,transcriber's ,note3 ,this e-text includes characters that will only display in ,,utf-#h 7,unicode7 text readers1 including a few words of ,greek3 ,taktik overline or tilde to show following -n or -m leaf symbol2 pointing-finger symbol double-ended dagger1 used in size notations 7below7 ,if any of these characters do not display properly-- in particular1 if the diacritic does not appear directly above the letter-- or if the quotation marks in this paragraph appear as garbage1 make sure your text reader's 8character set0 or 8file encoding0 is set to ,unicode 7,,utf-#h74 ,you may also need to change the default font4 ,as a last resort1 use the ,,ascii version of this file instead4 ,some aspects of the original book had to be modified for all versions of this plain-text file4 ,superscript letters are shown with 3 ye1 yt4 ,marginal quotation marks are shown inline as 8 and 01 approximating the beginning and end of the marked passage4 ,in the original text1 no quotation marks were printed inline4 ,paragraphs are broken up for sidenotes1 with blank lines before and after4 ,original paragraph breaks are shown as two blank lines4 ,brackets within the body text are in the original4 ,all sidenotes except the one beginning 8,this noble ,earle0 were printed in italics2 markup has been omitted to reduce visual clutter4 ,at least four sizes of text were used1 often in combination with italics4 ,the variants are shown here as3 very large larger smaller ,further errors and anomalies are listed at the end of the text1 along with those ,euclid citations identified by number4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,,the ,,elements ,,of ,,geometrie of the most auncient ,philosopher ,,evclide of ,megara4 ,faithfully 7now first7 translated into the ,englishe toung1 by ,h4 ,billingsley1 ,citizen of ,london4 ,whereunto are annexed certaine ,scholies1 ,annotations1 and ,inuentions1 of the best ,mathematiciens1 both of time past1 and in this our age4 ,with a very fruitfull ,prface made by ,m4 ,i4 ,dee1 specifying the chiefe ,mathematicall ,scices1 what they are1 and wherunto commodious3 where1 also1 are disclosed certaine new ,secrets ,mathematicall and ,mechanicall1 vntill these our daies1 greatly missed4 ,imprinted at ,london by ,iohn ,daye4 ,the ,translator to the ,reader4 ,there is 7gentle ,reader7 nothing 7the word of ,god onely set apart7 which so much beautifieth and adorneth the soule and minde of m1 as doth the knowledge of good artes and sciences3 as the knowledge of naturall and morall ,philosophie4 ,the one setteth before our eyes1 the creatures of ,god1 both in the heauens aboue1 and in the earth beneath3 in which as in a glasse1 we beholde the exceding maiestie and wisedome of ,god1 in adorning and beautifying them as we see3 in geuing vnto them such wonderfull and manifolde proprieties1 and naturall workinges1 and that so diuersly and in such varietie3 farther in maintaining and conseruing them continually1 whereby to praise and adore him1 as by ,s4 ,paule we are taught4 ,the other teacheth vs rules and preceptes of vertue1 how1 in common life amongest men1 we ought to walke vprightly3 what dueties pertaine to our selues1 what pertaine to the gouernment or good order both of an housholde1 and also of a citie or common wealth4 ,the reading likewise of histories1 conduceth not a litle1 to the adorning of the soule minde of man1 a studie of all men cmended3 by it are seene and knowen the artes and doinges of infinite wise men gone before vs4 ,in histories are contained infinite examples of heroicall vertues to be of vs followed1 and horrible examples of vices to be of vs eschewed4 ,many other artes also there are which beautifie the minde of man3 but of all other none do more garnishe beautifie it1 then those artes which are called ,mathematicall4 ,unto the knowledge of which no man can attaine1 without the perfecte knowledge and instruction of the principles1 groundes1 and ,elementes of ,geometrie4 ,but perfectly to be instructed in them1 requireth diligent studie and reading of olde auncient authors4 ,amongest which1 none for a beginner is to be preferred before the most auncient ,philosopher ,euclide of ,megara4 ,for of all others he hath in a true methode and iuste order1 gathered together whatsoeuer any before him had of these ,elementes written3 inuenting also and adding many thinges of his owne3 wherby he hath in due forme accomplished the arte3 first geuing definitions1 principles1 groundes1 wherof he deduceth his ,propositions or conclusions1 in such wonderfull wise1 that that which goeth before1 is of necessitie required to the proufe of that which followeth4 ,so that without the diligent studie of ,euclides ,elementes1 it is impossible to attaine vnto the perfecte knowledge of ,geometrie1 and consequently of any of the other ,mathematicall sciences4 ,wherefore considering the want lacke of such good authors hitherto in our ,englishe tounge1 lamenting also the negligence1 and lacke of zeale to their countrey in those of our nation1 to whom ,god hath geuen both knowledge1 also abilitie to translate into our tounge1 and to publishe abroad such good authors1 and bookes 7the chiefe instrumentes of all learninges73 seing moreouer that many good wittes both of gentlemen and of others of all degrees1 much desirous and studious of these artes1 and seeking for them as much as they can1 sparing no paines1 and yet frustrate of their intent1 by no meanes attaining to that which they seeke3 ,i haue for their sakes1 with some charge great trauaile1 faithfully translated into our vulgare toge1 set abroad in ,print1 this booke of ,euclide4 ,whereunto ,i haue added easie and plaine declarations and examples by figures1 of the definitions4 ,in which booke also ye shall in due place finde manifolde additions1 ,scholies1 ,annotations1 and ,inuentions3 which ,i haue gathered out of many of the most famous chiefe ,mathematicis1 both of old time1 and in our age3 as by diligent reading it in course1 ye shall well perceaue4 ,the fruite and gaine which ,i require for these my paines and trauaile1 shall be nothing els1 but onely that thou gentle reader1 will gratefully accept the same3 and that thou mayest thereby receaue some profite3 and moreouer to excite and stirre vp others learned1 to do the like1 to take paines in that behalfe4 ,by meanes wherof1 our ,englishe tounge shall no lesse be enriched with good ,authors1 then are other straunge tounges3 as the ,dutch1 ,french1 ,italian1 and ,spanishe3 in which are red all good authors in a maner1 found amongest the ,grekes or ,latines4 ,which is the chiefest cause1 that amongest th do florishe so many cunning and skilfull men1 in the inuentions of straunge and wonderfull thinges1 as in these our daies we see there do4 ,which fruite and gaine if ,i attaine vnto1 it shall encourage me hereafter1 in such like sort to translate1 and set abroad some other good authors1 both pertaining to religion 7as partly ,i haue already done7 and also pertaining to the ,mathematicall ,artes4 ,thus gentle reader farewell4 787 ,decoration ,,to ,,the ,,vnfained ,,lovers of truthe1 and constant ,studentes of ,noble ,sciences1 ,,iohn ,,dee of ,london1 hartily wisheth grace from heauen1 and most prosperous successe in all their honest attemptes and exercises4 ,diuine ,plato1 the great ,master of many worthy ,philosophers1 and the constant auoucher1 and pithy perswader of ,vnum1 ,bonum1 and ,ens3 in his ,schole and ,academie1 sundry times 7besides his ordinary ,scholers7 was visited of a certaine kinde of men1 allured by the noble fame of ,plato1 and the great commendation of hys profound and profitable doctrine4 ,but when such ,hearers1 after long harkening to him1 perceaued1 that the drift of his discourses issued out1 to conclude1 this ,vnum1 ,bonum1 and ,ens1 to be ,spirituall1 ,infinite1 ternall1 ,omnipotent1 c4 ,nothyng beyng alledged or expressed1 ,how1 worldly goods3 how1 worldly dignitie3 how1 health1 ,strgth or lustines of body3 nor yet the meanes1 how a merueilous sensible and bodyly blysse and felicitie hereafter1 might be atteyned3 ,straightway1 the fantasies of those hearers1 were dampt3 their opinion of ,plato1 was clene chaunged3 yea his doctrine was by them despised3 and his schole1 no more of them visited4 ,which thing1 his ,scholer1 ,aristotle1 narrowly csidering1 founde the cause therof1 to be1 8,for that they had no forwarnyng and information1 in generall10 whereto his doctrine tended4 ,for1 so1 might they haue had occasion1 either to haue forborne his schole hauntyng3 7if they1 then1 had misliked his ,scope and purpose7 or constantly to haue continued therin3 to their full satisfaction3 if such his finall scope intent1 had ben to their desire4 ,wherfore1 ,aristotle1 euer1 after that1 vsed in brief1 to forewarne his owne ,scholers and hearers1 8both of what matter1 and also to what ende1 he tooke in hand to speake1 or teach40 ,while ,i consider the diuerse trades of these two excellent ,philosophers 7and am most sure1 both1 that ,plato right well1 otherwise could teach3 and that ,aristotle mought boldely1 with his hearers1 haue dealt in like sorte as ,plato did7 ,i am in no little pang of perplexitie3 ,bycause1 that1 which ,i mislike1 is most easy for me to performe 7and to haue ,plato for my exple47 ,and that1 which ,i know to be most commendable3 and 7in this first bringyng1 into common handling1 the ,artes ,mathematicall7 to be most necessary3 is full of great difficultie and sundry daungers4 ,yet1 neither do ,i think it mete1 for so straunge matter 7as now is ment to be published7 and to so straunge an audience1 to be bluntly1 at first1 put forth1 without a peculiar ,preface3 ,nor 7,imitatyng ,aristotle7 well can ,i hope1 that accordyng to the amplenes and dignitie of the ,state ,mathematicall1 ,i am able1 either playnly to prescribe the materiall boundes3 or precisely to expresse the chief purposes1 and most wonderfull applications therof4 ,and though ,i am sure1 that such as did shrinke from ,plato his schole1 after they had perceiued his finall conclusion1 would in these thinges haue ben his most diligent hearers 7so infinitely mought their desires1 in fine and at length1 by our ,artes ,mathematicall be satisfied7 yet1 by this my ,prface forewarnyng1 ,aswell all such1 may 7to their great behofe7 the soner1 hither be allured3 as also the ,pythagoricall1 and ,platonicall perfect scholer1 and the constant profound ,philosopher1 with more ease and spede1 may 7like the ,bee17 gather1 hereby1 both wax and hony4 ,the intent of this ,preface4 ,wherfore1 seyng ,i finde great occasion 7for the causes alleged1 and farder1 in respect of my ,art ,mathematike generall7 to vse 8a certaine forewarnyng and ,prface1 whose content shalbe1 that mighty1 most plesaunt1 and frutefull ,mathematicall ,tree1 with his chief armes and second 7grifted7 braunches3 ,both1 what euery one is1 and also1 what commodity1 in generall1 is to be looked for1 aswell of griff as stocke3 ,and forasmuch as this enterprise is so great1 that1 to this our tyme1 it neuer was 7to my knowledge7 by any achieued3 ,and also it is most hard1 in these our drery dayes1 to such rare and straunge ,artes1 to wyn due and common credit30 ,neuertheles1 if1 for my sincere endeuour to satisfie your honest expectation1 you will but lend me your thkefull mynde a while3 and1 to such matter as1 for this time1 my penne 7with spede7 is hable to deliuer1 apply your eye or eare attentifely3 perchaunce1 at once1 and for the first salutyng1 this ,preface you will finde a lesson long enough4 ,and either you will1 for a second 7by this7 be made much the apter3 or shortly become1 well hable your selues1 of the lyons claw1 to coniecture his royall symmetrie1 and farder propertie4 ,now then1 gentle1 my frendes1 and countrey men1 ,turne your eyes1 and bend your myndes to that doctrine1 which for our present purpose1 my simple talent is hable to yeld you4 ,all thinges which are1 haue beyng1 are found vnder a triple diuersitie generall4 ,for1 either1 they are demed ,supernaturall1 ,naturall1 or1 of a third being4 ,thinges ,supernaturall1 are immateriall1 simple1 indiuisible1 incorruptible1 vnchangeable4 ,things ,naturall1 are materiall1 compounded1 diuisible1 corruptible1 and chaungeable4 ,thinges ,supernaturall1 are1 of the minde onely1 comprehended3 ,things ,naturall1 of the sense exterior1 ar hable to be perceiued4 ,in thinges ,naturall1 probabilitie and coniecture hath place3 ,but in things ,supernaturall1 chief demstration1 most sure ,science is to be had4 ,by which properties comparasons of these two1 more easily may be described1 the state1 condition1 nature and property of those thinges1 which1 we before termed of a third being3 which1 by a peculier name also1 are called ,thynges ,mathematicall4 ,for1 these1 beyng 7in a maner7 middle1 betwene thinges supernaturall and naturall3 are not so absolute and excellent1 as thinges supernatural3 ,nor yet so base and grosse1 as things naturall3 ,but are thinges immateriall3 and neuerthelesse1 by materiall things hable somewhat to be signified4 ,and though their particular ,images1 by ,art1 are aggregable and diuisible3 yet the generall ,formes1 notwithstandyng1 are constant1 vnchaungeable1 vntrsformable1 and incorruptible4 ,neither of the sense1 can they1 at any tyme1 be perceiued or iudged4 ,nor yet1 for all that1 in the royall mynde of man1 first conceiued4 ,but1 surmountyng the imperfecti of coniecture1 weenyng and opinion3 and commyng short of high intellectuall ccepti1 are the ,mercurial fruite of ,dianticall discourse1 in perfect imagination subsistyng4 ,a meruaylous newtralitie haue these thinges ,mathematicall1 and also a straunge participati betwene thinges supernaturall1 immortall1 intellectual1 simple and indiuisible3 and thynges naturall1 mortall1 sensible1 compounded and diuisible4 ,probabilitie and sensible prose1 may well serue in thinges naturall3 and is commendable3 ,in ,mathematicall reasoninges1 a probable ,argument1 is nothyng regarded3 nor yet the testimony of sense1 any whit credited3 ,but onely a perfect demonstration1 of truthes certaine1 necessary1 and inuincible3 vniuersally and necessaryly concluded3 is allowed as sufficient for 8an ,argument exactly and purely ,mathematical40 ,note the worde1 ,vnit1 to expresse the ,greke ,monas1 not ,vnitie3 as we haue all1 commonly1 till now1 vsed4 ,of ,mathematicall thinges1 are two principall kindes3 namely1 ,number1 and ,magnitude4 ,number4 ,number1 we define1 to be1 a certayne ,mathematicall ,sme1 of ,vnits4 ,and1 an ,vnit1 is that thing ,mathematicall1 ,indiuisible1 by participation of some likenes of whose property1 any thing1 which is in deede1 or is counted ,one1 may resonably be called ,one4 ,we account an ,vnit1 a thing ,mathematicall1 though it be no ,number1 and also indiuisible3 because1 of it1 materially1 ,number doth consist3 which1 principally1 is a thing ,mathematicall4 ,magnitude4 ,magnitude is a thing ,mathematicall1 by participation of some likenes of whose nature1 any thing is iudged long1 broade1 or thicke4 8,a thicke ,magnitude we call a ,solide1 or a ,body4 ,what ,magnitude so euer1 is ,solide or ,thicke1 is also broade1 long4 ,a broade magnitude1 we call a ,superficies or a ,plaine4 ,euery playne magnitude1 hath also length4 ,a long magnitude1 we terme a ,line4 ,a ,line is neither thicke nor broade1 but onely long3 ,euery certayne ,line1 hath two endes3 ,a point4 ,the endes of a line1 are ,pointes called4 ,a ,point1 is a thing ,mathematicall1 indiuisible1 which may haue a certayne determined situation40 ,if a ,poynt moue from a determined situation1 the way wherein it moued1 is also a ,line3 mathematically produced1 whereupon1 of the auncient ,mathematiciens1 ,a ,line4 a ,line is called the race or course of a ,point4 ,a ,poynt we define1 by the name of a thing ,mathematicall3 though it be no ,magnitude1 and indiuisible3 because it is the propre ende1 and bound of a ,line3 which is a true ,magnitude4 ,magnitude4 ,and ,magnitude we may define to be that thing ,mathematicall1 which is diuisible for euer1 in partes diuisible1 long1 broade or thicke4 ,therefore though a ,poynt be no ,magnitude1 yet ,terminatiuely1 we recken it a thing ,mathematicall 7as ,i sayd7 by reason it is properly the end1 and bound of a line4 ,neither ,number1 nor ,magnitude1 haue any ,materialitie4 ,first1 we will consider of ,number1 and of the ,science ,mathematicall1 to it appropriate1 called ,arithmetike3 and afterward of ,magnitude1 and his ,science1 called ,geometrie4 ,but that name contenteth me not3 whereof a word or two hereafter shall be sayd4 ,how ,immateriall and free from all matter1 ,number is1 who doth not perceaue8 yea1 who doth not wonderfully wder at it8 ,for1 neither pure ,element1 nor ,aristoteles1 ,quinta ,essentia1 is hable to serue for ,number1 as his propre matter4 ,nor yet the puritie and simplenes of ,substance ,spirituall or ,angelicall1 will be found propre enough thereto4 ,and therefore the great godly ,philosopher ,anitius ,boetius1 sayd3 ,omnia qucunque a primua rerum natura constructa sunt1 ,numerorum videntur ratione formata4 ,hoc enim fuit principale in animo ,conditoris ,exemplar4 ,that is3 ,all thinges 7which from the very first originall being of thinges1 haue bene framed and made7 do appeare to be ,formed by the reason of ,numbers4 ,for this was the principall example or patterne in the minde of the ,creator4 ,o comfortable allurement1 ,o rauishing perswasion1 to deale with a ,science1 whose ,subiect1 is so ,auncient1 so pure1 so excellent1 so surmounting all creatures1 so vsed of the ,almighty and incomprehensible wisdome of the ,creator1 in the distinct creation of all creatures3 in all their distinct partes1 properties1 natures1 and vertues1 by order1 and most absolute number1 brought1 from ,nothing1 to the ,formalitie of their being and state4 ,by ,numbers propertie therefore1 of vs1 by all possible meanes1 7to the perfection of the ,science7 learned1 we may both winde and draw our selues into the inward and deepe search and vew1 of all creatures distinct vertues1 natures1 properties1 and ,formes3 ,and also1 farder1 arise1 clime1 ascend1 and mount vp 7with ,speculatiue winges7 in spirit1 to behold in the ,glas of ,creation1 the ,forme of ,formes1 the ,exemplar ,number of all thinges ,numerable3 both visible and inuisible1 mortall and immortall1 ,corporall and ,spirituall4 ,part of this profound and diuine ,science1 had ,ioachim the ,prophesier atteyned vnto3 by ,numbers ,formall1 ,naturall1 and ,rationall1 forseyng1 concludyng1 and forshewyng great particular euents1 long before their comming4 ,his bookes yet remainyng1 hereof1 are good profe3 ,and the noble ,earle of ,mirandula1 7besides that17 a sufficient witnesse3 that ,ioachim1 in his prophesies1 proceded by no other way1 then by ,numbers ,formall4 ,and this ,earle hym selfe1 in ,rome1 ,ano4 #adhh4 99 set vp #ijj4 ,conclusions1 in all kinde of ,sciences1 openly to be disputed of3 and among the rest1 in his ,conclusions ,mathematicall1 7in the eleuenth ,conclusion7 hath in ,latin1 this ,english sentence4 ,by ,numbers1 a way is had1 to the searchyng out1 and vnderstandyng of euery thyng1 hable to be knowen4 ,for the verifying of which ,conclusion1 ,i promise to aunswere to the #gd4 ,questions1 vnder written1 by the way of ,numbers4 ,which ,cclusions1 ,i omit here to rehearse3 aswell auoidyng superfluous prolixitie3 as1 bycause ,ioannes ,picus1 workes1 are commonly had4 ,but1 in any case1 ,i would wish that those ,conclusions were red diligently1 and perceiued of such1 as are earnest ,obseruers and ,considerers of the constant law of nbers3 which is planted in thyngs ,naturall and ,supernaturall3 and is prescribed to all ,creatures1 inuiolably to be kept4 ,for1 so1 besides many other thinges1 in those ,conclusions to be marked1 it would apeare1 how sincerely1 within my boundes1 ,i disclose the wonderfull mysteries1 by numbers1 to be atteyned vnto4 ,of my former wordes1 easy it is to be gathered1 that ,number hath a treble state3 ,one1 in the ,creator3 an other in euery ,creature 7in respect of his complete constitution37 and the third1 in ,spirituall and ,angelicall ,myndes1 and in the ,soule of m4 ,in the first and third state1 ,number1 is termed ,number ,numbryng4 ,but in all ,creatures1 otherwise1 ,number1 is termed ,nber ,numbred4 ,and in our ,soule1 ,nber beareth such a swaye1 and hath such an affinitie therwith3 that some of the old ,philosophers taught1 ,mans ,soule1 to be a ,number mouyng it selfe4 ,and in dede1 in vs1 though it be a very ,accident3 yet such an ,accident it is1 that before all ,creatures it had perfect beyng1 in the ,creator1 ,sempiternally4 ,number ,numbryng therfore1 is the discretion discerning1 and distincting of thinges4 ,but in ,god the ,creator1 ,this discretion1 in the beginnyng1 produced orderly and distinctly all thinges4 ,for his ,numbryng1 then1 was his ,creatyng of all thinges4 ,and his ,continuall ,numbryng1 of all thinges1 is the ,conseruation of them in being3 ,and1 where and when he will lacke an ,vnit3 there and then1 that particular thyng shalbe ,discreated4 ,here ,i stay4 ,but our ,seuerallyng1 distinctyng1 and ,numbryng1 createth nothyng3 but of ,multitude considered1 maketh certaine and distinct determination4 ,and albeit these thynges be waighty and truthes of great importance1 yet 7by the infinite goodnes of the ,almighty ,ternarie17 ,artificiall ,methods and easy wayes are made1 by which the zelous ,philosopher1 may wyn nere this ,riuerish ,ida1 this ,mountayne of ,contemplation3 and more then ,contemplation4 ,and also1 though ,number1 be a thyng so ,immateriall1 so diuine1 and ternall3 yet by degrees1 by litle and litle1 stretchyng forth1 and applying some likenes of it1 as first1 to thinges ,spirituall3 and then1 bryngyng it lower1 to thynges sensibly perceiued3 as of a momentanye sounde iterated3 then to the least thynges that may be seen1 numerable3 ,and at length1 7most grossely17 to a multitude of any corporall thynges seen1 or felt3 and so1 of these grosse and sensible thynges1 we are trayned to learne a certaine ,image or likenes of numbers3 and to vse ,arte in them to our pleasure and proffit4 ,so grosse is our conuersation1 and dull is our apprehension3 while mortall ,sense1 in vs1 ruleth the common wealth of our litle world4 ,hereby we say1 ,three ,lyons1 are three3 or a ,ternarie4 ,three ,egles1 are three1 or a ,ternarie4 ,which 99 ,ternaries1 are eche1 the ,vnion1 knot1 and ,vniformitie1 of three discrete and distinct ,vnits4 ,that is1 we may in eche ,ternarie1 thrise1 seuerally pointe1 and shew a part1 ,one1 ,one1 and ,one4 ,where1 in ,numbryng1 we say ,one1 two1 ,three4 ,but how farre1 these visible ,ones1 do differre from our ,indiuisible ,vnits 7in pure ,arithmetike1 principally considered7 no man is ignorant4 ,yet from these grosse and materiall thynges1 may we be led vpward1 by degrees1 so1 informyng our rude ,imagination1 toward the cceiuyng of ,numbers1 absolutely 73,not supposing1 nor admixtyng any thyng created1 ,corporall or ,spirituall1 to support1 conteyne1 or represent those ,numbers imagined37 that at length1 we may be hable1 to finde the number of our owne name1 gloriously exemplified and registred in the booke of the ,trinitie most blessed and ternall4 ,but farder vnderstand1 that vulgar ,practisers1 haue ,numbers1 otherwise1 in sundry ,considerations3 and extend their name farder1 then to ,numbers1 whose least part is an ,vnit4 ,for the common ,logist1 ,reckenmaster1 or ,arithmeticien1 in hys vsing of ,numbers3 of an ,vnit1 imagineth lesse partes3 and calleth them ,fractions4 ,as of an ,vnit1 he maketh an halfe1 and thus noteth