, did
wrongfully limite, measure, encroach, or challenge (by pretence of iust
content, and measure) those landes and groundes: great losse,
disquietnes, murder, and warre did (full oft) ensue: Till, by Gods
mercy, and mans Industrie, The perfect Science of Lines, Plaines, and
Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. The
people then, by this art pleasured, and greatly relieued, in their
landes iust measuring: & other Philosophers, writing Rules for land
measuring: betwene them both, thus, confirmed the name of _Geometria_,
that is, (according to the very etimologie of the word) Land measuring.
Wherin, the people knew no farder, of Magnitudes vse, but in Plaines:
and the Philosophers, of thẽ, had no feet hearers, or Scholers: farder
to disclose vnto, then of flat, plaine _Geometrie_. And though, these
Philosophers, knew of farder vse, and best vnderstode the etymologye of
the worde, yet this name _Geometria_, was of them applyed generally to
all sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and
_Pythagoras_: When they would precisely declare their owne doctrine.
Then, was

    [* Plato. 7. de Rep.]

* _Geometria_, with them, _Studium quod circa planum versatur_. But,
well you may perceiue by _Euclides Elementes_, that more ample is our
Science, then to measure Plaines: and nothyng lesse therin is tought (of
purpose) then how to measure Land. An other name, therfore, must nedes
be had, for our Mathematicall Science of Magnitudes: which regardeth
neither clod, nor turff: neither hill, nor dale: neither earth nor
heauen: but is absolute _Megethologia_: not creping on ground, and
dasseling the eye, with pole perche, rod or lyne: but “liftyng the hart
aboue the heauens, by inuisible lines, and

    [☞]

immortall beames meteth with the reflexions, of the light
incomprehensible: and so procureth Ioye, and perfection vnspeakable.” Of
which true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_
seemed to haue good taste, and iudgement: and (by the name of
_Geometrie_) so noted it: and warned his Scholers therof: as, in hys
seuenth _Dialog_, of the Common wealth, may euidently be sene. Where (in
Latin) thus it is: right well translated: _Profecto, nobis hoc non
negabunt, Quicun[que] vel paululum quid Geometriæ gustârunt, quin hæc
Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa
versantur._ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuer
haue, (but euen very litle) tasted of Geometrie, will not denye vnto vs,
this: but that this Science, is of an other condicion, quite contrary to
that, which they that are exercised in it, do speake of it._+ And there
it followeth, of our _Geometrie_, _Quòd quæritur cognoscendi illius
gratia, quod semper est, non & eius quod oritur quando[que] & interit.
Geometria, eius quod est semper, Cognitio est. Attollet igitur
(ô Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum
preparabit cogitationem, vt ad supera conuertamus: quæ, nunc, contra
quàm decet, ad inferiora deijcimus. &c. Quàm maximè igitur præcipiendum
est, vt qui præclarissimam hanc habitãt Civitatem, nullo modo,
Geometriam spernant. Nam & quæ præter ipsius propositum, quodam modo
esse videntur, haud exigua sunt. &c._ It must nedes be confessed (saith
_Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, which
is euer: and not of that, which, in tyme, both is bred and is brought to
an ende. &c. Geometrie is the knowledge of that which is euerlastyng. It
will lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by
that meanes, it will prepare the Thought, to the Philosophicall loue of
wisdome: that we may turne or conuert, toward heauenly thinges =[both
mynde and thought]= which now, otherwise then becommeth vs, we cast down
on base or inferior things. &c. Chiefly, therfore, Commaundement must be
giuen, that such as do inhabit this most honorable Citie, by no meanes,
despise Geometrie. For euen those thinges =[done by it]= which, in
manner, seame to be, beside the purpose of Geometrie: are of no small
importance. &c._+ And besides the manifold vses of _Geometrie_, in
matters appertainyng to warre, he addeth more, of second vnpurposed
frute, and commoditye, arrising by _Geometrie_: saying: _Scimus quin
etiam, ad Disciplinas omnes facilius per discendas, interesse omnino,
attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam,
secundo loco discendam Iuuenibus statuamus._ That is. +_But, also, we
know, that for the more easy learnyng of all Artes, it importeth much,
whether one haue any knowledge in Geometrie, or no. &c. Let vs therfore
make an ordinance or decree, that this Science, of young men shall be
learned in the second place._+ This was _Diuine Plato_ his Iudgement,
both of the purposed, chief, and perfect vse of _Geometrie_: and of his
second, dependyng, deriuatiue commodities. And for vs, Christen men,
a thousand thousand mo occasions are, to haue nede of the helpe of *

    [I. D.
    * Herein, I would gladly shake of, the earthly name,
    of Geometrie.]

_Megethologicall_ Contemplations: wherby, to trayne our Imaginations and
Myndes, by litle and litle, to forsake and abandon, the grosse and
corruptible Obiectes, of our vtward senses: and to apprehend, by sure
doctrine demonstratiue, Things Mathematicall. And by them, readily to be
holpen and conducted to conceiue, discourse, and conclude of things
Intellectual, Spirituall, æternall, and such as concerne our Blisse
euerlasting: which, otherwise (without Speciall priuiledge of
Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally)
is hable to reach vnto, or to Compasse. And, veryly, by my small Talent
(from aboue) I am hable to proue and testifie, that the litterall Text,
and order of our diuine Law, Oracles, and Mysteries, require more skill
in Numbers, and Magnitudes: then (commonly) the expositors haue vttered:
but rather onely (at the most) so warned: & shewed their own want
therin. (To name any, is nedeles: and to note the places, is, here, no
place: But if I be duely asked, my answere is ready.) And without the
litterall, Grammaticall, Mathematicall or Naturall verities of such
places, by good and certaine Arte, perceiued, no Spirituall sense
(propre to those places, by Absolute _Theologie_) will thereon depend.

    [☞]

“No man, therfore, can doute, but toward the atteyning of knowledge
incomparable, and Heauenly Wisedome: Mathematicall Speculations, both of
Numbers and Magnitudes: are meanes, aydes, and guides: ready, certaine,
and necessary.” From henceforth, in this my Preface, will I frame my
talke, to _Plato_ his fugitiue Scholers: or, rather, to such, who well
can, (and also wil,) vse their vtward senses, to the glory of God, the
benefite of their Countrey, and their owne secret contentation, or
honest preferment, on this earthly Scaffold. To them, I will orderly
recite, describe & declare a great Number of Artes, from our two
Mathematicall fountaines, deriued into the fieldes of _Nature_. Wherby,
such Sedes, and Rotes, as lye depe hyd in the groũd of _Nature_, are
refreshed, quickened, and prouoked to grow, shote vp, floure, and giue
frute, infinite, and incredible. And these Artes, shalbe such, as vpon
Magnitudes properties do depende, more, then vpon Number. And by good
reason we may call them Artes, and Artes Mathematicall Deriuatiue: for
(at this tyme) I Define

    [An Arte.]

+An Arte, to be a Methodicall cõplete Doctrine, hauing abundancy of
sufficient, and peculier matter to deale with, by the allowance of the
Metaphisicall Philosopher: the knowledge whereof, to humaine state is
necessarye.+ And that I account,

    [Art Mathematicall Deriuatiue.]

+An Art Mathematicall deriuatiue, which by Mathematicall demonstratiue
Method, in Nũbers, or Magnitudes, ordreth and confirmeth his doctrine,
as much & as perfectly, as the matter subiect will admit.+ And for that,
I entend to vse the name and propertie of a

    [A Mechanitien.]

_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke it
good, (for distinction sake) to giue you also a brief description, what
I meane therby. +A Mechanicien, or a Mechanicall workman is he, whose
skill is, without knowledge of Mathematicall demonstration, perfectly to
worke and finishe any sensible worke, by the Mathematicien principall or
deriuatiue, demonstrated or demonstrable.+ Full well I know, that he
which inuenteth, or maketh these demonstrations, is generally called _A
speculatiue Mechanicien_: which differreth nothyng from a _Mechanicall
Mathematicien_. So, in respect of diuerse actions, one man may haue the
name of sundry artes: as, some tyme, of a Logicien, some tymes (in the
same matter otherwise handled) of a Rethoricien. Of these trifles,
I make, (as now, in respect of my Preface,) small account: to fyle thẽ
for the fine handlyng of subtile curious disputers. In other places,
they may commaunde me, to giue good reason: and yet, here, I will not be
vnreasonable.


    [+1.+]

First, then, from the puritie, absolutenes, and Immaterialitie of
Principall _Geometrie_, is that kinde of _Geometrie_ deriued, which
vulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible
magnitudes, their iust quãtities and contentes.+

    [Geometrie vulgar.]

This, teacheth to measure, either at hand: and the practiser, to be by
the thing Measured: and so, by due applying of Cumpase, Rule, Squire,
Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to
the Length, Plaine, or Solide measured,

    [1.]

* to be certified, either of the length, perimetry, or distance lineall:
and this is called, _Mecometrie_. Or

    [2.]

* to be certified of the content of any plaine Superficies: whether it
be in ground Surueyed, Borde, or Glasse measured, or such like thing:
which measuring, is named _Embadometrie_.

    [3.]

* Or els to vnderstand the Soliditie, and content of any bodily thing:
as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels,
small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale
vessells, &c, the Measuring, commonly, hath a peculier name: and is
called _Gaging_. And the generall name of these Solide measures, is
_Stereometrie_.

    [+2.+]

Or els, this _vulgar Geometrie_, hath consideration to teach the
practiser, how to measure things, with good distance betwene him and the
thing measured: and to vnderstand thereby, either

    [1.]

* how Farre, a thing seene (on land or water) is from the measurer: and
this may be called _Apomecometrie_:

    [2.]

Or, how High or depe, aboue or vnder the leuel of the measurers stãding,
any thing is, which is sene on land or water, called _Hypsometrie_.

    [3.]

* Or, it informeth the measurer, how Broad any thing is, which is in the
measurers vew: so it be on Land or Water, situated: and may be called
_Platometrie_. Though I vse here to condition, the thing measured, to be
on Land, or Water Situated:

    [Note.]

yet, know for certaine, that the sundry heigthe of Cloudes, blasing
Starres, and of the Mone, may (by these meanes) haue their distances
from the earth: and, of the blasing Starres and Mone, the Soliditie
(aswell as distances) to be measured: But because, neither these things
are vulgarly taught: nor of a common practiser so ready to be executed:
I, rather, let such measures be reckened incident to some of our other
Artes, dealing with thinges on high, more purposely, then this vulgar
Land measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c._


Of these Feates (farther applied) is Sprong the Feate of _Geodesie_, or
Land Measuring: more cunningly to measure & Suruey Land, Woods, and
Waters, a farre of. More cunningly, I say: But God knoweth (hitherto) in
these Realmes of England and Ireland (whether through ignorance or
fraude, I can not tell, in euery particular)

    [Note.]

how great wrong and iniurie hath (in my time) bene committed by vntrue
measuring and surueying of Land or Woods, any way. And, this I am sure:
that the Value of the difference, betwene the truth and such Surueyes,
would haue bene hable to haue foũd (for euer) in eche of our two
Vniuersities, an excellent Mathematicall Reader: to eche, allowing
(yearly) a hundred Markes of lawfull money of this realme: which, in
dede, would seme requisit, here, to be had (though by other wayes
prouided for) as well, as, the famous Vniuersitie of Paris, hath two
Mathematicall Readers: and eche, two hundreth French Crownes yearly, of
the French Kinges magnificent liberalitie onely. Now, againe, to our
purpose returning: Moreouer, of the former knowledge Geometricall, are
growen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and
_Stratarithmetrie_.


“+‡Geographie‡+ teacheth wayes, by which, in sũdry formes, (as
_Sphærike_, _Plaine_ or other), the Situation of Cities, Townes,
Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, &
such other things, vpõ the outface of the earthly Globe (either in the
whole, or in some principall mẽber and portion therof cõtayned) may be
described and designed, in cõmensurations Analogicall to Nature and
veritie: and most aptly to our vew, may be represented.” Of this Arte
how great pleasure, and how manifolde commodities do come vnto vs, daily
and hourely: of most men, is perceaued. While, some, to beautifie their
Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other
some, for thinges past, as battels fought, earthquakes, heauenly
fyringes, & such occurentes, in histories mentioned: therby liuely, as
it were, to vewe the place, the region adioyning, the distance from vs:
and such other circumstances. Some other, presently to vewe the large
dominion of the Turke: the wide Empire of the Moschouite: and the litle
morsell of ground, where Christendome (by profession) is certainly
knowen. Litle, I say, in respecte of the rest. &c. Some, either for
their owne iorneyes directing into farre landes: or to vnderstand of
other mens trauailes. To conclude, some, for one purpose: and some, for
an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, &
Geographicall Globes. Of whose vse, to speake sufficiently, would
require a booke peculier.


+‡Chorographie‡+ seemeth to be an vnderling, and a twig, of
_Geographie_: and yet neuerthelesse, is in practise manifolde, and in
vse very ample. “This teacheth Analogically to describe a small portion
or circuite of ground, with the contentes: not regarding what
commensuration it hath to the whole, or any parcell, without it,
contained. But in the territory or parcell of ground which it taketh in
hand to make description of, it leaueth out (or vndescribed) no notable,
or odde thing, aboue the ground visible. Yea and sometimes, of thinges
vnder ground, geueth some peculier marke: or warning: as of Mettall
mines, Cole pittes, Stone quarries. &c.” Thus, a Dukedome, a Shiere,
a Lordship, or lesse, may be described distinctly. But marueilous
pleasant, and profitable it is, in the exhibiting to our eye, and
commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true
Symmetry: not approching to any of them: and out of Gunne shot. &c.
Hereby, the _Architect_ may furnishe him selfe, with store of what
patterns he liketh: to his great instruction: euen in those thinges
which outwardly are proportioned: either simply in them selues: or
respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also,
terme this particular description of places, _Topographie_.


“+‡Hydrographie‡+, deliuereth to our knowledge, on Globe or in Plaine,
the perfect Analogicall description of the Ocean Sea coastes, through
the whole world: or in the chiefe and principall partes thereof:” with
the Iles and chiefe particular places of daungers, conteyned within the
boundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes,
Rockes, Races, Countertides, Whorlepooles. &c. This, dealeth with the
Element of the water chiefly: as _Geographie_ did principally take the
Element of the Earthes description (with his appertenances) to taske.
And besides thys, _Hydrographie_, requireth a particular Register of
certaine Landmarkes (where markes may be had) from the sea, well hable
to be skried, in what point of the Seacumpase they appeare, and what
apparent forme, Situation, and bignes they haue, in respecte of any
daungerous place in the sea, or nere vnto it, assigned: And in all
Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes,
come and go, the _Hydrographer_ ought to recorde. The Soundinges
likewise: and the Chanels wayes: their number, and depthes ordinarily,
at ebbe and flud, ought the _Hydrographer_, by obseruation and diligence
of _Measuring_, to haue certainly knowen. And many other pointes, are
belonging to perfecte _Hydrographie_, and for to make a _Rutter_, by: of
which, I nede not here speake: as of the describing, in any place, vpon
Globe or Plaine, the 32. pointes of the Compase, truely: (wherof,
scarsly foure, in England, haue right knowledge: bycause, the lines
therof, are no straight lines, nor Circles.) Of making due proiection of
a Sphere in plaine. Of the Variacion of the Compas, from true Northe:
And such like matters (of great importance, all) I leaue to speake of,
in this place: bycause, I may seame (al ready) to haue enlarged the
boundes, and duety of an Hydographer, much more, then any man (to this
day) hath noted, or prescribed. Yet am I well hable to proue, all these
thinges, to appertaine, and also to be proper to the Hydrographer. The
chief vse and ende of this Art, is the Art of Nauigation: but it hath
other diuerse vses: euen by them to be enioyed, that neuer lacke sight
of land.


+‡Stratarithmetrie‡+, is the Skill, (appertainyng to the warre,) by
which a man can set in figure, analogicall to any _Geometricall_ figure
appointed, any certaine number or summe of men: of such a figure
capable: (by reason of the vsuall spaces betwene Souldiers allowed: and
for that, of men, can be made no Fractions. Yet, neuertheles, he can
order the giuen summe of men, for the greatest such figure, that of
them, cã be ordred) and certifie, of the ouerplus: (if any be) and of
the next certaine summe, which, with the ouerplus, will admit a figure
exactly proportionall to the figure assigned. By which Skill, also, of
any army or company of men: (the figure & sides of whose orderly
standing, or array, is knowen) he is able to expresse the iust number of
men, within that figure conteined: or (orderly) able to be conteined.

