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 Scholics , Annotations , and Inuentions , of the best Mathematicians , both of time past , and i● this our age .

With a very fruitfull Praeface made by M. I. Dee , specifying the chiefe Mathematicall Sciences , what they are , and wherunto commodious● where , also , are disclosed certaine new Secrets . Mathematicall and Mech●●icall , vntill these our daies greatly missed :

Imprinted at London by Iohn Daye .

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

DIuine Plato , the great Master of many worthy Philosophers , and the constant auoucher , and pithy perswader of Vnum , Bonum , and Ens : in his Schole and Academie , sundry times ( besides his ordinary Scholers ) was visited of a certaine kinde of men , allured by the noble fame of Plato , and the great commendation of hys profound and profitable doctrine . But when such Hearers , after long harkening to him , perceaued , that the drift of his discourses issued out , to conclude , this Vnum , Bonum , and Ens , to be Spirituall , Infinite , AEternall , Omnipotent , &c. Nothyng beyng alledged or expressed , How , worldly goods : how , worldly dignitie : how , health , Strength or Iustines of body : nor yet the meanes , how a merue●lous sensible and bodyly blysse and felicitie hereafter , might be atteyned : Straightway , the fantasies of those hearers , were dampt : their opinion of Plato , was clene chaunged : yea his doctrine was by them despised : and his schole , no more of ●hem visited . Which thing , his Scholer , Aristotle , narrowly considering , founde the cause therof , to be , For that they had no forwarnyng and information , in generall , whereto his doctrine tended . For , so , might they haue had occasion , either to haue forborne his schole hauntyng : ( if they , then , had misliked his Scope and purpose ) or constantly to haue continued therin : to their full satisfaction : if such finall scope & intent , had ben to their desire . Wherfore , Aristotle , euer , after that , vsed in brief , to forewarne his owne Scholers and hearers , both of what matter , and also to what ende , he tooke in hand to speake , or teach . While I consider the diuerse trades of these two excellent Philosophers ( and am most sure , both , that Plato right well , otherwise could teach : and that Aristotle mought boldely , with his hearers , haue dealt in like sorte as Plato did ) I am in no little pang of perplexitie : Bycause , that , which I mislike , is most easy for me to performe ( and to haue Plato for my example . ) And that , which I know to be most commendable : and ( in this first bringyng , into common handling , the Artes Mathematicall ) to be most necessary : is full of great difficultie and sundry daungers . Yet● neither do I think it me●e , 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 s●hole , after they had perc●iued his finall conclusion , would in these thinges haue ben his most diligent hearers ) so infinitely mought their desires , in fine and at length , by our Artes Mathematicall be satisfied ) yet , by this my Praeface & forewarnyng , Aswell all such , may ( to their great behofe ) the soner , hither be allured : as also the Pythagoricall , and Platonicall perfect scholer , and the constant profound Philosopher , with more ease and spede , may ( like the Bee , ) gather , hereby , both wax and hony .