it1 4 and so of other1 7infinitely diuerse7 partes of an ,vnit4 ,yea and farder1 hath1 ,fractions of ,fractions4 c4 ,and1 forasmuch1 as1 ,addition1 ,substraction1 ,multiplication1 ,diuision and ,extraction of ,rotes1 are the chief1 and sufficient partes of ,arithmetike3 ,arithmetike4 which is1 the ,science that demonstrateth the properties1 of ,numbers1 and all operatis1 in numbers to be performed3 ,note4 8,how often1 therfore1 these fiue sundry sortes of ,operations1 do1 for the most part1 of their execution1 differre from the fiue operations of like generall property and name1 in our ,whole numbers practisable1 ,so often1 7for a more distinct doctrine7 we1 vulgarly account and name it1 an other kynde of ,arithmetike40 ,and by this reason3 #a4 the ,consideration1 doctrine1 and working1 in whole numbers onely3 where1 of an ,vnit1 is no lesse part to be allowed3 is named 7as it were7 an ,arithmetike by it selfe4 ,and so of the ,arithmetike of ,fractions4 #b4 ,in lyke sorte1 the necessary1 wonderfull and ,secret doctrine of ,proportion1 and proportionalytie hath purchased vnto it selfe a peculier maner of handlyng and workyng3 and so may seme an other forme of ,arithmetike4 #c4 ,moreouer1 the ,astronomers1 for spede and more commodious calculation1 haue deuised a peculier maner of orderyng nbers1 about theyr circular motions1 by ,sexagenes1 and ,sexagesmes4 ,by ,signes1 ,degrees and ,minutes c4 which commonly is called the ,arithmetike of ,astronomical or ,phisicall ,fractions4 ,that1 haue ,i briefly noted1 by the name of ,arithmetike ,circular4 ,bycause it is also vsed in circles1 not ,astronomicall4 c4 #d4 ,practise hath led ,numbers farder1 and hath framed them1 to take vpon them1 the shew of ,magnitudes propertie3 ,which is ,incommensurabilitie and ,irrationalitie4 7,for in pure ,arithmetike1 an ,vnit1 is the common ,measure of all ,numbers47 ,and1 here1 ,nbers are become1 as ,lynes1 ,playnes and ,solides3 some tymes ,rationall1 some tymes ,irrationall4 ,and haue propre and peculier characters1 7as 4 4 and so of other4 ,which is to signifie ,rote ,square1 ,rote ,cubik3 and so forth37 propre and peculier fashions in the fiue principall partes3 ,wherfore the practiser1 estemeth this1 a diuerse ,arithmetike from the other4 ,practise bryngeth in1 here1 diuerse compoundyng of ,numbers3 as some tyme1 two1 three1 foure 7or more7 ,radicall nbers1 diuersly knit1 by signes1 of ,more ,lesse3 as thus #ab #ae4 ,or thus #ai #ab - #b4 c4 ,and some tyme with whole numbers1 or fractions of whole ,number1 amg them3 as #bj #bd4 #af #cc - #aj4 #dd #ab #i4 ,and so1 infinitely1 may hap the varietie4 ,after this3 ,both the one and the other hath fractions incident3 and so is this ,arithmetike greately enlarged1 by diuerse exhibityng and vse of ,compositions and mixtynges4 ,consider how1 ,i 7beyng desirous to deliuer the student from error and ,cauillation7 do giue to this ,practise1 the name of the ,arithmetike of ,radicall numbers3 ,not1 of ,irrationall or ,surd ,numbers3 which other while1 are ,rationall3 though they haue the ,signe of a ,rote before them1 which1 ,arithmetike of whole ,numbers most vsuall1 would say they had no such ,roote3 and so account them ,surd ,numbers3 which1 generally spok1 is vntrue3 as ,euclides tenth booke may teach you4 ,therfore to call them1 generally1 ,radicall ,numbers1 7by reason of the signe 4 prefixed17 is a sure way3 and a sufficient generall distinction from all other ordryng and vsing of ,numbers3 ,and yet 7beside all this7 ,consider3 the infinite desire of knowledge1 and incredible power of mans ,search and ,capacitye3 how1 they1 ioyntly haue waded farder 7by mixtyng of speculation and practise7 and haue found out1 and atteyned to the very chief perfection 7almost7 of ,numbers ,practicall vse4 ,which thing1 is well to be perceiued in that great ,arithmeticall ,arte of quation3 commonly called the ,rule of ,coss4 or ,algebra4 ,the ,latines termed it1 ,regulam ,rei ,census1 that is1 the ,rule of the thyng and his value4 ,with an apt name3 comprehendyng the first and last pointes of the worke4 ,and the vulgar names1 both in ,italian1 ,frenche and ,spanish1 depend 7in namyng it17 vpon the signification of the ,latin word1 ,res3 ,a thing3 vnleast they vse the name of ,algebra4 ,and therin 7commonly7 is a dubble error4 ,the one1 of them1 which thinke it to be of ,geber his inuentyng3 the other of such as call it ,algebra4 ,for1 first1 though ,geber for his great skill in ,numbers1 ,geometry1 ,astronomy1 and other maruailous ,artes1 mought haue semed hable to haue first deuised the sayd ,rule3 and also the name carryeth with it a very nere likenes of ,geber his name3 yet true it is1 that a ,greke ,philosopher and ,mathematicien1 named ,diophantus1 before ,geber his tyme1 wrote #ac4 bookes therof 7of which1 six are yet extant3 and ,i had them to 99vse1 99 ,anno4 #aeej4 of the famous ,mathematicien1 and my great frende1 ,petrus ,montaureus37 ,and secondly1 the very name1 is ,algiebar1 and not ,algebra3 as by the ,arabien ,auicen1 may be proued3 who hath these precise wordes in ,latine1 by ,andreas ,alpagus 7most perfect in the ,arabik tung7 so translated4 ,scientia faciendi ,algiebar ,almachabel4 i4 ,scientia inueniendi numerum ignotum1 per additionem ,numeri1 diuisionem quationem4 ,which is to say3 ,the ,science of workyng ,algiebar and ,almachabel1 that is1 the ,science of findyng an vnknowen number1 by ,addyng of a ,number1 ,diuision quation4 ,here haue you the name3 and also the principall partes of the ,rule1 touched4 ,to name it1 ,the rule1 or ,art of quation1 doth signifie the middle part and the ,state of the ,rule4 ,this ,rule1 hath his peculier ,characters3 #e4 and the principal partes of ,arithmetike1 to it appertayning1 do differre from the other ,arithmeticall operations4 ,this ,arithmetike1 hath ,nbers ,simple1 ,cpound1 ,mixt3 and ,fractions1 accordingly4 ,this ,rule1 and ,arithmetike of ,algiebar1 is so profound1 so generall and so 7in maner7 conteyneth the whole power of ,numbers ,application practicall3 that mans witt1 can deale with nothyng1 more proffitable about numbers3 nor match1 with a thyng1 more mete for the diuine force of the ,soule1 7in humane ,studies1 affaires1 or exercises7 to be tryed in4 ,perchaunce you looked for1 7long ere now17 to haue had some particular profe1 or euident testimony of the vse1 proffit and ,commodity of ,arithmetike vulgar1 in the ,common lyfe and trade of men4 ,therto1 then1 ,i will now frame my selfe3 ,but herein great care ,i haue1 least length of sundry profes1 might make you deme1 that either ,i did misdoute your zelous mynde to vertues schole3 or els mistrust your hable witts1 by some1 to gesse much more4 ,a profe then1 foure1 fiue1 or six1 such1 will ,i bryng1 as any reasonable man1 therwith may be persuaded1 to loue honor1 yea learne and exercise the excellent ,science of ,arithmetike4 ,and first3 who1 nerer at hand1 can be a better witnesse of the frute receiued by ,arithmetike1 then all kynde of ,marchants8 ,though not all1 alike1 either nede it1 or vse it4 ,how could they forbeare the vse and helpe of the ,rule1 called the ,golden ,rule8 ,simple and ,compounde3 both forward and backward8 ,how might they misse ,arithmeticall helpe in the ,rules of ,felowshyp3 either without tyme1 or with tyme8 and betwene the ,marchant his ,factor8 ,the ,rules of ,bartering in wares onely3 or part in wares1 and part in money1 would they gladly want8 ,our ,marchant venturers1 and ,trauaylers ouer ,sea1 how could they order their doynges iustly and without losse1 vnleast certaine and generall ,rules for ,exchage of money1 and ,rechaunge1 were1 for their vse1 deuised8 ,the ,rule of ,alligation1 in how sundry cases1 doth it conclude for them1 such precise verities1 as neither by naturall witt1 nor other experience1 they1 were hable1 els1 to know8 ,and 7with the ,marchant then to make an end7 how ample wonderfull is the ,rule of ,false positions8 especially as it is now1 by two excellent ,mathematiciens 7of my familier acquayntance in their life time7 enlarged8 ,i meane ,gemma ,frisius1 and ,simon ,iacob4 ,who can either in brief conclude1 the generall and ,capitall ,rules8 or who can ,imagine the ,myriades of sundry ,cases1 and particular examples1 in ,act and earnest1 continually wrought1 tried and concluded by the forenamed ,rules1 onely8 ,how sundry other ,arithmeticall practises1 are commonly in ,marchantes handes1 and knowledge3 ,they them selues1 can1 at large1 testifie4 ,the ,mintmaster1 and ,goldsmith1 in their ,mixture of ,metals1 either of diuerse kindes1 or diuerse values3 how are they1 or may they1 exactly be directed1 and meruailously pleasured1 if ,arithmetike be their guide8 ,and the honorable ,phisicis1 will gladly confesse them selues1 much beholding to the ,science of ,arithmetike1 and that sundry wayes3 ,but chiefly in their ,art of ,graduation1 and compounde ,medicines4 ,and though ,galenus1 ,auerrois1 ,arnoldus1 ,lullus1 and other haue published their positions1 aswell in the quantities of the ,degrees aboue ,temperament1 as in the ,rules1 concluding the new ,forme resulting3 yet a more precise1 commodious1 and easy ,method1 is extant3 by a ,countreyman of ours ,r4 ,b4 7aboue #bjj4 yeares ago7 inuented4 ,and forasmuch as ,i am vncertaine1 who hath the same3 or when that litle ,latin treatise1 7as the ,author writ it17 shall come to be ,printed3 7,both to declare the desire ,i haue to pleasure my ,countrey1 wherin ,i may3 and also1 for very good profe of ,numbers vse1 in this most subtile and frutefull1 ,philosophicall ,conclusion17 ,i entend in the meane while1 most briefly1 and with my farder helpe1 to communicate the pith therof vnto you4 ,first describe a circle3 whose diameter let be an inch4 ,diuide the ,circumference into foure equall partes4 ,fr the ,center1 by those #d4 sections1 extend #d4 right lines3 eche of #d4 inches and a halfe long3 or of as many as you liste1 aboue #d4 without the circumference of the circle3 ,so that they shall be of #d4 inches long 7at the least7 without the ,circle4 ,make good euident markes1 at euery inches end4 ,if you list1 you may subdiuide the inches againe into #aj4 or #ab4 smaller partes1 equall4 ,at the endes of the lines1 write the names of the #d4 principall elementall ,qualities4 ,hote and ,colde1 one against the other4 ,and likewise ,moyst and ,dry1 one against the other4 ,and in the ,circle write ,temperate4 ,which ,temperature hath a good ,latitude3 as appeareth by the ,complexion of man4 ,and therefore we haue allowed vnto it1 the foresayd ,circle3 and not a point ,mathematicall or ,physicall4 99 ,take some part of ,lullus counsayle in his booke de ,q4 ,essentia4 ,now1 when you haue two thinges ,miscible1 whose degrees are 99 truely knowen3 ,of necessitie1 either they are of one ,quantitie and waight1 or of diuerse4 ,if they be of one ,quantitie and waight3 whether their formes1 be ,contrary ,qualities1 or of one kinde 7but of diuerse intentions and degrees7 or a ,temperate1 and a ,contrary1 ,the forme resulting of their ,mixture1 is in the ,middle betwene the degrees of the formes mixt4 ,as for example1 let ,a1 be ,moist in the first degree3 and ,b1 ,dry in the third degree4 ,adde #a4 and #c4 that maketh #d3 the halfe or middle of #d4 is #b4 ,this #b4 is the middle1 equally distant from ,a and ,b 99 ,note4 7for the 99 ,temperament is counted none4 ,and for it1 you must put a ,ciphre1 if at any time1 it be in mixture74 ,,hote ,c ,e ,,moist ,a ,,temperate ,b ,,drye ---------------------------------------- -------- ,d ,,cold ,counting then from ,b1 #b4 degrees1 toward ,a3 you finde it to be ,dry in the first degree3 ,so is the ,forme resulting of the ,mixture of ,a1 and ,b1 in our example4 ,i will geue you an other example4 ,suppose1 you haue two thinges1 as ,c1 and ,d3 and of ,c1 the ,heate to be in the #d4 degree3 and of ,d1 the ,colde1 to be remisse1 euen vnto the ,temperament4 ,now1 for ,c1 you take #d3 and for ,d1 you take a ,ciphre3 which1 added vnto #d1 yeldeth onely #d4 ,the middle1 or halfe1 whereof1 is #b4 ,wherefore the ,forme resulting of ,c1 and ,d1 is ,hote in the second degree3 for1 #b4 degrees1 accounted from ,c1 toward ,d1 ende iuste in the #b4 degree of heate4 ,of the third maner1 ,i will geue also an example3 which let be this3 ,note4 ,i haue a liquid ,medicine whose ,qualitie of heate is in the #d4 degree exalted3 as was ,c1 in the example foregoing3 and an other liquid ,medicine ,i haue3 whose ,qualitie1 is heate1 in the first degree4 ,of eche of these1 ,i mixt a like quantitie3 ,subtract here1 the lesse fr the more3 and the residue diuide into two equall partes3 whereof1 the one part1 either added to the lesse1 or subtracted from the higher degree1 doth produce the degree of the ,forme resulting1 by this mixture of ,c1 and ,e4 ,as1 if from #d4 ye abate #a4 there resteth #c4 the halfe of #c4 is #a3 ,adde to #a4 this #a3 you haue #b4 ,or subtract from #d4 this #a3 you haue likewise #b remayning4 ,which declareth1 the ,forme resulting1 to be ,heate1 in the middle of the third degree4 ,the ,second ,rule4 8,but if the ,quantities of two thinges ,commixt1 be diuerse1 and the ,intensions 7of their ,formes ,miscible7 be in diuerse degrees1 and heigthes4 7,whether those ,formes be of one kinde1 or of ,contrary kindes1 or of a ,temperate and a ,contrary1 ,what proportion is of the lesse quantitie to the greater1 the same shall be of the difference1 which is betwene the degree of the ,forme resulting1 and the degree of the greater quantitie of the thing miscible1 to the difference1 which is betwene the same degree of the ,forme resulting1 and the degree of the lesse quantitie4 ,as for example4 ,let two pound of ,liquor be geuen1 hote in the #d4 degree3 one pound of ,liquor be geuen1 hote in the third degree40 ,i would gladly know the ,forme resulting1 in the ,mixture of these two ,liquors4 ,set downe your nbers in order1 thus4 ,p4 #b4 ,hote4 #d4 ,p4 #a4 ,hote4 #c4 ,now by the rule of ,algiebar1 haue ,i deuised a very easie1 briefe1 and generall maner of working in this case4 ,let vs first1 suppose that ,middle ,forme resulting1 to be #ax3 as that ,rule teacheth4 ,and because 7by our ,rule1 here geuen7 as the waight of #a4 is to #b3 ,so is the difference betwene #d4 7the degree of the greater quantitie7 and #ax3 to the difference betwene #ax and #c3 7the degree of the thing1 in lesse qutitie4 ,and with all1 #ax1 being alwayes in a certaine middell1 betwene the two heigthes or degrees74 ,for the first difference1 ,i set #d-ax3 and for the second1 ,i set #ax-#c4 ,and1 now againe1 ,i say1 as #a4 is to #b4 so is #d-ax to #ax-#c4 ,wherfore1 of these foure proportionall numbers1 the first and the fourth ,multiplied1 one by the other1 do make as much1 as the second and the third ,multiplied the one by the other4 ,let these ,multiplications be made accordingly4 ,and of the first and the fourth1 we haue #ax-#c4 and of the second the third1 #h-bx4 ,wherfore1 our quation is betwene #ax-#c3 and #h-bx4 ,which may be reduced1 according to the ,arte of ,algiebar3 as1 here1 adding #c4 to eche part1 geueth the quation1 thus1 #ax#aa-bx4 ,and yet againe1 contracting1 or ,reducing it3 ,adde to eche part1 #bx3 ,then haue you #cx quall to #aa3 thus represented #cx#aa4 ,wherefore1 diuiding #aa4 by #c3 the ,quotient is #c3 the ,valew of our #ax1 ,coss1 or ,thing1 first supposed4 ,and that is the heigth1 or ,intension of the ,forme resulting3 which is1 ,heate1 in two thirdes of the fourth degree3 ,and here ,i set the shew of the worke in conclusion1 thus4 ,the proufe hereof is easie3 by subtracting #c4 from #c1 resteth 4 ,subtracte the same heigth of the ,forme resulting1 7which is #c7 fr #d3 then resteth 3 ,you see1 that is double to 3 as #b4,p4 is double to #a4,p4 ,so should it be3 by the rule here geuen4 ,note4 ,as you added to eche part of the quation1 #c3 so if ye first added to eche part #bx1 it would stand1 #cx-#ch4 ,and now adding to eche part #c3 you haue 7as afore7 #cx#aa4 ,p4 #b4 ,hote4 #d4 ,the forme #c resulting4 ,p4 #a4 ,hote4 #c4 ,and though ,i1 here1 speake onely of two thyngs ,miscible3 and most commonly mo then three1 foure1 fiue or six1 7c47 are to be ,mixed3 7and in one ,compound to be reduced3 the ,forme resultyng of the same1 to serue the turne7 yet these ,rules are sufficient3 duely repeated and iterated4 ,note4 ,in procedyng first1 with any two3 and then1 with the ,forme ,resulting1 and an other3 so forth3 ,for1 the last worke1 concludeth the ,forme resultyng of them all3 ,i nede nothing to speake1 of the ,mixture 7here supposed7 what it is4 ,common ,philosophie hath defined it1 saying1 ,mixtio est miscibilium1 alteratorum1 per minima coniunctorum1 ,vnio4 ,euery word in the definition1 is of great importance4 ,i nede not also spend any time1 to shew1 how1 the other manner of distributing of degrees1 doth agree to these ,rules4 ,neither nede ,i of the farder vse belonging to the ,crosse of ,graduation 7before described7 in this place declare1 vnto such as are capable of that1 which ,i haue all ready sayd4 ,neither yet with examples specifie the ,manifold varieties1 by the foresayd two generall ,rules1 to be ordered4 ,the witty and ,studious1 here1 haue sufficient3 ,and they which are not hable to atteine to this1 without liuely teaching1 and more in particular3 would haue larger discoursing1 then is mete in this place to be dealt withall3 ,and other 7perchaunce7 with a proude snuffe will disdaine this litle3 and would be vnthankefull for much more4 ,i1 therfore conclude3 and wish such as haue modest and earnest ,philosophicall mindes1 to laude ,god highly for this3 and to ,meruayle1 that the profoundest and subtilest point1 concerning ,mixture of ,formes and ,qualities ,naturall1 is so ,matcht and maryed with the most simple1 easie1 and short way of the noble ,rule of ,algiebar4 ,who can remaine1 therfore vnpersuaded1 to loue1 alow1 and honor the excellent ,science of ,arithmetike8 ,for1 here1 you may perceiue that the litle finger of ,arithmetike1 is of more might and contriuing1 then a hunderd thousand mens wittes1 of the middle sorte1 are hable to perfourme1 or truely to conclude1 with out helpe thereof4 ,now will we farder1 by the wise and valiant ,capitaine1 be certified1 what helpe he hath1 by the ,rules of ,arithmetike3 in one of the ,artes to him appertaining3 ,and of the ,grekes named 4 4 8,that is1 the ,skill of ,ordring ,souldiers in ,battell ray after the best maner to all purposes40 ,this ,art so much dependeth vppon ,numbers vse1 and the ,mathematicals1 that lianus 7the best writer therof17 in his worke1 to the ,emperour ,hadrianus1 by his perfection1 in the ,mathematicals1 7beyng greater1 then other before him had17 thinketh his booke to passe all other the excellent workes1 written of that ,art1 vnto his dayes4 ,for1 of it1 had written neas3 ,cyneas of ,thessaly3 ,pyrrhus ,epirota3 and ,alexander his sonne3 ,clearchus3 ,pausanias3 ,euangelus3 ,polybius1 familier frende to ,scipio3 ,eupolemus3 ,iphicrates1 ,possidonius3 and very many other worthy ,capitaines1 ,philosophers and ,princes of ,immortall fame and memory3 ,whose fayrest floure of their garland 7in this feat7 was ,arithmetike3 and a litle perceiuerance1 in ,geometricall ,figures4 ,but in many other cases doth ,arithmetike stand the ,capitaine in great stede4 ,as in proportionyng of vittayles1 for the ,army1 either remaining at a stay3 or suddenly to be encreased with a certaine number of ,souldiers3 and for a certain tyme4 ,or by good ,art to diminish his company1 to make the victuals1 longer to serue the remanent1 for a certaine determined tyme3 if nede so require4 ,and so in sundry his other accountes1 ,reckeninges1 ,measurynges1 and proportionynges1 the wise1 expert1 and ,circumspect ,capitaine will affirme the ,science of ,arithmetike1 to be one of his chief ,counsaylors1 directers and aiders4 ,which thing 7by good meanes7 was euident to the ,noble1 the ,couragious1 the loyall1 and ,curteous ,iohn1 late ,earle of ,warwicke4 ,who was a yong ,gentleman1 throughly knowne to very few4 ,albeit his lusty valiantnes1 force1 and ,skill in ,chiualrous feates and exercises3 his humblenes1 and frendelynes to all men1 were thinges1 openly1 of the world perceiued4 ,but what rotes 7otherwise17 vertue had fastened in his brest1 what ,rules of godly and honorable life he had framed to him selfe3 what vices1 7in some then liuing7 notable1 he tooke great care to eschew3 what manly vertues1 in other noble men1 7florishing before his eyes17 he ,sythingly aspired after3 what prowesses he purposed and ment to achieue3 with what feats and ,artes1 he began to furnish and fraught him selfe1 for the better seruice of his ,kyng and ,countrey1 both in peace warre4 ,these 7,i say7 his ,heroicall ,meditations1 forecastinges and determinations1 no twayne1 7,i thinke7 beside my selfe1 can so perfectly1 and truely report4 ,and therfore1 in ,conscience1 ,i count it my part1 for the honor1 preferment1 procuring of vertue 7thus1 briefly7 to haue put his ,name1 in the ,register of ,fame ,immortall4 ,to our purpose4 ,this ,iohn1 by one of his actes 7besides many other3 both in ,england and ,fraunce1 by me1 in him noted47 did disclose his harty loue to vertuous ,sciences3 and his noble intent1 to excell in ,martiall prowesse3 ,when he1 with humble request1 and instant ,solliciting3 got the best ,rules 7either in time past by ,greke or ,romaine1 or in our time vsed3 and new ,stratagemes therin deuised7 for ordring of all ,companies1 summes and ,numbers of m1 7,many1 or few7 with one kinde of weapon1 or mo1 appointed3 with ,artillery1 or without3 on horsebacke1 or on fote3 to giue1 or take onset3 to seem many1 being few3 to seem few1 being many4 ,to marche in battaile or ,iornay3 with many such feates1 to ,foughten field1 ,skarmoush1 or ,ambushe appartaining3 ,this noble ,earle1 dyed ,anno4 #aeed4 skarse of #bd4 yeares of age3 hauing no issue by his wife3 ,daughter to the ,duke of ,somerset4 ,and of all these1 liuely designementes 7most curiously7 to be in velame parchement described3 with ,notes peculier markes1 as the ,arte requireth3 and all these ,rules1 and descriptions ,arithmeticall1 inclosed in a riche ,case of ,gold1 he vsed to weare about his necke3 as his ,iuell most precious1 and ,counsaylour most trusty4 ,thus1 ,arithmetike1 of him1 was shryned in gold3 ,of ,numbers frute1 he had good hope4 ,now1 ,numbers therfore innumerable1 in ,numbers prayse1 his shryne shall finde4 ,what nede ,i1 7for farder profe to you7 of the ,scholemasters of ,iustice1 to require testimony3 how nedefull1 how frutefull1 how skillfull a thing ,arithmetike is8 ,i meane1 the ,lawyers of all sortes4 ,vndoubtedly1 the ,ciuilians1 can meruaylously declare3 how1 neither the ,auncient ,romaine lawes1 without good knowledge of ,numbers art1 can be perceiued3 ,nor 7,iustice in infinite ,cases7 without due proportion1 7narrowly considered17 is hable to be executed4 ,how ,iustly1 with great knowledge of ,arte1 did ,papinianus institute a law of partition1 and allowance1 betwene man and wife after a diuorce8 ,but how ,accursius1 ,baldus1 ,bartolus1 ,iason1 ,alexander1 and finally ,alciatus1 7being otherwise1 notably well learned7 do iumble1 gesse1 and erre1 from the quity1 art and ,intent of the lawmaker3 ,arithmetike can detect1 and conuince3 and clerely1 make the truth to shine4 ,good ,bartolus1 tyred in the examining proportioning of the matter3 and with ,accursius ,glosse1 much cumbred3 burst out1 and sayd3 ,nulla est in toto libro1 hac glossa difficilior3 ,cuius computationem nec ,scholastici nec ,doctores intelligunt4 c4 ,that is3 ,in the whole booke1 there is no ,glosse harder then this3 ,whose accoumpt or