    [* Note.]

* And this figure, and sides therof, he is hable to know: either beyng
by, and at hand: or a farre of. Thus farre, stretcheth the description
and property of _Stratarithmetrie_: sufficient for this tyme and place.

    [The difference betwene Stratarithmetrie and Tacticie.]

“It differreth from the Feate _Tacticall_, _De aciebus instruendis._
bycause, there, is necessary the wisedome and foresight, to what purpose
he so ordreth the men: and Skillfull hability, also, for any occasion,
or purpose, to deuise and vse the aptest and most necessary order, array
and figure of his Company and Summe of men.” By figure, I meane: as,
either of a _Perfect Square_, _Triangle_, _Circle_, _Ouale_, _long
square_, (of the Grekes it is called _Eteromekes_) _Rhombe_, _Rhomboïd_,
_Lunular_, _Ryng_, _Serpentine_, and such other Geometricall figures:
Which, in warres, haue ben, and are to be vsed: for commodiousnes,
necessity, and auauntage &c. And no small skill ought he to haue, that
should make true report, or nere the truth, of the numbers and Summes,
of footemen or horsemen, in the Enemyes ordring. A farre of, to make an
estimate, betwene nere termes of More and Lesse, is not a thyng very
rife, among those that gladly would do it.

    [I. D.
    Frende, you will finde it hard, to performe my description
    of this Feate. But by Chorographie, you may helpe your selfe
    some what: where the Figures knowne (in Sides and Angles)
    are not Regular: And where, Resolution into Triangles can
    serue. &c. And yet you will finde it strange to deale thus
    generally with Arithmeticall figures: and, that for Battayle
    ray. Their contentes, differ so much from like Geometricall
    Figures.]

Great pollicy may be vsed of the Capitaines, (at tymes fete, and in
places conuenient) as to vse Figures, which make greatest shew, of so
many as he hath: and vsing the aduauntage of the three kindes of vsuall
spaces: (betwene footemen or horsemen) to take the largest: or when he
would seme to haue few, (beyng many:) contrarywise, in Figure, and
space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer
is carefull to come nere the truth herein, besides the Iudgement of his
expert eye, his skill of Ordering _Tacticall_, the helpe of his
Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously
framed for cariage and vse) He may wonderfully helpe him selfe, by
perspectiue Glasses. In which, (I trust) our posterity will proue more
skillfull and expert, and to greater purposes, then in these dayes, can
(almost) be credited to be possible.


Thus haue I lightly passed ouer the Artificiall Feates, chiefly
dependyng vpon vulgar _Geometrie_: & commonly and generally reckened
vnder the name of _Geometrie_. But there are other (very many)
_Methodicall Artes_, which, declyning from the purity, simplicitie, and
Immateriality, of our Principall Science of _Magnitudes_: do yet
neuertheles vse the great ayde, direction, and Method of the sayd
principall Science, and haue propre names, and distinct: both from the
Science of _Geometrie_, (from which they are deriued) and one from the
other. As +Perspectiue, Astronomie, Musike, Cosmographie, Astrologie,
Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie,
Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture,
Nauigation, Thaumaturgike+ and +Archemastrie+. I thinke it necessary,
orderly, of these to giue some peculier descriptions: and withall, to
touch some of their commodious vses, and so to make this Preface, to be
a little swete, pleasant Nosegaye for you: to comfort your Spirites,
beyng almost out of courage, and in despayre, (through brutish brute)
Weenyng that _Geometrie_, had but serued for buildyng of an house, or a
curious bridge, or the roufe of Westminster hall, or some witty pretty
deuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That the
thing is farre otherwise, then the world, (commonly) to this day, hath
demed, by worde and worke, good profe wilbe made.


Among these Artes, by good reason, +‡Perspectiue‡+ ought to be had, ere
of _Astronomicall Apparences_, perfect knowledge can be atteyned. And
bycause of the prerogatiue of _Light_, beyng the first of _Gods
Creatures_: and the eye, the light of our body, and his Sense most
mighty, and his organ most Artificiall and _Geometricall_: At
_Perspectiue_, we will begyn therfore. +Perspectiue, is an Art
Mathematicall, which demonstrateth the maner, and properties, of all
Radiations Direct, Broken, and Reflected.+ This Description, or
Notation, is brief: but it reacheth so farre, as the world is wyde. It
concerneth all Creatures, all Actions, and passions, by Emanation of
beames perfourmed. Beames, or naturall lines, (here) I meane, not of
light onely, or of colour (though they, to eye, giue shew, witnes, and
profe, wherby to ground the Arte vpon) but also of other _Formes_, both
_Substantiall_, and _Accidentall_, the certaine and determined actiue
Radiall emanations. By this Art (omitting to speake of the highest
pointes) we may vse our eyes, and the light, with greater pleasure: and
perfecter Iudgement: both of things, in light seen, & of other: which by
like order of Lightes Radiations, worke and produce their effectes. We
may be ashamed to be ignorant of the cause, why so sundry wayes our eye
is deceiued, and abused: as, while the eye weeneth a roũd Globe or
Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise
iudgeth a plaine Square, to be roũd: supposeth walles parallels, to
approche, a farre of: rofe and floure parallels, the one to bend
downward, the other to rise vpward, at a little distance from you.
Againe, of thinges being in like swiftnes of mouing, to thinke the
nerer, to moue faster: and the farder, much slower. Nay, of two thinges,
wherof the one (incomparably) doth moue swifter then the other, to deme
the slower to moue very swift, & the other to stand: what an error is
this, of our eye? Of the Raynbow, both of his Colours, of the order of
the colours, of the bignes of it, the place and heith of it, (&c) to
know the causes demonstratiue, is it not pleasant, is it not necessary?
of two or three Sonnes appearing: of Blasing Sterres: and such like
thinges: by naturall causes, brought to passe, (and yet neuertheles, of
farder matter, Significatiue) is it not commodious for man to know the
very true cause, & occasion Naturall? Yea, rather, is it not, greatly,
against the Souerainty of Mans nature, to be so ouershot and abused,
with thinges (at hand) before his eyes? as with a Pecockes tayle, and a
Doues necke: or a whole ore, in water, holden, to seme broken. Thynges,
farre of, to seeme nere: and nere, to seme farre of. Small thinges, to
seme great: and great, to seme small. One man, to seme an Army. Or a man
to be curstly affrayed of his owne shaddow. Yea, so much, to feare,
that, if you, being (alone) nere a certaine glasse, and proffer, with
dagger or sword, to foyne at the glasse, you shall suddenly be moued to
giue backe (in maner) by reason of an Image,

    [☞ A marueilous Glasse.]

appearing in the ayre, betwene you & the glasse, with like hand, sword
or dagger, & with like quicknes, foyning at your very eye, likewise as
you do at the Glasse. Straunge, this is, to heare of: but more
meruailous to behold, then these my wordes can signifie. And
neuerthelesse by demonstration Opticall, the order and cause therof, is
certified: euen so, as the effect is consequent. Yea, thus much more,
dare I take vpon me, toward the satisfying of the noble courrage, that
longeth ardently for the wisedome of Causes Naturall: as to let him
vnderstand, that, in London, he may with his owne eyes, haue profe of
that, which I haue sayd herein. A Gentleman, (which, for his good
seruice, done to his Countrey, is famous and honorable:

    [S. W. P.]

and for skill in the Mathematicall Sciences, and Languages, is the Od
man of this land. &c.) euen he, is hable: and (I am sure) will, very
willingly, let the Glasse, and profe be sene: and so I (here) request
him: for the encrease of wisedome, in the honorable: and for the
stopping of the mouthes malicious: and repressing the arrogancy of the
ignorant. Ye may easily gesse, what I meane. This Art of _Perspectiue_,
is of that excellency, and may be led, to the certifying, and executing
of such thinges, as no man would easily beleue: without Actuall profe
perceiued. I speake nothing of _Naturall Philosophie_, which, without
_Perspectiue_, can not be fully vnderstanded, nor perfectly atteined
vnto. Nor, of _Astronomie_: which, without _Perspectiue_, can not well
be grounded: Nor _Astrologie_, naturally Verified, and auouched. That
part hereof, which dealeth with Glasses (which name, Glasse, is a
generall name, in this Arte, for any thing, from which, a Beame
reboundeth) is called _Catoptrike_: and hath so many vses, both
merueilous, and proffitable: that, both, it would hold me to long, to
note therin the principall conclusions, all ready knowne: And also
(perchaunce) some thinges, might lacke due credite with you: And I,
therby, to leese my labor: and you, to slip into light Iudgement,

    [* ☞]

Before you haue learned sufficiently the powre of Nature and Arte.


Now, to procede: +‡Astronomie‡, is an Arte Mathematicall, which
demonstrateth the distance, magnitudes, and all naturall motions,
apparences, and passions propre to the Planets and fixed Sterres: for
any time past, present and to come: in respect of a certaine Horizon, or
without respect of any Horizon.+ By this Arte we are certified of the
distance of the Starry Skye, and of eche _Planete_ from the Centre of
the Earth: and of the greatnes of any Fixed starre sene, or _Planete_,
in respect of the Earthes greatnes. As, we are sure (by this Arte) that
the Solidity, Massines and Body of the _Sonne_, conteineth the quantitie
of the whole Earth and Sea, a hundred thre score and two times, lesse by
⅛ one eight parte of the earth. But the Body of the whole earthly globe
and Sea, is bigger then the body of the Mone, three and forty times
lesse by ⅛ of the Mone. Wherfore the _Sonne_ is bigger then the _Mone_,
7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then
the _Mone_. And yet the vnskillfull man, would iudge them a like bigge.
Wherfore, of Necessity, the one is much farder from vs, then the other.
The _Sonne_, when he is fardest from the earth (which, now, in our age,
is, when he is in the 8. degree, of Cancer) is, 1179 Semidiameters of
the Earth, distante. And the _Mone_ when she is fardest from the earth,
is 68 Semidiameters of the earth and ⅓ The nerest, that the _Mone_
commeth to the earth, is Semidiameters 52¼ The distance of the Starry
Skye is, frõ vs, in Semidiameters of the earth 20081½ Twenty thousand
fourescore, one, and almost a halfe. Subtract from this, the _Mones_
nerest distance, from the Earth: and therof remaineth Semidiameters of
the earth 20029¼ Twenty thousand nine and twenty and a quarter.

    [Note.]

So thicke is the heauenly Palace, that the _Planetes_ haue all their
exercise in, and most meruailously perfourme the Commaũdement and Charge
to them giuen by the omnipotent Maiestie of the king of kings. This is
that, which in _Genesis_ is called _Ha Rakia_. Consider it well. The
Semidiameter of the earth, cõteineth of our common miles 3436 4/11 three
thousand, foure hundred thirty six and foure eleuenth partes of one
myle: Such as the whole earth and Sea, round about, is 21600. One and
twenty thousand six hundred of our myles. Allowyng for euery degree of
the greatest circle, thre score myles. Now if you way well with your
selfe but this litle parcell of frute _Astronomicall_, as concerning the
bignesse, Distances of _Sonne_, _Mone_, _Sterry Sky_, and the huge
massines of _Ha Rakia_, will you not finde your Consciences moued, with
the kingly Prophet, to sing the confession of Gods Glory, and say, +_The
Heauens declare the glory of God, and the Firmament =[Ha Rakia]= sheweth
forth the workes of his handes_+. And so forth, for those fiue first
staues, of that kingly Psalme. Well, well, It is time for some to lay
hold on wisedome, and to Iudge truly of thinges: and notso to expound
the Holy word, all by Allegories: as to Neglect the wisedome, powre and
Goodnes of God, in, and by his Creatures, and Creation to be seen and
learned. By parables and Analogies of whose natures and properties, the
course of the Holy Scripture, also, declareth to vs very many Mysteries.
The whole Frame of Gods Creatures, (which is the whole world,) is to vs,
a bright glasse: from which, by reflexion, reboundeth to our knowledge
and perceiuerance, Beames, and Radiations: representing the Image of his
Infinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taught
and persuaded to Glorifie our Creator, as God: and be thankefull
therfore. Could the Heathenistes finde these vses, of these most pure,
beawtifull, and Mighty Corporall Creatures: and shall we, after that the
true _Sonne_ of rightwisenesse is risen aboue the _Horizon_, of our
temporall _Hemisphærie_, and hath so abundantly streamed into our
hartes, the direct beames of his goodnes, mercy, and grace: Whose heat
All Creatures feele: Spirituall and Corporall: Visible and Inuisible.
Shall we (I say) looke vpon the _Heauen_, _Sterres_, and _Planets_, as
an Oxe and an Asse doth: no furder carefull or inquisitiue, what they
are: why were they Created, How do they execute that they were Created
for? Seing, All Creatures, were for our sake created: and both we, and
they, Created, chiefly to glorifie the Almighty Creator: and that, by
all meanes, to vs possible. _Nolite ignorare_ (saith _Plato in
Epinomis_) _Astronomiam, Sapientissimũ quiddam esse._ +_Be ye not
ignorant, Astronomie to be a thyng of excellent wisedome._+
_Astronomie_, was to vs, from the beginning commended, and in maner
commaunded by God him selfe. In asmuch as he made the _Sonne_, _Mone_,
and _Sterres_, to be to vs, for _Signes_, and knowledge of Seasons, and
for Distinctions of Dayes, and yeares. Many wordes nede not. But I wish,
euery man should way this word, _Signes_. And besides that, conferre it
also with the tenth Chapter of _Hieremie_. And though Some thinke, that
there, they haue found a rod: Yet Modest Reason, will be indifferent
Iudge, who ought to be beaten therwith, in respect of our purpose.
Leauing that: I pray you vnderstand this: that without great diligence
of Obseruation, examination and Calculation, their periods and courses
(wherby _Distinction_ of Seasons, yeares, and New Mones might precisely
be knowne) could not exactely be certified. Which thing to performe, is
that _Art_, which we here haue Defined to be _Astronomie_. Wherby, we
may haue the distinct Course of Times, dayes, yeares, and Ages: aswell
for Consideratiõ of Sacred Prophesies, accomplished in due time,
foretold: as for high Mysticall Solemnities holding: And for all other
humaine affaires, Conditions, and couenantes, vpon certaine time,
betwene man and man: with many other great vses: Wherin, (verely), would
be great incertainty, Confusion, vntruth, and brutish Barbarousnes:
without the wonderfull diligence and skill of this Arte: continually
learning, and determining Times, and periodes of Time, by the Record of
the heauenly booke, wherin all times are written: and to be read with an
_Astronomicall staffe_, in stede of a festue.


+‡Musike‡+, of Motion, hath his Originall cause: Therfore, after the
motions most swift, and most Slow, which are in the Firmament, of Nature
performed: and vnder the _Astronomers Consideration_: now I will Speake
of an other kinde of _Motion_, producing sound, audible, and of Man
numerable. _Musike_ I call here that _Science_, which of the Grekes is
called _Harmonice_. Not medling with the Controuersie betwene the
auncient _Harmonistes_, and _Canonistes_. +Musike is a Mathematicall
Science, which teacheth, by sense and reason, perfectly to iudge, and
order the diuersities of soundes, hye and low.+ _Astronomie_ and
_Musike_ are Sisters, saith _Plato_. As, for _Astronomie_, the eyes: So,
for _Harmonious Motion_, the eares were made. But as _Astronomie_ hath a
more diuine Contemplation, and cõmodity, then mortall eye can perceiue:
So, is _Musike_ to be considered,

    [1.]

that the * Minde may be preferred, before the eare. And from audible
sound, we ought to ascende, to the examination: which numbers are
_Harmonious_, and which not. And why, either, the one are: or the other
are not. I could at large,

    [2.]

in the heauenly * motions and distances, describe a meruailous Harmonie,
of _Pythagoras_ Harpe

    [3.]

with eight stringes. Also, somwhat might be sayd of _Mercurius_ * two
Harpes,

    [4.]

eche of foure Stringes Elementall. And very straunge matter, might be
alledged of the _Harmonie_,

    [5.]

to our * Spirituall part appropriate. As in _Ptolomaus_ third boke, in
the fourth and sixth Chapters may appeare. *

    [6.]