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

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

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

Of Mathematicall thinges , are two principall kinde●● namely , Number , Number . and Magnitude . Number , we define , to be , a certayne Mathematicall Summe , of Vnits . Note the worde , Vnit , to expresse the Greke Mona● , & not Vnitie : as we hau● all , commonly , till now , vsed . And , an Vnit , is that thing Mathematicall , Indiuisible , by participation of some likenes of whose property , any thing , which is in de●de , or is counted One , may resonably be called One. We account an Vnit , a thing Mathematicall , though it be no Number , and also indiuisible● because , of it , materially , Number doth consist● which , principally , is a thing Mathematicall . Magnitude is a thing Mathematicall , Magnitude . by participation of some likenes of whose nature , any thing is iudged long , broade , or thicke . A thicke Magnitude we call a Solide , or a Body . What Magnitude so euer , is Solide or " Thicke , is also broade , & long . A broade magnitude , we call a Superficies or a Plaine . Euery playne magnitude , hath also length . A long magnitude , we terme a Line . A Line is neither thicke nor broade , but onely long : Euery certayne Line , hath two endes : The endes of a line , are Pointes called . A point . A Point , is a thing Mathematicall , indiuisible , which may haue a certayne determined situation . If a Poynt moue from a " determined situation , the way wherein it moued , is also a Line : mathematically produced . whereupon , of the auncient Mathematiciens , a Line is called the race or course of a Point . A Line . 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 Mathe●a●icall , 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 ? ye● , who doth not wonderfully wonder at it : For , neither pure Element , nor Aristoteles , Quinta Essentia , is hable to serue for Number , as his propre matter . Nor yet the puritie and simplenes of Substance Spirituall or Angelicall , will be found propre enough thereto . And therefore the great & godly Philosopher Anitius Boetius , sayd : Omnia quaecunque a primaua rerum na●●ra constructa sunt , Numerorum videntur ratione formata . Hoc enim fuit principale in animo Conditoris Exemplar . That is : All thinges ( which from the very first originall be●ng of thinges , haue bene framed and made ) do appeare to be Formed by the reason of Numbers . For this was the principall example or patterne in the minde of the Creator . O comfortable alluremen● , O rauishing perswasion , to deale with a Science , whose Subiect , i● so Auncient , so pure , so excellent , so surmounting all creatures , so vsed of the 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 v●w , of all creatures distinct vertues , natures , properties , and Former : 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. Quaestions , vnder written , by the way of Numbers . Which Conclusions , I omit here to rehearse : aswell auoidyng superfluous prolixitie : as , bycause Ioannes Picus , workes , are commonly had . But , in any case , I would wish that those Conclusions were red diligently , and perceiued of such , as are earnest Obseruers and Considerers of the constant law of numbers : which is planted in thyngs Naturall and Supernaturall : and is prescribed to all Creatures , inuiolably to be kept . For , so , besides many other thinges , in those Conclusions to be marked , it would apeare , how sincerely , & within my boundes , I disclose the wonderfull mysteries , by numbers , to be atteyned vnto .

Of my former wordes , easy it is to be gathered , that Number hath a treble state : One , in the Creator : an other in euery Creature ( in respect of his complete constitution : ) and the third , in Spirituall and Angelicall Myndes , and in the Soule of man . In the first and third state , Number , is termed Number Numbryng . But in all Creatures , otherwise , Number , is termed Number Numbred . And in our Soule , Number beareth such a swaye , and hath such an affinitie therwith : that some of the old Philosophers taught , Mans Soule , to be a Number mouyng it selfe . And in dede , in vs , though it be a very Accident : yet such an Accident it is , that before all Creatures it had perfect beyng , in the Creator , Sempiternally . Number Numbryng therfore , is the discretion discerning , and distincting of thinges . But in God the Creator , This discretion , in the beginnyng , produced orderly and distinctly all thinges . For his Numbryng , then , was his Creatyng of all thinges . And his Continuall Numbryng , of all thinges , is the Conseruation of them in being : And , where and when he will lacke an Vnit : there and then , that particular thyng shal be Discreated . Here I stay . But our Seuerallyng , distinctyng , and Numbryng , createth nothyng● but of Multitude considered , maketh certaine and distinct determination . And albeit these thynges be waighty and truthes of great importance , yet ( by the infinite goodnes of the Almighty Ternarie , ) Artificiall Methods and easy wayes are made , by which the zelous Philosopher , may wyn nere this Riuerish Ida ● this Mountayne of Contemplation : and more then Contemplation . And also , though Number , be a thyng so Immateriall , so diuine , and aeternall : yet by degrees , by litle and litle , stretchyng forth , and applying some likenes of it , as first , to thinges Spirituall : and then , bryngyng it lower , to thynges sensibly perceiued : as of a momentanye sounde iterated : then to the least thynges that may be seen , numerable : And at length , ( most grossely , ) to a multitude of any corporall thynges seen , or felt : and so , of these grosse and sensible thynges , we are trayned to learne a certaine Image or likenes of numbers : and to vse Arte in them to our pleasure and proffit . So grosse is our conuersation , and dull is our apprehension : while mortall Sense , in vs , ruleth the common wealth of our litle world . Hereby we say , Three Lyons , are three : or a Ternarie . Three Egles , are three , or a Ternarie . Which ☞ Ternaries , are eche , the Vnion , knot , and Vniformitie , of three discrete and distinct Vnits . That is , we may in eche Ternarie , thrise , seuerally pointe , and shew a part , One , One , and One. Where , in Numbryng , we say One , two , Three . But how farre , these visible Ones , do differre from our Indiuisible Vnits ( in pure Arithmetike , principally considered ) no man is ignorant . Yet from these grosse and materiall thynges , may we be led vpward , by degrees , so , informyng our rude Imagination , toward the coceiuyng of Numbers , absolutely ( Not supposing , nor admixtyng any thyng created , Corporall or Spirituall , to support , conteyne , or represent those Numbers imagined : ) that at length , we may be hable , to finde the number of our owne name , gloriously exemplified and registred in the booke of the Trinitie most blessed and aeternall .