reckenyng1 neither the ,scholers1 nor the ,doctours vnderstand4 c4 ,what can they say of ,iulianus law1 ,si ita ,scriptum4 c4 ,of the ,testators will iustly performing1 betwene the wife1 ,sonne and daughter8 ,how can they perceiue the quitie of ,aphricanus1 ,arithmeticall ,reckening1 where he treateth of ,lex ,falcidia8 ,how can they deliuer him1 from his ,reprouers3 and their maintainers3 as ,ioannes1 ,accursius ,hypolitus and ,alciatus8 ,how ,iustly and artificially1 was ,africanus reckening made8 ,proportionating to the ,sommes bequeathed1 the ,contributions of eche part8 ,namely1 for the hundred presently receiued1 #ag-a_/g4 ,and for the hundred1 receiued after ten monethes1 #ab-f_/g3 which make the #cj3 which were to be ctributed by the legataries to the heire4 ,for1 what proportion1 #ajj hath to #ge3 the same hath #ag-a_/g to #ab-f_/g3 ,which is ,sesquitertia3 that is1 as #d1 to #c4 which make #g4 ,wonderfull many places1 in the ,ciuile law1 require an expert ,arithmeticien1 to vnderstand the deepe ,iudgemt1 ,iust determinati of the ,auncient ,romaine ,lawmakers4 ,but much more expert ought he to be1 who should be hable1 to decide with quitie1 the infinite varietie of ,cases1 which do1 or may happen1 vnder euery one of those lawes and ordinances ,ciuile4 ,hereby1 easely1 ye may now coniecture3 that in the ,canon law3 and in the lawes of the ,realme 7which with vs1 beare the chief ,authoritie71 ,iustice and equity might be greately preferred1 and skilfully executed1 through due skill of ,arithmetike1 and proportions appertainyng4 ,the worthy ,philosophers1 and prudent lawmakers 7who haue written many bookes ,de ,republica3 ,how the best state of ,common wealthes might be procured and mainteined17 haue very well determined of ,iustice3 7which1 not onely1 is the ,base and foundacion of ,common weales3 but also the totall perfection of all our workes1 words1 and thoughtes37 defining it1 ,iustice4 8to be that vertue1 by which1 to euery one1 is rendred1 that to him appertaineth40 ,god challengeth this at our handes1 to be honored as ,god3 to be loued1 as a father3 to be feared as a ,lord master4 ,our neighbours proporti1 is also prescribed of the ,almighty lawmaker3 which is1 to do to other1 euen as we would be done vnto4 ,these proportions1 are in ,iustice necessary3 in duety1 commendable3 and of ,common wealthes1 the life1 strength1 stay and florishing4 ,aristotle in his ,ethikes 7to fatch the sede of ,iustice1 and light of direction1 to vse and execute the same7 was fayne to fly to the perfection1 and power of ,numbers3 for proportions ,arithmeticall and ,geometricall4 ,plato in his booke called ,epinomis 7which boke1 is the ,threasury of all his doctrine7 where1 his purpose is1 to seke a ,science1 which1 when a man had it1 perfectly3 he might seme1 and so be1 in dede1 ,wise4 ,he1 briefly1 of other ,sciences discoursing1 findeth them1 not hable to bring it to passe3 ,but of the ,science of ,numbers1 he sayth4 ,illa1 qu numerum mortalium generi dedit1 id profecto efficiet4 ,deum autem aliquem1 magis quam fortunam1 ad salutem nostram1 hoc munus nobis arbitror contulisse4 c4 ,nam ipsum bonorum omnium ,authorem1 cur non maximi boni1 ,prudenti dico1 causam arbitramur8 ,that ,science1 verely1 which hath taught mankynde number1 shall be able to bryng it to passe4 ,and1 ,i thinke1 a certaine ,god1 rather then fortune1 to haue giuen vs this gift1 for our blisse4 ,for1 why should we not ,iudge him1 who is the ,author of all good things1 to be also the cause of the greatest good thyng1 namely1 ,wisedome8 ,there1 at length1 he proueth ,wisedome to be atteyned1 by good ,skill of ,numbers4 ,with which great ,testimony1 and the manifold profes1 and reasons1 before expressed1 you may be sufficiently and fully persuaded3 of the perfect ,science of ,arithmetike1 to make this accounte3 ,that of all ,sciences1 next to ,theologie1 it is most diuine1 most pure1 most ample and generall1 most profounde1 most subtile1 most commodious and most necessary4 ,whose next ,sister1 is the ,absolute ,science of ,magnitudes3 of which 7by the ,direction and aide of him1 whose ,magnitude is ,infinite1 and of vs ,incomprehensible7 ,i now entend1 so to write1 that both with the ,multitude1 and also with the ,magnitude of ,meruaylous and frutefull verities1 you 7my frendes and ,countreymen7 may be stird vp1 and awaked1 to behold what certaine ,artes and ,sciences1 7to our vnspeakable behofe7 our heauenly father1 hath for vs prepared1 and reuealed1 by sundry ,philosophers and ,mathematiciens4 ,both1 ,number and ,magnitude1 haue a certaine ,originall sede1 7as it were7 of an incredible property3 and of man1 neuer hable1 ,fully1 to be declared4 ,of ,number1 an ,vnit1 and of ,magnitude1 a ,poynte1 doo seeme to be much like ,originall causes3 ,but the diuersitie neuerthelesse1 is great4 ,we defined an ,vnit1 to be a thing ,mathematicall ,indiuisible3 ,a ,point1 likewise1 we sayd to be a ,mathematicall thing ,indiuisible4 ,and farder1 that a ,point may haue a certaine determined ,situation3 that is1 that we may assigne1 and prescribe a ,point1 to be here1 there1 yonder4 c4 ,herein1 7behold7 our ,vnit is free1 and can abyde no bondage1 or to be tyed to any place1 or seat3 diuisible or indiuisible4 ,agayne1 by reason1 a ,point may haue a ,situation limited to him3 a certaine motion1 therfore 7to a place1 and from a place7 is to a ,point incident and appertainyng4 ,but an ,vnit1 can not be imagined to haue any motion4 ,a ,point1 by his motion1 produceth1 ,mathematically1 a line3 7as we sayd before7 which is the first kinde of ,magnitudes1 and most simple3 ,an ,vnit1 can not produce any number4 ,a ,line1 though it be produced of a ,point moued1 yet1 it doth not consist of pointes3 ,number1 though it be not produced of an ,vnit1 yet doth it ,consist of vnits1 as a materiall cause4 ,but formally1 ,number4 ,number1 is the ,vnion1 and ,vnitie of ,vnits4 ,which vnyting and knitting1 is the workemanship of our minde3 which1 of distinct and discrete ,vnits1 maketh a ,number3 by vniformitie1 resulting of a certaine multitude of ,vnits4 ,and so1 euery number1 may haue his least part1 giuen3 namely1 an ,vnit3 ,but not of a ,magnitude1 7no1 not of a ,lyne17 the least part can be giu3 by cause1 infinitly1 diuision therof1 may be conceiued4 ,all ,magnitude1 is either a ,line1 a ,plaine1 or a ,solid4 ,which ,line1 ,plaine1 or ,solid1 of no ,sense1 can be perceiued1 nor exactly by hd 7any way7 represented3 nor of ,nature produced3 ,but1 as 7by degrees7 ,number did come to our perceiuerance3 ,so1 by visible formes1 we are holpen to imagine1 what our ,line ,mathematicall1 is4 ,what our ,point1 is4 ,so precise1 are our ,magnitudes1 that one ,line is no broader then an other3 for they haue no bredth3 ,nor our ,plaines haue any thicknes4 ,nor yet our ,bodies1 any weight3 be they neuer so large of dimensi4 ,our ,bodyes1 we can haue ,smaller1 then either ,arte or ,nature can produce any3 and ,greater also1 then all the world can comprehend4 ,our least ,magnitudes1 can be diuided into so many partes1 as the greatest4 ,as1 a ,line of an inch long1 7with vs7 may be diuided into as many partes1 as may the diameter of the whole world1 from ,east to ,west3 or any way extended3 ,what priuiledges1 aboue all manual ,arte1 and ,natures might1 haue our two ,sciences ,mathematicall8 to exhibite1 and to deale with thinges of such power1 liberty1 simplicity1 puritie1 and perfection8 ,and in them1 so certainly1 so orderly1 so precisely to procede3 as1 excellent is that workem ,mechanicall ,iudged1 who nerest can approche to the representing of workes1 ,mathematically demonstrated8 ,and our two ,sciences1 remaining pure1 and absolute1 in their proper termes1 and in their owne ,matter3 to haue1 and allowe1 onely such ,demonstrations1 as are plaine1 certaine1 vniuersall1 and of an ternall veritye8 ,geometrie4 ,this ,science of ,magnitude1 his properties1 conditions1 and appertenances3 commonly1 now is1 and from the beginnyng1 hath of all ,philosophers1 ben called ,geometrie4 ,but1 veryly1 with a name to base and scant1 for a ,science of such dignitie and amplenes4 ,and1 perchaunce1 that name1 by cmon and secret consent1 of all wisemen1 hitherto hath ben suffred to remayne3 that it might carry with it a perpetuall memorye1 of the first and notablest benefite1 by that ,science1 to common people shewed3 ,which was1 when ,boundes and meres of land and ground were lost1 and confounded 7as in ,egypt1 yearely1 with the ouerflowyng of ,nilus1 the greatest and longest riuer in the world7 or1 that ground bequeathed1 were to be assigned3 or1 ground sold1 were to be layd out3 or 7when disorder preuailed7 that ,comms were distributed into seueralties4 ,for1 where1 vpon these such like occasis1 ,some by ignorce1 some by negligce1 ,some by fraude1 and some by violence1 did wrongfully limite1 measure1 encroach1 or challenge 7by pretence of iust content1 and measure7 those landes and groundes3 great losse1 disquietnes1 murder1 and warre did 7full oft7 ensue3 ,till1 by ,gods mercy1 and mans ,industrie1 ,the perfect ,science of ,lines1 ,plaines1 and ,solides 7like a diuine ,iusticier17 gaue vnto euery man1 his owne4 ,the people then1 by this art pleasured1 and greatly relieued1 in their landes iust measuring3 other ,philosophers1 writing ,rules for land measuring3 betwene them both1 thus1 confirmed the name of ,geometria1 that is1 7according to the very etimologie of the word7 ,land measuring4 ,wherin1 the people knew no farder1 of ,magnitudes vse1 but in ,plaines3 and the ,philosophers1 of th1 had no feet hearers1 or ,scholers3 farder to disclose vnto1 then of flat1 plaine ,geometrie4 ,and though1 these ,philosophers1 knew of farder vse1 and best vnderstode the etymologye of the worde1 yet this name ,geometria1 was of them applyed generally to all sortes of ,magnitudes3 vnleast1 otherwhile1 of ,plato1 and ,pythagoras3 ,when they would precisely declare their owne doctrine4 ,then1 was 99 ,plato4 #g4 de ,rep4 99 ,geometria1 with them1 ,studium quod circa planum versatur4 ,but1 well you may perceiue by ,euclides ,elementes1 that more ample is our ,science1 then to measure ,plaines3 and nothyng lesse therin is tought 7of purpose7 then how to measure ,land4 ,an other name1 therfore1 must nedes be had1 for our ,mathematicall ,science of ,magnitudes3 which regardeth neither clod1 nor turff3 neither hill1 nor dale3 neither earth nor heauen3 but is absolute ,megethologia3 not creping on ground1 and dasseling the eye1 with pole perche1 rod or lyne3 but 8liftyng the hart aboue the heauens1 by inuisible lines1 and immortall beames meteth with the reflexions1 of the light incomprehensible3 and so procureth ,ioye1 and perfection vnspeakable40 ,of which true vse of our ,megethica1 or ,megethologia1 ,diuine ,plato seemed to haue good taste1 and iudgement3 and 7by the name of ,geometrie7 so noted it3 and warned his ,scholers therof3 as1 in hys seuenth ,dialog1 of the ,common wealth1 may euidently be sene4 ,where 7in ,latin7 thus it is3 right well translated3 ,profecto1 nobis hoc non negabunt1 ,quicunque vel paululum quid ,geometri gustrunt1 quin hc ,scientia1 contr1 omnino se habeat1 qum de ea loquuntur1 qui in ipsa versantur4 ,in ,english1 thus4 ,verely 7sayth ,plato7 whosoeuer haue1 7but euen very litle7 tasted of ,geometrie1 will not denye vnto vs1 this3 but that this ,science1 is of an other condicion1 quite contrary to that1 which they that are exercised in it1 do speake of it4 ,and there it followeth1 of our ,geometrie1 ,qud quritur cognoscendi illius gratia1 quod semper est1 non eius quod oritur quandoque interit4 ,geometria1 eius quod est semper1 ,cognitio est4 ,attollet igitur 7 ,generose vir7 ad ,veritatem1 animum3 atque ita1 ad ,philosophandum preparabit cogitationem1 vt ad supera conuertamus3 qu1 nunc1 contra qum decet1 ad inferiora deijcimus4 c4 ,qum maxim igitur prcipiendum est1 vt qui prclarissimam hanc habitt ,civitatem1 nullo modo1 ,geometriam spernant4 ,nam qu prter ipsius propositum1 quodam modo esse videntur1 haud exigua sunt4 c4 ,it must nedes be confessed 7saith ,plato7 ,that ,geometrie is learned1 for the knowyng of that1 which is euer3 and not of that1 which1 in tyme1 both is bred and is brought to an ende4 c4 ,geometrie is the knowledge of that which is euerlastyng4 ,it will lift vp therfore 7,o ,gentle ,syr7 our mynde to the ,veritie3 and by that meanes1 it will prepare the ,thought1 to the ,philosophicall loue of wisdome3 that we may turne or conuert1 toward heauenly thinges both mynde and thought which now1 otherwise then becommeth vs1 we cast down on base or inferior things4 c4 ,chiefly1 therfore1 ,commaundement must be giuen1 that such as do inhabit this most honorable ,citie1 by no meanes1 despise ,geometrie4 ,for euen those thinges done by it which1 in manner1 seame to be1 beside the purpose of ,geometrie3 are of no small importance4 c4 ,and besides the manifold vses of ,geometrie1 in matters appertainyng to warre1 he addeth more1 of second vnpurposed frute1 and commoditye1 arrising by ,geometrie3 saying3 ,scimus quin etiam1 ad ,disciplinas omnes facilius per discendas1 interesse omnino1 attigerit ne ,geometriam aliquis1 an non4 c4 ,hanc ergo ,doctrinam1 secundo loco discendam ,iuuenibus statuamus4 ,that is4 ,but1 also1 we know1 that for the more easy learnyng of all ,artes1 it importeth much1 whether one haue any knowledge in ,geometrie1 or no4 c4 ,let vs therfore make an ordinance or decree1 that this ,science1 of young men shall be learned in the second place4 ,this was ,diuine ,plato his ,iudgement1 both of the purposed1 chief1 and perfect vse of ,geometrie3 and of his second1 dependyng1 deriuatiue commodities4 ,and for vs1 ,christen men1 a thousand thousand mo occasions are1 to haue nede of the helpe of 99 ,i4 ,d4 99 ,herein1 ,i would gladly shake of1 the earthly name1 of ,geometrie4 ,megethologicall ,contemplations3 wherby1 to trayne our ,imaginations and ,myndes1 by litle and litle1 to forsake and abandon1 the grosse and corruptible ,obiectes1 of our vtward senses3 and to apprehend1 by sure doctrine demonstratiue1 ,things ,mathematicall4 ,and by them1 readily to be holpen and conducted to conceiue1 discourse1 and conclude of things ,intellectual1 ,spirituall1 ternall1 and such as concerne our ,blisse euerlasting3 which1 otherwise 7without ,speciall priuiledge of ,illumination1 or ,reuelation fr heauen7 ,no mortall mans wyt 7naturally7 is hable to reach vnto1 or to ,compasse4 ,and1 veryly1 by my small ,talent 7from aboue7 ,i am hable to proue and testifie1 that the litterall ,text1 and order of our diuine ,law1 ,oracles1 and ,mysteries1 require more skill in ,numbers1 and ,magnitudes3 then 7commonly7 the expositors haue vttered3 but rather onely 7at the most7 so warned3 shewed their own want therin4 7,to name any1 is nedeles3 and to note the places1 is1 here1 no place3 ,but if ,i be duely asked1 my answere is ready47 ,and without the litterall1 ,grammaticall1 ,mathematicall or ,naturall verities of such places1 by good and certaine ,arte1 perceiued1 no ,spirituall sense 7propre to those places1 by ,absolute ,theologie7 will thereon depend4 8,no man1 therfore1 can doute1 but toward the atteyning of knowledge incomparable1 and ,heauenly ,wisedome3 ,mathematicall ,speculations1 both of ,numbers and ,magnitudes3 are meanes1 aydes1 and guides3 ready1 certaine1 and necessary40 ,from henceforth1 in this my ,preface1 will ,i frame my talke1 to ,plato his fugitiue ,scholers3 or1 rather1 to such1 who well can1 7and also wil17 vse their vtward senses1 to the glory of ,god1 the benefite of their ,countrey1 and their owne secret contentation1 or honest preferment1 on this earthly ,scaffold4 ,to them1 ,i will orderly recite1 describe declare a great ,number of ,artes1 from our two ,mathematicall fountaines1 deriued into the fieldes of ,nature4 ,wherby1 such ,sedes1 and ,rotes1 as lye depe hyd in the grod of ,nature1 are refreshed1 quickened1 and prouoked to grow1 shote vp1 floure1 and giue frute1 infinite1 and incredible4 ,and these ,artes1 shalbe such1 as vpon ,magnitudes properties do depende1 more1 then vpon ,number4 ,and by good reason we may call them ,artes1 and ,artes ,mathematicall ,deriuatiue3 for 7at this tyme7 ,i ,define ,an ,arte4 ,an ,arte1 to be a ,methodicall cplete ,doctrine1 hauing abundancy of sufficient1 and peculier matter to deale with1 by the allowance of the ,metaphisicall ,philosopher3 the knowledge whereof1 to humaine state is necessarye4 ,and that ,i account1 ,art ,mathematicall ,deriuatiue4 ,an ,art ,mathematicall deriuatiue1 which by ,mathematicall demonstratiue ,method1 in ,nbers1 or ,magnitudes1 ordreth and confirmeth his doctrine1 as much as perfectly1 as the matter subiect will admit4 ,and for that1 ,i entend to vse the name and propertie of a ,a ,mechanitien4 ,mechanicien1 otherwise1 then 7hitherto7 it hath ben vsed1 ,i thinke it good1 7for distinction sake7 to giue you also a brief description1 what ,i meane therby4 ,a ,mechanicien1 or a ,mechanicall workman is he1 whose skill is1 without knowledge of ,mathematicall demonstration1 perfectly to worke and finishe any sensible worke1 by the ,mathematicien principall or deriuatiue1 demonstrated or demonstrable4 ,full well ,i know1 that he which inuenteth1 or maketh these demonstrations1 is generally called ,a speculatiue ,mechanicien3 which differreth nothyng from a ,mechanicall ,mathematicien4 ,so1 in respect of diuerse actions1 one man may haue the name of sundry artes3 as1 some tyme1 of a ,logicien1 some tymes 7in the same matter otherwise handled7 of a ,rethoricien4 ,of these trifles1 ,i make1 7as now1 in respect of my ,preface17 small account3 to fyle th for the fine handlyng of subtile curious disputers4 ,in other places1 they may commaunde me1 to giue good reason3 and yet1 here1 ,i will not be vnreasonable4 #a4 ,first1 then1 from the puritie1 absolutenes1 and ,immaterialitie of ,principall ,geometrie1 is that kinde of ,geometrie deriued1 which vulgarly is counted ,geometrie3 and is the ,arte of ,measuring sensible magnitudes1 their iust qutities and contentes4 ,geometrie vulgar4 ,this1 teacheth to measure1 either at hand3 and the practiser1 to be by the thing ,measured3 and so1 by due applying of ,cumpase1 ,rule1 ,squire1 ,yarde1 ,ell1 ,perch1 ,pole1 ,line1 ,gaging rod1 7or such like instrument7 to the ,length1 ,plaine1 or ,solide measured1 #a4 99 to be certified1 either of the length1 perimetry1 or distance lineall3 and this is called1 ,mecometrie4 ,or #b4 99 to be certified of the content of any plaine ,superficies3 whether it be in ground ,surueyed1 ,borde1 or ,glasse measured1 or such like thing3 which measuring1 is named ,embadometrie4 #c4 99 ,or els to vnderstand the ,soliditie1 and content of any bodily thing3 as of ,tymber and ,stone1 or the content of ,pits1 ,pondes1 ,wells1 ,vessels1 small great1 of all fashions4 ,where1 of ,wine1 ,oyle1 ,beere1 or ,ale vessells1 c1 the ,measuring1 commonly1 hath a peculier name3 and is called ,gaging4 ,and the generall name of these ,solide measures1 is ,stereometrie4 #b4 ,or els1 this vulgar ,geometrie1 hath consideration to teach the practiser1 how to measure things1 with good distance betwene him and the thing measured3 and to vnderstand thereby1 either #a4 99 how ,farre1 a thing seene 7on land or water7 is from the measurer3 and this may be called ,apomecometrie3 #b4 ,or1 how ,high or depe1 aboue or vnder the leuel of the measurers stding1 any thing is1 which is sene on land or water1 called ,hypsometrie4 #c4 99 ,or1 it informeth the measurer1 how ,broad any thing is1 which is in the measurers vew3 so it be on ,land or ,water1 situated3 and may be called ,platometrie4 ,though ,i vse here to condition1 the thing measured1 to be on ,land1 or ,water ,situated3 ,note4 yet1 know for certaine1 that the sundry heigthe of ,cloudes1 blasing ,starres1 and of the ,mone1 may 7by these meanes7 haue their distances from the earth3 and1 of the blasing ,starres and ,mone1 the ,soliditie 7aswell as distances7 to be measured3 ,but because1 neither these things are vulgarly taught3 nor of a common practiser so ready to be executed3 ,i1 rather1 let such measures be reckened incident to some of our other ,artes1 dealing with thinges on high1 more purposely1 then this vulgar ,land measuring ,geometrie doth3 as in ,perspectiue and ,astronomie1 c4 ,of these ,feates 7farther applied7 is ,sprong the ,feate of ,geodesie1 or ,land ,measuring3 more cunningly to measure ,suruey ,land1 ,woods1 and ,waters1 a farre of4 ,more cunningly1 ,i say3 ,but ,god knoweth 7hitherto7 in these ,realmes of ,england and ,ireland 7whether through ignorance or fraude1 ,i can not tell1 in euery particular7 ,note4 how great wrong and iniurie hath 7in my time7 bene committed by vntrue measuring and surueying of ,land or ,woods1 any way4 ,and1 this ,i am sure3 that the ,value of the difference1 betwene the truth and such ,surueyes1 would haue bene hable to haue fod 7for euer7 in eche of our two ,vniuersities1 an excellent ,mathematicall ,reader3 to eche1 allowing 7yearly7 a hundred ,markes of lawfull money of this realme3 which1 in dede1 would seme requisit1 here1 to be had 7though by other wayes prouided for7 as well1 as1 the famous ,vniuersitie of ,paris1 hath two ,mathematicall ,readers3 and eche1 two hundreth ,french ,crownes yearly1 of the ,french ,kinges magnificent liberalitie onely4 ,now1 againe1 to our purpose returning3 ,moreouer1 of the former knowledge ,geometricall1 are growen the ,skills of ,geographie1 ,chorographie1 ,hydrographie1 and ,stratarithmetrie4 8,geographie teacheth wayes1 by which1 in sdry formes1 7as ,sphrike1 ,plaine or other71 the ,situation of ,cities1 ,townes1 ,villages1 ,fortes1 ,castells1 ,mountaines1 ,woods1 ,hauens1 ,riuers1 ,crekes1 such other things1 vp the outface of the earthly ,globe 7either in the whole1 or in some principall mber and portion therof ctayned7 may be described and designed1 in cmensurations ,analogicall to ,nature and veritie3 and most aptly to our vew1 may be represented40 ,of this ,arte how great pleasure1 and how manifolde commodities do come vnto vs1 daily and hourely3 of most men1 is perceaued4 ,while1 some1 to beautifie their ,halls1 ,parlers1 ,chambers1 ,galeries1 ,studies1 or ,libraries with3 other some1 for thinges past1 as battels fought1 earthquakes1 heauenly fyringes1 such occurentes1 in histories mentioned3 therby liuely1 as it were1 to vewe the place1 the region adioyning1 the distance from vs3 and such other circumstances4 ,some other1 presently to vewe the large dominion of the ,turke3 the wide ,empire of the ,moschouite3 and the litle morsell of ground1 where ,christendome 7by profession7 is certainly knowen4 ,litle1 ,i say1 in respecte of the rest4 c4 ,some1 either for their owne iorneyes directing into farre landes3 or to vnderstand of other mens trauailes4 ,to conclude1 some1 for one purpose3 and some1 for an other1 liketh1 loueth1 getteth1 and vseth1 ,mappes1 ,chartes1 ,geographicall ,globes4 ,of whose vse1 to speake sufficiently1 would require a booke peculier4 ,chorographie seemeth to be an vnderling1 and a twig1 of ,geographie3 and yet neuerthelesse1 is in practise manifolde1 and in vse very ample4 8,this teacheth ,analogically to describe a small portion or circuite of ground1 with the contentes3 not regarding what commensuration it hath to the whole1 or any parcell1 without it1 contained4 ,but in the territory or parcell of ground which it taketh in hand to make description of1 it leaueth out 7or vndescribed7 no notable1 or odde thing1 aboue the ground visible4 ,yea and sometimes1 of thinges vnder ground1 geueth some peculier marke3 or warning3 as of ,mettall mines1 ,cole pittes1 ,stone quarries4 