And what is the cause of the apt bonde, and frendly felowship, of the
Intellectuall and Mentall part of vs, with our grosse & corruptible
body: but a certaine Meane, and _Harmonious Spiritualitie, with both
participatyng, & of both (in a maner) resultynge In

    [7.]

the * Tune of Mans voyce, and also

    [8.]

* the sound of Instrument_, what might be sayd, of _Harmonie_: No common
Musicien would lightly beleue.

    [I. D.
    Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7.
    chapters. Where you shall haue some occasion farder to thinke
    of Musike, than commonly is thought.]

But of the sundry Mixture (as I may terme it) and concurse, diuerse
collation, and Application of these _Harmonies_: as of thre, foure,
fiue, or mo: Maruailous haue the effectes ben: and yet may be founde,
and produced the like: with some proportionall consideration for our
time, and being: in respect of the State, of the thinges then: in which,
and by which, the wondrous effectes were wrought. _Democritus_ and
_Theophrastus_ affirmed, that, by _Musike_, griefes and diseases of the
Minde, and body might be cured, or inferred. And we finde in Recorde,
that _Terpander_, _Arion_, _Ismenias_, _Orpheus_, _Amphion_, _Dauid_,
_Pythagoras_, _Empedocles_, _Asclepiades_ and _Timotheus_, by
_Harmonicall_ Consonãcy, haue done, and brought to pas, thinges, more
then meruailous, to here of. Of them then, making no farder discourse,
in this place: Sure I am, that Common _Musike_, commonly vsed, is found
to the _Musiciens_ and Hearers, to be so Commodious and pleasant, That
if I would say and dispute, but thus much: That it were to be otherwise
vsed, then it is, I should finde more repreeuers, then I could finde
priuy, or skilfull of my meaning. In thinges therfore euident, and
better knowen, then I can expresse: and so allowed and liked of, (as I
would wish, some other thinges, had the like hap) I will spare to
enlarge my lines any farder, but consequently follow my purpose.


+‡Of Cosmographie‡+, I appointed briefly in this place, to geue you some
intelligence. +Cosmographie, is the whole and perfect description of the
heauenly, and also elementall parte of the world, and their homologall
application, and mutuall collation necessarie.+ This Art, requireth
_Astronomie_, _Geographie_, _Hydrographie_ and _Musike_. Therfore, it is
no small Arte, nor so simple, as in common practise, it is (slightly)
considered. This matcheth Heauen, and the Earth, in one frame, and aptly
applieth parts Correspõdent: So, as, the Heauenly Globe, may (in
practise) be duely described vpon the Geographicall, and Hydrographicall
Globe. And there, for vs to consider an _Æquonoctiall Circle_, _an
Ecliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes,
Latitudes, Declinations, and Verticalitie: also Climes, and Parallels:
and by an _Horizon_ annexed, and reuolution of the earthly Globe (as the
Heauen, is, by the _Primouant_, caried about in 24. æquall Houres) to
learne the Risinges and Settinges of Sterres (of _Virgill_ in his
_Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sphære_,
to Perdicca King of the Macedonians: of _Diocles_, to King _Antigonus_,
and of other famous _Philosophers_ prescribed) a thing necessary, for
due manuring of the earth, for _Nauigation_, for the Alteration of mans
body: being, whole, Sicke, wounded, or brused. By the Reuolution, also,
or mouing of the Globe Cosmographicall, the Rising and Setting of the
Sonne: the Lengthes, of dayes and nightes: the Houres and times (both
night and day) are knowne: with very many other pleasant and necessary
vses: Wherof, some are knowne: but better remaine, for such to know and
vse:

    [☞]

who of a sparke of true fire, can make a wonderfull bonfire, by applying
of due matter, duely.


+‡Of Astrologie‡+, here I make an Arte, seuerall from _Astronomie_: not
by new deuise, but by good reason and authoritie: for, +Astrologie, is
an Arte Mathematicall, which reasonably demonstrateth the operations and
effectes, of the naturall beames, of light, and secrete influence: of
the Sterres and Planets: in euery element and elementall body: at all
times, in any Horizon assigned.+ This Arte is furnished with many other
great Artes and experiences: As with perfecte _Perspectiue_,
_Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4.
Elementes, the Arte of Graduation, and some good vnderstãding in
_Musike_: and yet moreouer, with an other great Arte, hereafter
following, though I, here, set this before, for some considerations me
mouing. Sufficient (you see) is the stuffe, to make this rare and
secrete Arte, of: and hard enough to frame to the Conclusion
Syllogisticall. Yet both the manifolde and continuall trauailes of the
most auncient and wise Philosophers, for the atteyning of this Arte: and
by examples of effectes, to confirme the same: hath left vnto vs
sufficient proufe and witnesse: and we, also, daily may perceaue, That
mans body, and all other Elementall bodies, are altered, disposed,
ordred, pleasured, and displeasured, by the Influentiall working of the
_Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth
_Aristotle_, in the first of his _Meteorologicall_ bookes, in the second
Chapter: _Est autem necessariò Mundus iste, supernis lationibus ferè
continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causà prima
putanda omnibus est, vnde motus principium existit._ That is: +_This
=[Elementall]= World is of necessitie, almost, next adioyning, to the
heauenly motions: That, from thence, all his vertue or force may be
gouerned. For, that is to be thought the first Cause vnto all: from
which, the beginning of motion, is._+ And againe, in the tenth Chapter.
_Oportet igitur & horum principia sumamus, & causas omnium similiter.
Principium igitur vt mouens, præcipuum[que] & omnium primum, Circulus
ille est, in quo manifeste Solis latio, &c._ And so forth. His
_Meteorologicall_ bookes, are full of argumentes, and effectuall
demonstrations, of the vertue, operation, and power of the heauenly
bodies, in and vpon the fower Elementes, and other bodies, of them
(either perfectly, or vnperfectly) composed. And in his second booke,
_De Generatione & Corruptione_, in the tenth Chapter. _Quocirca & prima
latio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea
nam[que] & continua est, & duobus motibus fit:_ In Englishe, thus.
+_Wherefore the vppermost motion, is not the cause of Generation and
Corruption, but the motion of the Zodiake: for, that, both, is
continuall, and is caused of two mouinges._+ And in his second booke,
and second Chapter of hys _Physikes_. _Homo nam[que] generat hominem,
at[que] Sol._ +_For Man (sayth he) and the Sonne, are cause of mans
generation._+ Authorities may be brought, very many: both of 1000. 2000.
yea and 3000. yeares Antiquitie: of great _Philosophers_, _Expert_,
_Wise_, and godly men, for that Conclusion: which, daily and hourely, we
men, may discerne and perceaue by sense and reason: All beastes do
feele, and simply shew, by their actions and passions, outward and
inward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally,
the Elementes, and all thinges of the Elementes composed, do geue
Testimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions,
vertues, and naturall motions, depend of the Actiuitie of the heauenly
motions and Influences. Whereby, beside the specificall order and forme,
due to euery seede: and beside the Nature, propre to the Indiuiduall
Matrix, of the thing produced: What shall be the heauenly Impression,
the perfect and circumspecte Astrologien hath to Conclude._+ Not onely
(by _Apotelesmes_) τὸ ὁτὶ]. but by Naturall and Mathematicall
demonstration τὸ διότι. Whereunto, what Sciences are requisite (without
exception) I partly haue here warned: And in my _Propædeumes_ (besides
other matter there disclosed) I haue Mathematically furnished vp the
whole Method: To this our age, not so carefully handled by any, that
euer I saw, or heard of. I was,

    [* Anno. 1548 and 1549. in Louayn.]

(for * 21. yeares ago) by certaine earnest disputations, of the Learned
_Gerardus Mercator_, and _Antonius Gogaua_, (and other,) therto so
prouoked: and (by my constant and inuincible zeale to the veritie) in
obseruations of Heauenly Influencies (to the Minute of time,) than, so
diligent: And chiefly by the Supernaturall influence, from the Starre of
Iacob, so directed: That any Modest and Sober Student, carefully and
diligently seking for the Truth, will both finde & cõfesse, therin, to
be the Veritie, of these my wordes: And also become a Reasonable
Reformer, of three Sortes of people: about these Influentiall
Operations, greatly erring from the truth.

    [Note.]

Wherof, the one, is +Light Beleuers+, the other, +Light Despisers+, and
the third +Light Practisers+. The first, & most cõmon Sort, thinke the
Heauen and Sterres, to be answerable to any their doutes or desires:

    [1.]

which is not so: and, in dede, they, to much, ouer reache. The Second
sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare
any Sway in Generation

    [2.]

and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and
_Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so
farre in distance, so manifold in their motions, so constant in their
periodes. &c.) they assigne a sleight, simple office or two, and so
allow vnto thẽ (according to their capacities) as much vertue, and power
Influentiall, as to the Signe of the _Sunne_, _Mone_, and seuen Sterres,
hanged vp (for Signes) in London, for distinction of houses, & such
grosse helpes, in our worldly affaires: And they vnderstand not (or will
not vnderstand) of the other workinges, and vertues of the Heauenly
_Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husband
man: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, as
the Porpentine doth: nor will allow these perfect, and incorruptible
mighty bodies, so much vertuall Radiation, & Force, as they see in a
litle peece of a _Magnes stone_: which, at great distance, sheweth his
operation. And perchaunce they thinke, the Sea & Riuers (as the Thames)
to be some quicke thing, and so to ebbe, and flow, run in and out, of
them selues, at their owne fantasies. God helpe, God helpe. Surely,
these men, come to short: and either are to dull: or willfully blind:
or, perhaps, to malicious. The third man, is the common and vulgare
_Astrologien_, or Practiser: who, being not duely, artificially, and
perfectly

    [3.]

furnished: yet, either for vaine glory, or gayne: or like a simple dolt,
& blinde Bayard, both in matter and maner, erreth: to the discredit of
the _Wary_, and modest _Astrologien_: and to the robbing of those most
noble corporall Creatures, of their Naturall Vertue: being most mighty:
most beneficiall to all elementall Generation, Corruption and the
appartenances: and most Harmonious in their Monarchie: For which
thinges, being knowen, and modestly vsed: we might highly, and
continually glorifie God, with the princely Prophet, saying. +_The
Heauens declare the Glorie of God: who made the Heauẽs in his wisedome:
who made the Sonne, for to haue dominion of the day: the Mone and
Sterres to haue dominion of the nyght: whereby, Day to day vttereth
talke: and night, to night declareth knowledge. Prayse him, all ye
Sterres, and Light. Amen._+


In order, now foloweth, of +‡Statike‡+, somewhat to say, what we meane
by that name: and what commodity, doth, on such Art, depend. +Statike,
is an Arte Mathematicall, which demonstrateth the causes of heauynes,
and lightnes of all thynges: and of motions and properties, to heauynes
and lightnes, belonging.+ And for asmuch as, by the Bilanx, or Balance
(as the chief sensible Instrument,) Experience of these demonstrations
may be had: we call this Art, _Statike:_ that is, _the Experimentes of
the Balance_. Oh, that men wist, what proffit, (all maner of wayes) by
this Arte might grow, to the hable examiner, and diligent practiser.
“Thou onely, knowest all thinges precisely (O God) who hast made weight
and Balance, thy Iudgement: who hast created all thinges in _Number,
Waight, and Measure_: and hast wayed the mountaines and hils in a
Balance: who hast peysed in thy hand, both Heauen and earth. We therfore
warned by the Sacred word, to Consider thy Creatures: and by that
consideration, to wynne a glyms (as it were,) or shaddow of
perceiuerance, that thy wisedome, might, and goodnes is infinite, and
vnspeakable, in thy Creatures declared: And being farder aduertised, by
thy mercifull goodnes, that, three principall wayes, were, of the, vsed
in Creation of all thy Creatures, namely, _Number_, _Waight_ and
_Measure_, And for as much as, of _Number_ and _Measure_, the two Artes
(auncient, famous, and to humaine vses most necessary,) are, all ready,
sufficiently knowen and extant: This third key, we beseche thee (through
thy accustomed goodnes,) that it may come to the nedefull and sufficient
knowledge, of such thy Seruauntes, as in thy workemanship, would gladly
finde, thy true occasions (purposely of the vsed) whereby we should
glorifie thy name, and shew forth (to the weaklinges in faith) thy
wondrous wisedome and Goodnes. Amen.”


Meruaile nothing at this pang (godly frend, you Gentle and zelous
Student.) An other day, perchaunce, you will perceiue, what occasion
moued me. Here, as now, I will giue you some ground, and withall some
shew, of certaine commodities, by this Arte arising. And bycause this
Arte is rare, my wordes and practises might be to darke: vnleast you had
some light, holden before the matter: and that, best will be, in giuing
you, out of _Archimedes_ demonstrations, a few principal Conclusions, as
foloweth.

  +1.+

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

  +2.+

  +If Solide Magnitudes, being of the same bignes, or quãtitie, that
  any Liquor is, and hauyng also the same Waight: be let downe into
  the same Liquor, they will settle downeward, so, that no parte of
  them, shall be aboue the Superficies of the Liquor: and yet
  neuertheles, they will not sinke vtterly downe, or drowne.+

  +3.+

  +If any Solide Magnitude beyng Lighter then a Liquor, be let downe
  into the same Liquor, it will settle downe, so farre into the same
  Liquor, that so great a quantitie of that Liquor, as is the parte of
  the Solid Magnitude, settled downe into the same Liquor: is in
  Waight, æquall, to the waight of the whole Solid Magnitude.+

  +4.+

  +Any Solide Magnitude, Lighter then a Liquor, forced downe into the
  same Liquor, will moue vpward, with so great a power, by how much,
  the Liquor hauyng æquall quantitie to the whole Magnitude, is
  heauyer then the same Magnitude.+

  +5.+

  +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into
  the same Liquor, will sinke downe vtterly: And wilbe in that Liquor,
  Lighter by so much, as is the waight or heauynes of the Liquor,
  hauing bygnes or quantitie, æquall to the Solid Magnitude.+

  +6.+

      [I. D.
      The Cutting of a Sphære according to any proportion assigned
      may by this proposition be done Mechanically by tempering
      Liquor to a certayne waight in respect of the waight of the
      Sphære therein Swymming.]

  +If any Solide Magnitude, Lighter then a Liquor, be let downe into
  the same Liquor, the waight of the same Magnitude, will be, to the
  Waight of the Liquor. (Which is æquall in quantitie to the whole
  Magnitude,) in that proportion, that the parte, of the Magnitude
  settled downe, is to the whole Magnitude.+


By these verities, great Errors may be reformed, in Opinion of the
Naturall Motion of thinges, Light and Heauy. Which errors, are in
Naturall Philosophie (almost) of all mẽ allowed: to much trusting to
Authority: and false Suppositions. As, +Of any two bodyes, the heauyer,
to moue downward faster then the lighter.+

    [A common error, noted.]

This error, is not first by me, Noted: but by one _Iohn Baptist de
Benedictis_. The chief of his propositions, is this: which seemeth a
Paradox.


+If there be two bodyes of one forme, and of one kynde, æquall in
quantitie or vnæquall,

    [A paradox.]

they will moue by æquall space, in æquall tyme: So that both theyr
mouynges be in ayre, or both in water: or in any one Middle.+


Hereupon, in the feate of +Gunnyng+,

    [N. T.]

certaine good discourses (otherwise) may receiue great amendement, and
furderance.

    [The wonderfull vse of these Propositions.]

In the entended purpose, also, allowing somwhat to the imperfection of
Nature: not aunswerable to the precisenes of demonstration. Moreouer, by
the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth,
the Fire, may be nerely, knowen, how light or heauy they are (Naturally)
in their assigned partes: or in the whole. And then, to thinges
Elementall, turning your practise: you may deale for the proportion of
the Elementes, in the thinges Compounded. Then, to the proportions of
the Humours in Man: their waightes: and the waight of his bones, and
flesh. &c. Than, by waight, to haue consideration of the Force of man,
any maner of way: in whole or in part. Then, may you, of Ships water
drawing, diuersly, in the Sea and in fresh water, haue pleasant
consideration: and of waying vp of any thing, sonken in Sea or in fresh
water &c. And (to lift vp your head a loft:) by waight, you may, as
precisely, as by any instrument els, measure the Diameters of _Sonne_
and _Mone. &c._ Frende, I pray you, way these thinges, with the iust
Balance of Reason. And you will finde Meruailes vpon Meruailes: And
esteme one Drop of Truth (yea in Naturall Philosophie) more worth, then
whole Libraries of Opinions, vndemonstrated: or not aunswering to
Natures Law, and your experience. Leauing these thinges, thus: I will
giue you two or three, light practises, to great purpose: and so finish
my Annotation _Staticall_. In Mathematicall matters, by the Mechaniciens
ayde, we will behold, here, the Commodity of waight.

