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* * * * *
* * * *
* * * * *
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.