c40 ,thus1 a ,dukedome1 a ,shiere1 a ,lordship1 or lesse1 may be described distinctly4 ,but marueilous pleasant1 and profitable it is1 in the exhibiting to our eye1 and commensuration1 the plat of a ,citie1 ,towne1 ,forte1 or ,pallace1 in true ,symmetry3 not approching to any of them3 and out of ,gunne shot4 c4 ,hereby1 the ,architect may furnishe him selfe1 with store of what patterns he liketh3 to his great instruction3 euen in those thinges which outwardly are proportioned3 either simply in them selues3 or respectiuely1 to ,hilles1 ,riuers1 ,hauens1 and ,woods adioyning4 ,some also1 terme this particular description of places1 ,topographie4 8,hydrographie1 deliuereth to our knowledge1 on ,globe or in ,plaine1 the perfect ,analogicall description of the ,ocean ,sea coastes1 through the whole world3 or in the chiefe and principall partes thereof30 with the ,iles and chiefe particular places of daungers1 conteyned within the boundes1 and ,sea coastes described3 as1 of ,quicksandes1 ,bankes1 ,pittes1 ,rockes1 ,races1 ,countertides1 ,whorlepooles4 c4 ,this1 dealeth with the ,element of the water chiefly3 as ,geographie did principally take the ,element of the ,earthes description 7with his appertenances7 to taske4 ,and besides thys1 ,hydrographie1 requireth a particular ,register of certaine ,landmarkes 7where markes may be had7 from the sea1 well hable to be skried1 in what point of the ,seacumpase they appeare1 and what apparent forme1 ,situation1 and bignes they haue1 in respecte of any daungerous place in the sea1 or nere vnto it1 assigned3 ,and in all ,coastes1 what ,mone1 maketh full ,sea3 and what way1 the ,tides and ,ebbes1 come and go1 the ,hydrographer ought to recorde4 ,the ,soundinges likewise3 and the ,chanels wayes3 their number1 and depthes ordinarily1 at ebbe and flud1 ought the ,hydrographer1 by obseruation and diligence of ,measuring1 to haue certainly knowen4 ,and many other pointes1 are belonging to perfecte ,hydrographie1 and for to make a ,rutter1 by3 of which1 ,i nede not here speake3 as of the describing1 in any place1 vpon ,globe or ,plaine1 the #cb4 pointes of the ,compase1 truely3 7wherof1 scarsly foure1 in ,england1 haue right knowledge3 bycause1 the lines therof1 are no straight lines1 nor ,circles47 ,of making due proiection of a ,sphere in plaine4 ,of the ,variacion of the ,compas1 from true ,northe3 ,and such like matters 7of great importance1 all7 ,i leaue to speake of1 in this place3 bycause1 ,i may seame 7al ready7 to haue enlarged the boundes1 and duety of an ,hydographer1 much more1 then any man 7to this day7 hath noted1 or prescribed4 ,yet am ,i well hable to proue1 all these thinges1 to appertaine1 and also to be proper to the ,hydrographer4 ,the chief vse and ende of this ,art1 is the ,art of ,nauigation3 but it hath other diuerse vses3 euen by them to be enioyed1 that neuer lacke sight of land4 ,stratarithmetrie1 is the ,skill1 7appertainyng to the warre17 by which a man can set in figure1 analogicall to any ,geometricall figure appointed1 any certaine number or summe of men3 of such a figure capable3 7by reason of the vsuall spaces betwene ,souldiers allowed3 and for that1 of men1 can be made no ,fractions4 ,yet1 neuertheles1 he can order the giuen summe of men1 for the greatest such figure1 that of them1 c be ordred7 and certifie1 of the ouerplus3 7if any be7 and of the next certaine summe1 which1 with the ouerplus1 will admit a figure exactly proportionall to the figure assigned4 ,by which ,skill1 also1 of any army or company of men3 7the figure sides of whose orderly standing1 or array1 is knowen7 he is able to expresse the iust number of men1 within that figure conteined3 or 7orderly7 able to be conteined4 99 ,note4 99 ,and this figure1 and sides therof1 he is hable to know3 either beyng by1 and at hand3 or a farre of4 ,thus farre1 stretcheth the description and property of ,stratarithmetrie3 sufficient for this tyme and place4 ,the difference betwene ,stratarithmetrie and ,tacticie4 8,it differreth from the ,feate ,tacticall1 ,de aciebus instruendis4 bycause1 there1 is necessary the wisedome and foresight1 to what purpose he so ordreth the men3 and ,skillfull hability1 also1 for any occasion1 or purpose1 to deuise and vse the aptest and most necessary order1 array and figure of his ,company and ,summe of men40 ,by figure1 ,i meane3 as1 either of a ,perfect ,square1 ,triangle1 ,circle1 ,ouale1 long square1 7of the ,grekes it is called ,eteromekes7 ,rhombe1 ,rhombod1 ,lunular1 ,ryng1 ,serpentine1 and such other ,geometricall figures3 ,which1 in warres1 haue ben1 and are to be vsed3 for commodiousnes1 necessity1 and auauntage c4 ,and no small skill ought he to haue1 that should make true report1 or nere the truth1 of the numbers and ,summes1 of footemen or horsemen1 in the ,enemyes ordring4 ,a farre of1 to make an estimate1 betwene nere termes of ,more and ,lesse1 is not a thyng very rife1 among those that gladly would do it4 ,i4 ,d4 ,frende1 you will finde it hard1 to performe my description of this ,feate4 ,but by ,chorographie1 you may helpe your selfe some what3 where the ,figures knowne 7in ,sides and ,angles7 are not ,regular3 ,and where1 ,resolution into ,triangles can serue4 c4 ,and yet you will finde it strange to deale thus generally with ,arithmeticall figures3 and1 that for ,battayle ray4 ,their contentes1 differ so much from like ,geometricall ,figures4 ,great pollicy may be vsed of the ,capitaines1 7at tymes fete1 and in places conuenient7 as to vse ,figures1 which make greatest shew1 of so many as he hath3 and vsing the aduauntage of the three kindes of vsuall spaces3 7betwene footemen or horsemen7 to take the largest3 or when he would seme to haue few1 7beyng many37 contrarywise1 in ,figure1 and space4 ,the ,herald1 ,purseuant1 ,sergeant ,royall1 ,capitaine1 or who soeuer is carefull to come nere the truth herein1 besides the ,iudgement of his expert eye1 his skill of ,ordering ,tacticall1 the helpe of his ,geometricall instrument3 ,ring1 or ,staffe ,astronomicall3 7commodiously framed for cariage and vse7 ,he may wonderfully helpe him selfe1 by perspectiue ,glasses4 ,in which1 7,i trust7 our posterity will proue more skillfull and expert1 and to greater purposes1 then in these dayes1 can 7almost7 be credited to be possible4 ,thus haue ,i lightly passed ouer the ,artificiall ,feates1 chiefly dependyng vpon vulgar ,geometrie3 commonly and generally reckened vnder the name of ,geometrie4 ,but there are other 7very many7 ,methodicall ,artes1 which1 declyning from the purity1 simplicitie1 and ,immateriality1 of our ,principall ,science of ,magnitudes3 do yet neuertheles vse the great ayde1 direction1 and ,method of the sayd principall ,science1 and haue propre names1 and distinct3 both from the ,science of ,geometrie1 7from which they are deriued7 and one from the other4 ,as ,perspectiue1 ,astronomie1 ,musike1 ,cosmographie1 ,astrologie1 ,statike1 ,anthropographie1 ,trochilike1 ,helicosophie1 ,pneumatithmie1 ,menadrie1 ,hypogeiodie1 ,hydragogie1 ,horometrie1 ,zographie1 ,architecture1 ,nauigation1 ,thaumaturgike and ,archemastrie4 ,i thinke it necessary1 orderly1 of these to giue some peculier descriptions3 and withall1 to touch some of their commodious vses1 and so to make this ,preface1 to be a little swete1 pleasant ,nosegaye for you3 to comfort your ,spirites1 beyng almost out of courage1 and in despayre1 7through brutish brute7 ,weenyng that ,geometrie1 had but serued for buildyng of an house1 or a curious bridge1 or the roufe of ,westminster hall1 or some witty pretty deuise1 or engyn1 appropriate to a ,carpenter1 or a ,ioyner c4 ,that the thing is farre otherwise1 then the world1 7commonly7 to this day1 hath demed1 by worde and worke1 good profe wilbe made4 ,among these ,artes1 by good reason1 ,perspectiue ought to be had1 ere of ,astronomicall ,apparences1 perfect knowledge can be atteyned4 ,and bycause of the prerogatiue of ,light1 beyng the first of ,gods ,creatures3 and the eye1 the light of our body1 and his ,sense most mighty1 and his organ most ,artificiall and ,geometricall3 ,at ,perspectiue1 we will begyn therfore4 ,perspectiue1 is an ,art ,mathematicall1 which demonstrateth the maner1 and properties1 of all ,radiations ,direct1 ,broken1 and ,reflected4 ,this ,description1 or ,notation1 is brief3 but it reacheth so farre1 as the world is wyde4 ,it concerneth all ,creatures1 all ,actions1 and passions1 by ,emanation of beames perfourmed4 ,beames1 or naturall lines1 7here7 ,i meane1 not of light onely1 or of colour 7though they1 to eye1 giue shew1 witnes1 and profe1 wherby to ground the ,arte vpon7 but also of other ,formes1 both ,substantiall1 and ,accidentall1 the certaine and determined actiue ,radiall emanations4 ,by this ,art 7omitting to speake of the highest pointes7 we may vse our eyes1 and the light1 with greater pleasure3 and perfecter ,iudgement3 both of things1 in light seen1 of other3 which by like order of ,lightes ,radiations1 worke and produce their effectes4 ,we may be ashamed to be ignorant of the cause1 why so sundry wayes our eye is deceiued1 and abused3 as1 while the eye weeneth a rod ,globe or ,sphere 7beyng farre of7 to be a flat and plaine ,circle1 and so likewise iudgeth a plaine ,square1 to be rod3 supposeth walles parallels1 to approche1 a farre of3 rofe and floure parallels1 the one to bend downward1 the other to rise vpward1 at a little distance from you4 ,againe1 of thinges being in like swiftnes of mouing1 to thinke the nerer1 to moue faster3 and the farder1 much slower4 ,nay1 of two thinges1 wherof the one 7incomparably7 doth moue swifter then the other1 to deme the slower to moue very swift1 the other to stand3 what an error is this1 of our eye8 ,of the ,raynbow1 both of his ,colours1 of the order of the colours1 of the bignes of it1 the place and heith of it1 7c7 to know the causes demonstratiue1 is it not pleasant1 is it not necessary8 of two or three ,sonnes appearing3 of ,blasing ,sterres3 and such like thinges3 by naturall causes1 brought to passe1 7and yet neuertheles1 of farder matter1 ,significatiue7 is it not commodious for man to know the very true cause1 occasion ,naturall8 ,yea1 rather1 is it not1 greatly1 against the ,souerainty of ,mans nature1 to be so ouershot and abused1 with thinges 7at hand7 before his eyes8 as with a ,pecockes tayle1 and a ,doues necke3 or a whole ore1 in water1 holden1 to seme broken4 ,thynges1 farre of1 to seeme nere3 and nere1 to seme farre of4 ,small thinges1 to seme great3 and great1 to seme small4 ,one man1 to seme an ,army4 ,or a man to be curstly affrayed of his owne shaddow4 ,yea1 so much1 to feare1 that1 if you1 being 7alone7 nere a certaine glasse1 and proffer1 with dagger or sword1 to foyne at the glasse1 you shall suddenly be moued to giue backe 7in maner7 by reason of an ,image1 ,a marueilous ,glasse4 appearing in the ayre1 betwene you the glasse1 with like hand1 sword or dagger1 with like quicknes1 foyning at your very eye1 likewise as you do at the ,glasse4 ,straunge1 this is1 to heare of3 but more meruailous to behold1 then these my wordes can signifie4 ,and neuerthelesse by demonstration ,opticall1 the order and cause therof1 is certified3 euen so1 as the effect is consequent4 ,yea1 thus much more1 dare ,i take vpon me1 toward the satisfying of the noble courrage1 that longeth ardently for the wisedome of ,causes ,naturall3 as to let him vnderstand1 that1 in ,london1 he may with his owne eyes1 haue profe of that1 which ,i haue sayd herein4 ,a ,gentleman1 7which1 for his good seruice1 done to his ,countrey1 is famous and honorable3 ,s4 ,w4 ,p4 and for skill in the ,mathematicall ,sciences1 and ,languages1 is the ,od man of this land4 c47 euen he1 is hable3 and 7,i am sure7 will1 very willingly1 let the ,glasse1 and profe be sene3 and so ,i 7here7 request him3 for the encrease of wisedome1 in the honorable3 and for the stopping of the mouthes malicious3 and repressing the arrogancy of the ignorant4 ,ye may easily gesse1 what ,i meane4 ,this ,art of ,perspectiue1 is of that excellency1 and may be led1 to the certifying1 and executing of such thinges1 as no man would easily beleue3 without ,actuall profe perceiued4 ,i speake nothing of ,naturall ,philosophie1 which1 without ,perspectiue1 can not be fully vnderstanded1 nor perfectly atteined vnto4 ,nor1 of ,astronomie3 which1 without ,perspectiue1 can not well be grounded3 ,nor ,astrologie1 naturally ,verified1 and auouched4 ,that part hereof1 which dealeth with ,glasses 7which name1 ,glasse1 is a generall name1 in this ,arte1 for any thing1 from which1 a ,beame reboundeth7 is called ,catoptrike3 and hath so many vses1 both merueilous1 and proffitable3 that1 both1 it would hold me to long1 to note therin the principall conclusions1 all ready knowne3 ,and also 7perchaunce7 some thinges1 might lacke due credite with you3 ,and ,i1 therby1 to leese my labor3 and you1 to slip into light ,iudgement1 99 ,before you haue learned sufficiently the powre of ,nature and ,arte4 ,now1 to procede3 ,astronomie1 is an ,arte ,mathematicall1 which demonstrateth the distance1 magnitudes1 and all naturall motions1 apparences1 and passions propre to the ,planets and fixed ,sterres3 for any time past1 present and to come3 in respect of a certaine ,horizon1 or without respect of any ,horizon4 ,by this ,arte we are certified of the distance of the ,starry ,skye1 and of eche ,planete from the ,centre of the ,earth3 and of the greatnes of any ,fixed starre sene1 or ,planete1 in respect of the ,earthes greatnes4 ,as1 we are sure 7by this ,arte7 that the ,solidity1 ,massines and ,body of the ,sonne1 conteineth the quantitie of the whole ,earth and ,sea1 a hundred thre score and two times1 lesse by one eight parte of the earth4 ,but the ,body of the whole earthly globe and ,sea1 is bigger then the body of the ,mone1 three and forty times lesse by of the ,mone4 ,wherfore the ,sonne is bigger then the ,mone1 #gjjj times1 lesse1 by #ei #ci_/fd that is1 precisely #fidj #be_/fd bigger then the ,mone4 ,and yet the vnskillfull man1 would iudge them a like bigge4 ,wherfore1 of ,necessity1 the one is much farder from vs1 then the other4 ,the ,sonne1 when he is fardest from the earth 7which1 now1 in our age1 is1 when he is in the #h4 degree1 of ,cancer7 is1 #aagi ,semidiameters of the ,earth1 distante4 ,and the ,mone when she is fardest from the earth1 is #fh ,semidiameters of the earth and ,the nerest1 that the ,mone commeth to the earth1 is ,semidiameters #eb ,the distance of the ,starry ,skye is1 fr vs1 in ,semidiameters of the earth #bjjha ,twenty thousand fourescore1 one1 and almost a halfe4 ,subtract from this1 the ,mones nerest distance1 from the ,earth3 and therof remaineth ,semidiameters of the earth #bjjbi ,twenty thousand nine and twenty and a quarter4 ,note4 ,so thicke is the heauenly ,palace1 that the ,planetes haue all their exercise in1 and most meruailously perfourme the ,commadement and ,charge to them giuen by the omnipotent ,maiestie of the king of kings4 ,this is that1 which in ,genesis is called ,ha ,rakia4 ,consider it well4 ,the ,semidiameter of the earth1 cteineth of our common miles #cdcf #d_/aa three thousand1 foure hundred thirty six and foure eleuenth partes of one myle3 ,such as the whole earth and ,sea1 round about1 is #bafjj4 ,one and twenty thousand six hundred of our myles4 ,allowyng for euery degree of the greatest circle1 thre score myles4 ,now if you way well with your selfe but this litle parcell of frute ,astronomicall1 as concerning the bignesse1 ,distances of ,sonne1 ,mone1 ,sterry ,sky1 and the huge massines of ,ha ,rakia1 will you not finde your ,consciences moued1 with the kingly ,prophet1 to sing the confession of ,gods ,glory1 and say1 ,the ,heauens declare the glory of ,god1 and the ,firmament ,ha ,rakia sheweth forth the workes of his handes4 ,and so forth1 for those fiue first staues1 of that kingly ,psalme4 ,well1 well1 ,it is time for some to lay hold on wisedome1 and to ,iudge truly of thinges3 and notso to expound the ,holy word1 all by ,allegories3 as to ,neglect the wisedome1 powre and ,goodnes of ,god1 in1 and by his ,creatures1 and ,creation to be seen and learned4 ,by parables and ,analogies of whose natures and properties1 the course of the ,holy ,scripture1 also1 declareth to vs very many ,mysteries4 ,the whole ,frame of ,gods ,creatures1 7which is the whole world17 is to vs1 a bright glasse3 from which1 by reflexion1 reboundeth to our knowledge and perceiuerance1 ,beames1 and ,radiations3 representing the ,image of his ,infinite goodnes1 ,omnipotcy1 and wisedome4 ,and we therby1 are taught and persuaded to ,glorifie our ,creator1 as ,god3 and be thankefull therfore4 ,could the ,heathenistes finde these vses1 of these most pure1 beawtifull1 and ,mighty ,corporall ,creatures3 and shall we1 after that the true ,sonne of rightwisenesse is risen aboue the ,horizon1 of our temporall ,hemisphrie1 and hath so abundantly streamed into our hartes1 the direct beames of his goodnes1 mercy1 and grace3 ,whose heat ,all ,creatures feele3 ,spirituall and ,corporall3 ,visible and ,inuisible4 ,shall we 7,i say7 looke vpon the ,heauen1 ,sterres1 and ,planets1 as an ,oxe and an ,asse doth3 no furder carefull or inquisitiue1 what they are3 why were they ,created1 ,how do they execute that they were ,created for8 ,seing1 ,all ,creatures1 were for our sake created3 and both we1 and they1 ,created1 chiefly to glorifie the ,almighty ,creator3 and that1 by all meanes1 to vs possible4 ,nolite ignorare 7saith ,plato in ,epinomis7 ,astronomiam1 ,sapientissim quiddam esse4 ,be ye not ignorant1 ,astronomie to be a thyng of excellent wisedome4 ,astronomie1 was to vs1 from the beginning commended1 and in maner commaunded by ,god him selfe4 ,in asmuch as he made the ,sonne1 ,mone1 and ,sterres1 to be to vs1 for ,signes1 and knowledge of ,seasons1 and for ,distinctions of ,dayes1 and yeares4 ,many wordes nede not4 ,but ,i wish1 euery man should way this word1 ,signes4 ,and besides that1 conferre it also with the tenth ,chapter of ,hieremie4 ,and though ,some thinke1 that there1 they haue found a rod3 ,yet ,modest ,reason1 will be indifferent ,iudge1 who ought to be beaten therwith1 in respect of our purpose4 ,leauing that3 ,i pray you vnderstand this3 that without great diligence of ,obseruation1 examination and ,calculation1 their periods and courses 7wherby ,distinction of ,seasons1 yeares1 and ,new ,mones might precisely be knowne7 could not exactely be certified4 ,which thing to performe1 is that ,art1 which we here haue ,defined to be ,astronomie4 ,wherby1 we may haue the distinct ,course of ,times1 dayes1 yeares1 and ,ages3 aswell for ,considerati of ,sacred ,prophesies1 accomplished in due time1 foretold3 as for high ,mysticall ,solemnities holding3 ,and for all other humaine affaires1 ,conditions1 and couenantes1 vpon certaine time1 betwene man and man3 with many other great vses3 ,wherin1 7verely71 would be great incertainty1 ,confusion1 vntruth1 and brutish ,barbarousnes3 without the wonderfull diligence and skill of this ,arte3 continually learning1 and determining ,times1 and periodes of ,time1 by the ,record of the heauenly booke1 wherin all times are written3 and to be read with an ,astronomicall staffe1 in stede of a festue4 ,musike1 of ,motion1 hath his ,originall cause3 ,therfore1 after the motions most swift1 and most ,slow1 which are in the ,firmament1 of ,nature performed3 and vnder the ,astronomers ,consideration3 now ,i will ,speake of an other kinde of ,motion1 producing sound1 audible1 and of ,man numerable4 ,musike ,i call here that ,science1 which of the ,grekes is called ,harmonice4 ,not medling with the ,controuersie betwene the auncient ,harmonistes1 and ,canonistes4 ,musike is a ,mathematicall ,science1 which teacheth1 by sense and reason1 perfectly to iudge1 and order the diuersities of soundes1 hye and low4 ,astronomie and ,musike are ,sisters1 saith ,plato4 ,as1 for ,astronomie1 the eyes3 ,so1 for ,harmonious ,motion1 the eares were made4 ,but as ,astronomie hath a more diuine ,contemplation1 and cmodity1 then mortall eye can perceiue3 ,so1 is ,musike to be considered1 #a4 that the 99 ,minde may be preferred1 before the eare4 ,and from audible sound1 we ought to ascende1 to the examination3 which numbers are ,harmonious1 and which not4 ,and why1 either1 the one are3 or the other are not4 ,i could at large1 #b4 in the heauenly 99 motions and distances1 describe a meruailous ,harmonie1 of ,pythagoras ,harpe #c4 with eight stringes4 ,also1 somwhat might be sayd of ,mercurius 99 two ,harpes1 #d4 eche of foure ,stringes ,elementall4 ,and very straunge matter1 might be alledged of the ,harmonie1 #e4 to our 99 ,spirituall part appropriate4 ,as in ,ptolomaus third boke1 in the fourth and sixth ,chapters may appeare4 99 #f4 ,and what is the cause of the apt bonde1 and frendly felowship1 of the ,intellectuall and ,mentall part of vs1 with our grosse corruptible body3 but a certaine ,meane1 and ,harmonious ,spiritualitie1 with both participatyng1 of both 7in a maner7 resultynge ,in #g4 the 99 ,tune of ,mans voyce1 and also #h4 99 the sound of ,instrument1 what might be sayd1 of ,harmonie3 ,no common ,musicien would lightly beleue4 ,i4 ,d4 ,read in ,aristotle his #h4 booke of ,politikes3 the #e1 #f1 and #g4 chapters4 ,where you shall haue some occasion farder to thinke of ,musike1 than commonly is thought4 ,but of the sundry ,mixture 7as ,i may terme it7 and concurse1 diuerse collation1 and ,application of these ,harmonies3 as of thre1 foure1 fiue1 or mo3 ,maruailous haue the effectes ben3 and yet may be founde1 and produced the like3 with some proportionall consideration for our time1 and being3 in respect of the ,state1 of the thinges then3 in which1 and by which1 the wondrous effectes were wrought4 ,democritus and ,theophrastus affirmed1 that1 by ,musike1 griefes and diseases of the ,minde1 and body might be cured1 or inferred4 ,and we finde in ,recorde1 that ,terpander1 ,arion1 ,ismenias1 ,orpheus1 ,amphion1 ,dauid1 ,pythagoras1 ,empedocles1 ,asclepiades and ,timotheus1 by ,harmonicall ,consoncy1 haue done1 and brought to pas1 thinges1 more then meruailous1 to here of4 ,of them then1 making no farder discourse1 in this place3 ,sure ,i am1 that ,common ,musike1 commonly vsed1 is found to the ,musiciens and ,hearers1 to be so ,commodious and pleasant1 ,that if ,i would say and dispute1 but thus much3 ,that it were to be otherwise vsed1 then it is1 ,i should finde more repreeuers1 then ,i could finde priuy1 or skilfull of my meaning4 ,in thinges therfore euident1 and better knowen1 then ,i can expresse3 and so allowed and liked of1 7as ,i would wish1 some other thinges1 had the like hap7 ,i will spare to enlarge my lines any farder1 but consequently follow my purpose4 ,of ,cosmographie1 ,i appointed briefly in this place1 to geue you some intelligence4 ,cosmographie1 is the whole and perfect description of the heauenly1 and also elementall parte of the world1 and their homologall application1 and mutuall collation necessarie4 ,this ,art1 requireth ,astronomie1 ,geographie1 ,hydrographie and ,musike4 ,therfore1 it is no small ,arte1 nor so simple1 as in common practise1 it is 7slightly7 considered4 ,this matcheth ,heauen1 and the ,earth1 in one frame1 and aptly applieth parts ,correspdent3 ,so1 as1 the ,heauenly ,globe1 may 7in practise7 be duely described vpon the ,geographicall1 and ,hydrographicall ,globe4 ,and there1 for vs to consider an quonoctiall ,circle1 an ,ecliptike line1 ,colures1 ,poles1 ,sterres in their true ,longitudes1 ,latitudes1 ,declinations1 and ,verticalitie3 also ,climes1 and ,parallels3 and by an ,horizon annexed1 and reuolution of