    [The practise Staticall, to know the proportion, betwene
    the Cube, and the Sphære.]

Make a Cube, of any one Vniforme: and through like heauy stuffe: of the
same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall to
the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne,
Lead, Siluer. &c. (being, as I sayd, of like nature, condition, and like
waight throughout.) And you may, by Say Balance, haue prepared a great
number of the smallest waightes: which, by those Balance can be
discerned or tryed: and so, haue proceded to make you a perfect Pyle,
company & Number of waightes: to the waight of six, eight, or twelue
pound waight: most diligently tryed, all. And of euery one, the Content
knowen, in your least waight, that is wayable. [They that can not haue
these waightes of precisenes: may, by Sand, Vniforme, and well dusted,
make them a number of waightes, somewhat nere precisenes: by halfing
euer the Sand: they shall, at length, come to a least common waight.
Therein, I leaue the farder matter, to their discretion, whom nede shall
pinche.] The _Venetians_ consideration of waight, may seme precise
enough: by eight descentes progressionall, * halfing, from a grayne.

    [I. D.
    * For, so, haue you .256. partes of a Graine.]

Your Cube, Sphære, apt Balance, and conuenient waightes, being ready:
fall to worke.❉. First, way your Cube. Note the Number of the waight.
Way, after that, your Sphære. Note likewise, the Nũber of the waight. If
you now find the waight of your Cube, to be to the waight of the Sphære,
as 21. is to 11: Then you see, how the Mechanicien and _Experimenter_,
without Geometrie and Demonstration, are (as nerely in effect) tought
the proportion of the Cube to the Sphere: as I haue demonstrated it, in
the end of the twelfth boke of _Euclide_. Often, try with the same Cube
and Sphære. Then, chaunge, your Sphære and Cube, to an other matter: or
to an other bignes: till you haue made a perfect vniuersall Experience
of it. Possible it is, that you shall wynne to nerer termes, in the
proportion.


When you haue found this one certaine Drop of Naturall veritie, procede
on, to Inferre, and duely to make assay, of matter depending. As,
bycause it is well demonstrated, that a Cylinder, whose heith, and
Diameter of his base, is æquall to the Diameter of the Sphære, is
Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of
the waight of the Sphære, adde halfe so much, as it is: and so haue you
the number of the waight of that Cylinder. Which is also Comprehended of
our former Cube: So, that the base of that Cylinder, is a Circle
described in the Square, which is the base of our Cube. But the Cube and
the Cylinder, being both of one heith, haue their Bases in the same
proportion, in the which, they are, one to an other, in their Massines
or Soliditie. But, before, we haue two numbers, expressing their
Massines, Solidities, and Quantities, by waight: wherfore,

    [* =The proportion of the Square to the Circle inscribed.=]

we haue * the proportion of the Square, to the Circle, inscribed in the
same Square. And so are we fallen into the knowledge sensible, and
Experimentall of _Archimedes_ great Secret: of him, by great trauaile of
minde, sought and found. Wherfore, to any Circle giuen, you can giue a
Square æquall:

    [* =The Squaring of the Circle, Mechanically.=]

* as I haue taught, in my Annotation, vpon the first proposition of the
twelfth boke, And likewise, to any Square giuen, you may giue a Circle
æquall:

    [* =To any Square geuen, to geue a Circle, equall.=]

* If you describe a Circle, which shall be in that proportion, to your
Circle inscribed, as the Square is to the same Circle: This, you may do,
by my Annotations, vpon the second proposition of the twelfth boke of
_Euclide_, in my third Probleme there. Your diligence may come to a
proportion, of the Square to the Circle inscribed, nerer the truth, then
is the proportion of 14. to 11. And consider, that you may begyn at the
Circle and Square, and so come to conclude of the Sphære, & the Cube,
what their proportion is: as now, you came from the Sphære to the
Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough
one hole drawẽ, as the maner is) if you make a Square figure & way it:
and then, describing theron, the Circle inscribed: & cut of, & file
away, precisely (to the Circle) the ouerplus of the Square: you shall
then, waying your Circle, see, whether the waight of the Square, be to
your Circle, as 14. to 11. As I haue Noted, in the beginning of
_Euclides_ twelfth boke. &c. after this resort to my last proposition,
vpon the last of the twelfth. And there, helpe your selfe, to the end.
And, here, Note this, by the way.

    [Note Squaring of the Circle without knowledge of the
    proportion betwene Circumference and Diameter.]

That we may Square the Circle, without hauing knowledge of the
proportion, of the Circumference to the Diameter: as you haue here
perceiued. And otherwayes also, I can demonstrate it. So that, many haue
cumberd them selues superfluously, by trauailing in that point first,
which was not of necessitie, first: and also very intricate. And easily,
you may, (and that diuersly) come to the knowledge of the Circumference:
the Circles Quantitie, being first knowen. Which thing, I leaue to your
consideration: making hast to despatch an other Magistrall Probleme: and
to bring it, nerer to your knowledge, and readier dealing with, then the
world (before this day,) had it for you, that I can tell of. And that
is, _A Mechanicall Dubblyng of the Cube: &c._ Which may, thus, be done:

    [To Dubble the Cube redily: by Art Mechanicall: depending
    vppon Demonstration Mathematicall.]

+Make of Copper plates, or Tyn plates, a foursquare vpright Pyramis, or
a Cone: perfectly fashioned in the holow, within. Wherin, let great
diligence be vsed, to approche (as nere as may be) to the Mathematicall
perfection of those figures. At their bases, let them be all open: euery
where, els, most close, and iust to. From the vertex, to the
Circumference of the base of the Cone: & to the sides of the base of the
Pyramis:+

    [=I. D.=
    =The 4. sides of this Pyramis must be 4. Isosceles
    Triangles alike and æquall.=]

+Let 4. straight lines be drawen, in the inside of the Cone and Pyramis:
makyng at their fall, on the perimeters of the bases, equall angles on
both sides them selues, with the sayd perimeters. These 4. lines (in the
Pyramis: and as many, in the Cone) diuide: one, in 12. æquall partes:
and an other, in 24. an other, in 60, and an other, in 100. (reckenyng
vp from the vertex.) Or vse other numbers of diuision, as experience
shall teach you.+

    [=I. D.=
    =* In all workinges with this Pyramis or Cone, Let their
    Situations be in all Pointes and Conditions, alike,
    or all one: while you are about one Worke. Els you
    will erre.=]

+Then, * set your Cone or Pyramis, with the vertex downward,
perpendicularly, in respect of the Base. (Though it be otherwayes, it
hindreth nothyng.) So let thẽ most stedily be stayed.+ Now, if there be
a Cube, which you wold haue Dubbled. Make you a prety Cube of Copper,
Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or
Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so
proportiõ in respect of your Pyramis or Cone, that the Pyramis or Cone,
will be hable to conteine the waight of them, in water, 3. or 4. times:
at the least: what stuff so euer they be made of. Let not your Solid
angle, at the vertex, be to sharpe: but that the water may come with
ease, to the very vertex, of your hollow Cone or Pyramis. Put one of
your Solid Cubes in a Balance apt: take the waight therof exactly in
water. Powre that water, (without losse) into the hollow Pyramis or
Cone, quietly. Marke in your lines, what numbers the water Cutteth: Take
the waight of the same Cube againe: in the same kinde of water, which
you had before:

    [=I. D.=
    =* Consider well whan you must put your waters togyther:
    and whan, you must empty your first water, out of your
    Pyramis or Cone. Els you will erre.=]

put that* also, into the Pyramis or Cone, where you did put the first.
Marke now againe, in what number or place of the lines, the water
Cutteth them. Two wayes you may conclude your purpose: it is to wete,
either by numbers or lines. By numbers: as, if you diuide the side of
your Fundamentall Cube into so many æquall partes, as it is capable of,
conueniently, with your ease, and precisenes of the diuision. For, as
the number of your first and lesse line (in your hollow Pyramis or
Cone,) is to the second or greater (both being counted from the vertex)
so shall the number of the side of your Fundamentall Cube, be to the
nũber belonging to the Radicall side, of the Cube, dubble to your
Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it
selfe, whether it be dubble or no, to the Cubik number of your
Fundamentall Cube. By lines, thus: As your lesse and first line, (in
your hollow Pyramis or Cone,) is to the second or greater, so let the
Radical side of your Fundamẽtall Cube, be to a fourth proportionall
line, by the 12. proposition, of the sixth boke of _Euclide_. Which
fourth line, shall be the Rote Cubik, or Radicall side of the Cube,
dubble to your Fundamentall Cube: which is the thing we desired.

    [☞ God be thanked for this Inuention, & the fruite ensuing.]

For this, may I (with ioy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking the
holy and glorious Trinity: hauing greater cause therto, then

    [* Vitruuius. Lib. 9. Cap. 3.]

* _Archimedes_ had (for finding the fraude vsed in the Kinges Crowne, of
Gold): as all men may easily Iudge: by the diuersitie of the frute
following of the one, and the other. Where I spake before, of a hollow
Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it
with water, precisely full, and poure that water into your Pyramis or
Cone. And here note the lines cutting in your Pyramis or Cone. Againe,
fill your coffen, like as you did before. Put that Water, also, to the
first. Marke the second cutting of your lines. Now, as you proceded
before, so must you here procede.

    [* Note.]

* And if the Cube, which you should Double, be neuer so great: you haue,
thus, the proportion (in small) betwene your two litle Cubes: And then,
the side, of that great Cube (to be doubled) being the third, will haue
the fourth, found, to it proportionall: by the 12. of the sixth of
Euclide.


    [Note, as concerning the Sphæricall Superficies of the Water.]

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

    [☞]

“Square the Semidiameter, from the Centre of the earth, to your first
Waters Superficies. Square then, halfe the Subtendent of that watry
Superficies (which Subtendent must haue the equall partes of his
measure, all one, with those of the Semidiameter of the earth to your
watry Superficies): Subtracte this square, from the first: Of the
residue, take the Rote Square. That Rote, Subtracte from your first
Semidiameter of the earth to your watry Superficies: that, which
remaineth, is the heith of the water, in the middle, aboue the leuell.”
Which, you will finde, to be a thing insensible. And though it were
greatly sensible, *

    [* Note.]

yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides
twelfth booke, noted: you may reduce all, to a true Leuell. But, farther
diligence, of you is to be vsed, against accidentall causes of the
waters swelling: as by hauing (somwhat) with a moyst Sponge, before,
made moyst your hollow Pyramis or Cone, will preuent an accidentall
cause of Swelling, &c. Experience will teach you abundantly: with great
ease, pleasure, and cõmoditie.


Thus, may you Double the Cube Mechanically, Treble it, and so forth, in
any proportion.

    [Note this Abridgement of Dubbling the Cube. &c.]

Now will I Abridge your paine, cost, and Care herein. Without all
preparing of your Fundamentall Cubes: you may (alike) worke this
Conclusion. For, that, was rather a kinde of Experimentall demõstration,
then the shortest way: and all, vpon one Mathematicall Demonstration
depending. “Take water (as much as conueniently will serue your turne:
as I warned before of your Fundamentall Cubes bignes) Way it precisely.
Put that water, into your Pyramis or Cone. Of the same kinde of water,
then take againe, the same waight you had before: put that likewise into
the Pyramis or Cone. For, in eche time, your marking of the lines, how
the Water doth cut them, shall geue you the proportion betwen the
Radicall sides, of any two Cubes, wherof the one is Double to the other:
working as before I haue taught you:

    [* ☞ Note.]

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


Yet farther proceding with our droppe of Naturall truth:

    [To giue Cubes one to the other in any proportion,
    Rationall or Irrationall.]

+you may (now) geue Cubes, one to the other, in any proportiõ geuẽ:
Rationall or Irrationall+: on this maner. Make a hollow Parallelipipedon
of Copper or Tinne: with one Base wãting, or open: as in our Cubike
Coffen. Frõ the bottome of that Parallelipipedon, raise vp, many
perpendiculars, in euery of his fower sides. Now if any proportion be
assigned you, in right lines: Cut one of your perpendiculars (or a line
equall to it, or lesse then it) likewise: by the 10. of the sixth of
Euclide. And those two partes, set in two sundry lines of those
perpendiculars (or you may set them both, in one line) making their
beginninges, to be, at the base: and so their lengthes to extend vpward.
Now, set your hollow Parallelipipedon, vpright, perpendicularly,
steadie. Poure in water, handsomly, to the heith of your shorter line.
Poure that water, into the hollow Pyramis or Cone. Marke the place of
the rising. Settle your hollow Parallelipipedon againe. Poure water into
it: vnto the heith of the second line, exactly.

    [* Emptying the first.]

Poure that water * duely into the hollow Pyramis or Cone: Marke now
againe, where the water cutteth the same line which you marked before.
For, there, as the first marked line, is to the second: So shall the two
Radicall sides be, one to the other, of any two Cubes: which, in their
Soliditie, shall haue the same proportion, which was at the first
assigned: were it Rationall or Irrationall.


Thus, in sundry waies you may furnishe your selfe with such straunge and
profitable matter: which, long hath bene wished for. And though it be
Naturally done and Mechanically: yet hath it a good Demonstration
Mathematicall.

    [=The demonstrations of this Dubbling of the Cube, and of the
rest.=]

Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones,
in the proportions assigned: and like Pyramids or Cones, are in
proportion, one to the other, in the proportion of their Homologall
sides (or lines) tripled. Wherefore, if to the first, and second lines,
found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in
continuall proportion: that fourth line, shall be to the first, as the
greater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of
Euclide. If Pyramis to Pyramis, or Cone to Cone, be double,

    [I. D.
    = * Hereby, helpe your self to become a præcise practiser.
    And so consider, how, nothing at all, you are hindred
    (sensibly) by the Conuexitie of the water.=]

then shall * Line to Line, be also double, &c. But, as our first line,
is to the second, so is the Radicall side of our Fundamentall Cube, to
the Radicall side of the Cube to be made, or to be doubled: and
therefore, to those twaine also, a third and a fourth line, in
continuall proportion, ioyned: will geue the fourth line in that
proportion to the first, as our fourth Pyramidall, or Conike line, was
to his first: but that was double, or treble, &c. as the Pyramids or
Cones were, one to an other (as we haue proued) therfore, this fourth,
shalbe also double or treble to the first, as the Pyramids or Cones were
one to an other: But our made Cube, is described of the second in
proportion, of the fower proportionall lines:

    [= * By the 33. of the eleuenth booke of Euclide.=]

therfore * as the fourth line, is to the first, so is that Cube, to the
first Cube: and we haue proued the fourth line, to be to the first, as
the Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is to
the Cube, as Pyramis is to Pyramis, or Cone is to Cone.

    [I. D.
    = * And your diligence in practise, can so (in waight of
    water) performe it: Therefore, now, you are able to geue
    good reason of your whole doing.=]

But we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double or
treble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was
to be demonstrated. And of the Parallelipipedõ, it is euidẽt, that the
water Solide Parallelipipedons, are one to the other, as their heithes
are, seing they haue one base. Wherfore the Pyramids or Cones, made of
those water Parallelipipedons, are one to the other, as the lines are
(one to the other) betwene which, our proportion was assigned. But the
Cubes made of lines, after the proportiõ of the Pyramidal or Conik
_homologall_ lines, are one to the other, as the Pyramides or Cones are,
one to the other (as we before did proue) therfore, the Cubes made,
shalbe one to the other, as the lines assigned, are one to the other:
Which was to be demonstrated. Note.

    [* _Note this Corollary._]

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

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

* Now may you, +Betwene two lines giuen, finde two middle proportionals,
in Continuall proportion: by the hollow Parallelipipedon, and the hollow
Pyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre
right lines may be found, proportionall in any proportion assigned, of
which, shal be produced a Parallelipipedon, æquall to the
Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36.
proposition, of the 11. boke of _Euclide_. Now, all those thinges, which
_Vitruuius_ in his Architecture, specified hable to be done, by dubbling
of the Cube: Or, by finding of two middle proportionall lines, betwene
two lines giuen, may easely be performed. Now, that Probleme, which I
noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke
of _Euclide_, is proued possible. Now, may any regular body, be
Transformed into an other, &c. Now, any regular body: any Sphere, yea
any Mixt Solid: and (that more is) Irregular Solides, may be made (in
any proportiõ assigned) like vnto the body, first giuen. Thus, of a
_Manneken_, (as the _Dutch_ Painters terme it) in the same _Symmetrie_,
may a Giant be made: and that, with any gesture, by the Manneken vsed:
and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make
one, of the same Mould (in any assigned proportion) bigger or lesser.