But farder vnderstand , that vulgar Practisers , haue Numbers , otherwise , in 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 operations , in numbers to be performed . How often , therfore , Note . 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 : 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 Ari●hmetike by it selfe . And so of the Arithmetike of Fractions . 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 . Moreouer , the Astronomers , for spede and more commodious calculation , haue deuised a peculier maner of orderyng numbers , about theyr circular motions , by Sexagenes , and Sexagesmes . By Signes , Degrees and Minutes &c. which commonly is called the Arithmetike of Astronomical or Phisicall Fractions . That , haue I briefly noted , by the name of Arithmetike Circular . By cause it is also vsed in circles , not Astronomicall . &c. 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 pu●e Arithmetike , an Vnit , is the common Measure of all Numbers . ) And , here , Numbers are become , as Lynes , Playnes and Solides : some tymes Rationall , some tymes Irrationall ● And haue propre and peculier characters , ( as √ ● . √ ● . and so of other . Which is to signifie Ro●e Square , Rote Cubik : and so forth : ) & propre and peculier fashions in the fiue principall partes : Wherfore the practiser , estemeth this , a diuerse Arithmetike from the other . Practise bryngeth in , here , diuerse compoundyng of Numbers : as some tyme , two , three , foure ( or more ) Radicall numbers , diuersly knit , by signes , o● More & Lesse : as thus √ ● 12 + √ ● 15. Or ●hus √ ●● 19 + √ ● 12 - √ ● 2● &c. And some tyme with whole numbers , or fractions of whole Number , among them as 20 + √ ●●4● √ ● + 33 - √ ● 10 , √ ●● 44 + 12 - + √ ●9 . And so , infinitely , may hap the varie●ie . After this : Both the one and the other hath fractions incident : and so is this Arithmetike greately enlarged , by diuerse exhibityng and vse of Compositions and mixtynges . Consider how● I ( beyng desirous to deliuer the student from error and Cauillation ) do giue to this Practise , the name of the Arithmetike of Radicall numbers : Not of Irrationall or Surd Numbers● which other while , are Rationall : though they haue the Signe of a Rote before them , which , Arithmetike of whole Numbers most vsuall , would say they had no such Roote : and so account them Surd Numbers : which generally spoken , is vntrue : as Euclides tenth booke may teach you . Therfore to call them , generally , Radicall Numbers , ( by reason of the signe √ . prefixed , ) is a sure way : and a sufficient generall distinction from all other ordryng and vsing of Numbers : And yet ( beside all this ) Consider : the infinite desire of knowledge , and incredible power of mans Search and Capacitye : how , they , ioyntly haue waded farder ( by mixtyng of speculation and practise ) and haue found out , and atteyned to the very chief perfection ( almost ) of Numbers Practicall vse . Which thing , is well to be perceiued in that great Arithmeticall Arte of AEquation : commonly called the Rule of Coss. or Algebra . The Latines termed it , Regulam Rei & Census , that is , the Rule of the thyng and his value . With an apt name : comprehendyng the first and last pointes of the worke . And the vulgar names , both in Italian , Frenche and Spanish , depend ( in namyng it , ) vpon the signification of the Latin word , Res : A thing : vnleast they vse the name of Algebra . And therin ( commonly ) is a dubble error . The one , of them , which thinke it to be of Geber his inuentyng : the other of such as call it Algebra . For , first , though Geber for his great skill in Numbers , Geometry , Astronomy , and other maruailous Artes , mought haue semed hable to haue first deuised the sayd Rule : and also the name carryeth with it a very nere likenes of Geber his name : yet true it is , that a Greke Philosopher and Mathematicien , named Diophantus , before Geber his tyme , wrote 13. bookes therof ( of which , six are yet extant : and I had them to * * Anno. 1550. vse , of the famous Mathematicien , and my great frende , Petrus Mon●aureus : ) And secondly , the very name , is Algiebar , and not Algebra : as by the Arabien Auicen , may be proued : who hath these precise wordes in Latine , by Andreas Alpagus ( most perfect in the Arabik tung ) so translated . Scientia faciendi Algiebar & Almachabel . i. Scientia inueniendi numerum ignotum , per additionem Numeri , & diuisionem & aequationem . Which is to say : The Science of workyng Algiebar and Almachabel , that is , the Science of findyng an vnknowen number , by Addyng of a Number , & Diuision & aequation . Here haue you the name : and also the principall partes of the Rule , touched . To name it , The rule , or Art of AEquation , doth signi●ie the middle part and the State of the Rule . This Rule , hath his peculier Characters : and the principal partes of Arithmetike , to it appertayning , do differe from the other Arithmeticall operations . This Arithmetike , hath Numbers Simple , Compound , Mixt : and Fractions , accordingly . This Rule , and Arithmetike of Algiebar , is so profound , so generall and so ( in maner ) conteyneth the whole power of Numbers Application practicall : that mans witt , can deale with nothyng● more proffitable about numbers : nor match , with a thyng , more mete for the diuine force of the Soule , ( in humane Studies , affaires , or exercises ) to be tryed in . Perchaunce you looked for , ( long ere now , ) to haue had some particular profe , or euident testimony of the vse , proffit and Commodity of Arithmetike vulgar , in the Common lyfe and trade of men . Therto , then , I will now frame my selfe : But herein great care I haue , least length of sundry profes , might make you deme , that either I did misdoute your zelous mynde to vertues schole : or els mistrust your hable witts , by some , to gesse much more . A profe then , foure , fiue , or six , such , will I bryng , as any reasonable man , therwith may be persuaded , to loue & honor , yea learne and exercise the excellent Science of Arithmetike .