the earthly ,globe 7as the ,heauen1 is1 by the ,primouant1 caried about in #bd4 quall ,houres7 to learne the ,risinges and ,settinges of ,sterres 7of ,virgill in his ,georgikes3 of ,hesiod3 of ,hippocrates in his ,medicinall ,sphre1 to ,perdicca ,king of the ,macedonians3 of ,diocles1 to ,king ,antigonus1 and of other famous ,philosophers prescribed7 a thing necessary1 for due manuring of the earth1 for ,nauigation1 for the ,alteration of mans body3 being1 whole1 ,sicke1 wounded1 or brused4 ,by the ,reuolution1 also1 or mouing of the ,globe ,cosmographicall1 the ,rising and ,setting of the ,sonne3 the ,lengthes1 of dayes and nightes3 the ,houres and times 7both night and day7 are knowne3 with very many other pleasant and necessary vses3 ,wherof1 some are knowne3 but better remaine1 for such to know and vse3 who of a sparke of true fire1 can make a wonderfull bonfire1 by applying of due matter1 duely4 ,of ,astrologie1 here ,i make an ,arte1 seuerall from ,astronomie3 not by new deuise1 but by good reason and authoritie3 for1 ,astrologie1 is an ,arte ,mathematicall1 which reasonably demonstrateth the operations and effectes1 of the naturall beames1 of light1 and secrete influence3 of the ,sterres and ,planets3 in euery element and elementall body3 at all times1 in any ,horizon assigned4 ,this ,arte is furnished with many other great ,artes and experiences3 ,as with perfecte ,perspectiue1 ,astronomie1 ,cosmographie1 ,naturall ,philosophie of the #d4 ,elementes1 the ,arte of ,graduation1 and some good vnderstding in ,musike3 and yet moreouer1 with an other great ,arte1 hereafter following1 though ,i1 here1 set this before1 for some considerations me mouing4 ,sufficient 7you see7 is the stuffe1 to make this rare and secrete ,arte1 of3 and hard enough to frame to the ,conclusion ,syllogisticall4 ,yet both the manifolde and continuall trauailes of the most auncient and wise ,philosophers1 for the atteyning of this ,arte3 and by examples of effectes1 to confirme the same3 hath left vnto vs sufficient proufe and witnesse3 and we1 also1 daily may perceaue1 ,that mans body1 and all other ,elementall bodies1 are altered1 disposed1 ordred1 pleasured1 and displeasured1 by the ,influentiall working of the ,sunne1 ,mone1 and the other ,starres and ,planets4 ,and therfore1 sayth ,aristotle1 in the first of his ,meteorologicall bookes1 in the second ,chapter3 ,est autem necessari ,mundus iste1 supernis lationibus fer continuus4 ,vt1 inde1 vis eius vniuersa regatur4 ,ea siquidem ,caus prima putanda omnibus est1 vnde motus principium existit4 ,that is3 ,this ,elementall ,world is of necessitie1 almost1 next adioyning1 to the heauenly motions3 ,that1 from thence1 all his vertue or force may be gouerned4 ,for1 that is to be thought the first ,cause vnto all3 from which1 the beginning of motion1 is4 ,and againe1 in the tenth ,chapter4 ,oportet igitur horum principia sumamus1 causas omnium similiter4 ,principium igitur vt mouens1 prcipuumque omnium primum1 ,circulus ille est1 in quo manifeste ,solis latio1 c4 ,and so forth4 ,his ,meteorologicall bookes1 are full of argumentes1 and effectuall demonstrations1 of the vertue1 operation1 and power of the heauenly bodies1 in and vpon the fower ,elementes1 and other bodies1 of them 7either perfectly1 or vnperfectly7 composed4 ,and in his second booke1 ,de ,generatione ,corruptione1 in the tenth ,chapter4 ,quocirca prima latio1 ,ortus ,interitus causa non est3 ,sed obliqui ,circuli latio3 ea namque continua est1 duobus motibus fit3 ,in ,englishe1 thus4 ,wherefore the vppermost motion1 is not the cause of ,generation and ,corruption1 but the motion of the ,zodiake3 for1 that1 both1 is continuall1 and is caused of two mouinges4 ,and in his second booke1 and second ,chapter of hys ,physikes4 ,homo namque generat hominem1 atque ,sol4 ,for ,man 7sayth he7 and the ,sonne1 are cause of mans generation4 ,authorities may be brought1 very many3 both of #ajjj4 #bjjj4 yea and #cjjj4 yeares ,antiquitie3 of great ,philosophers1 ,expert1 ,wise1 and godly men1 for that ,conclusion3 which1 daily and hourely1 we men1 may discerne and perceaue by sense and reason3 ,all beastes do feele1 and simply shew1 by their actions and passions1 outward and inward3 ,all ,plants1 ,herbes1 ,trees1 ,flowers1 and ,fruites4 ,and finally1 the ,elementes1 and all thinges of the ,elementes composed1 do geue ,testimonie 7as ,aristotle sayd7 that theyr ,whole ,dispositions1 vertues1 and naturall motions1 depend of the ,actiuitie of the heauenly motions and ,influences4 ,whereby1 beside the specificall order and forme1 due to euery seede3 and beside the ,nature1 propre to the ,indiuiduall ,matrix1 of the thing produced3 ,what shall be the heauenly ,impression1 the perfect and circumspecte ,astrologien hath to ,conclude4 ,not onely 7by ,apotelesmes7 4 but by ,naturall and ,mathematicall demonstration 4 ,whereunto1 what ,sciences are requisite 7without exception7 ,i partly haue here warned3 ,and in my ,propdeumes 7besides other matter there disclosed7 ,i haue ,mathematically furnished vp the whole ,method3 ,to this our age1 not so carefully handled by any1 that euer ,i saw1 or heard of4 ,i was1 99 ,anno4 #aedh and #aedi4 in ,louayn4 7for 99 #ba4 yeares ago7 by certaine earnest disputations1 of the ,learned ,gerardus ,mercator1 and ,antonius ,gogaua1 7and other17 therto so prouoked3 and 7by my constant and inuincible zeale to the veritie7 in obseruations of ,heauenly ,influencies 7to the ,minute of time17 than1 so diligent3 ,and chiefly by the ,supernaturall influence1 from the ,starre of ,iacob1 so directed3 ,that any ,modest and ,sober ,student1 carefully and diligently seking for the ,truth1 will both finde cfesse1 therin1 to be the ,veritie1 of these my wordes3 ,and also become a ,reasonable ,reformer1 of three ,sortes of people3 about these ,influentiall ,operations1 greatly erring from the truth4 ,note4 ,wherof1 the one1 is ,light ,beleuers1 the other1 ,light ,despisers1 and the third ,light ,practisers4 ,the first1 most cmon ,sort1 thinke the ,heauen and ,sterres1 to be answerable to any their doutes or desires3 #a4 which is not so3 and1 in dede1 they1 to much1 ouer reache4 ,the ,second sorte thinke no ,influentiall vertue 7fr the heauenly bodies7 to beare any ,sway in ,generation #b4 and ,corruption1 in this ,elementall world4 ,and to the ,sunne1 ,mone and ,sterres 7being so many1 so pure1 so bright1 so wonderfull bigge1 so farre in distance1 so manifold in their motions1 so constant in their periodes4 c47 they assigne a sleight1 simple office or two1 and so allow vnto th 7according to their capacities7 as much vertue1 and power ,influentiall1 as to the ,signe of the ,sunne1 ,mone1 and seuen ,sterres1 hanged vp 7for ,signes7 in ,london1 for distinction of houses1 such grosse helpes1 in our worldly affaires3 ,and they vnderstand not 7or will not vnderstand7 of the other workinges1 and vertues of the ,heauenly ,sunne1 ,mone1 and ,sterres3 not so much1 as the ,mariner1 or ,husband man3 no1 not so much1 as the ,elephant doth1 as the ,cynocephalus1 as the ,porpentine doth3 nor will allow these perfect1 and incorruptible mighty bodies1 so much vertuall ,radiation1 ,force1 as they see in a litle peece of a ,magnes stone3 which1 at great distance1 sheweth his operation4 ,and perchaunce they thinke1 the ,sea ,riuers 7as the ,thames7 to be some quicke thing1 and so to ebbe1 and flow1 run in and out1 of them selues1 at their owne fantasies4 ,god helpe1 ,god helpe4 ,surely1 these men1 come to short3 and either are to dull3 or willfully blind3 or1 perhaps1 to malicious4 ,the third man1 is the common and vulgare ,astrologien1 or ,practiser3 who1 being not duely1 artificially1 and perfectly #c4 furnished3 yet1 either for vaine glory1 or gayne3 or like a simple dolt1 blinde ,bayard1 both in matter and maner1 erreth3 to the discredit of the ,wary1 and modest ,astrologien3 and to the robbing of those most noble corporall ,creatures1 of their ,naturall ,vertue3 being most mighty3 most beneficiall to all elementall ,generation1 ,corruption and the appartenances3 and most ,harmonious in their ,monarchie3 ,for which thinges1 being knowen1 and modestly vsed3 we might highly1 and continually glorifie ,god1 with the princely ,prophet1 saying4 ,the ,heauens declare the ,glorie of ,god3 who made the ,heaus in his wisedome3 who made the ,sonne1 for to haue dominion of the day3 the ,mone and ,sterres to haue dominion of the nyght3 whereby1 ,day to day vttereth talke3 and night1 to night declareth knowledge4 ,prayse him1 all ye ,sterres1 and ,light4 ,amen4 ,in order1 now foloweth1 of ,statike1 somewhat to say1 what we meane by that name3 and what commodity1 doth1 on such ,art1 depend4 ,statike1 is an ,arte ,mathematicall1 which demonstrateth the causes of heauynes1 and lightnes of all thynges3 and of motions and properties1 to heauynes and lightnes1 belonging4 ,and for asmuch as1 by the ,bilanx1 or ,balance 7as the chief sensible ,instrument17 ,experience of these demonstrations may be had3 we call this ,art1 ,statike3 that is1 the ,experimentes of the ,balance4 ,oh1 that men wist1 what proffit1 7all maner of wayes7 by this ,arte might grow1 to the hable examiner1 and diligent practiser4 8,thou onely1 knowest all thinges precisely 7,o ,god7 who hast made weight and ,balance1 thy ,iudgement3 who hast created all thinges in ,number1 ,waight1 and ,measure3 and hast wayed the mountaines and hils in a ,balance3 who hast peysed in thy hand1 both ,heauen and earth4 ,we therfore warned by the ,sacred word1 to ,consider thy ,creatures3 and by that consideration1 to wynne a glyms 7as it were17 or shaddow of perceiuerance1 that thy wisedome1 might1 and goodnes is infinite1 and vnspeakable1 in thy ,creatures declared3 ,and being farder aduertised1 by thy mercifull goodnes1 that1 three principall wayes1 were1 of the1 vsed in ,creation of all thy ,creatures1 namely1 ,number1 ,waight and ,measure1 ,and for as much as1 of ,number and ,measure1 the two ,artes 7auncient1 famous1 and to humaine vses most necessary17 are1 all ready1 sufficiently knowen and extant3 ,this third key1 we beseche thee 7through thy accustomed goodnes17 that it may come to the nedefull and sufficient knowledge1 of such thy ,seruauntes1 as in thy workemanship1 would gladly finde1 thy true occasions 7purposely of the vsed7 whereby we should glorifie thy name1 and shew forth 7to the weaklinges in faith7 thy wondrous wisedome and ,goodnes4 ,amen40 ,meruaile nothing at this pang 7godly frend1 you ,gentle and zelous ,student47 ,an other day1 perchaunce1 you will perceiue1 what occasion moued me4 ,here1 as now1 ,i will giue you some ground1 and withall some shew1 of certaine commodities1 by this ,arte arising4 ,and bycause this ,arte is rare1 my wordes and practises might be to darke3 vnleast you had some light1 holden before the matter3 and that1 best will be1 in giuing you1 out of ,archimedes demonstrations1 a few principal ,conclusions1 as foloweth4 #a4 ,the ,superficies of euery ,liquor1 by it selfe consistyng1 and in quyet1 is ,sphricall3 the centre whereof1 is the same1 which is the centre of the ,earth4 #b4 ,if ,solide ,magnitudes1 being of the same bignes1 or qutitie1 that any ,liquor is1 and hauyng also the same ,waight3 be let downe into the same ,liquor1 they will settle downeward1 so1 that no parte of them1 shall be aboue the ,superficies of the ,liquor3 and yet neuertheles1 they will not sinke vtterly downe1 or drowne4 #c4 ,if any ,solide ,magnitude beyng ,lighter then a ,liquor1 be let downe into the same ,liquor1 it will settle downe1 so farre into the same ,liquor1 that so great a quantitie of that ,liquor1 as is the parte of the ,solid ,magnitude1 settled downe into the same ,liquor3 is in ,waight1 quall1 to the waight of the whole ,solid ,magnitude4 #d4 ,any ,solide ,magnitude1 ,lighter then a ,liquor1 forced downe into the same ,liquor1 will moue vpward1 with so great a power1 by how much1 the ,liquor hauyng quall quantitie to the whole ,magnitude1 is heauyer then the same ,magnitude4 #e4 ,any ,solid ,magnitude1 heauyer then a ,liquor1 beyng let downe into the same ,liquor1 will sinke downe vtterly3 ,and wilbe in that ,liquor1 ,lighter by so much1 as is the waight or heauynes of the ,liquor1 hauing bygnes or quantitie1 quall to the ,solid ,magnitude4 #f4 ,i4 ,d4 ,the ,cutting of a ,sphre 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 ,sphre therein ,swymming4 ,if any ,solide ,magnitude1 ,lighter then a ,liquor1 be let downe into the same ,liquor1 the waight of the same ,magnitude1 will be1 to the ,waight of the ,liquor4 7,which is quall in quantitie to the whole ,magnitude17 in that proportion1 that the parte1 of the ,magnitude settled downe1 is to the whole ,magnitude4 ,by these verities1 great ,errors may be reformed1 in ,opinion of the ,naturall ,motion of thinges1 ,light and ,heauy4 ,which errors1 are in ,naturall ,philosophie 7almost7 of all m allowed3 to much trusting to ,authority3 and false ,suppositions4 ,as1 ,of any two bodyes1 the heauyer1 to moue downward faster then the lighter4 ,a common error1 noted4 ,this error1 is not first by me1 ,noted3 but by one ,iohn ,baptist de ,benedictis4 ,the chief of his propositions1 is this3 which seemeth a ,paradox4 ,if there be two bodyes of one forme1 and of one kynde1 quall in quantitie or vnquall1 ,a paradox4 they will moue by quall space1 in quall tyme3 ,so that both theyr mouynges be in ayre1 or both in water3 or in any one ,middle4 ,hereupon1 in the feate of ,gunnyng1 ,n4 ,t4 certaine good discourses 7otherwise7 may receiue great amendement1 and furderance4 ,the wonderfull vse of these ,propositions4 ,in the entended purpose1 also1 allowing somwhat to the imperfection of ,nature3 not aunswerable to the precisenes of demonstration4 ,moreouer1 by the foresaid propositions 7wisely vsed47 ,the ,ayre1 the water1 the ,earth1 the ,fire1 may be nerely1 knowen1 how light or heauy they are 7,naturally7 in their assigned partes3 or in the whole4 ,and then1 to thinges ,elementall1 turning your practise3 you may deale for the proportion of the ,elementes1 in the thinges ,compounded4 ,then1 to the proportions of the ,humours in ,man3 their waightes3 and the waight of his bones1 and flesh4 c4 ,than1 by waight1 to haue consideration of the ,force of man1 any maner of way3 in whole or in part4 ,then1 may you1 of ,ships water drawing1 diuersly1 in the ,sea and in fresh water1 haue pleasant consideration3 and of waying vp of any thing1 sonken in ,sea or in fresh water c4 ,and 7to lift vp your head a loft37 by waight1 you may1 as precisely1 as by any instrument els1 measure the ,diameters of ,sonne and ,mone4 c4 ,frende1 ,i pray you1 way these thinges1 with the iust ,balance of ,reason4 ,and you will finde ,meruailes vpon ,meruailes3 ,and esteme one ,drop of ,truth 7yea in ,naturall ,philosophie7 more worth1 then whole ,libraries of ,opinions1 vndemonstrated3 or not aunswering to ,natures ,law1 and your experience4 ,leauing these thinges1 thus3 ,i will giue you two or three1 light practises1 to great purpose3 and so finish my ,annotation ,staticall4 ,in ,mathematicall matters1 by the ,mechaniciens ayde1 we will behold1 here1 the ,commodity of waight4 ,the practise ,staticall1 to know the proportion1 betwene the ,cube1 and the ,sphre4 ,make a ,cube1 of any one ,vniforme3 and through like heauy stuffe3 of the same ,stuffe1 make a ,sphre or ,globe1 precisely1 of a ,diameter quall to the ,radicall side of the ,cube4 ,your stuffe1 may be wood1 ,copper1 ,tinne1 ,lead1 ,siluer4 c4 7being1 as ,i sayd1 of like nature1 condition1 and like waight throughout47 ,and you may1 by ,say ,balance1 haue prepared a great number of the smallest waightes3 which1 by those ,balance can be discerned or tryed3 and so1 haue proceded to make you a perfect ,pyle1 company ,number of waightes3 to the waight of six1 eight1 or twelue pound waight3 most diligently tryed1 all4 ,and of euery one1 the ,content knowen1 in your least waight1 that is wayable4 ,they that can not haue these waightes of precisenes3 may1 by ,sand1 ,vniforme1 and well dusted1 make them a number of waightes1 somewhat nere precisenes3 by halfing euer the ,sand3 they shall1 at length1 come to a least common waight4 ,therein1 ,i leaue the farder matter1 to their discretion1 whom nede shall pinche4 ,the ,venetians consideration of waight1 may seme precise enough3 by eight descentes progressionall1 99 halfing1 from a grayne4 ,i4 ,d4 99 ,for1 so1 haue you 4#bef4 partes of a ,graine4 ,your ,cube1 ,sphre1 apt ,balance1 and conuenient waightes1 being ready3 fall to worke44 ,first1 way your ,cube4 ,note the ,number of the waight4 ,way1 after that1 your ,sphre4 ,note likewise1 the ,nber of the waight4 ,if you now find the waight of your ,cube1 to be to the waight of the ,sphre1 as #ba4 is to #aa3 ,then you see1 how the ,mechanicien and ,experimenter1 without ,geometrie and ,demonstration1 are 7as nerely in effect7 tought the proportion of the ,cube to the ,sphere3 as ,i haue demonstrated it1 in the end of the twelfth boke of ,euclide4 ,often1 try with the same ,cube and ,sphre4 ,then1 chaunge1 your ,sphre and ,cube1 to an other matter3 or to an other bignes3 till you haue made a perfect vniuersall ,experience of it4 ,possible it is1 that you shall wynne to nerer termes1 in the proportion4 ,when you haue found this one certaine ,drop of ,naturall veritie1 procede on1 to ,inferre1 and duely to make assay1 of matter depending4 ,as1 bycause it is well demonstrated1 that a ,cylinder1 whose heith1 and ,diameter of his base1 is quall to the ,diameter of the ,sphre1 is ,sesquialter to the same ,sphre 7that is1 as #c4 to #b37 ,to the number of the waight of the ,sphre1 adde halfe so much1 as it is3 and so haue you the number of the waight of that ,cylinder4 ,which is also ,comprehended of our former ,cube3 ,so1 that the base of that ,cylinder1 is a ,circle described in the ,square1 which is the base of our ,cube4 ,but the ,cube and the ,cylinder1 being both of one heith1 haue their ,bases in the same proportion1 in the which1 they are1 one to an other1 in their ,massines or ,soliditie4 ,but1 before1 we haue two numbers1 expressing their ,massines1 ,solidities1 and ,quantities1 by waight3 wherfore1 99 ,the proportion of the ,square to the ,circle inscribed4 we haue 99 the proportion of the ,square1 to the ,circle1 inscribed in the same ,square4 ,and so are we fallen into the knowledge sensible1 and ,experimentall of ,archimedes great ,secret3 of him1 by great trauaile of minde1 sought and found4 ,wherfore1 to any ,circle giuen1 you can giue a ,square quall3 99 ,the ,squaring of the ,circle1 ,mechanically4 99 as ,i haue taught1 in my ,annotation1 vpon the first proposition of the twelfth boke1 ,and likewise1 to any ,square giuen1 you may giue a ,circle quall3 99 ,to any ,square geuen1 to geue a ,circle1 equall4 99 ,if you describe a ,circle1 which shall be in that proportion1 to your ,circle inscribed1 as the ,square is to the same ,circle3 ,this1 you may do1 by my ,annotations1 vpon the second proposition of the twelfth boke of ,euclide1 in my third ,probleme there4 ,your diligence may come to a proportion1 of the ,square to the ,circle inscribed1 nerer the truth1 then is the proportion of #ad4 to #aa4 ,and consider1 that you may begyn at the ,circle and ,square1 and so come to conclude of the ,sphre1 the ,cube1 what their proportion is3 as now1 you came from the ,sphre to the ,circle4 ,for1 of ,siluer1 or ,gold1 or ,latton ,lamyns or plates 7thorough one hole draw1 as the maner is7 if you make a ,square figure way it3 and then1 describing theron1 the ,circle inscribed3 cut of1 file away1 precisely 7to the ,circle7 the ouerplus of the ,square3 you shall then1 waying your ,circle1 see1 whether the waight of the ,square1 be to your ,circle1 as #ad4 to #aa4 ,as ,i haue ,noted1 in the beginning of ,euclides twelfth boke4 c4 after this resort to my last proposition1 vpon the last of the twelfth4 ,and there1 helpe your selfe1 to the end4 ,and1 here1 ,note this1 by the way4 ,note ,squaring of the ,circle without knowledge of the proportion betwene ,circumference and ,diameter4 ,that we may ,square the ,circle1 without hauing knowledge of the proportion1 of the ,circumference to the ,diameter3 as you haue here perceiued4 ,and otherwayes also1 ,i can demonstrate it4 ,so that1 many haue cumberd them selues superfluously1 by trauailing in that point first1 which was not of necessitie1 first3 and also very intricate4 ,and easily1 you may1 7and that diuersly7 come to the knowledge of the ,circumference3 the ,circles ,quantitie1 being first knowen4 ,which thing1 ,i leaue to your consideration3 making hast to despatch an other ,magistrall ,probleme3 and to bring it1 nerer to your knowledge1 and readier dealing with1 then the world 7before this day17 had it for you1 that ,i can tell of4 ,and that is1 ,a ,mechanicall ,dubblyng of the ,cube3 c4 ,which may1 thus1 be done3 ,to ,dubble the ,cube redily3 by ,art ,mechanicall3 depending vppon ,demonstration ,mathematicall4 ,make of ,copper plates1 or ,tyn plates1 a foursquare vpright ,pyramis1 or a ,cone3 perfectly fashioned in the holow1 within4 ,wherin1 let great diligence be vsed1 to approche 7as nere as may be7 to the ,mathematicall perfection of those figures4 ,at their bases1 let them be all open3 euery where1 els1 most close1 and iust to4 ,from the vertex1 to the ,circumference of the base of the ,cone3 to the sides of the base of the ,pyramis3 ,i4 ,d4 ,the #d4 sides of this ,pyramis must be #d4 ,isosceles ,triangles alike and quall4 ,let #d4 straight lines be drawen1 in the inside of the ,cone and ,pyramis3 makyng at their fall1 on the perimeters of the bases1 equall angles on both sides them selues1 with the sayd perimeters4 ,these #d4 lines 7in the ,pyramis3 and as many1 in the ,cone7 diuide3 one1 in #ab4 quall partes3 and an other1 in #bd4 an other1 in #fj1 and an other1 in #ajj4 7reckenyng vp from the vertex47 ,or vse other numbers of diuision1 as experience shall teach you4 ,i4 ,d4 99 ,in all workinges with this ,pyramis or ,cone1 ,let their ,situations be in all ,pointes and ,conditions1 alike1 or all one3 while you are about one ,worke4 ,els you will erre4 ,then1 99 set your ,cone or ,pyramis1 with the vertex downward1 perpendicularly1 in respect of the ,base4 7,though it be otherwayes1 it hindreth nothyng47 ,so let th most stedily be stayed4 ,now1 if there be a ,cube1 which you wold haue ,dubbled4 ,make you a prety ,cube of ,copper1 ,siluer1 ,lead1 ,tynne1 ,wood1 ,stone1 or ,bone4 ,or els make a hollow ,cube1 or ,cubik coffen1 of ,copper1 ,siluer1 ,tynne1 or ,wood c4 ,these1 you may so proporti in respect of your ,pyramis or ,cone1 that the ,pyramis or ,cone1 will be hable to conteine the waight of them1 in water1 #c4 or #d4 times3 at the least3 what stuff so euer they be made of4 ,let not your ,solid angle1 at the vertex1 be to sharpe3 but that the water may come with ease1 to the very vertex1 of your hollow ,cone or ,pyramis4 ,put one of your ,solid ,cubes in a ,balance apt3 take the waight therof exactly in water4 ,powre that water1 7without losse7 into the hollow ,pyramis or ,cone1 quietly4 ,marke in your lines1 what numbers the water ,cutteth3 ,take the waight of the same ,cube againe3 in the same kinde of water1 which you had before3 ,i4 ,d4 99 ,consider well whan you must put your waters togyther3 and whan1 you must empty your first water1 out of your ,pyramis or ,cone4 ,els you will erre4 put that99 also1 into the ,pyramis or ,cone1 where you did put the first4 ,marke now againe1 in what number or place of the lines1 the water ,cutteth them4 ,two wayes you may conclude your purpose3 it is to wete1 either by numbers or lines4 ,by