    [* ☞]

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

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

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


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


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

    [MAN is the Lesse World.]

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

    [Micro Cosmus.]

Microcosmicall constitution.

    [* ☞]

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

    [* Lib. 3. Cap. 1.]

Looke in * _Vitruuius_, whether I deale sincerely for your behoufe, or
no. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Looke
in the 27. and 28. Chapters, of the second booke, _De occulta
Philosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther.
Remember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ so
long agoe pronounced: of so many a Philosopher repeated: and of the
_Wisest_ attempted: And then, you will perceaue, how long agoe, you haue
bene called to the Schole, where this Arte might be learned. Well. I am
nothing affrayde, of the disdayne of some such, as thinke Sciences and
Artes, to be but Seuen. Perhaps, those Such, may, with ignorance, and
shame enough, come short of them Seuen also: and yet neuerthelesse they
can not prescribe a certaine number of Artes: and in eche, certaine
vnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly
rise vp: and there was no such order taken, that,

    [☞]

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


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

    [Saw Milles.]

As Milles, to Saw great and very long Deale bordes, no man being by.
Such haue I seene in Germany: and in the Citie of Prage: in the kingdome
of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all
maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight,
Spring, Man or Beast, moued. Take in your hand, _Agricola De re
Metallica:_ and then shall you (in all Mines) perceaue, how great nede
is, of Whele worke. By Wheles, straunge workes and incredible, are done:
as will, in other Artes hereafter, appeare. A wonderfull example of
farther possibilitie, and present commoditie, was sene in my time, in a
certaine Instrument: which by the Inuenter and Artificer (before) was
solde for xx. Talentes of Golde: and then had (by misfortune) receaued
some iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend the
same, and presented it vnto the Emperour _Charles_ the fifth.
_Hieronymus Cardanus_, can be my witnesse, that therein, was one Whele,
which moued, and that, in such rate, that, in 7000. yeares onely, his
owne periode should be finished. A thing almost incredible: But how
farre, I keepe me within my boundes: very many men (yet aliue) can tell.


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

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

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

    [Proclus. Pag. 18.]

Wherat, the King wondring: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει
λέγοντι πιϛευτεόν. _From this day, forward_ (said the King) _Credit
ought to be giuen to Archimedes, what soeuer he sayth._


+‡Pneumatithmie‡ demonstrateth by close hollow Geometricall Figures,
(regular and irregular) the straunge properties (in motion or stay) of
the Water, Ayre, Smoke, and Fire, in theyr cõtinuitie, and as they are
ioyned to the Elementes next them.+ This Arte, to the Naturall
Philosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_
is not in the world. And that, all Nature, abhorreth it so much: that,
contrary to ordinary law, the Elementes will moue or stand. As, Water to
ascend: rather then betwene him and Ayre, Space or place should be left,
more then (naturally) that quãtitie of Ayre requireth, or can fill.
Againe, Water to hang, and not descend: rather then by descending, to
leaue Emptines at his backe. The like, is of Fire and Ayre: they will
descend: when, either, their Cõtinuitie should be dissolued: or their
next Element forced from them. And as they will not be extended, to
discontinuitie: So, will they not, nor yet of mans force, can be prest
or pent, in space, not sufficient and aunswerable to their bodily
substance. Great force and violence will they vse, to enioy their
naturall right and libertie.

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

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


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

    [=Plutarchus in Marco Marcello.=]

_Marcus Marcellus the Consul_, being their Generall Capitaine.

    [=Synesius in Epistolis.=]

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

    [=Polybius.=]

    [=Plinius.=]

    [=Quintilianus.=]

    [=T. Liuius.=]

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

    [=* Athenæus.=]

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

    [= * Galenus.=]

    [=Anthemius.=]

what with his * Burning Glasses: by which he fired their other Shippes a
far-of: what, with his other pollicies, deuises, and engines, he so
manfully acquit him selfe: that all the Force, courage, and pollicie of
the Romaines (for a great season) could nothing preuaile, for the
winning of Syracusa. Wherupon, the Romanes named _Archimedes_,
_Briareus_, and _Centimanus_. _Zonaras_ maketh mention of one _Proclus_,
who so well had perceiued _Archimedes_ Arte of _Menadrie_, and had so
well inuented of his owne, that with his Burning Glasses,

    [Burning Glasses.]

being placed vpon the walles of Bysance, he multiplied so the heate of
the Sunne, and directed the beames of the same against his enemies Nauie
with such force, and so sodeinly (like lightening) that he burned and
destroyed both man and ship. And _Dion_specifieth of _Priscus_,
a _Geometricien_ in Bysance, who inuented and vsed sondry Engins, of
Force multiplied: Which was cause, that the _Emperour Seuerus_ pardoned
him, his life, after he had wonne Bysance: Bycause he honored the Arte,
wytt, and rare industrie of _Priscus_. But nothing inferior to the
inuention of these engines of Force, was the inuention of Gunnes.

    [Gunnes.]

Which, from an English man, had the occasion and order of first
inuenting: though in an other land, and by other men, it was first
executed. And they that should see the record, where the occasion and
order generall, of Gunning, is first discoursed of, would thinke: that,
“small thinges, slight, and cõmon: comming to wise mens consideration,
and industrious mens handling, may grow to be of force incredible.”


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


+‡Hydragogie‡, demonstrateth the possible leading of Water, by Natures
lawe, and by artificiall helpe, from any head (being a Spring, standing,
or running Water) to any other place assigned.+ Long, hath this Arte
bene in vse: and much thereof written: and very marueilous workes
therein, performed: as may yet appeare, in Italy: by the Ruynes
remaining of the Aqueductes. In other places, of Riuers leading through
the Maine land, Nauigable many a Mile. And in other places, of the
marueilous forcinges of Water to Ascend. which all, declare the great
skill, to be required of him, who should in this Arte be perfecte, for
all occasions of waters possible leading. To speake of the allowance of
the Fall, for euery hundred foote: or of the Ventills (if the waters
labour be farre, and great) I neede not: Seing, at hand (about vs) many
expert men can sufficiently testifie, in effecte, the order: though the
Demonstration of the Necessitie thereof, they know not: Nor yet, if they
should be led, vp and downe, and about Mountaines, from the head of the
Spring: and then, a place being assigned: and of them, to be demaunded,
how low or high, that last place is, in respecte of the head, from which
(so crokedly, and vp and downe) they be come: Perhaps, they would not,
or could not, very redily, or nerely assoyle that question. _Geometrie_
therefore, is necessary to _Hydragogie_. Of the sundry wayes to force
water to ascend, eyther by _Tympane_, _Kettell mills_, _Skrue_,
_Ctesibike_, or such like: in _Vitruuius_, _Agricola_, (and other,)
fully, the maner may appeare. And so, thereby, also be most euident, how
the Artes, of _Pneumatithmie_, _Helicosophie_, _Statike_, _Trochilike_,
and _Menadrie_, come to the furniture of this, in Speculation, and to
the Commoditie of the Common Wealth, in practise.


+‡Horometrie‡, is an Arte Mathematicall, which demõstrateth, how, at all
times appointed, the precise vsuall denominatiõ of time, may be knowen,
for any place assigned.+ These wordes, are smoth and plaine easie
Englishe, but the reach of their meaning, is farther, then you woulde
lightly imagine. Some part of this Arte, was called in olde time,
_Gnomonice_: and of late, _Horologiographia_: and in Englishe, may be
termed, _Dialling_. Auncient is the vse, and more auncient, is the
Inuention. The vse, doth well appeare to haue bene (at the least) aboue
two thousand and three hundred yeare agoe:

    [4. Reg. 20.]

in * King _Achaz_ Diall, then, by the Sunne, shewing the distinction of
time. By Sunne, Mone, and Sterres, this Dialling may be performed, and
the precise Time of day or night knowen. But the demonstratiue
delineation of these Dialls, of all sortes, requireth good skill, both
of _Astronomie_, and _Geometrie_ Elementall, Sphæricall, Phænomenall,
and Conikall. Then, to vse the groundes of the Arte, for any regular
Superficies, in any place offred: and (in any possible apt position
therof) theron, to describe (all maner of wayes) how, vsuall howers, may
be (by the _Sunnes_ shadow) truely determined: will be found no sleight
Painters worke. So to Paint, and prescribe the Sunnes Motion, to the
breadth of a heare. In this Feate (in my youth) I Inuented a way, +How
in any Horizontall, Murall, or Æquinoctiall Diall, &c. At all howers
(the Sunne shining) the Signe and Degree ascendent, may be knowen.+
Which is a thing very necessary for the Rising of those fixed Sterres:
whose Operation in the Ayre, is of great might, euidently. I speake no
further, of the vse hereof. Bur forasmuch as, Mans affaires require
knowledge of Times & Momentes, when, neither Sunne, Mone, or Sterre, can
be sene: Therefore, by Industrie Mechanicall, was inuented, first, how,
by Water, running orderly, the Time and howers might be knowen: whereof,
the famous _Ctesibius_, was Inuentor: a man, of _Vitruuius_, to the Skie
(iustly) extolled. Then, after that, by Sand running, were howers
measured: Then, by _Trochilike_ with waight: And of late time, by
_Trochilike_ with Spring: without waight. All these, by Sunne or Sterres
direction (in certaine time) require ouersight and reformation,
according to the heauenly Æquinoctiall Motion: besides the inæqualitie
of their owne Operation. There remayneth (without parabolicall meaning
herein) among the Philosophers,

    [A perpetuall Motion.]

a more excellent, more commodious, and more marueilous way, then all
these: of hauing the motion of the Primouant (or first æquinoctiall
motion,) by Nature and Arte, Imitated: which you shall (by furder search
in waightier studyes) hereafter, vnderstand more of. And so, it is tyme
to finish this Annotation, of Tymes distinction, vsed in our common, and
priuate affaires: The commoditie wherof, no man would want, that can
tell, how to bestow his tyme.


+‡Zographie‡, is an Arte Mathematicall, which teacheth and
demonstrateth, how, the Intersection of all visuall Pyramides, made by
any playne assigned, (the Centre, distance, and lightes, beyng
determined) may be, by lynes, and due propre colours, represented.+ A
notable Arte, is this: and would require a whole Volume, to declare the
property thereof: and the Commodities ensuyng. Great skill of
_Geometrie_, _Arithmetike_, _Perspectiue_, and _Anthropographie_, with
many other particular Artes, hath the _Zographer_, nede of, for his
perfection. For, the most excellent Painter, (who is but the propre
Mechanicien, & Imitator sensible, of the Zographer) hath atteined to
such perfection, that Sense of Man and beast, haue iudged thinges
painted, to be things naturall, and not artificiall: aliue, and not
dead. This Mechanicall Zographer (commonly called the Painter) is
meruailous in his skill: and seemeth to haue a certaine diuine power:
As, of frendes absent, to make a frendly, present comfort: yea, and of
frendes dead, to giue a continuall, silent presence: not onely with vs,
but with our posteritie, for many Ages. And so procedyng, Consider, How,
in Winter, he can shew you, the liuely vew of Sommers Ioy, and riches:
and in Sommer, exhibite the countenance of Winters dolefull State, and
nakednes. Cities, Townes, Fortes, Woodes, Armyes, yea whole Kingdomes
(be they neuer so farre, or greate) can he, with ease, bring with him,
home (to any mans Iudgement) as Paternes liuely, of the thinges
rehearsed. In one little house, can he, enclose (with great pleasure of
the beholders,) the portrayture liuely, of all visible Creatures, either
on earth, or in the earth, liuing: or in the waters lying, Creping,
slyding, or swimming: or of any foule, or fly, in the ayre flying. Nay,
in respect of the Starres, the Skie, the Cloudes: yea, in the shew of
the very light it selfe (that Diuine Creature) can he match our eyes
Iudgement, most nerely. What a thing is this? thinges not yet being, he
can represent so, as, at their being, the Picture shall seame (in maner)
to haue Created them. To what Artificer, is not Picture, a great
pleasure and Commoditie? Which of them all, will refuse the Direction
and ayde of Picture? The Architect, the Goldsmith, and the Arras Weauer:
of Picture, make great account. Our liuely Herbals, our portraitures of
birdes, beastes, and fishes: and our curious Anatomies, which way, are
they most perfectly made, or with most pleasure, of vs beholden? Is it
not, by Picture onely? And if Picture, by the Industry of the Painter,
be thus commodious and meruailous: what shall be thought of _Zographie_,
the Scholemaster of Picture, and chief gouernor? Though I mencion not
_Sculpture_, in my Table of Artes Mathematicall: yet may all men
perceiue, How, that _Picture_ and _Sculpture_, are Sisters germaine: and
both, right profitable, in a Commõ wealth. and of _Sculpture_, aswell as
of Picture, excellent Artificers haue written great bokes in
commendation. Witnesse I take, of _Georgio Vasari_, _Pittore Aretino_:
of _Pomponius Gauricus_: and other. To these two Artes, (with other,) is
a certaine od Arte, called _Althalmasat_, much beholdyng: more, then the
common _Sculptor_, _Entayler_, _Keruer_, _Cutter_, _Grauer_, _Founder_,
or _Paynter (&c)_ know their Arte, to be commodious.


    [An objection.]

+‡Architecture‡+, to many may seme not worthy, or not mete, to be
reckned among the _Artes Mathematicall_. To whom, I thinke good, to giue
some account of my so doyng. Not worthy, (will they say,) bycause it is
but for building, of a house, Pallace, Church, Forte, or such like,
grosse workes. And you, also, defined the _Artes Mathematicall_, to be
such, as dealed with no Materiall or corruptible thing: and also did
demonstratiuely procede in their faculty, by Number or Magnitude. First,

    [The Answer.]

you see, that I count, here, _Architecture_, among those _Artes
Mathematicall_, which are Deriued from the Principals: and you know,
that such, may deale with Naturall thinges, and sensible matter. Of
which, “some draw nerer, to the Simple and absolute Mathematicall
Speculation, then other do.