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

The Mintmaster , and Goldsmith , in their Mixture of Metals , either of diuerse kindes , or diuerse values : how are they , or may they , exactly be directed , and meruailously pleasured , if Arithmetike be their guide ? And the honorable Phisicians , will gladly confesse them selues , much beholding to the Science of Arithmetike , and that sundry wayes : But chiefly in their Art of Graduation , and compounde Medicines . And though Galenus , Auer●ois , 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 : R. B. by a Countreyman of ours ( aboue 200 yeares ago ) inuented . And forasmuch as I am vncertaine , who hath the same : or when that litle Latin treatise , ( as the Author writ it , ) shall come to be Printed : ( Both to declare the desire I haue to pleasure my Countrey , wherin I may : and also , for very good profe of Numbers vse , in this most subtile and frutefull , Philosophicall Conclusion , ) I entend in the meane while , most briefly , and with my farder helpe , to communicate the pith therof vnto you .

First describe a circle : whose diameter let be an inch . Diuide the Circumference into foure equall partes . From the Center , by those 4. sections , extend 4. right lines : eche of 4. inches and a halfe long : or of as many as you liste , aboue 4. without the circumference of the circle : So that they shall be of 4. inches long ( at the lea●t ) without the Circle● Make good euident markes , at euery inches end . If you list , you may subdiuide the inches againe into 10. or 12. smaller partes , equall . At the endes of the lines , write the names of the 4. principall elementall Qualities . Hote and Cold● , one against the other . And likewise Moyst and Dry , one against the other . And in the Circle write Temperate . Which Temperature hath a good 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 .