numbers3 as1 if you diuide the side of your ,fundamentall ,cube into so many quall partes1 as it is capable of1 conueniently1 with your ease1 and precisenes of the diuision4 ,for1 as the number of your first and lesse line 7in your hollow ,pyramis or ,cone17 is to the second or greater 7both being counted from the vertex7 so shall the number of the side of your ,fundamentall ,cube1 be to the nber belonging to the ,radicall side1 of the ,cube1 dubble to your ,fundamentall ,cube3 ,which being multiplied ,cubik wise1 will sone shew it selfe1 whether it be dubble or no1 to the ,cubik number of your ,fundamentall ,cube4 ,by lines1 thus3 ,as your lesse and first line1 7in your hollow ,pyramis or ,cone17 is to the second or greater1 so let the ,radical side of your ,fundamtall ,cube1 be to a fourth proportionall line1 by the #ab4 proposition1 of the sixth boke of ,euclide4 ,which fourth line1 shall be the ,rote ,cubik1 or ,radicall side of the ,cube1 dubble to your ,fundamentall ,cube3 which is the thing we desired4 ,god be thanked for this ,inuention1 the fruite ensuing4 ,for this1 may ,i 7with ioy7 say1 1 1 3 thanking the holy and glorious ,trinity3 hauing greater cause therto1 then 99 ,vitruuius4 ,lib4 #i4 ,cap4 #c4 99 ,archimedes had 7for finding the fraude vsed in the ,kinges ,crowne1 of ,gold73 as all men may easily ,iudge3 by the diuersitie of the frute following of the one1 and the other4 ,where ,i spake before1 of a hollow ,cubik ,coffen3 the like vse1 is of it3 and without waight4 ,thus4 ,fill it with water1 precisely full1 and poure that water into your ,pyramis or ,cone4 ,and here note the lines cutting in your ,pyramis or ,cone4 ,againe1 fill your coffen1 like as you did before4 ,put that ,water1 also1 to the first4 ,marke the second cutting of your lines4 ,now1 as you proceded before1 so must you here procede4 99 ,note4 99 ,and if the ,cube1 which you should ,double1 be neuer so great3 you haue1 thus1 the proportion 7in small7 betwene your two litle ,cubes3 ,and then1 the side1 of that great ,cube 7to be doubled7 being the third1 will haue the fourth1 found1 to it proportionall3 by the #ab4 of the sixth of ,euclide4 ,note1 as concerning the ,sphricall ,superficies of the ,water4 ,note1 that all this while1 ,i forget not my first ,proposition ,staticall1 here rehearsed3 that1 the ,superficies of the water1 is ,sphricall4 ,wherein1 vse your discretion3 to the first line1 adding a small heare breadth1 more3 and to the second1 halfe a heare breadth more1 to his length4 ,for1 you will easily perceaue1 that the difference can be no greater1 in any ,pyramis or ,cone1 of you to be handled4 ,which you shall thus trye4 ,for finding the swelling of the water aboue leuell4 8,square the ,semidiameter1 from the ,centre of the earth1 to your first ,waters ,superficies4 ,square then1 halfe the ,subtendent of that watry ,superficies 7which ,subtendent must haue the equall partes of his measure1 all one1 with those of the ,semidiameter of the earth to your watry ,superficies73 ,subtracte this square1 from the first3 ,of the residue1 take the ,rote ,square4 ,that ,rote1 ,subtracte from your first ,semidiameter of the earth to your watry ,superficies3 that1 which remaineth1 is the heith of the water1 in the middle1 aboue the leuell40 ,which1 you will finde1 to be a thing insensible4 ,and though it were greatly sensible1 99 99 ,note4 yet1 by helpe of my sixt ,theoreme vpon the last ,proposition of ,euclides twelfth booke1 noted3 you may reduce all1 to a true ,leuell4 ,but1 farther diligence1 of you is to be vsed1 against accidentall causes of the waters swelling3 as by hauing 7somwhat7 with a moyst ,sponge1 before1 made moyst your hollow ,pyramis or ,cone1 will preuent an accidentall cause of ,swelling1 c4 ,experience will teach you abundantly3 with great ease1 pleasure1 and cmoditie4 ,thus1 may you ,double the ,cube ,mechanically1 ,treble it1 and so forth1 in any proportion4 ,note this ,abridgement of ,dubbling the ,cube4 c4 ,now will ,i ,abridge your paine1 cost1 and ,care herein4 ,without all preparing of your ,fundamentall ,cubes3 you may 7alike7 worke this ,conclusion4 ,for1 that1 was rather a kinde of ,experimentall demstration1 then the shortest way3 and all1 vpon one ,mathematicall ,demonstration depending4 8,take water 7as much as conueniently will serue your turne3 as ,i warned before of your ,fundamentall ,cubes bignes7 ,way it precisely4 ,put that water1 into your ,pyramis or ,cone4 ,of the same kinde of water1 then take againe1 the same waight you had before3 put that likewise into the ,pyramis or ,cone4 ,for1 in eche time1 your marking of the lines1 how the ,water doth cut them1 shall geue you the proportion betwen the ,radicall sides1 of any two ,cubes1 wherof the one is ,double to the other3 working as before ,i haue taught you3 99 ,note4 99 sauing that for you ,fundamentall ,cube his ,radicall side3 here1 you may take a right line1 at pleasure40 ,yet farther proceding with our droppe of ,naturall truth3 ,to giue ,cubes one to the other in any proportion1 ,rationall or ,irrationall4 you may 7now7 geue ,cubes1 one to the other1 in any proporti geu3 ,rationall or ,irrationall3 on this maner4 ,make a hollow ,parallelipipedon of ,copper or ,tinne3 with one ,base wting1 or open3 as in our ,cubike ,coffen4 ,fr the bottome of that ,parallelipipedon1 raise vp1 many perpendiculars1 in euery of his fower sides4 ,now if any proportion be assigned you1 in right lines3 ,cut one of your perpendiculars 7or a line equall to it1 or lesse then it7 likewise3 by the #aj4 of the sixth of ,euclide4 ,and those two partes1 set in two sundry lines of those perpendiculars 7or you may set them both1 in one line7 making their beginninges1 to be1 at the base3 and so their lengthes to extend vpward4 ,now1 set your hollow ,parallelipipedon1 vpright1 perpendicularly1 steadie4 ,poure in water1 handsomly1 to the heith of your shorter line4 ,poure that water1 into the hollow ,pyramis or ,cone4 ,marke the place of the rising4 ,settle your hollow ,parallelipipedon againe4 ,poure water into it3 vnto the heith of the second line1 exactly4 99 ,emptying the first4 ,poure that water 99 duely into the hollow ,pyramis or ,cone3 ,marke now againe1 where the water cutteth the same line which you marked before4 ,for1 there1 as the first marked line1 is to the second3 ,so shall the two ,radicall sides be1 one to the other1 of any two ,cubes3 which1 in their ,soliditie1 shall haue the same proportion1 which was at the first assigned3 were it ,rationall or ,irrationall4 ,thus1 in sundry waies you may furnishe your selfe with such straunge and profitable matter3 which1 long hath bene wished for4 ,and though it be ,naturally done and ,mechanically3 yet hath it a good ,demonstration ,mathematicall4 ,the demonstrations of this ,dubbling of the ,cube1 and of the rest4 ,which is this3 ,alwaies1 you haue two ,like ,pyramids3 or two ,like ,cones1 in the proportions assigned3 and like ,pyramids or ,cones1 are in proportion1 one to the other1 in the proportion of their ,homologall sides 7or lines7 tripled4 ,wherefore1 if to the first1 and second lines1 found in your hollow ,pyramis or ,cone1 you ioyne a third and a fourth1 in continuall proportion3 that fourth line1 shall be to the first1 as the greater ,pyramis or ,cone1 is to the lesse3 by the #cc4 of the eleuenth of ,euclide4 ,if ,pyramis to ,pyramis1 or ,cone to ,cone1 be double1 ,i4 ,d4 99 ,hereby1 helpe your self to become a prcise practiser4 ,and so consider1 how1 nothing at all1 you are hindred 7sensibly7 by the ,conuexitie of the water4 then shall 99 ,line to ,line1 be also double1 c4 ,but1 as our first line1 is to the second1 so is the ,radicall side of our ,fundamentall ,cube1 to the ,radicall side of the ,cube to be made1 or to be doubled3 and therefore1 to those twaine also1 a third and a fourth line1 in continuall proportion1 ioyned3 will geue the fourth line in that proportion to the first1 as our fourth ,pyramidall1 or ,conike line1 was to his first3 but that was double1 or treble1 c4 as the ,pyramids or ,cones were1 one to an other 7as we haue proued7 therfore1 this fourth1 shalbe also double or treble to the first1 as the ,pyramids or ,cones were one to an other3 ,but our made ,cube1 is described of the second in proportion1 of the fower proportionall lines3 99 ,by the #cc4 of the eleuenth booke of ,euclide4 therfore 99 as the fourth line1 is to the first1 so is that ,cube1 to the first ,cube3 and we haue proued the fourth line1 to be to the first1 as the ,pyramis or ,cone1 is to the ,pyramis or ,cone3 ,wherefore the ,cube is to the ,cube1 as ,pyramis is to ,pyramis1 or ,cone is to ,cone4 ,i4 ,d4 99 ,and your diligence in practise1 can so 7in waight of water7 performe it3 ,therefore1 now1 you are able to geue good reason of your whole doing4 ,but we 99 ,suppose ,pyramis to ,pyramis1 or ,cone to ,cone1 to be double or treble4 c4 ,therfore ,cube1 is to ,cube1 double1 or treble1 c4 ,which was to be demonstrated4 ,and of the ,parallelipiped1 it is euidt1 that the water ,solide ,parallelipipedons1 are one to the other1 as their heithes are1 seing they haue one base4 ,wherfore the ,pyramids or ,cones1 made of those water ,parallelipipedons1 are one to the other1 as the lines are 7one to the other7 betwene which1 our proportion was assigned4 ,but the ,cubes made of lines1 after the proporti of the ,pyramidal or ,conik homologall lines1 are one to the other1 as the ,pyramides or ,cones are1 one to the other 7as we before did proue7 therfore1 the ,cubes made1 shalbe one to the other1 as the lines assigned1 are one to the other3 ,which was to be demonstrated4 ,note4 99 ,note this ,corollary4 99 ,this1 my ,demonstrati is more generall1 then onely in ,square ,pyramis or ,cone3 ,consider well4 ,thus1 haue ,i1 both ,mathematically and ,mechanically1 ben very long in wordes3 yet 7,i trust7 nothing tedious to them1 who1 to these thinges1 are well affected4 ,and verily ,i am forced 7auoiding prolixitie7 to omit sundry such things1 easie to be practised3 which to the ,mathematicien1 would be a great ,threasure3 and to the ,mechanicien1 no small gaine4 99 ,the great ,commodities following of these new ,inuentions4 99 ,now may you1 ,betwene two lines giuen1 finde two middle proportionals1 in ,continuall proportion3 by the hollow ,parallelipipedon1 and the hollow ,pyramis1 or ,cone4 ,now1 any ,parallelipipedon rectangle being giuen3 thre right lines may be found1 proportionall in any proportion assigned1 of which1 shal be produced a ,parallelipipedon1 quall to the ,parallelipipedon giuen4 ,hereof1 ,i noted somwhat1 vpon the #cf4 proposition1 of the #aa4 boke of ,euclide4 ,now1 all those thinges1 which ,vitruuius in his ,architecture1 specified hable to be done1 by dubbling of the ,cube3 ,or1 by finding of two middle proportionall lines1 betwene two lines giuen1 may easely be performed4 ,now1 that ,probleme1 which ,i noted vnto you1 in the end of my ,addition1 vpon the #cd4 of the #aa4 boke of ,euclide1 is proued possible4 ,now1 may any regular body1 be ,transformed into an other1 c4 ,now1 any regular body3 any ,sphere1 yea any ,mixt ,solid3 and 7that more is7 ,irregular ,solides1 may be made 7in any proporti assigned7 like vnto the body1 first giuen4 ,thus1 of a ,manneken1 7as the ,dutch ,painters terme it7 in the same ,symmetrie1 may a ,giant be made3 and that1 with any gesture1 by the ,manneken vsed3 and contrarywise4 ,now1 may you1 of any ,mould1 or ,modell of a ,ship1 make one1 of the same ,mould 7in any assigned proportion7 bigger or lesser4 99 ,now1 may you1 of any 99 ,gunne1 or little peece of ordinace1 make an other1 with the same ,symmetrie 7in all pointes7 as great1 and as little1 as you will4 ,marke that3 and thinke on it4 ,infinitely1 may you apply this1 so long sought for1 and now so easily concluded3 and withall1 so willingly and frankly communicated to such1 as faithfully deale with vertuous studies4 ,such is the ,fruite of the ,mathematicall ,sciences and ,artes4 ,thus1 can the ,mathematicall minde1 deale ,speculatiuely in his own ,arte3 and by good meanes1 ,mount aboue the cloudes and sterres3 ,and thirdly1 he can1 by order1 ,descend1 to frame ,naturall thinges1 to wonderfull vses3 and when he list1 retire home into his owne ,centre3 and there1 prepare more ,meanes1 to ,ascend or ,descend by3 and1 all1 to the glory of ,god1 and our honest delectation in earth4 ,although1 the ,printer1 hath looked for this ,prface1 a day or two1 yet could ,i not bring my pen from the paper1 before ,i had giuen you comfortable warning1 and brief instructions1 of some of the ,commodities1 by ,statike1 hable to be reaped3 ,in the rest1 ,i will therfore1 be as brief1 as it is possible3 and with all1 describing them1 somwhat accordingly4 ,and that1 you shall perceiue1 by this1 which in order commeth next4 ,for1 wheras1 it is so ample and wonderfull1 that1 an whole yeare long1 one might finde fruitfull matter therin1 to speake of3 and also in practise1 is a ,threasure endeles3 yet will ,i glanse ouer it1 with wordes very few4 ,this do ,i call ,anthropographie4 ,which is an ,art restored1 and of my preferment to your ,seruice4 ,i pray you1 thinke of it1 as of one of the chief pointes1 of ,humane knowledge4 ,although it be1 but now1 first ,cfirmed1 with this new name3 yet the matter1 hath from the beginning1 ben in consideration of all perfect ,philosophers4 ,anthropographie1 is the description of the ,number1 ,measure1 ,waight1 figure1 ,situation1 and colour of euery diuerse thing1 conteyned in the perfect body of ,,man3 with certain knowledge of the ,symmetrie1 figure1 waight1 ,characterization1 and due locall motion1 of any parcell of the sayd body1 assigned3 and of ,nbers1 to the sayd parcell appertainyng4 ,this1 is the one part of the ,definition1 mete for this place3 ,sufficient to notifie1 the particularitie1 and excellency of the ,arte3 and why it is1 here1 ascribed to the ,mathematicals4 ,yf the description of the heauenly part of the world1 had a peculier ,art1 called ,astronomie3 ,if the description of the earthly ,globe1 hath his peculier arte1 called ,geographie4 ,if the ,matching of both1 hath his peculier ,arte1 called ,cosmographie3 ,which is the ,descripti of the whole1 and vniuersall frame of the world3 ,why should not the description of ,,man is the ,lesse ,world4 him1 who is the ,lesse world3 and1 fr the beginning1 called ,microcosmus 7that is4 ,the ,lesse ,world47 ,and for whose sake1 and seruice1 all bodily creatures els1 were created3 ,who1 also1 participateth with ,spirites1 and ,angels3 and is made to the ,image and similitude of ,god3 haue his peculier ,art8 and be called the ,arte of ,artes3 rather1 then1 either to want a name1 or to haue to base and impropre a name8 ,you must of sundry professions1 borow or challenge home1 peculier partes hereof3 and farder procede3 as1 ,god1 ,nature1 ,reason and ,experience shall informe you4 ,the ,anatomistes will restore to you1 some part3 ,the ,physiognomistes1 some3 ,the ,chyromantistes some4 ,the ,metaposcopistes1 some3 ,the excellent1 ,albert ,durer1 a good part3 the ,arte of ,perspectiue1 will somwhat1 for the ,eye1 helpe forward3 ,pythagoras1 ,hipocrates1 ,plato1 ,galenus1 ,meletius1 many other 7in certaine thinges7 will be ,contributaries4 ,and farder1 the ,heauen1 the ,earth1 and all other ,creatures1 will eche shew1 and offer their ,harmonious seruice1 to fill vp1 that1 which wanteth hereof3 and with your own ,experience1 concluding3 you may ,methodically register the whole1 for the posteritie3 ,whereby1 good profe will be had1 of our ,harmonious1 and ,micro ,cosmus4 ,microcosmicall constitution4 99 ,the outward ,image1 and vew hereof3 to the ,art of ,zographie and ,painting1 to ,sculpture1 and ,architecture3 7for ,church1 ,house1 ,fort1 or ,ship7 is most necessary and profitable3 for that1 it is the chiefe base and foundation of them4 99 ,lib4 #c4 ,cap4 #a4 ,looke in 99 ,vitruuius1 whether ,i deale sincerely for your behoufe1 or no4 ,looke in ,albertus ,durerus1 ,de ,symmetria humani ,corporis4 ,looke in the #bg4 and #bh4 ,chapters1 of the second booke1 ,de occulta ,philosophia4 ,consider the ,arke of ,noe4 ,and by that1 wade farther4 ,remember the ,delphicall ,oracle ,,nosce ,,teipsvm 7,knowe thy selfe7 so long agoe pronounced3 of so many a ,philosopher repeated3 and of the ,wisest attempted3 ,and then1 you will perceaue1 how long agoe1 you haue bene called to the ,schole1 where this ,arte might be learned4 ,well4 ,i am nothing affrayde1 of the disdayne of some such1 as thinke ,sciences and ,artes1 to be but ,seuen4 ,perhaps1 those ,such1 may1 with ignorance1 and shame enough1 come short of them ,seuen also3 and yet neuerthelesse they can not prescribe a certaine number of ,artes3 and in eche1 certaine vnpassable boundes1 to ,god1 ,nature1 and mans ,industrie4 ,new ,artes1 dayly rise vp3 and there was no such order taken1 that1 ,all ,artes1 should in one age1 or in one land1 or of one man1 be made knowen to the world4 ,let vs embrace the giftes of ,god1 and wayes to wisedome1 in this time of grace1 from aboue1 continually bestowed on them1 who thankefully will receiue them3 ,et bonis ,omnia ,cooperabuntur in bonum4 ,trochilike1 is that ,art ,mathematicall1 which demonstrateth the properties of all ,circular motions1 ,simple and ,compounde4 ,and bycause the frute hereof1 vulgarly receiued1 is in ,wheles1 it hath the name of ,trochilike3 as a man would say1 ,whele ,art4 ,by this art1 a ,whele may be geuen which shall moue ones about1 in any tyme assigned4 ,two ,wheles may be giuen1 whose turnynges about in one and the same tyme1 7or equall tymes71 shall haue1 one to the other1 any proportion appointed4 ,by ,wheles1 may a straight line be described3 ,likewise1 a ,spirall line in plaine1 ,conicall ,section lines1 and other ,irregular lines1 at pleasure1 may be drawen4 ,these1 and such like1 are principall ,conclusions of this ,arte3 and helpe forward many pleasant and profitable ,mechanicall workes3 ,saw ,milles4 ,as ,milles1 to ,saw great and very long ,deale bordes1 no man being by4 ,such haue ,i seene in ,germany3 and in the ,citie of ,prage3 in the kingdome of ,bohemia3 ,coyning ,milles1 ,hand ,milles for ,corne grinding3 ,and all maner of ,milles1 and ,whele worke3 ,by ,winde1 ,smoke1 ,water1 ,waight1 ,spring1 ,man or ,beast1 moued4 ,take in your hand1 ,agricola ,de re ,metallica3 and then shall you 7in all ,mines7 perceaue1 how great nede is1 of ,whele worke4 ,by ,wheles1 straunge workes and incredible1 are done3 as will1 in other ,artes hereafter1 appeare4 ,a wonderfull example of farther possibilitie1 and present commoditie1 was sene in my time1 in a certaine ,instrument3 which by the ,inuenter and ,artificer 7before7 was solde for xx4 ,talentes of ,golde3 and then had 7by misfortune7 receaued some iniurie and hurt3 ,and one ,ianellus of ,cremona did mend the same1 and presented it vnto the ,emperour ,charles the fifth4 ,hieronymus ,cardanus1 can be my witnesse1 that therein1 was one ,whele1 which moued1 and that1 in such rate1 that1 in #gjjj4 yeares onely1 his owne periode should be finished4 ,a thing almost incredible3 ,but how farre1 ,i keepe me within my boundes3 very many men 7yet aliue7 can tell4 ,helicosophie1 is nere ,sister to ,trochilike3 and is1 ,an ,arte ,mathematicall1 which demonstrateth the designing of all ,spirall lines in ,plaine1 on ,cylinder1 ,cone1 ,sphre1 ,conoid1 and ,sphroid1 and their properties appertayning4 ,the vse hereof1 in ,architecture1 and diuerse ,instrumentes and ,engines1 is most necessary4 ,for1 in many thinges1 the ,skrue worketh the feate1 which1 els1 could not be performed4 ,by helpe hereof1 99 ,atheneus ,lib4 #e4 cap4 #h4 it is 99 recorded1 that1 where all the power of the ,citie of ,syracusa1 was not hable to moue a certaine ,ship 7being on ground7 mightie ,archimedes1 setting to1 his ,skruish ,engine1 caused ,hiero the king1 by him self1 at ease1 to remoue her1 as he would4 ,proclus4 ,pag4 #ah4 ,wherat1 the ,king wondring3 1 1 4 ,from this day1 forward 7said the ,king7 ,credit ought to be giuen to ,archimedes1 what soeuer he sayth4 ,pneumatithmie demonstrateth by close hollow ,geometricall ,figures1 7regular and irregular7 the straunge properties 7in motion or stay7 of the ,water1 ,ayre1 ,smoke1 and ,fire1 in theyr ctinuitie1 and as they are ioyned to the ,elementes next them4 ,this ,arte1 to the ,naturall ,philosopher1 is very proffitable3 to proue1 that ,vacuum1 or ,emptines is not in the world4 ,and that1 all ,nature1 abhorreth it so much3 that1 contrary to ordinary law1 the ,elementes will moue or stand4 ,as1 ,water to ascend3 rather then betwene him and ,ayre1 ,space or place should be left1 more then 7naturally7 that qutitie of ,ayre requireth1 or can fill4 ,againe1 ,water to hang1 and not descend3 rather then by descending1 to leaue ,emptines at his backe4 ,the like1 is of ,fire and ,ayre3 they will descend3 when1 either1 their ,ctinuitie should be dissolued3 or their next ,element forced from them4 ,and as they will not be extended1 to discontinuitie3 ,so1 will they not1 nor yet of mans force1 can be prest or pent1 in space1 not sufficient and aunswerable to their bodily substance4 ,great force and violence will they vse1 to enioy their naturall right and libertie4 ,to go to the bottom of the ,sea without daunger4 ,hereupon1 two or three men together1 by keping ,ayre vnder a great ,cauldron1 and forcyng the same downe1 orderly1 may without harme descend to the ,sea bottome3 and continue there a tyme c4 ,where1 ,note1 how the thicker ,element 7as the ,water7 giueth place to the thynner 7as1 is the ayre37 and receiueth violence of the thinner1 in maner4 c4 ,pumps and all maner of ,bellowes1 haue their ground of this ,art3 and many other straunge deuises4 ,as1 ,hydraulica1 ,organes goyng by water4 c4 ,of this ,feat1 7called commonly ,pneumatica17 goodly workes are extant1 both in ,greke1 and ,latin4 ,with old and learned ,schole men1 it is called ,scientia de pleno vacuo4 ,menadrie1 is an ,arte ,mathematicall1 which demonstrateth1 how1 aboue ,natures vertue and power simple3 ,vertue and force may be multiplied3 and so1 to direct1 to lift1 to pull to1 and to put or cast fro1 any multiplied or simple1 determined ,vertue1 ,waight or ,force3 naturally1 not1 so1 directible or moueable4 ,very much is this ,art furdred by other ,artes3 as1 in some pointes1 by ,perspectiue3 in some1 by ,statike3 in some1 by ,trochilike3 and in other1 by ,helicosophie3 and ,pneumatithmie4 ,by this ,art1 all ,cranes1 ,gybbettes1 ,ingines to lift vp1 or to force any thing1 any maner way1 are ordred3 and the certaine cause of their force1 is knowne3 ,as1 the force which one man hath with the ,duche waghen ,racke3 therwith1 to set vp agayne1 a mighty waghen laden1 being ouerthrowne4 ,the force of the ,crossebow ,racke1 is certainly1 here1 demonstrated4 ,the reason1 why one m1 doth with a leauer1 lift that1 which ,sixe men1 with their handes onely1 could not1 so easily do4 ,by this ,arte1 in our common ,cranes in ,london1 where powre is to ,crane vp1 the waight of #bjjj4 pound3 by two ,wheles more 7by good order added7 ,arte concludeth1 that there may be ,craned vp #bjjjjj4 pound waight c4 ,so well knew ,archimedes this ,arte3 that he alone1 with his deuises and engynes1 7twise or thrise7 spoyled and discomfited the whole ,army and ,hoste of the ,romaines1 besieging ,syracusa1 ,plutarchus in ,marco ,marcello4 ,marcus ,marcellus the ,consul1 being their ,generall ,capitaine4 ,synesius in ,epistolis4 ,such huge ,stones1 so many1 with such force1 and so farre1 did he with his engynes hayle among them1 out of the ,citie4 ,polybius4 ,plinius4 ,quintilianus4 ,t4 ,liuius4 ,and by ,sea likewise3 