    [☞]

And though, the _Architect_ procureth, enformeth, & directeth, the
_Mechanicien_, to handworke, & the building actuall, of house, Castell,
or Pallace, and is chief Iudge of the same: yet, with him selfe (as
chief _Master_ and _Architect_,) remaineth the Demonstratiue reason and
cause, of the Mechaniciens worke: in Lyne, plaine, and Solid: by
_Geometricall_, _Arithmeticall_, _Opticall_, _Musicall_,
_Astronomicall_, _Cosmographicall_” (& to be brief) by all the former
Deriued _Artes Mathematicall_, and other Naturall Artes, hable to be
confirmed and stablished. If this be so: then, may you thinke, that
_Architecture_, hath good and due allowance, in this honest Company of
_Artes Mathematicall_ Deriuatiue. I will, herein, craue Iudgement of two
most perfect _Architectes_: the one, being _Vitruuius_, the Romaine: who
did write ten bookes thereof, to the Emperour _Augustus_ (in whose daies
our Heauenly Archemaster, was borne): and the other, _Leo Baptista
Albertus_, a Florentine: who also published ten bookes therof.
_Architectura_ (sayth _Vitruuius_) _est Scientia pluribus disciplinis &
varijs eruditionibus ornata: cuius Iudicio probantur omnia, quæ ab
cæteris Artificibus perficiuntur opera._ That is. +Architecture, is a
Science garnished with many doctrines & diuerse instructions: by whose
Iudgement, all workes, by other workmen finished, are Iudged.+ It
followeth. _Ea nascitur ex Fabrica, & Ratiocinatione. &c. Ratiocinatio
autem est, quæ, res fabricatas, Solertia ac ratione proportionis,
demonstrare at[que] explicare potest. +Architecture, groweth of Framing,
and Reasoning. &c. Reasoning, is that, which of thinges framed, with
forecast, and proportion: can make demonstration, and manifest
declaration.+_ Againe. _Cùm, in omnibus enim rebus, tùm maximè etiam in
Architectura, hæc duo insunt: quod significatur, & quod significat.
Significatur proposita res, de qua dicitur: hanc autem Significat
Demonstratio, rationibus doctrinarum explicata. +Forasmuch as, in all
thinges: therefore chiefly in Architecture, these two thinges are: the
thing signified: and that which signifieth. The thing propounded,
whereof we speake, is the thing Signified. But Demonstration, expressed
with the reasons of diuerse doctrines, doth signifie the same thing.+_
After that. _Vt literatus sit, peritus Graphidos, eruditus Geometriæ, &
Optices non ignarus: instructus Arithmetica: historias complures
nouerit, Philosophos diligenter audiuerit: Musicam sciuerit: Medicinæ
non sit ignarus, responsa Iurisperitorũ nouerit: Astrologiam, Cæli[que]
rationes cognitas habeat. +An Architect+_ (sayth he) +_ought to
vnderstand Languages, to be skilfull of Painting, well instructed in
Geometrie, not ignorant of Perspectiue, furnished with Arithmetike, haue
knowledge of many histories, and diligently haue heard Philosophers,
haue skill of Musike, not ignorant of Physike, know the aunsweres of
Lawyers, and haue Astronomie, and the courses Cælestiall, in good
knowledge._+ He geueth reason, orderly, wherefore all these Artes,
Doctrines, and Instructions, are requisite in an excellent _Architect_.
And (for breuitie) omitting the Latin text, thus he hath. +_Secondly, it
is behofefull for an Architect to haue the knowledge of Painting: that
he may the more easilie fashion out, in patternes painted, the forme of
what worke he liketh. And Geometrie, geueth to Architecture many helpes:
and first teacheth the Vse of the Rule, and the Cumpasse: wherby
(chiefly and easilie) the descriptions of Buildinges, are despatched in
Groundplats: and the directions of Squires, Leuells, and Lines.
Likewise, by Perspectiue, the Lightes of the heauen, are well led, in
the buildinges: from certaine quarters of the world. By Arithmetike, the
charges of Buildinges are summed together: the measures are expressed,
and the hard questions of Symmetries, are by Geometricall Meanes and
Methods discoursed on. &c. Besides this, of the Nature of thinges (which
in Greke is called φυσιολογία) Philosophie doth make declaration. Which,
it is necessary, for an Architect, with diligence to haue learned:
because it hath many and diuers naturall questions: as specially, in
Aqueductes. For in their courses, leadinges about, in the leuell ground,
and in the mountinges, the naturall Spirites or breathes are ingendred
diuers wayes: The hindrances, which they cause, no man can helpe, but
he, which out of Philosophie, hath learned the originall causes of
thinges. Likewise, who soeuer shall read Ctesibius, or Archimedes
bookes, (and of others, who haue written such Rules) can not thinke, as
they do: vnlesse he shall haue receaued of Philosophers, instructions in
these thinges. And Musike he must nedes know: that he may haue
vnderstanding, both of Regular and Mathematicall Musike: that he may
temper well his Balistes, Catapultes, and Scorpions. &c. Moreouer, the
Brasen Vessels, which in Theatres, are placed by Mathematicall order, in
ambries, vnder the steppes: and the diuersities of the soundes (which
y^e Grecians call ηχεῖα) are ordred according to Musicall Symphonies &
Harmonies: being distributed in y^e Circuites, by Diatessaron, Diapente,
and Diapason. That the conuenient voyce, of the players sound, whẽ it
came to these preparations, made in order, there being increased: with
y^t increasing, might come more cleare & pleasant, to y^e eares of the
lokers on. &c. And of Astronomie, is knowẽ y^e East, West, South, and
North. The fashion of the heauen, the Æquinox, the Solsticie, and the
course of the sterres. Which thinges, vnleast one know: he can not
perceiue, any thyng at all, the reason of Horologies. Seyng therfore
this ample Science, is garnished, beautified and stored, with so many
and sundry skils and knowledges: I thinke, that none can iustly account
them selues Architectes, of the suddeyne. But they onely, who from their
childes yeares, ascendyng by these degrees of knowledges, beyng fostered
vp with the atteynyng of many Languages and Artes, haue wonne to the
high Tabernacle of Architecture. &c. And to whom Nature hath giuen such
quicke Circumspection, sharpnes of witt, and Memorie, that they may be
very absolutely skillfull in Geometrie, Astronomie, Musike, and the rest
of the Artes Mathematicall: Such, surmount and passe the callyng, and
state, of Architectes:

    [A Mathematicien.]

and are become Mathematiciens. &c. And they are found, seldome. As, in
tymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes:
Apollonius Pergęus: Eratosthenes Cyreneus: Archimedes, and Scopas,
Syracusians. Who also, left to theyr posteritie, many Engines and
Gnomonicall workes: by numbers and naturall meanes, inuented and
declared._+


Thus much, and the same wordes (in sense) in one onely Chapter of this
Incõparable _Architect Vitruuius_, shall you finde. And if you should,
but take his boke in your hand, and slightly loke thorough it, you would
say straight way:

    [Vitruuius.]

This is _Geometrie_, _Arithmetike_, _Astronomie_, _Musike_,
_Anthropographie_, _Hydragogie_, _Horometrie_. _&c_. and (to cõclude)
the Storehouse of all workmãship. Now, let vs listen to our other Iudge,
our Florentine, _Leo Baptista_: and narrowly consider, how he doth
determine of _Architecture_. _Sed ante[que] vltra progrediar. &c. +But
before I procede any further +_(sayth he) +_I thinke, that I ought to
expresse, what man I would haue to bee allowed an Architect. For, I will
not bryng in place a Carpenter: as though you might Compare him to the
Chief Masters of other Artes. For the hand of the Carpenter, is the
Architectes Instrument._+

    [VVho is an Architect.]

+_But I will appoint the Architect to be “that man, who hath the skill,
(by a certaine and meruailous meanes and way,) both in minde and
Imagination to determine and also in worke to finish: what workes so
euer, by motion of waight, and cuppling and framyng together of bodyes,
may most aptly be Commodious for the worthiest Vses of Man.” And that he
may be able to performe these thinges, he hath nede of atteynyng and
knowledge of the best, and most worthy thynges. &c. The whole Feate of
Architecture in buildyng, consisteth in Lineamentes, and in Framyng. And
the whole power and skill of Lineamentes, tendeth to this: that the
right and absolute way may be had, of Coaptyng and ioyning Lines and
angles: by which, the face of the buildyng or frame, may be comprehended
and concluded. And it is the property of Lineamentes, to prescribe vnto
buildynges, and euery part of them, an apt place, & certaine nũber:
a worthy maner, and a semely order: that, so, y^e whole forme and figure
of the buildyng, may rest in the very Lineamentes. &c. And we may
prescribe in mynde and imagination the whole formes, *

    [* The Immaterialitie of perfect Architecture.]

all material stuffe beyng secluded. Which point we shall atteyne, by
Notyng and forepointyng the angles, and lines, by a sure and certaine
direction and connexion. Seyng then, these thinges, are thus:_+

    [What, Lineament is.]

+_Lineamente, shalbe the certaine and constant prescribyng, conceiued in
mynde: made in lines and angles: and finished with a learned minde and
wyt._+ “We thanke you Master _Baptist_, that you haue so aptly brought
your Arte, and phrase therof, to haue some Mathematicall perfection:

    [Note.]

by certaine order, nũber, forme, figure, and _Symmetrie_ mentall:” all
naturall & sensible stuffe set a part. Now, then, it is euident, (Gentle
reader) how aptely and worthely, I haue preferred _Architecture_, to be
bred and fostered vp in the Dominion of the pereles _Princesse_,
_Mathematica_: and to be a naturall Subiect of hers. And the name of
_Architecture_, is of the principalitie, which this Science hath, aboue
all other Artes. And _Plato_ affirmeth, the _Architect_ to be _Master_
ouer all, that make any worke. Wherupon, he is neither Smith, nor
Builder: nor, separately, any Artificer: but the Hed, the Prouost, the
Directer, and Iudge of all Artificiall workes, and all Artificers. For,
the true _Architect_, is hable to teach, Demonstrate, distribute,
describe, and Iudge all workes wrought. And he, onely, searcheth out the
causes and reasons of all Artificiall thynges. Thus excellent, is
_Architecture_: though few (in our dayes) atteyne thereto: yet may not
the Arte, be otherwise thought on, then in very dede it is worthy. Nor
we may not, of auncient Artes, make new and imperfect Definitions in our
dayes: for scarsitie of Artificers: No more, than we may pynche in, the
Definitions of _Wisedome_, or _Honestie_, or of _Frendeshyp_ or of
_Iustice_. No more will I consent, to Diminish any whit, of the
perfection and dignitie, (by iust cause) allowed to absolute
_Architecture_. Vnder the Direction of this Arte, are thre principall,
necessary _Mechanicall Artes_. Namely, _Howsing_, _Fortification_, and
_Naupegie_. _Howsing_, I vnderstand, both for Diuine Seruice, and Mans
common vsage: publike, and priuate. Of _Fortification_ and _Naupegie_,
straunge matter might be told you: But perchaunce, some will be tyred,
with this Bederoll, all ready rehearsed: and other some, will nycely nip
my grosse and homely discoursing with you: made in post hast: for feare
you should wante this true and frendly warnyng, and tast giuyng, of the
_Power Mathematicall_. Lyfe is short, and vncertaine: Tymes are
perilouse: &c. And still the Printer awayting, for my pen staying: All
these thinges, with farder matter of Ingratefulnes, giue me occasion to
passe away, to the other Artes remainyng, with all spede possible.


+The Arte of ‡Nauigation‡, demonstrateth how, by the shortest good way,
by the aptest Directiõ, & in the shortest time, a sufficient Ship,
betwene any two places (in passage Nauigable,) assigned: may be
cõducted: and in all stormes, & naturall disturbances chauncyng, how, to
vse the best possible meanes, whereby to recouer the place first
assigned.+ What nede, the _Master Pilote_, hath of other Artes, here
before recited, it is easie to know: as, of _Hydrographie_,
_Astronomie_, _Astrologie_, and _Horometrie_. Presupposing continually,
the common Base, and foundacion of all: namely _Arithmetike_ and
_Geometrie_. So that, he be hable to vnderstand, and Iudge his own
necessary Instrumentes, and furniture Necessary: Whether they be
perfectly made or no: and also can, (if nede be) make them, hym selfe.
As Quadrantes, The Astronomers Ryng, The Astronomers staffe, The
Astrolabe vniuersall. An Hydrographicall Globe. Charts Hydrographicall,
true, (not with parallell Meridians). The Common Sea Compas: The Compas
of variacion: The Proportionall, and Paradoxall Compasses

    [Anno. 1559.]

(of me Inuented, for our two Moscouy Master Pilotes, at the request of
the Company) Clockes with spryng: houre, halfe houre, and three houre
Sandglasses: & sundry other Instrumẽtes: And also, be hable, on Globe,
or Playne to describe the Paradoxall Compasse: and duely to vse the
same, to all maner of purposes, whereto it was inuented. And also, be
hable to Calculate the Planetes places for all tymes.


Moreouer, with Sonne Mone or Sterre (or without) be hable to define the
Longitude & Latitude of the place, which he is in: So that, the
Longitude & Latitude of the place, from which he sayled, be giuen: or by
him, be knowne. whereto, appertayneth expert meanes, to be certified
euer, of the Ships way. &c. And by foreseing the Rising, Settyng,
Nonestedyng, or Midnightyng of certaine tempestuous fixed Sterres: or
their Coniunctions, and Anglynges with the Planetes, &c. he ought to
haue expert coniecture of Stormes, Tempestes, and Spoutes: and such lyke
Meteorologicall effectes, daungerous on Sea. For (as _Plato_ sayth,)
_Mutationes, opportunitates[que] temporum presentire, non minus rei
militari, quàm Agriculturæ, Nauigationi[que] conuenit. +To foresee the
alterations and opportunities of tymes, is conuenient, no lesse to the
Art of Warre, then to Husbandry and Nauigation.+_ And besides such
cunnyng meanes, more euident tokens in Sonne and Mone, ought of hym to
be knowen: such as (the Philosophicall Poëte) _Virgilius_ teacheth, in
hys _Georgikes_. Where he sayth,

    [Sidenote: Georgic. 1.]

  _Sol quo[que] & exoriens & quum se condet in vndas,
  Signa dabit, Solem certissima signa sequuntur. &c.
    -------- Nam sæpe videmus,
  Ipsius in vultu varios errare colores.
  Cæruleus, pluuiam denunciat, igneus Euros.
  Sin maculæ incipient rutilo immiscerier igni,
  Omnia tum pariter vento, nimbis[que] videbis
  Feruere: non illa quisquam me nocte per altum
  Ire, ne[que] a terra moueat conuellere funem. &c.
  Sol tibi signa dabit. Solem quis dicere falsum
  Audeat? -------- &c._

And so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and
Beastes, and of many thynges els, a certaine Sympathicall forewarnyng
may be had: sometymes to great pleasure and proffit, both on Sea and
Land. Sufficiently, for my present purpose, it doth appeare, by the
premisses, how _Mathematicall_, the _Arte_ of _Nauigation_, is: and how
it nedeth and also vseth other _Mathematicall Artes_: And now, if I
would go about to speake of the manifold Commodities, commyng to this
Land, and others, by Shypps and _Nauigation_, you might thinke, that I
catch at occasions, to vse many wordes, where no nede is.


Yet, this one thyng may I, (iustly) say. In _Nauigation_, none ought to
haue greater care, to be skillfull, then our English Pylotes. And
perchaunce, Some, would more attempt: And other Some, more willingly
would be aydyng, it they wist certainely, What Priuiledge, God had
endued this Iland with, by reason of Situation, most commodious for
_Nauigation_, to Places most Famous & Riche. And though,

    [* Anno. 1567 S. H. G.]

(of * Late) a young Gentleman, a Courragious Capitaine, was in a great
readynes, with good hope, and great causes of persuasion, to haue
ventured, for a Discouerye, (either _Westerly_, by _Cape de Paramantia_:
or _Esterly_, aboue _Noua Zemla_, and the _Cyremisses_) and was, at the
very nere tyme of Attemptyng, called and employed otherwise (both then,
and since,) in great good seruice to his Countrey, as the Irish Rebels
haue * tasted:

    [* Anno. 1569]

Yet, I say, (though the same Gentleman, doo not hereafter, deale
therewith) Some one, or other, should listen to the Matter: and by good
aduise, and discrete Circumspection, by little, and little, wynne to the
sufficient knowledge of that +Trade+ and +Voyage+: Which, now, I would
be sory, (through Carelesnesse, want of Skill, and Courrage,) should
remayne Vnknowne and vnheard of. Seyng, also, we are herein, halfe
Challenged, by the learned, by halfe request, published. Therof, verely,
might grow Commoditye, to this Land chiefly, and to the rest of the
Christen Common wealth, farre passing all riches and worldly Threasure.


+‡Thaumaturgike‡, is that Art Mathematicall, which giueth certaine order
to make straunge workes, of the sense to be perceiued, and of men
greatly to be wondred at.+ By sundry meanes, this _Wonder-worke_ is
wrought. Some, by _Pneumatithmie_. As the workes of _Ctesibius_ and
_Hero_, Some by waight. wherof _Timæus_ speaketh. Some, by Stringes
strayned, or Springs, therwith Imitating liuely Motions. Some, by other
meanes, as the Images of Mercurie: and the brasen hed, made by _Albertus
Magnus_, which dyd seme to speake. _Boethius_ was excellent in these
feates. To whom, _Cassiodorus_ writyng, sayth. +_Your purpose is to know
profound thynges: and to shew meruayles. By the disposition of your
Arte, Metals do low: Diomedes of brasse, doth blow a Trumpet loude:
a brasen Serpent hisseth: byrdes made, sing swetely. Small thynges we
rehearse of you, who can Imitate the heauen. &c._+ Of the straunge
Selfmouyng, which, at Saint Denys, by Paris,

    [* Anno. 1551]

* I saw, ones or twise (_Orontius_ beyng then with me, in Company) it
were to straunge to tell. But some haue written it. And yet, (I hope) it
is there, of other to be sene. And by _Perspectiue_ also straunge
thinges, are done. As partly (before) I gaue you to vnderstand in
_Perspectiue_. As, to see in the Ayre, a loft, the lyuely Image of an
other man, either walkyng to and fro: or standyng still. Likewise, to
come into an house, and there to see the liuely shew of Gold, Siluer or
precious stones: and commyng to take them in your hand, to finde nought
but Ayre. Hereby, haue some men (in all other matters counted wise)
fouly ouershot thẽ selues: misdeaming of the meanes. Therfore sayd
_Claudius Cælestinus_.

    [De his quæ Mundo mirabiliter eueniunt. cap. 8.]

_Hodie magnæ literaturæ viros & magna reputationis videmus, opera quedam
quasi miranda, supra Naturã putare: de quibus in Perspectiua doctus
causam faciliter reddidisset._ That is. +_Now a dayes, we see some men,
yea of great learnyng and reputation, to Iudge certain workes as
meruaylous, aboue the power of Nature: Of which workes, one that were
skillfull in Perspectiue might easely haue giuen the Cause._+ Of
_Archimedes Sphære_, _Cicero_ witnesseth.

    [Tusc. 1.]

Which is very straunge to thinke on. +_For when Archimedes_+ (sayth he)
+_did fasten in a Sphære, the mouynges of the Sonne, Mone, and of the
fiue other Planets, he did, as the God, which (in Timæus of Plato) did
make the world. That, one turnyng, should rule motions most vnlike in
slownes, and swiftnes._+ But a greater cause of meruayling we haue by
_Claudianus_ report hereof. Who affirmeth this _Archimedes worke_, to
haue ben of Glasse. And discourseth of it more at large: which I omit.
The Doue of wood, which the _Mathematicien Archytas_ did make to flye,
is by _Agellius_ spoken of. Of _Dædalus_ straunge Images, _Plato_
reporteth. _Homere_ of _Vulcans Selfmouers_, (by secret wheles) leaueth
in writyng. _Aristotle_, in hys _Politikes_, of both, maketh mention.
Meruaylous was the workemanshyp, of late dayes, performed by good skill
of _Trochilike. &c._ For in Noremberge, A flye of Iern, beyng let out of
the Artificers hand, did (as it were) fly about by the gestes, at the
table, and at length, as though it were weary, retourne to his masters
hand agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of
the same Towne, a mighty way, and that a loft in the Ayre, toward the
Emperour comming thether: and followed hym, beyng come to the gate of
the towne. *

    [* ☞]

Thus, you see, what, Arte Mathematicall can performe, when Skill, will,
Industry, and Hability, are duely applyed to profe.


    [A Digression.]

And for these, and such like marueilous Actes and Feates, Naturally,
Mathematically, and Mechanically, wrought and contriued:

    [Apologeticall.]

ought any honest Student, and Modest Christian Philosopher, be counted,
& called a +Coniurer+? Shall the folly of Idiotes, and the Mallice of
the Scornfull, so much preuaile, that He, who seeketh no worldly gaine
or glory at their handes: But onely, of God, the threasor of heauenly
wisedome, & knowledge of pure veritie: Shall he (I say) in the meane
space, be robbed and spoiled of his honest name and fame? He that seketh
(by S. Paules aduertisement) in the Creatures Properties, and wonderfull
vertues, to finde iuste cause, to glorifie the Æternall, and Almightie
Creator by: Shall that man, be (in hugger mugger) condemned, as a
Companion of the Helhoundes, and a Caller, and Coniurer of wicked and
damned Spirites? He that bewaileth his great want of time, sufficient
(to his contentation) for learning of Godly wisdome, and Godly Verities
in: and onely therin setteth all his delight: Will that mã leese and
abuse his time, in dealing with the Chiefe enemie of Christ our Redemer:
the deadly foe of all mankinde: the subtile and impudent peruerter of
Godly Veritie: the Hypocriticall Crocodile: the Enuious Basiliske,
continually desirous, in the twinke of an eye, to destroy all Mankinde,
both in Body and Soule, æternally? Surely (for my part, somewhat to say
herein) I haue not learned to make so brutish, and so wicked a Bargaine.
Should I, for my xx. or xxv. yeares Studie: for two or three thousand
Markes spending: seuen or eight thousand Miles going and trauailing,
onely for good learninges sake: And that, in all maner of wethers: in
all maner of waies and passages: both early and late: in daunger of
violence by man: in daunger of destruction by wilde beastes: in hunger:
in thirst: in perilous heates by day, with toyle on foote: in daungerous
dampes of colde, by night, almost bereuing life: (as God knoweth): with
lodginges, oft times, to small ease: and somtime to lesse securitie. And
for much more (then all this) done & suffred, for Learning and attaining
of Wisedome: Should I (I pray you) for all this, no otherwise, nor more
warily: or (by Gods mercifulnes) no more luckily, haue fished, with so
large, and costly, a Nette, so long time in drawing (and that with the
helpe and aduise of Lady Philosophie, & Queene Theologie): but at
length, to haue catched, and drawen vp, * a Frog?

    [* A prouerb. Fayre fisht, and caught a Frog.]

Nay, a Deuill? For, so, doth the Common peuish Pratler Imagine and
Iangle: And, so, doth the Malicious skorner, secretly wishe, & brauely
and boldly face down, behinde my backe. Ah, what a miserable thing, is
this kinde of Men? How great is the blindnes & boldnes, of the
Multitude, in thinges aboue their Capacitie? What a Land: what a People:
what Maners: what Times are these? Are they become Deuils, them selues:
and, by false witnesse bearing against their Neighbour, would they also,
become Murderers? Doth God, so long geue them respite, to reclaime them
selues in, from this horrible slaundering of the giltlesse: contrary to
their owne Consciences: and yet will they not cease? Doth the Innocent,
forbeare the calling of them, Iuridically to aunswere him, according to
the rigour of the Lawes: and will they despise his Charitable pacience?
As they, against him, by name, do forge, fable, rage, and raise
slaunder, by Worde & Print: Will they prouoke him, by worde and Print,
likewise, to Note their Names to the World: with their particular
deuises, fables, beastly Imaginations, and vnchristen-like slaunders?
Well: Well. O (you such) my vnkinde Countrey men. O vnnaturall Countrey
men. O vnthankfull Countrey men. O Brainsicke, Rashe, Spitefull, and
Disdainfull Countrey men. Why oppresse you me, thus violently, with your
slaundering of me: Contrary to Veritie: and contrary to your owne
Consciences? And I, to this hower, neither by worde, deede, or thought,
haue bene, any way, hurtfull, damageable, or iniurious to you, or yours?
Haue I, so long, so dearly, so farre, so carefully, so painfully, so
daungerously sought & trauailed for the learning of Wisedome, &
atteyning of Vertue: And in the end (in your iudgemẽt) am I become,
worse, then when I begã? Worse, thẽ a Mad man? A dangerous Member in the
Common Wealth: and no Member of the Church of Christ? Call you this, to
be Learned? Call you this, to be a Philosopher? and a louer of Wisedome?
To forsake the straight heauenly way: and to wallow in the broad way of
damnation? To forsake the light of heauenly Wisedome: and to lurke in
the dungeon of the Prince of darkenesse? To forsake the Veritie of God,
& his Creatures: and to fawne vpon the Impudent, Craftie, Obstinate
Lier, and continuall disgracer of Gods Veritie, to the vttermost of his
power? To forsake the Life & Blisse Æternall: and to cleaue vnto the
Author of Death euerlasting? that Murderous Tyrant, most gredily
awaiting the Pray of Mans Soule? Well: I thanke God and our Lorde Iesus
Christ, for the Comfort which I haue by the Examples of other men,
before my time: To whom, neither in godlines of life, nor in perfection
of learning, I am worthy to be compared: and yet, they sustained the
very like Iniuries, that I do: or rather, greater. Pacient _Socrates_,
his _Apologie_ will testifie: _Apuleius_ his _Apologies_, will declare
the Brutishnesse of the Multitude. _Ioannes Picus_, Earle of Mirandula,
his _Apologie_ will teach you, of the Raging slaunder of the Malicious
Ignorant against him. _Ioannes Trithemius_, his _Apologie_ will
specifie, how he had occasion to make publike Protestation: as well by
reason of the Rude Simple: as also, in respect of such, as were counted
to be of the wisest sort of men. “Many could I recite: But I deferre the
precise and determined handling of this matter: being loth to detect the
Folly & Mallice of my Natiue Countrey men. *

    [* ☞]

Who, so hardly, can disgest or like any extraordinary course of
Philosophicall Studies: not falling within the Cumpasse of their
Capacitie: or where they are not made priuie of the true and secrete
cause, of such wonderfull Philosophicall Feates.” These men, are of
fower sortes, chiefly. The first, I may name, _Vaine pratling busie
bodies_: The second, _Fond Frendes_: The third, _Imperfectly zelous_:
and the fourth, _Malicious Ignorant_. To eche of these (briefly, and in
charitie) I will say a word or two, and so returne to my Præface.

    [1.]

_Vaine pratling busie bodies_, vse your idle assemblies, and
conferences, otherwise, then in talke of matter, either aboue your
Capacities, for hardnesse: or contrary to your Consciences, in Veritie.

    [2.]

_Fonde Frendes_, leaue of, so to commend your vnacquainted frend, vpon
blinde affection: As, because he knoweth more, then the common Student:
that, therfore, he must needes be skilfull, and a doer, in such matter
and maner, as you terme _Coniuring_. Weening, thereby, you aduaunce his
fame: and that you make other men, great marueilers of your hap, to haue
such a learned frend. Cease to ascribe Impietie, where you pretend
Amitie. For, if your tounges were true, then were that your frend,
_Vntrue_, both to God, and his Soueraigne. Such _Frendes_ and
_Fondlinges_, I shake of, and renounce you: Shake you of, your Folly.

    [3.]

_Imperfectly zelous_, to you, do I say: that (perhaps) well, do you
Meane: But farre you misse the Marke: If a Lambe you will kill, to feede
the flocke with his bloud. Sheepe, with Lambes bloud, haue no naturall
sustenaunce: No more, is Christes flocke, with horrible slaunders, duely
ædified. Nor your faire pretense, by such rashe ragged Rhetorike, any
whit, well graced. But such, as so vse me, will finde a fowle Cracke in
their Credite. Speake that you know: And know, as you ought: Know not,
by Heare say, when life lieth in daunger. Search to the quicke, & let
Charitie be your guide.

    [4.]

_Malicious Ignorant_, what shall I say to thee? _Prohibe linguam tuam a
malo. A detractione parcite linguæ. +Cause thy toung to refraine frõ
euill. Refraine your toung from slaunder.+_ Though your tounges be
sharpned, Serpent like, & Adders poyson lye in your lippes:

    [Psal. 140.]

yet take heede, and thinke, betimes, with your selfe, _Vir linguosus non
stabilietur in terra. Virum violentum venabitur malum, donec
præcipitetur._ For, sure I am, _Quia faciet Dominus Iudicium afflicti:
& vindictam pauperum._


Thus, I require you, my assured frendes, and Countrey men (you
Mathematiciens, Mechaniciens, and Philosophers, Charitable and discrete)
to deale in my behalf, with the light & vntrue tounged, my enuious
Aduersaries, or Fond frends. And farther, I would wishe, that at leysor,
you would consider, how _Basilius Magnus_, layeth _Moses_ and _Daniel_,
before the eyes of those, which count all such Studies Philosophicall
(as mine hath bene) to be vngodly, or vnprofitable. Waye well
_S. Stephen_ his witnesse of _Moses_.

    [Act. 7. C.]

_Eruditus est Moses omni Sapientia Ægyptiorũ: & erat potens in verbis &
operibus suis. +Moses was instructed in all maner of wisedome of the
Ægyptians: and he was of power both in his wordes, and workes.+_ You see
this Philosophicall Power & Wisedome, which _Moses_ had, to be nothing
misliked of the Holy Ghost. Yet _Plinius_ hath recorded, _Moses_ to be a
wicked _Magicien_. And that (of force) must be, either for this
Philosophicall wisedome, learned, before his calling to the leading of
the Children of _Israel_: or for those his wonders, wrought before King
_Pharao_, after he had the conducting of the _Israelites_. As concerning
the first, you perceaue, how _S. Stephen_, at his Martyrdome (being full
of the Holy Ghost) in his Recapitulation of the olde Testament, hath
made mention of _Moses_ Philosophie: with good liking of it: And
_Basilius Magnus_ also, auoucheth it, to haue bene to _Moses_ profitable
(and therefore, I say, to the Church of God, necessary). But as
cõcerning _Moses_ wonders, done before King _Pharao_: God, him selfe,
sayd: _Vide vt omnia ostenta, quæ posui in manu tua, facias coram
Pharaone. +See that thou do all those wonders before Pharao, which I
haue put in thy hand.+_ Thus, you euidently perceaue, how rashly,
_Plinius_ hath slaundered _Moses_,

    [Lib. 30. Cap. 1.]

of vayne fraudulent _Magike_, saying: _Est & alia Magices Factio,
a Mose, Iamne, & Iotape, Iudæis pendens: sed multis millibus annorum
post Zoroastrem. &c._

    [1.]

Let all such, therefore, who, in Iudgement and Skill of Philosophie, are
farre Inferior to _Plinie_, “take good heede, least they ouershoote them
selues rashly,” in

    [☞]

Iudging of _Philosophers straunge Actes_: and the Meanes, how they are
done.

    [2.]

But, much more, ought they to beware of forging, deuising, and imagining
monstrous feates, and wonderfull workes, when and where, no such were
done: no, not any sparke or likelihode, of such, as they, without all
shame, do report.

    [3.]

And (to conclude) most of all, let them be ashamed of Man, and afraide
of the dreadfull and Iuste Iudge: both Folishly or Maliciously to
deuise: and then, deuilishly to father their new fond Monsters on me:
Innocent, in hand and hart: for trespacing either against the lawe of
God, or Man, in any my Studies or Exercises, Philosophicall, or
Mathematicall: As in due time, I hope, will be more manifest.


Now end I, with +‡Archemastrie‡+. Which name, is not so new, as this
Arte is rare. For an other Arte, vnder this, a degree (for skill and
power) hath bene indued with this English name before. And yet, this,
may serue for our purpose, sufficiently, at this present. +This Arte,
teacheth to bryng to actuall experience sensible, all worthy conclusions
by all the Artes Mathematicall purposed, & by true Naturall Philosophie
concluded: & both addeth to them a farder scope, in the termes of the
same Artes, & also by hys propre Method, and in peculier termes,
procedeth, with helpe of the foresayd Artes, to the performance of
complet Experiẽces, which of no particular Art, are hable (Formally) to
be challenged.+ If you remember, how we considered _Architecture_, in
respect of all common handworkes: some light may you haue, therby, to
vnderstand the Souerainty and propertie of this Science. _Science_ I may
call it, rather, then an Arte: for the excellency and Mastershyp it
hath, ouer so many, and so mighty Artes and Sciences. And bycause it
procedeth by _Experiences_, and searcheth forth the causes of
Conclusions, by _Experiences_: and also putteth the Conclusions them
selues, in _Experience_, it is named of some, _Scientia Experimentalis_.
The +_Experimentall Science_+. _Nicolaus Cusanus_ termeth it so, in hys
_Experimentes Statikall_, And an other _Philosopher_,

    [R. B.]

of this land Natiue (the floure of whose worthy fame, can neuer dye nor
wither) did write therof largely, at the request of _Clement the sixt_.
The Arte carrieth with it, a wonderfull Credit: By reason, it
certefieth, sensibly, fully, and completely to the vtmost power of
Nature, and Arte. This Arte, certifieth by _Experience_ complete and
absolute: and other Artes, with their Argumentes, and Demonstrations,
persuade: and in wordes, proue very well their Conclusions. *

    [☞]

But wordes, and Argumentes, are no sensible certifying: nor the full and
finall frute of Sciences practisable. And though some Artes, haue in
them, _Experiences_, yet they are not complete, and brought to the
vttermost, they may be stretched vnto, and applyed sensibly. As for
example: the Naturall Philosopher disputeth and maketh goodly shew of
reason: And the Astronomer, and the Opticall Mechanicien, put some
thynges in _Experience_: but neither, all, that they may: nor yet
sufficiently, and to the vtmost, those, which they do, There, then, the
_Archemaster_ steppeth in, and leadeth forth on, the _Experiences_, by
order of his doctrine _Experimentall_, to the chief and finall power of
Naturall and Mathematicall Artes. Of two or three men, in whom, this
Description of _Archemastry_ was _Experimentally_, verified, I haue read
and hard: and good record, is of their such perfection. So that, this
Art, is no fantasticall Imagination: as some Sophister, might, _Cum suis
Insolubilibus_, make a florish: and dassell your Imagination: and dash
your honest desire and Courage, from beleuing these thinges, so vnheard
of, so meruaylous, & of such Importance. Well: as you will. I haue
forewarned you. I haue done the part of a frende: I haue discharged my
Duety toward God: for my small Talent, at hys most mercyfull handes
receiued. To this Science, doth the _Science Alnirangiat_, great
Seruice. Muse nothyng of this name. I chaunge not the name, so vsed, and
in Print published by other: beyng a name, propre to the Science. Vnder
this, commeth _Ars Sintrillia_, by _Artephius_, briefly written. But the
chief Science, of the Archemaster, (in this world) as yet knowen, is an
other (as it were) OPTICAL Science: wherof, the name shall be told (God
willyng) when I shall haue some, (more iust) occasion, therof, to
Discourse.