though their ,ships might come to the walls of ,syracusa1 yet hee vtterly confounded the ,romaine ,nauye4 ,what with his mighty ,stones hurlyng3 99 ,athenus4 what with ,pikes of 99 #ah fote long1 made like shaftes3 which he forced almost a quarter of a myle4 ,what1 with his catchyng hold of their ,shyps1 and hoysing them vp aboue the water1 and suddenly letting them fall into the ,sea againe3 99 ,galenus4 ,anthemius4 what with his 99 ,burning ,glasses3 by which he fired their other ,shippes a far-of3 what1 with his other pollicies1 deuises1 and engines1 he so manfully acquit him selfe3 that all the ,force1 courage1 and pollicie of the ,romaines 7for a great season7 could nothing preuaile1 for the winning of ,syracusa4 ,wherupon1 the ,romanes named ,archimedes1 ,briareus1 and ,centimanus4 ,zonaras maketh mention of one ,proclus1 who so well had perceiued ,archimedes ,arte of ,menadrie1 and had so well inuented of his owne1 that with his ,burning ,glasses1 ,burning ,glasses4 being placed vpon the walles of ,bysance1 he multiplied so the heate of the ,sunne1 and directed the beames of the same against his enemies ,nauie with such force1 and so sodeinly 7like lightening7 that he burned and destroyed both man and ship4 ,and ,dionspecifieth of ,priscus1 a ,geometricien in ,bysance1 who inuented and vsed sondry ,engins1 of ,force multiplied3 ,which was cause1 that the ,emperour ,seuerus pardoned him1 his life1 after he had wonne ,bysance3 ,bycause he honored the ,arte1 wytt1 and rare industrie of ,priscus4 ,but nothing inferior to the inuention of these engines of ,force1 was the inuention of ,gunnes4 ,gunnes4 ,which1 from an ,english man1 had the occasion and order of first inuenting3 though in an other land1 and by other men1 it was first executed4 ,and they that should see the record1 where the occasion and order generall1 of ,gunning1 is first discoursed of1 would thinke3 that1 8small thinges1 slight1 and cmon3 comming to wise mens consideration1 and industrious mens handling1 may grow to be of force incredible40 ,hypogeiodie1 is an ,arte ,mathematicall1 demonstratyng1 how1 vnder the ,sphricall ,superficies of the earth1 at any depth1 to any perpendicular line assigned 7whose distance from the perpendicular of the entrance3 and the ,azimuth1 likewise1 in respect of the said entrance1 is knowen7 certaine way may be prscribed and gone3 ,and how1 any way aboue the ,superficies of the earth designed1 may vnder earth1 at any depth limited1 be kept3 goyng alwayes1 perpendicularly1 vnder the way1 on earth designed3 ,and1 contrarywise1 ,any way1 7straight or croked17 vnder the earth1 beyng giuen3 vppon the vtface1 or ,superficies of the earth1 to ,lyne out the same3 ,so1 as1 from the ,centre of the earth1 perpendiculars drawen to the ,sphricall ,superficies of the earth1 shall precisely fall in the ,correspondent pointes of those two wayes4 ,this1 with all other ,cases and circumstances herein1 and appertenances1 this ,arte demonstrateth4 ,this ,arte1 is very ample in varietie of ,conclusions3 and very profitable sundry wayes to the ,common ,wealth4 ,the occasion of my ,inuenting this ,arte1 was at the request of two ,gentlemen1 who had a certaine worke 7of gaine7 vnder ground3 and their groundes did ioyne ouer the worke3 and by reason of the crokednes1 diuers depthes1 and heithes of the way vnder ground1 they were in doubt1 and at controuersie1 vnder whose ground1 as then1 the worke was4 ,the name onely 7before this7 was of me published1 ,de ,itinere ,subterraneo3 ,the rest1 be at ,gods will4 ,for ,pioners1 ,miners1 ,diggers for ,mettalls1 ,stone1 ,cole1 and for secrete passages vnder ground1 betwene place and place 7as this land hath diuerse7 and for other purposes1 any man may easily perceaue1 both the great fruite of this ,arte1 and also in this ,arte1 the great aide of ,geometrie4 ,hydragogie1 demonstrateth the possible leading of ,water1 by ,natures lawe1 and by artificiall helpe1 from any head 7being a ,spring1 standing1 or running ,water7 to any other place assigned4 ,long1 hath this ,arte bene in vse3 and much thereof written3 and very marueilous workes therein1 performed3 as may yet appeare1 in ,italy3 by the ,ruynes remaining of the ,aqueductes4 ,in other places1 of ,riuers leading through the ,maine land1 ,nauigable many a ,mile4 ,and in other places1 of the marueilous forcinges of ,water to ,ascend4 which all1 declare the great skill1 to be required of him1 who should in this ,arte be perfecte1 for all occasions of waters possible leading4 ,to speake of the allowance of the ,fall1 for euery hundred foote3 or of the ,ventills 7if the waters labour be farre1 and great7 ,i neede not3 ,seing1 at hand 7about vs7 many expert men can sufficiently testifie1 in effecte1 the order3 though the ,demonstration of the ,necessitie thereof1 they know not3 ,nor yet1 if they should be led1 vp and downe1 and about ,mountaines1 from the head of the ,spring3 and then1 a place being assigned3 and of them1 to be demaunded1 how low or high1 that last place is1 in respecte of the head1 from which 7so crokedly1 and vp and downe7 they be come3 ,perhaps1 they would not1 or could not1 very redily1 or nerely assoyle that question4 ,geometrie therefore1 is necessary to ,hydragogie4 ,of the sundry wayes to force water to ascend1 eyther by ,tympane1 ,kettell mills1 ,skrue1 ,ctesibike1 or such like3 in ,vitruuius1 ,agricola1 7and other17 fully1 the maner may appeare4 ,and so1 thereby1 also be most euident1 how the ,artes1 of ,pneumatithmie1 ,helicosophie1 ,statike1 ,trochilike1 and ,menadrie1 come to the furniture of this1 in ,speculation1 and to the ,commoditie of the ,common ,wealth1 in practise4 ,horometrie1 is an ,arte ,mathematicall1 which demstrateth1 how1 at all times appointed1 the precise vsuall denominati of time1 may be knowen1 for any place assigned4 ,these wordes1 are smoth and plaine easie ,englishe1 but the reach of their meaning1 is farther1 then you woulde lightly imagine4 ,some part of this ,arte1 was called in olde time1 ,gnomonice3 and of late1 ,horologiographia3 and in ,englishe1 may be termed1 ,dialling4 ,auncient is the vse1 and more auncient1 is the ,inuention4 ,the vse1 doth well appeare to haue bene 7at the least7 aboue two thousand and three hundred yeare agoe3 #d4 ,reg4 #bj4 in 99 ,king ,achaz ,diall1 then1 by the ,sunne1 shewing the distinction of time4 ,by ,sunne1 ,mone1 and ,sterres1 this ,dialling may be performed1 and the precise ,time of day or night knowen4 ,but the demonstratiue delineation of these ,dialls1 of all sortes1 requireth good skill1 both of ,astronomie1 and ,geometrie ,elementall1 ,sphricall1 ,phnomenall1 and ,conikall4 ,then1 to vse the groundes of the ,arte1 for any regular ,superficies1 in any place offred3 and 7in any possible apt position therof7 theron1 to describe 7all maner of wayes7 how1 vsuall howers1 may be 7by the ,sunnes shadow7 truely determined3 will be found no sleight ,painters worke4 ,so to ,paint1 and prescribe the ,sunnes ,motion1 to the breadth of a heare4 ,in this ,feate 7in my youth7 ,i ,inuented a way1 ,how in any ,horizontall1 ,murall1 or quinoctiall ,diall1 c4 ,at all howers 7the ,sunne shining7 the ,signe and ,degree ascendent1 may be knowen4 ,which is a thing very necessary for the ,rising of those fixed ,sterres3 whose ,operation in the ,ayre1 is of great might1 euidently4 ,i speake no further1 of the vse hereof4 ,bur forasmuch as1 ,mans affaires require knowledge of ,times ,momentes1 when1 neither ,sunne1 ,mone1 or ,sterre1 can be sene3 ,therefore1 by ,industrie ,mechanicall1 was inuented1 first1 how1 by ,water1 running orderly1 the ,time and howers might be knowen3 whereof1 the famous ,ctesibius1 was ,inuentor3 a man1 of ,vitruuius1 to the ,skie 7iustly7 extolled4 ,then1 after that1 by ,sand running1 were howers measured3 ,then1 by ,trochilike with waight3 ,and of late time1 by ,trochilike with ,spring3 without waight4 ,all these1 by ,sunne or ,sterres direction 7in certaine time7 require ouersight and reformation1 according to the heauenly quinoctiall ,motion3 besides the inqualitie of their owne ,operation4 ,there remayneth 7without parabolicall meaning herein7 among the ,philosophers1 ,a perpetuall ,motion4 a more excellent1 more commodious1 and more marueilous way1 then all these3 of hauing the motion of the ,primouant 7or first quinoctiall motion17 by ,nature and ,arte1 ,imitated3 which you shall 7by furder search in waightier studyes7 hereafter1 vnderstand more of4 ,and so1 it is tyme to finish this ,annotation1 of ,tymes distinction1 vsed in our common1 and priuate affaires3 ,the commoditie wherof1 no man would want1 that can tell1 how to bestow his tyme4 ,zographie1 is an ,arte ,mathematicall1 which teacheth and demonstrateth1 how1 the ,intersection of all visuall ,pyramides1 made by any playne assigned1 7the ,centre1 distance1 and lightes1 beyng determined7 may be1 by lynes1 and due propre colours1 represented4 ,a notable ,arte1 is this3 and would require a whole ,volume1 to declare the property thereof3 and the ,commodities ensuyng4 ,great skill of ,geometrie1 ,arithmetike1 ,perspectiue1 and ,anthropographie1 with many other particular ,artes1 hath the ,zographer1 nede of1 for his perfection4 ,for1 the most excellent ,painter1 7who is but the propre ,mechanicien1 ,imitator sensible1 of the ,zographer7 hath atteined to such perfection1 that ,sense of ,man and beast1 haue iudged thinges painted1 to be things naturall1 and not artificiall3 aliue1 and not dead4 ,this ,mechanicall ,zographer 7commonly called the ,painter7 is meruailous in his skill3 and seemeth to haue a certaine diuine power3 ,as1 of frendes absent1 to make a frendly1 present comfort3 yea1 and of frendes dead1 to giue a continuall1 silent presence3 not onely with vs1 but with our posteritie1 for many ,ages4 ,and so procedyng1 ,consider1 ,how1 in ,winter1 he can shew you1 the liuely vew of ,sommers ,ioy1 and riches3 and in ,sommer1 exhibite the countenance of ,winters dolefull ,state1 and nakednes4 ,cities1 ,townes1 ,fortes1 ,woodes1 ,armyes1 yea whole ,kingdomes 7be they neuer so farre1 or greate7 can he1 with ease1 bring with him1 home 7to any mans ,iudgement7 as ,paternes liuely1 of the thinges rehearsed4 ,in one little house1 can he1 enclose 7with great pleasure of the beholders17 the portrayture liuely1 of all visible ,creatures1 either on earth1 or in the earth1 liuing3 or in the waters lying1 ,creping1 slyding1 or swimming3 or of any foule1 or fly1 in the ayre flying4 ,nay1 in respect of the ,starres1 the ,skie1 the ,cloudes3 yea1 in the shew of the very light it selfe 7that ,diuine ,creature7 can he match our eyes ,iudgement1 most nerely4 ,what a thing is this8 thinges not yet being1 he can represent so1 as1 at their being1 the ,picture shall seame 7in maner7 to haue ,created them4 ,to what ,artificer1 is not ,picture1 a great pleasure and ,commoditie8 ,which of them all1 will refuse the ,direction and ayde of ,picture8 ,the ,architect1 the ,goldsmith1 and the ,arras ,weauer3 of ,picture1 make great account4 ,our liuely ,herbals1 our portraitures of birdes1 beastes1 and fishes3 and our curious ,anatomies1 which way1 are they most perfectly made1 or with most pleasure1 of vs beholden8 ,is it not1 by ,picture onely8 ,and if ,picture1 by the ,industry of the ,painter1 be thus commodious and meruailous3 what shall be thought of ,zographie1 the ,scholemaster of ,picture1 and chief gouernor8 ,though ,i mencion not ,sculpture1 in my ,table of ,artes ,mathematicall3 yet may all men perceiue1 ,how1 that ,picture and ,sculpture1 are ,sisters germaine3 and both1 right profitable1 in a ,comm wealth4 and of ,sculpture1 aswell as of ,picture1 excellent ,artificers haue written great bokes in commendation4 ,witnesse ,i take1 of ,georgio ,vasari1 ,pittore ,aretino3 of ,pomponius ,gauricus3 and other4 ,to these two ,artes1 7with other17 is a certaine od ,arte1 called ,althalmasat1 much beholdyng3 more1 then the common ,sculptor1 ,entayler1 ,keruer1 ,cutter1 ,grauer1 ,founder1 or ,paynter 7c7 know their ,arte1 to be commodious4 ,an objection4 ,architecture1 to many may seme not worthy1 or not mete1 to be reckned among the ,artes ,mathematicall4 ,to whom1 ,i thinke good1 to giue some account of my so doyng4 ,not worthy1 7will they say17 bycause it is but for building1 of a house1 ,pallace1 ,church1 ,forte1 or such like1 grosse workes4 ,and you1 also1 defined the ,artes ,mathematicall1 to be such1 as dealed with no ,materiall or corruptible thing3 and also did demonstratiuely procede in their faculty1 by ,number or ,magnitude4 ,first1 ,the ,answer4 you see1 that ,i count1 here1 ,architecture1 among those ,artes ,mathematicall1 which are ,deriued from the ,principals3 and you know1 that such1 may deale with ,naturall thinges1 and sensible matter4 ,of which1 8some draw nerer1 to the ,simple and absolute ,mathematicall ,speculation1 then other do4 ,and though1 the ,architect procureth1 enformeth1 directeth1 the ,mechanicien1 to handworke1 the building actuall1 of house1 ,castell1 or ,pallace1 and is chief ,iudge of the same3 yet1 with him selfe 7as chief ,master and ,architect17 remaineth the ,demonstratiue reason and cause1 of the ,mechaniciens worke3 in ,lyne1 plaine1 and ,solid3 by ,geometricall1 ,arithmeticall1 ,opticall1 ,musicall1 ,astronomicall1 ,cosmographicall0 7 to be brief7 by all the former ,deriued ,artes ,mathematicall1 and other ,naturall ,artes1 hable to be confirmed and stablished4 ,if this be so3 then1 may you thinke1 that ,architecture1 hath good and due allowance1 in this honest ,company of ,artes ,mathematicall ,deriuatiue4 ,i will1 herein1 craue ,iudgement of two most perfect ,architectes3 the one1 being ,vitruuius1 the ,romaine3 who did write ten bookes thereof1 to the ,emperour ,augustus 7in whose daies our ,heauenly ,archemaster1 was borne73 and the other1 ,leo ,baptista ,albertus1 a ,florentine3 who also published ten bookes therof4 ,architectura 7sayth ,vitruuius7 est ,scientia pluribus disciplinis varijs eruditionibus ornata3 cuius ,iudicio probantur omnia1 qu ab cteris ,artificibus perficiuntur opera4 ,that is4 ,architecture1 is a ,science garnished with many doctrines diuerse instructions3 by whose ,iudgement1 all workes1 by other workmen finished1 are ,iudged4 ,it followeth4 ,ea nascitur ex ,fabrica1 ,ratiocinatione4 c4 ,ratiocinatio autem est1 qu1 res fabricatas1 ,solertia ac ratione proportionis1 demonstrare atque explicare potest4 ,architecture1 groweth of ,framing1 and ,reasoning4 c4 ,reasoning1 is that1 which of thinges framed1 with forecast1 and proportion3 can make demonstration1 and manifest declaration4 ,againe4 ,cm1 in omnibus enim rebus1 tm maxim etiam in ,architectura1 hc duo insunt3 quod significatur1 quod significat4 ,significatur proposita res1 de qua dicitur3 hanc autem ,significat ,demonstratio1 rationibus doctrinarum explicata4 ,forasmuch as1 in all thinges3 therefore chiefly in ,architecture1 these two thinges are3 the thing signified3 and that which signifieth4 ,the thing propounded1 whereof we speake1 is the thing ,signified4 ,but ,demonstration1 expressed with the reasons of diuerse doctrines1 doth signifie the same thing4 ,after that4 ,vt literatus sit1 peritus ,graphidos1 eruditus ,geometri1 ,optices non ignarus3 instructus ,arithmetica3 historias complures nouerit1 ,philosophos diligenter audiuerit3 ,musicam sciuerit3 ,medicin non sit ignarus1 responsa ,iurisperitor nouerit3 ,astrologiam1 ,clique rationes cognitas habeat4 ,an ,architect 7sayth he7 ought to vnderstand ,languages1 to be skilfull of ,painting1 well instructed in ,geometrie1 not ignorant of ,perspectiue1 furnished with ,arithmetike1 haue knowledge of many histories1 and diligently haue heard ,philosophers1 haue skill of ,musike1 not ignorant of ,physike1 know the aunsweres of ,lawyers1 and haue ,astronomie1 and the courses ,clestiall1 in good knowledge4 ,he geueth reason1 orderly1 wherefore all these ,artes1 ,doctrines1 and ,instructions1 are requisite in an excellent ,architect4 ,and 7for breuitie7 omitting the ,latin text1 thus he hath4 ,secondly1 it is behofefull for an ,architect to haue the knowledge of ,painting3 that he may the more easilie fashion out1 in patternes painted1 the forme of what worke he liketh4 ,and ,geometrie1 geueth to ,architecture many helpes3 and first teacheth the ,vse of the ,rule1 and the ,cumpasse3 wherby 7chiefly and easilie7 the descriptions of ,buildinges1 are despatched in ,groundplats3 and the directions of ,squires1 ,leuells1 and ,lines4 ,likewise1 by ,perspectiue1 the ,lightes of the heauen1 are well led1 in the buildinges3 from certaine quarters of the world4 ,by ,arithmetike1 the charges of ,buildinges are summed together3 the measures are expressed1 and the hard questions of ,symmetries1 are by ,geometricall ,meanes and ,methods discoursed on4 c4 ,besides this1 of the ,nature of thinges 7which in ,greke is called 7 ,philosophie doth make declaration4 ,which1 it is necessary1 for an ,architect1 with diligence to haue learned3 because it hath many and diuers naturall questions3 as specially1 in ,aqueductes4 ,for in their courses1 leadinges about1 in the leuell ground1 and in the mountinges1 the naturall ,spirites or breathes are ingendred diuers wayes3 ,the hindrances1 which they cause1 no man can helpe1 but he1 which out of ,philosophie1 hath learned the originall causes of thinges4 ,likewise1 who soeuer shall read ,ctesibius1 or ,archimedes bookes1 7and of others1 who haue written such ,rules7 can not thinke1 as they do3 vnlesse he shall haue receaued of ,philosophers1 instructions in these thinges4 ,and ,musike he must nedes know3 that he may haue vnderstanding1 both of ,regular and ,mathematicall ,musike3 that he may temper well his ,balistes1 ,catapultes1 and ,scorpions4 c4 ,moreouer1 the ,brasen ,vessels1 which in ,theatres1 are placed by ,mathematicall order1 in ambries1 vnder the steppes3 and the diuersities of the soundes 7which ye ,grecians call 7 are ordred according to ,musicall ,symphonies ,harmonies3 being distributed in ye ,circuites1 by ,diatessaron1 ,diapente1 and ,diapason4 ,that the conuenient voyce1 of the players sound1 wh it came to these preparations1 made in order1 there being increased3 with yt increasing1 might come more cleare pleasant1 to ye eares of the lokers on4 c4 ,and of ,astronomie1 is know ye ,east1 ,west1 ,south1 and ,north4 ,the fashion of the heauen1 the quinox1 the ,solsticie1 and the course of the sterres4 ,which thinges1 vnleast one know3 he can not perceiue1 any thyng at all1 the reason of ,horologies4 ,seyng therfore this ample ,science1 is garnished1 beautified and stored1 with so many and sundry skils and knowledges3 ,i thinke1 that none can iustly account them selues ,architectes1 of the suddeyne4 ,but they onely1 who from their childes yeares1 ascendyng by these degrees of knowledges1 beyng fostered vp with the atteynyng of many ,languages and ,artes1 haue wonne to the high ,tabernacle of ,architecture4 c4 ,and to whom ,nature hath giuen such quicke ,circumspection1 sharpnes of witt1 and ,memorie1 that they may be very absolutely skillfull in ,geometrie1 ,astronomie1 ,musike1 and the rest of the ,artes ,mathematicall3 ,such1 surmount and passe the callyng1 and state1 of ,architectes3 ,a ,mathematicien4 and are become ,mathematiciens4 c4 ,and they are found1 seldome4 ,as1 in tymes past1 was ,aristarchus ,samius3 ,philolaus1 and ,archytas1 ,tarentynes3 ,apollonius ,pergus3 ,eratosthenes ,cyreneus3 ,archimedes1 and ,scopas1 ,syracusians4 ,who also1 left to theyr posteritie1 many ,engines and ,gnomonicall workes3 by numbers and naturall meanes1 inuented and declared4 ,thus much1 and the same wordes 7in sense7 in one onely ,chapter of this ,incparable ,architect ,vitruuius1 shall you finde4 ,and if you should1 but take his boke in your hand1 and slightly loke thorough it1 you would say straight way3 ,vitruuius4 ,this is ,geometrie1 ,arithmetike1 ,astronomie1 ,musike1 ,anthropographie1 ,hydragogie1 ,horometrie4 c4 and 7to cclude7 the ,storehouse of all workmship4 ,now1 let vs listen to our other ,iudge1 our ,florentine1 ,leo ,baptista3 and narrowly consider1 how he doth determine of ,architecture4 ,sed anteque vltra progrediar4 c4 ,but before ,i procede any further 7sayth he7 ,i thinke1 that ,i ought to expresse1 what man ,i would haue to bee allowed an ,architect4 ,for1 ,i will not bryng in place a ,carpenter3 as though you might ,compare him to the ,chief ,masters of other ,artes4 ,for the hand of the ,carpenter1 is the ,architectes ,instrument4 ,v,vho is an ,architect4 ,but ,i will appoint the ,architect to be 8that man1 who hath the skill1 7by a certaine and meruailous meanes and way17 both in minde and ,imagination to determine and also in worke to finish3 what workes so euer1 by motion of waight1 and cuppling and framyng together of bodyes1 may most aptly be ,commodious for the worthiest ,vses of ,man40 ,and that he may be able to performe these thinges1 he hath nede of atteynyng and knowledge of the best1 and most worthy thynges4 c4 ,the whole ,feate of ,architecture in buildyng1 consisteth in ,lineamentes1 and in ,framyng4 ,and the whole power and skill of ,lineamentes1 tendeth to this3 that the right and absolute way may be had1 of ,coaptyng and ioyning ,lines and angles3 by which1 the face of the buildyng or frame1 may be comprehended and concluded4 ,and it is the property of ,lineamentes1 to prescribe vnto buildynges1 and euery part of them1 an apt place1 certaine nber3 a worthy maner1 and a semely order3 that1 so1 ye whole forme and figure of the buildyng1 may rest in the very ,lineamentes4 c4 ,and we may prescribe in mynde and imagination the whole formes1 99 99 ,the ,immaterialitie of perfect ,architecture4 all material stuffe beyng secluded4 ,which point we shall atteyne1 by ,notyng and forepointyng the angles1 and lines1 by a sure and certaine direction and connexion4 ,seyng then1 these thinges1 are thus3 ,what1 ,lineament is4 ,lineamente1 shalbe the certaine and constant prescribyng1 conceiued in mynde3 made in lines and angles3 and finished with a learned minde and wyt4 8,we thanke you ,master ,baptist1 that you haue so aptly brought your ,arte1 and phrase therof1 to haue some ,mathematicall perfection3 ,note4 by certaine order1 nber1 forme1 figure1 and ,symmetrie mentall30 all naturall sensible stuffe set a part4 ,now1 then1 it is euident1 7,gentle reader7 how aptely and worthely1 ,i haue preferred ,architecture1 to be bred and fostered vp in the ,dominion of the pereles ,princesse1 ,mathematica3 and to be a naturall ,subiect of hers4 ,and the name of ,architecture1 is of the principalitie1 which this ,science hath1 aboue all other ,artes4 ,and ,plato affirmeth1 the ,architect to be ,master ouer all1 that make any worke4 ,wherupon1 he is neither ,smith1 nor ,builder3 nor1 separately1 any ,artificer3 but the ,hed1 the ,prouost1 the ,directer1 and ,iudge of all ,artificiall workes1 and all ,artificers4 ,for1 the true ,architect1 is hable to teach1 ,demonstrate1 distribute1 describe1 and ,iudge all workes wrought4 ,and he1 onely1 searcheth out the causes and reasons of all ,artificiall thynges4 ,thus excellent1 is ,architecture3 though few 7in our dayes7 atteyne thereto3 yet may not the ,arte1 be otherwise thought on1 then in very dede it is worthy4 ,nor we may not1 of auncient ,artes1 make new and imperfect ,definitions in our dayes3 for scarsitie of ,artificers3 ,no more1 than we may pynche in1 the ,definitions of ,wisedome1 or ,honestie1 or of ,frendeshyp or of ,iustice4 ,no more will ,i consent1 to ,diminish any whit1 of the perfection and dignitie1 7by iust cause7 allowed to absolute ,architecture4 ,vnder the ,direction of this ,arte1 