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


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


    [1.]

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


    [2.]

Neither are their Studies, hereby, any whit hindred. No more, then the
Italian _Vniuersities_, as _Academia Bononiensis_, _Ferrariensis_,
_Florentina_, _Mediolanensis_, _Patauina_, _Papiensis_, _Perusina_,
_Pisana_, _Romana_, _Senensis_, or any one of them, finde them selues,
any deale, disgraced, or their Studies any thing hindred, by _Frater
Lucas de Burgo_, or by _Nicolaus Tartalea_, who in vulgar Italian
language, haue published, not onely _Euclides Geometrie_, but of
_Archimedes_ somewhat: and in Arithmetike and Practicall Geometrie, very
large volumes, all in their vulgar speche. Nor in Germany haue the
famous _Vniuersities_, any thing bene discontent with _Albertus
Durerus_, his Geometricall Institutions in Dutch: or with _Gulielmus
Xylander_, his learned translation of the first sixe bookes of
_Euclide_, out of the Greke into the high Dutch. Nor with _Gualterus H.
Riffius_, his Geometricall Volume: very diligently translated into the
high Dutch tounge, and published. Nor yet the _Vniuersities_ of Spaine,
or Portugall, thinke their reputation to be decayed: or suppose any
their Studies to be hindred by the Excellent _P. Nonnius_, his
Mathematicall workes, in vulgare speche by him put forth. Haue you not,
likewise, in the French tounge, the whole Mathematicall Quadriuie? and
yet neither Paris, Orleance, or any of the other Vniuersities of
Fraunce, at any time, with the Translaters, or Publishers offended: or
any mans Studie thereby hindred?


    [3.]

And surely, the Common and Vulgar Scholer (much more, the Gramarian)
before his comming to the _Vniuersitie_, shall (or may) be, now
(according to _Plato_ his Counsell) sufficiently instructed in
_Arithmetike_ and _Geometrie_, for the better and easier learning of all
maner of _Philosophie_, _Academicall_, or _Peripateticall_. And by that
meanes, goe more cherefully, more skilfully, and spedily forwarde, in
his Studies, there to be learned. And, so, in lesse time, profite more,
then (otherwise) he should, or could do.


    [4.]

Also many good and pregnant Englishe wittes, of young Gentlemen, and of
other, who neuer intend to meddle with the profound search and Studie of
Philosophie (in the _Vniuersities_ to be learned) may neuerthelesse,
now, with more ease and libertie, haue good occasion, vertuously to
occupie the sharpnesse of their wittes: where, els (perchance)
otherwise, they would in fond exercises, spend (or rather leese) their
time: neither seruing God: nor furdering the Weale, common or priuate.


    [5.]

And great Comfort, with good hope, may the _Vniuersities_ haue, by
reason of this _Englishe_ +Geometrie, and Mathematicall Præface+, that
they (hereafter) shall be the more regarded, esteemed, and resorted
vnto. For, when it shall be knowen and reported, that of the
_Mathematicall Sciences_ onely, such great Commodities are ensuing (as I
haue specified): and that in dede, some of you vnlatined Studentes, can
be good witnesse, of such rare fruite by you enioyed (thereby): as
either, before this, was not heard of: or els, not so fully credited:
“Well, may all men coniecture, that farre greater ayde, and better
furniture, to winne to the Perfection of all Philosophie,

    [Vniuersities.]

may in the Vniuersities be had: being the Storehouses & Threasory of all
Sciences,

    [☞]

and all Artes, necessary for the best, and most noble State of Common
Wealthes.”

    [6.]

Besides this, how many a Common Artificer, is there, in these Realmes of
England and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who,
with their owne Skill and experience, already had, will be hable (by
these good helpes and informations) to finde out, and deuise, new
workes, straunge Engines, and Instrumentes: for sundry purposes in the
Common Wealth? or for priuate pleasure? and for the better maintayning
of their owne estate? I will not (therefore) fight against myne owne
shadowe. For, no man (I am sure) will open his mouth against this
Enterprise. No mã (I say) who either hath Charitie toward his brother
(and would be glad of his furtherance in vertuous knowledge): or that
hath any care & zeale for the bettering of the Cõmon state of this
Realme. Neither any, that make accompt, what the wiser sort of men (Sage
and Stayed) do thinke of them. To none (therefore) will I make any
_Apologie,_ for a vertuous acte doing: and for cõmending, or setting
forth, Profitable Artes to English men, in the English toung. “But, vnto
God our Creator, let vs all be thankefull: for that, +_As he, of his
Goodnes, by his Powre, and in his wisedome,

    [☞]

hath Created all thynges, in Number, Waight, and Measure_+: So, to vs,
of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne the
sufficient and necessary knowledge of the foresayd hys three principall
Instrumentes: Which Meanes, I haue abundantly proued vnto you, to be the
_Sciences_ and _Artes Mathematicall_.”


And though I haue ben pinched with straightnes of tyme: that, no way,
I could so pen downe the matter (in my Mynde) as I determined: hopyng of
conuenient laysure: Yet. if vertuous zeale, and honest Intent prouoke
and bryng you to the readyng and examinyng of this Compendious treatise,
I do not doute, but, as the veritie therof (accordyng to our purpose)
will be euident vnto you: So the pith and force therof, will persuade
you: and the wonderfull frute therof, highly pleasure you. And that you
may the easier perceiue, and better remember, the principall pointes,
whereof my Preface treateth,

    [The Ground platt of this Præface in a Table.]

I will giue you the +Groundplatt+ of my whole discourse, in a Table
annexed: from the first to the last, somewhat Methodically contriued.


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

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


                     _Written at my poore House
                     At Mortlake._

                      _Anno. 1570. February. 9._


  [Decoration]




  [Transcriber’s Note:

  The “Groundplat” was printed in the form of a stemma, or tree, on an
  oversized fold-out page. The layout was impossible to reproduce for
  this e-text, so the information has been rearranged in nested-list
  form. Size markings (see note at beginning of e-text) are relative
  within each paragraph.]


      _J. DEE_

  +‡Here haue you (according to my promisse) the Groundplat of‡+
    +my MATHEMATICALL Præface: annexed to _Euclide_ (now first)+
       published in our Englishe tounge. An. 1570. Febr. 3.


+‡Sciences, and Artes Mathematicall,‡ are, either+

  +‡Principall,‡ which are two, onely,+

    +Arithmetike.+

      +‡Simple‡+, Which dealeth with Numbers onely: and demonstrateth
        all their properties and appertenances: where, an Vnit, is
        Indiuisible.
      +‡Mixt‡+, Which with aide of Geometrie principall, demonstrateth
        some Arithmeticall Conclusion, or Purpose.

    +Geometrie.+

      +‡Simple‡+, Which dealeth with Magnitudes, onely: and
        demonstrateth all their properties, passions, and
        appertenances: whose Point, is Indiuisible.
      +‡Mixt‡+, Which with aide of Arithmetike principall,
        demonstrateth some Geometricall purpose, as +EVCLIDES
        ELEMENTES+.

    +‡The vse‡ whereof, is either,+

      In thinges Supernaturall, æternall, & Diuine: By Application,
        _Ascending_.
      In thinges Mathematicall: without farther Application.
      In thinges Naturall: both Substãtiall, & Accidentall, Visible,
        & Inuisible. &c. By Application: _Descending_.

    The like Vses and Applications are, (though in a degree lower) in
    the +Artes Mathematicall Deriuatiue+.

  +‡Deriuatiue‡ frõ the Principalls: of which, some haue+

    +‡The names of‡ the Principalls: as,+

      +_Arithmetike_, vulgar: which considereth+

        --Arithmetike of most vsuall whole numbers: And of Fractions to
        them appertaining.
        --Arithmetike of Proportions.
        --Arithmetike Circular.
        --Arithmetike of Radicall Nũbers: Simple, Compound, Mixt: And of
        their Fractions.
        --Arithmetike of Cossike Nũbers: with their Fractions: And the
        great Arte of Algiebar.

      +_Geometrie_, vulgar: which teacheth Measuring+

        +‡At hand‡+

          All Lengthes.--+Mecometrie.+
          All Plaines: As, Land, Borde, Glasse, &c.--+Embadometrie.+
          All Solids: As, Timber, Stone, Vessels, &c.--+Stereometrie.+

        +‡With distãce‡+ from the thing Measured, as,

          +‡How farre‡+, from the Measurer, any thing is: of him
            sene, on Land or Water: called +Apomecometrie+.
          +‡How high or deepe‡+, from the leuell of the Measurers
            standing, any thing is: Seene of hym, on Land or Water:
            called +Hypsometrie+.
          +‡How broad‡+, a thing is, which is in the Measurers view:
            so it be situated on Land or Water: called
            +Platometrie+.

        +‡Of which‡ are growen the Feates & Artes of+

          +Geodesie+: more cunningly to Measure and Suruey Landes,
            Woods, Waters. &c.
          +Geographie.+
          +Chorographie.+
          +Hydrographie.+
          +Stratarithmetrie.+

    +‡Propre names‡ as+,

      +Perspectiue,+--Which demonstrateth the maners and properties
      of all Radiations: Directe, Broken, and Reflected.

      +Astronomie,+--Which demonstrateth the Distances, Magnitudes,
      and all Naturall motions, Apparences, and Passions, proper to
      the Planets and fixed Starres: for any time, past, present, and
      to come: in respecte of a certaine Horizon, or without respecte
      of any Horizon.

      +Musike,+--Which demonstrateth by reason, and teacheth by
      sense, perfectly to iudge and order the diuersitie of Soundes,
      hie or low.

      +Cosmographie,+--Which, wholy and perfectly maketh description
      of the Heauenlym and also Elementall part of the World: and of
      these partes, maketh homologall application, and mutuall
      collation necessary.

      +Astrologie,+--Which reasonably demonstrateth the operations
      and effectes of the naturall beames of light, and secrete
      Influence of the Planets, and fixed Starres, in euery Element
      and Elementall body: at all times, in any Horizon assigned.

      +Statike,+--Which demonstrateth the causes of heauines and
      lightnes of all thinges: and of the motions and properties to
      heauines and lightnes belonging.

      +Anthropographie,+ Which describeth the Nũber, Measure, Waight,
      Figure, Situation, and colour of euery diuers thing contained in
      the perfecte body of MAN: and geueth certaine knowledge of the
      Figure, Symmetrie, Waight, Characterization, & due Locall motion
      of any percell of the said body assigned: and of numbers to the
      said percell appertaining.

      +Trochilike,+--Which demonstrateth the properties of all
      Circular motions: Simple and Compound.

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

      +Pneumatithmie,+--Which demonstrateth by close hollow
      Geometricall figures (Regular and Irregular) the straunge
      properties (in motion or stay) of the Water, Ayre, Smoke, and
      Fire, in their Continuitie, and as they are ioyned to the
      Elementes next them.

      +Menadrie,+--Which demonstrateth, how, aboue Natures Vertue,
      and power simple: Vertue and force, may be multiplied: and so
      to directe, to lift, to pull to, and to put or cast fro, any
      multiplied, or simple determined Vertue, Waight, or Force:
      naturally, not, so, directible, or moueable.

      +Hypogeiodie,+--Which demonstrateth, how, vnder the Sphæricall
      Superficies of the Earth, at any depth, to any perpendicular
      line assigned (whose distance from the perpendicular of the
      entrance: and the Azimuth likewise, in respecte of the sayd
      entrance, is knowen) certaine way, may be prescribed and gone,
      &c.

      +Hydragogie,+--Which demonstrateth the possible leading of
      water by Natures law, and by artificiall helpe, from any head
      (being Spring, standing, or running water) to any other place
      assigned.

      +Horometrie,+--Which demonstrateth, how, at all times
      appointed, the precise, vsuall denomination of time, may be
      knowen, for any place assigned.

      +Zographie,+--Which demonstrateth and teacheth, how, the
      Intersection of all visuall Pyramids, made by any plaine
      assigned (the Center, distance, and lightes being determined)
      may be, by lines, and proper colours represented.

      +Architecture,+--Which is a Science garnished with many
      doctrines, and diuers Instructions: by whose iudgement, all
      workes by other workmen finished, are iudged.

      +Nauigation,+--Which demonstrateth, how, by the Shortest good
      way, by the aptest direction, and in the shortest time:
      a sufficient Shippe, betwene any two places (in passage
      nauigable) assigned, may be conducted: and in all stormes and
      naturall disturbances chauncing, how to vse the best possible
      meanes, to recouer the place first assigned.

      +Thaumaturgike,+--Which geueth certaine order to make straunge
      workes, of the sense to be perceiued: and of men greatly to be
      wondred at.

      +Archemastrie,+--Which teacheth to bring to actuall experience
      sensible, all worthy conclusions, by all the Artes Mathematicall
      purposed: and by true Naturall philosophie, concluded: And both
      addeth to them a farder Scope, in the termes of the same Artes:
      and also, by his proper Method, and in peculiar termes,
      procedeth, with helpe of the forsayd Artes, to the performance
      of complete Experiences: which, of no particular Arte, are hable
      (Formally) to be challenged.


+¶ Imprinted by _Iohn Day_.+

An. 1570. Feb. 25.

       *       *       *       *       *
           *       *       *       *
       *       *       *       *       *

Errors and Anomalies:

Unless otherwise noted, spelling and punctuation are unchanged.
Errors are listed below, with the original form, if changed, shown in
[brackets]. Unusual words include “fatch” (probably used as a variant
of “fetch”) and the mathematical terms “sexagene” and “sexagesme”.

  How, worldly goods: how, worldly dignitie
    [_“o” in second “worldly” invisible_]
  his most diligent hearers (so infinitely mought  [hearers) so]
  the boundes, and duety of an Hydrographer  [Hydographer]
  of the Grekes it is called _Eteromekes_
    [_text unchanged: correct form is “Heteromekes”_]
  τὸ ὁτὶ  [_accent unchanged_]
  in our worldly affaires  [wordly]
  fall to worke.❉.
    [_Some text readers may not display the oversized-asterisk symbol._]
  _Emptying the first._  [Emptyting]
  Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει λέγοντι πιϛευτεόν
    [ἡμήρας ... πιϛευτέομ]
  of the suddeyne  [snddeyne]
  that the right and absolute way may be had  [he had]
  Georgic I: [_The quoted segments, each ending in “&c.”, are
    438-439; 451-457; 463-464._]

Additional Notes:

  The Greek letter η (eta) was consistently printed as if it were the
    ou-ligature ȣ.
  The Latin “-que” was written as an abbreviation resembling “-q´;”.
    It is shown here as [que].

  Mathematical symbols seen in the section accompanying the diagrams
  could not be reproduced. The following substitutions were made:
    --The curly “P” used for “Pounds” is shown as {P}.
    --The “potestas” symbol, used to represent “x” (the unknown),
    is shown as {x}.
    --All roots were expressed as the “root” sign √ combined with
    symbols for the power of 2 (doubled for power of 4, or fourth root)
    and 3. They are shown as ²√ ³√ ⁴√.

Euclid:

The following Propositions were identified by number.

6.12: (How) to find a fourth (line) proportional to three given straight
lines.

11.34: In equal parallelepipedal solids the bases are reciprocally
proportional to the heights; and those parallelepipedal solids in which
the bases are reciprocally proportional to the heights are equal.

11.36: If three straight lines are proportional, then the
parallelepipedal solid formed out of the three equals the
parallelepipedal solid on the mean which is equilateral, but equiangular
with the aforesaid solid.

12.1: Similar polygons inscribed in circles are to one another as the
squares on their diameters.

12.2: Circles are to one another as the squares on their diameters.

12.18 (“last”): Spheres are to one another in triplicate ratio of their
respective diameters.