are thre principall1 necessary ,mechanicall ,artes4 ,namely1 ,howsing1 ,fortification1 and ,naupegie4 ,howsing1 ,i vnderstand1 both for ,diuine ,seruice1 and ,mans common vsage3 publike1 and priuate4 ,of ,fortification and ,naupegie1 straunge matter might be told you3 ,but perchaunce1 some will be tyred1 with this ,bederoll1 all ready rehearsed3 and other some1 will nycely nip my grosse and homely discoursing with you3 made in post hast3 for feare you should wante this true and frendly warnyng1 and tast giuyng1 of the ,power ,mathematicall4 ,lyfe is short1 and vncertaine3 ,tymes are perilouse3 c4 ,and still the ,printer awayting1 for my pen staying3 ,all these thinges1 with farder matter of ,ingratefulnes1 giue me occasion to passe away1 to the other ,artes remainyng1 with all spede possible4 ,the ,arte of ,nauigation1 demonstrateth how1 by the shortest good way1 by the aptest ,directi1 in the shortest time1 a sufficient ,ship1 betwene any two places 7in passage ,nauigable17 assigned3 may be cducted3 and in all stormes1 naturall disturbances chauncyng1 how1 to vse the best possible meanes1 whereby to recouer the place first assigned4 ,what nede1 the ,master ,pilote1 hath of other ,artes1 here before recited1 it is easie to know3 as1 of ,hydrographie1 ,astronomie1 ,astrologie1 and ,horometrie4 ,presupposing continually1 the common ,base1 and foundacion of all3 namely ,arithmetike and ,geometrie4 ,so that1 he be hable to vnderstand1 and ,iudge his own necessary ,instrumentes1 and furniture ,necessary3 ,whether they be perfectly made or no3 and also can1 7if nede be7 make them1 hym selfe4 ,as ,quadrantes1 ,the ,astronomers ,ryng1 ,the ,astronomers staffe1 ,the ,astrolabe vniuersall4 ,an ,hydrographicall ,globe4 ,charts ,hydrographicall1 true1 7not with parallell ,meridians74 ,the ,common ,sea ,compas3 ,the ,compas of variacion3 ,the ,proportionall1 and ,paradoxall ,compasses ,anno4 #aeei4 7of me ,inuented1 for our two ,moscouy ,master ,pilotes1 at the request of the ,company7 ,clockes with spryng3 houre1 halfe houre1 and three houre ,sandglasses3 sundry other ,instrumtes3 ,and also1 be hable1 on ,globe1 or ,playne to describe the ,paradoxall ,compasse3 and duely to vse the same1 to all maner of purposes1 whereto it was inuented4 ,and also1 be hable to ,calculate the ,planetes places for all tymes4 ,moreouer1 with ,sonne ,mone or ,sterre 7or without7 be hable to define the ,longitude ,latitude of the place1 which he is in3 ,so that1 the ,longitude ,latitude of the place1 from which he sayled1 be giuen3 or by him1 be knowne4 whereto1 appertayneth expert meanes1 to be certified euer1 of the ,ships way4 c4 ,and by foreseing the ,rising1 ,settyng1 ,nonestedyng1 or ,midnightyng of certaine tempestuous fixed ,sterres3 or their ,coniunctions1 and ,anglynges with the ,planetes1 c4 he ought to haue expert coniecture of ,stormes1 ,tempestes1 and ,spoutes3 and such lyke ,meteorologicall effectes1 daungerous on ,sea4 ,for 7as ,plato sayth17 ,mutationes1 opportunitatesque temporum presentire1 non minus rei militari1 qum ,agricultur1 ,nauigationique conuenit4 ,to foresee the alterations and opportunities of tymes1 is conuenient1 no lesse to the ,art of ,warre1 then to ,husbandry and ,nauigation4 ,and besides such cunnyng meanes1 more euident tokens in ,sonne and ,mone1 ought of hym to be knowen3 such as 7the ,philosophicall ,pote7 ,virgilius teacheth1 in hys ,georgikes4 ,where he sayth1 ,sidenote3 ,georgic4 #a4 ,sol quoque exoriens quum se condet in vndas1 ,signa dabit1 ,solem certissima signa sequuntur4 c4 -------- ,nam spe videmus1 ,ipsius in vultu varios errare colores4 ,cruleus1 pluuiam denunciat1 igneus ,euros4 ,sin macul incipient rutilo immiscerier igni1 ,omnia tum pariter vento1 nimbisque videbis ,feruere3 non illa quisquam me nocte per altum ,ire1 neque a terra moueat conuellere funem4 c4 ,sol tibi signa dabit4 ,solem quis dicere falsum ,audeat8 -------- c4 ,and so of ,mone1 ,sterres1 ,water1 ,ayre1 ,fire1 ,wood1 ,stones1 ,birdes1 and ,beastes1 and of many thynges els1 a certaine ,sympathicall forewarnyng may be had3 sometymes to great pleasure and proffit1 both on ,sea and ,land4 ,sufficiently1 for my present purpose1 it doth appeare1 by the premisses1 how ,mathematicall1 the ,arte of ,nauigation1 is3 and how it nedeth and also vseth other ,mathematicall ,artes3 ,and now1 if ,i would go about to speake of the manifold ,commodities1 commyng to this ,land1 and others1 by ,shypps and ,nauigation1 you might thinke1 that ,i catch at occasions1 to vse many wordes1 where no nede is4 ,yet1 this one thyng may ,i1 7iustly7 say4 ,in ,nauigation1 none ought to haue greater care1 to be skillfull1 then our ,english ,pylotes4 ,and perchaunce1 ,some1 would more attempt3 ,and other ,some1 more willingly would be aydyng1 it they wist certainely1 ,what ,priuiledge1 ,god had endued this ,iland with1 by reason of ,situation1 most commodious for ,nauigation1 to ,places most ,famous ,riche4 ,and though1 99 ,anno4 #aefg ,s4 ,h4 ,g4 7of 99 ,late7 a young ,gentleman1 a ,courragious ,capitaine1 was in a great readynes1 with good hope1 and great causes of persuasion1 to haue ventured1 for a ,discouerye1 7either ,westerly1 by ,cape de ,paramantia3 or ,esterly1 aboue ,noua ,zemla1 and the ,cyremisses7 and was1 at the very nere tyme of ,attemptyng1 called and employed otherwise 7both then1 and since17 in great good seruice to his ,countrey1 as the ,irish ,rebels haue 99 tasted3 99 ,anno4 #aefi ,yet1 ,i say1 7though the same ,gentleman1 doo not hereafter1 deale therewith7 ,some one1 or other1 should listen to the ,matter3 and by good aduise1 and discrete ,circumspection1 by little1 and little1 wynne to the sufficient knowledge of that ,trade and ,voyage3 ,which1 now1 ,i would be sory1 7through ,carelesnesse1 want of ,skill1 and ,courrage17 should remayne ,vnknowne and vnheard of4 ,seyng1 also1 we are herein1 halfe ,challenged1 by the learned1 by halfe request1 published4 ,therof1 verely1 might grow ,commoditye1 to this ,land chiefly1 and to the rest of the ,christen ,common wealth1 farre passing all riches and worldly ,threasure4 ,thaumaturgike1 is that ,art ,mathematicall1 which giueth certaine order to make straunge workes1 of the sense to be perceiued1 and of men greatly to be wondred at4 ,by sundry meanes1 this ,wonder-worke is wrought4 ,some1 by ,pneumatithmie4 ,as the workes of ,ctesibius and ,hero1 ,some by waight4 wherof ,timus speaketh4 ,some1 by ,stringes strayned1 or ,springs1 therwith ,imitating liuely ,motions4 ,some1 by other meanes1 as the ,images of ,mercurie3 and the brasen hed1 made by ,albertus ,magnus1 which dyd seme to speake4 ,boethius was excellent in these feates4 ,to whom1 ,cassiodorus writyng1 sayth4 ,your purpose is to know profound thynges3 and to shew meruayles4 ,by the disposition of your ,arte1 ,metals do low3 ,diomedes of brasse1 doth blow a ,trumpet loude3 a brasen ,serpent hisseth3 byrdes made1 sing swetely4 ,small thynges we rehearse of you1 who can ,imitate the heauen4 c4 ,of the straunge ,selfmouyng1 which1 at ,saint ,denys1 by ,paris1 99 ,anno4 #aeea 99 ,i saw1 ones or twise 7,orontius beyng then with me1 in ,company7 it were to straunge to tell4 ,but some haue written it4 ,and yet1 7,i hope7 it is there1 of other to be sene4 ,and by ,perspectiue also straunge thinges1 are done4 ,as partly 7before7 ,i gaue you to vnderstand in ,perspectiue4 ,as1 to see in the ,ayre1 a loft1 the lyuely ,image of an other man1 either walkyng to and fro3 or standyng still4 ,likewise1 to come into an house1 and there to see the liuely shew of ,gold1 ,siluer or precious stones3 and commyng to take them in your hand1 to finde nought but ,ayre4 ,hereby1 haue some men 7in all other matters counted wise7 fouly ouershot th selues3 misdeaming of the meanes4 ,therfore sayd ,claudius ,clestinus4 ,de his qu ,mundo mirabiliter eueniunt4 cap4 #h4 ,hodie magn literatur viros magna reputationis videmus1 opera quedam quasi miranda1 supra ,natur putare3 de quibus in ,perspectiua doctus causam faciliter reddidisset4 ,that is4 ,now a dayes1 we see some men1 yea of great learnyng and reputation1 to ,iudge certain workes as meruaylous1 aboue the power of ,nature3 ,of which workes1 one that were skillfull in ,perspectiue might easely haue giuen the ,cause4 ,of ,archimedes ,sphre1 ,cicero witnesseth4 ,tusc4 #a4 ,which is very straunge to thinke on4 ,for when ,archimedes 7sayth he7 did fasten in a ,sphre1 the mouynges of the ,sonne1 ,mone1 and of the fiue other ,planets1 he did1 as the ,god1 which 7in ,timus of ,plato7 did make the world4 ,that1 one turnyng1 should rule motions most vnlike in slownes1 and swiftnes4 ,but a greater cause of meruayling we haue by ,claudianus report hereof4 ,who affirmeth this ,archimedes worke1 to haue ben of ,glasse4 ,and discourseth of it more at large3 which ,i omit4 ,the ,doue of wood1 which the ,mathematicien ,archytas did make to flye1 is by ,agellius spoken of4 ,of ,ddalus straunge ,images1 ,plato reporteth4 ,homere of ,vulcans ,selfmouers1 7by secret wheles7 leaueth in writyng4 ,aristotle1 in hys ,politikes1 of both1 maketh mention4 ,meruaylous was the workemanshyp1 of late dayes1 performed by good skill of ,trochilike4 c4 ,for in ,noremberge1 ,a flye of ,iern1 beyng let out of the ,artificers hand1 did 7as it were7 fly about by the gestes1 at the table1 and at length1 as though it were weary1 retourne to his masters hand agayne4 ,moreouer1 an ,artificiall ,egle1 was ordred1 to fly out of the same ,towne1 a mighty way1 and that a loft in the ,ayre1 toward the ,emperour comming thether3 and followed hym1 beyng come to the gate of the towne4 99 99 ,thus1 you see1 what1 ,arte ,mathematicall can performe1 when ,skill1 will1 ,industry1 and ,hability1 are duely applyed to profe4 ,a ,digression4 ,and for these1 and such like marueilous ,actes and ,feates1 ,naturally1 ,mathematically1 and ,mechanically1 wrought and contriued3 ,apologeticall4 ought any honest ,student1 and ,modest ,christian ,philosopher1 be counted1 called a ,coniurer8 ,shall the folly of ,idiotes1 and the ,mallice of the ,scornfull1 so much preuaile1 that ,he1 who seeketh no worldly gaine or glory at their handes3 ,but onely1 of ,god1 the threasor of heauenly wisedome1 knowledge of pure veritie3 ,shall he 7,i say7 in the meane space1 be robbed and spoiled of his honest name and fame8 ,he that seketh 7by ,s4 ,paules aduertisement7 in the ,creatures ,properties1 and wonderfull vertues1 to finde iuste cause1 to glorifie the ternall1 and ,almightie ,creator by3 ,shall that man1 be 7in hugger mugger7 condemned1 as a ,companion of the ,helhoundes1 and a ,caller1 and ,coniurer of wicked and damned ,spirites8 ,he that bewaileth his great want of time1 sufficient 7to his contentation7 for learning of ,godly wisdome1 and ,godly ,verities in3 and onely therin setteth all his delight3 ,will that m leese and abuse his time1 in dealing with the ,chiefe enemie of ,christ our ,redemer3 the deadly foe of all mankinde3 the subtile and impudent peruerter of ,godly ,veritie3 the ,hypocriticall ,crocodile3 the ,enuious ,basiliske1 continually desirous1 in the twinke of an eye1 to destroy all ,mankinde1 both in ,body and ,soule1 ternally8 ,surely 7for my part1 somewhat to say herein7 ,i haue not learned to make so brutish1 and so wicked a ,bargaine4 ,should ,i1 for my xx4 or xxv4 yeares ,studie3 for two or three thousand ,markes spending3 seuen or eight thousand ,miles going and trauailing1 onely for good learninges sake3 ,and that1 in all maner of wethers3 in all maner of waies and passages3 both early and late3 in daunger of violence by man3 in daunger of destruction by wilde beastes3 in hunger3 in thirst3 in perilous heates by day1 with toyle on foote3 in daungerous dampes of colde1 by night1 almost bereuing life3 7as ,god knoweth73 with lodginges1 oft times1 to small ease3 and somtime to lesse securitie4 ,and for much more 7then all this7 done suffred1 for ,learning and attaining of ,wisedome3 ,should ,i 7,i pray you7 for all this1 no otherwise1 nor more warily3 or 7by ,gods mercifulnes7 no more luckily1 haue fished1 with so large1 and costly1 a ,nette1 so long time in drawing 7and that with the helpe and aduise of ,lady ,philosophie1 ,queene ,theologie73 but at length1 to haue catched1 and drawen vp1 99 a ,frog8 99 ,a prouerb4 ,fayre fisht1 and caught a ,frog4 ,nay1 a ,deuill8 ,for1 so1 doth the ,common peuish ,pratler ,imagine and ,iangle3 ,and1 so1 doth the ,malicious skorner1 secretly wishe1 brauely and boldly face down1 behinde my backe4 ,ah1 what a miserable thing1 is this kinde of ,men8 ,how great is the blindnes boldnes1 of the ,multitude1 in thinges aboue their ,capacitie8 ,what a ,land3 what a ,people3 what ,maners3 what ,times are these8 ,are they become ,deuils1 them selues3 and1 by false witnesse bearing against their ,neighbour1 would they also1 become ,murderers8 ,doth ,god1 so long geue them respite1 to reclaime them selues in1 from this horrible slaundering of the giltlesse3 contrary to their owne ,consciences3 and yet will they not cease8 ,doth the ,innocent1 forbeare the calling of them1 ,iuridically to aunswere him1 according to the rigour of the ,lawes3 and will they despise his ,charitable pacience8 ,as they1 against him1 by name1 do forge1 fable1 rage1 and raise slaunder1 by ,worde ,print3 ,will they prouoke him1 by worde and ,print1 likewise1 to ,note their ,names to the ,world3 with their particular deuises1 fables1 beastly ,imaginations1 and vnchristen-like slaunders8 ,well3 ,well4 ,o 7you such7 my vnkinde ,countrey men4 ,o vnnaturall ,countrey men4 ,o vnthankfull ,countrey men4 ,o ,brainsicke1 ,rashe1 ,spitefull1 and ,disdainfull ,countrey men4 ,why oppresse you me1 thus violently1 with your slaundering of me3 ,contrary to ,veritie3 and contrary to your owne ,consciences8 ,and ,i1 to this hower1 neither by worde1 deede1 or thought1 haue bene1 any way1 hurtfull1 damageable1 or iniurious to you1 or yours8 ,haue ,i1 so long1 so dearly1 so farre1 so carefully1 so painfully1 so daungerously sought trauailed for the learning of ,wisedome1 atteyning of ,vertue3 ,and in the end 7in your iudgemt7 am ,i become1 worse1 then when ,i beg8 ,worse1 th a ,mad man8 ,a dangerous ,member in the ,common ,wealth3 and no ,member of the ,church of ,christ8 ,call you this1 to be ,learned8 ,call you this1 to be a ,philosopher8 and a louer of ,wisedome8 ,to forsake the straight heauenly way3 and to wallow in the broad way of damnation8 ,to forsake the light of heauenly ,wisedome3 and to lurke in the dungeon of the ,prince of darkenesse8 ,to forsake the ,veritie of ,god1 his ,creatures3 and to fawne vpon the ,impudent1 ,craftie1 ,obstinate ,lier1 and continuall disgracer of ,gods ,veritie1 to the vttermost of his power8 ,to forsake the ,life ,blisse ternall3 and to cleaue vnto the ,author of ,death euerlasting8 that ,murderous ,tyrant1 most gredily awaiting the ,pray of ,mans ,soule8 ,well3 ,i thanke ,god and our ,lorde ,iesus ,christ1 for the ,comfort which ,i haue by the ,examples of other men1 before my time3 ,to whom1 neither in godlines of life1 nor in perfection of learning1 ,i am worthy to be compared3 and yet1 they sustained the very like ,iniuries1 that ,i do3 or rather1 greater4 ,pacient ,socrates1 his ,apologie will testifie3 ,apuleius his ,apologies1 will declare the ,brutishnesse of the ,multitude4 ,ioannes ,picus1 ,earle of ,mirandula1 his ,apologie will teach you1 of the ,raging slaunder of the ,malicious ,ignorant against him4 ,ioannes ,trithemius1 his ,apologie will specifie1 how he had occasion to make publike ,protestation3 as well by reason of the ,rude ,simple3 as also1 in respect of such1 as were counted to be of the wisest sort of men4 8,many could ,i recite3 ,but ,i deferre the precise and determined handling of this matter3 being loth to detect the ,folly ,mallice of my ,natiue ,countrey men4 99 99 ,who1 so hardly1 can disgest or like any extraordinary course of ,philosophicall ,studies3 not falling within the ,cumpasse of their ,capacitie3 or where they are not made priuie of the true and secrete cause1 of such wonderfull ,philosophicall ,feates40 ,these men1 are of fower sortes1 chiefly4 ,the first1 ,i may name1 ,vaine pratling busie bodies3 ,the second1 ,fond ,frendes3 ,the third1 ,imperfectly zelous3 and the fourth1 ,malicious ,ignorant4 ,to eche of these 7briefly1 and in charitie7 ,i will say a word or two1 and so returne to my ,prface4 #a4 ,vaine pratling busie bodies1 vse your idle assemblies1 and conferences1 otherwise1 then in talke of matter1 either aboue your ,capacities1 for hardnesse3 or contrary to your ,consciences1 in ,veritie4 #b4 ,fonde ,frendes1 leaue of1 so to commend your vnacquainted frend1 vpon blinde affection3 ,as1 because he knoweth more1 then the common ,student3 that1 therfore1 he must needes be skilfull1 and a doer1 in such matter and maner1 as you terme ,coniuring4 ,weening1 thereby1 you aduaunce his fame3 and that you make other men1 great marueilers of your hap1 to haue such a learned frend4 ,cease to ascribe ,impietie1 where you pretend ,amitie4 ,for1 if your tounges were true1 then were that your frend1 ,vntrue1 both to ,god1 and his ,soueraigne4 ,such ,frendes and ,fondlinges1 ,i shake of1 and renounce you3 ,shake you of1 your ,folly4 #c4 ,imperfectly zelous1 to you1 do ,i say3 that 7perhaps7 well1 do you ,meane3 ,but farre you misse the ,marke3 ,if a ,lambe you will kill1 to feede the flocke with his bloud4 ,sheepe1 with ,lambes bloud1 haue no naturall sustenaunce3 ,no more1 is ,christes flocke1 with horrible slaunders1 duely dified4 ,nor your faire pretense1 by such rashe ragged ,rhetorike1 any whit1 well graced4 ,but such1 as so vse me1 will finde a fowle ,cracke in their ,credite4 ,speake that you know3 ,and know1 as you ought3 ,know not1 by ,heare say1 when life lieth in daunger4 ,search to the quicke1 let ,charitie be your guide4 #d4 ,malicious ,ignorant1 what shall ,i say to thee8 ,prohibe linguam tuam a malo4 ,a detractione parcite lingu4 ,cause thy toung to refraine fr euill4 ,refraine your toung from slaunder4 ,though your tounges be sharpned1 ,serpent like1 ,adders poyson lye in your lippes3 ,psal4 #adj4 yet take heede1 and thinke1 betimes1 with your selfe1 ,vir linguosus non stabilietur in terra4 ,virum violentum venabitur malum1 donec prcipitetur4 ,for1 sure ,i am1 ,quia faciet ,dominus ,iudicium afflicti3 vindictam pauperum4 ,thus1 ,i require you1 my assured frendes1 and ,countrey men 7you ,mathematiciens1 ,mechaniciens1 and ,philosophers1 ,charitable and discrete7 to deale in my behalf1 with the light vntrue tounged1 my enuious ,aduersaries1 or ,fond frends4 ,and farther1 ,i would wishe1 that at leysor1 you would consider1 how ,basilius ,magnus1 layeth ,moses and ,daniel1 before the eyes of those1 which count all such ,studies ,philosophicall 7as mine hath bene7 to be vngodly1 or vnprofitable4 ,waye well ,s4 ,stephen his witnesse of ,moses4 ,act4 #g4 ,c4 ,eruditus est ,moses omni ,sapientia gyptior3 erat potens in verbis operibus suis4 ,moses was instructed in all maner of wisedome of the gyptians3 and he was of power both in his wordes1 and workes4 ,you see this ,philosophicall ,power ,wisedome1 which ,moses had1 to be nothing misliked of the ,holy ,ghost4 ,yet ,plinius hath recorded1 ,moses to be a wicked ,magicien4 ,and that 7of force7 must be1 either for this ,philosophicall wisedome1 learned1 before his calling to the leading of the ,children of ,israel3 or for those his wonders1 wrought before ,king ,pharao1 after he had the conducting of the ,israelites4 ,as concerning the first1 you perceaue1 how ,s4 ,stephen1 at his ,martyrdome 7being full of the ,holy ,ghost7 in his ,recapitulation of the olde ,testament1 hath made mention of ,moses ,philosophie3 with good liking of it3 ,and ,basilius ,magnus also1 auoucheth it1 to haue bene to ,moses profitable 7and therefore1 ,i say1 to the ,church of ,god1 necessary74 ,but as ccerning ,moses wonders1 done before ,king ,pharao3 ,god1 him selfe1 sayd3 ,vide vt omnia ostenta1 qu posui in manu tua1 facias coram ,pharaone4 ,see that thou do all those wonders before ,pharao1 which ,i haue put in thy hand4 ,thus1 you euidently perceaue1 how rashly1 ,plinius hath slaundered ,moses1 ,lib4 #cj4 ,cap4 #a4 of vayne fraudulent ,magike1 saying3 ,est alia ,magices ,factio1 a ,mose1 ,iamne1 ,iotape1 ,iudis pendens3 sed multis millibus annorum post ,zoroastrem4 c4 #a4 ,let all such1 therefore1 who1 in ,iudgement and ,skill of ,philosophie1 are farre ,inferior to ,plinie1 8take good heede1 least they ouershoote them selues rashly10 in ,iudging of ,philosophers straunge ,actes3 and the ,meanes1 how they are done4 #b4 ,but1 much more1 ought they to beware of forging1 deuising1 and imagining monstrous feates1 and wonderfull workes1 when and where1 no such were done3 no1 not any sparke or likelihode1 of such1 as they1 without all shame1 do report4 #c4 ,and 7to conclude7 most of all1 let them be ashamed of ,man1 and afraide of the dreadfull and ,iuste ,iudge3 both ,folishly or ,maliciously to deuise3 and then1 deuilishly to father their new fond ,monsters on me3 ,innocent1 in hand and hart3 for trespacing either against the lawe of ,god1 or ,man1 in any my ,studies or ,exercises1 ,philosophicall1 or ,mathematicall3 ,as in due time1 ,i hope1 will be more manifest4 ,now end ,i1 with ,archemastrie4 ,which name1 is not so new1 as this ,arte is rare4 ,for an other ,arte1 vnder this1 a degree 7for skill and power7 hath bene indued with this ,english name before4 ,and yet1 this1 may serue for our purpose1 sufficiently1 at this present4 ,this ,arte1 teacheth to bryng to actuall experience sensible1 all worthy conclusions by all the ,artes ,mathematicall purposed1 by true ,naturall ,philosophie concluded3 both addeth to them a farder scope1 in the termes of the same ,artes1 also by hys propre ,method1 and in peculier termes1 procedeth1 with helpe of the foresayd ,artes1 to the performance of complet ,experices1 which of no particular ,art1 are hable 7,formally7 to be challenged4 ,if you remember1 how we considered ,architecture1 in respect of all common handworkes3 some light may you haue1 therby1 to vnderstand the ,souerainty and propertie of this ,science4 ,science ,i may call it1 rather1 then an ,arte3 for the excellency and ,mastershyp it hath1 ouer so many1 and so mighty ,artes and ,sciences4 ,and bycause it procedeth by ,experiences1 and searcheth forth the causes of ,conclusions1 by ,experiences3 and also putteth the ,conclusions them selues1 in ,experience1 it is named of some1 ,scientia ,experimentalis4 ,the ,experimentall ,science4 ,nicolaus ,cusanus termeth it so1 in hys ,experimentes ,statikall1 ,and an other ,philosopher1 ,r4 ,b4 of this land ,natiue 7the floure of whose worthy fame1 can neuer dye nor wither7 did write therof largely1 at the request of ,clement the sixt4 ,the ,arte carrieth with it1 a wonderfull ,credit3 ,by reason1 it certefieth1 sensibly1 fully1 and completely to the vtmost power of ,nature1 and ,arte4 ,this ,arte1 certifieth by ,experience complete and absolute3 and other ,artes1 with their ,argumentes1 and ,demonstrations1 persuade3 and in wordes1 proue very well their ,conclusions4 99 ,but wordes1 and ,argumentes1 are no sensible certifying3 nor the full and finall frute of ,sciences practisable4 ,and though some ,artes1 haue in them1 ,experiences1 yet they are not complete1 and brought to the vttermost1 they may be stretched vnto1 and applyed sensibly4 ,as for example3 the ,naturall ,philosopher disputeth and maketh goodly shew of reason3 ,and the ,astronomer1 and the ,opticall ,mechanicien1 put some thynges in ,experience3 but neither1 all1 that they may3 nor yet sufficiently1 and to the vtmost1 those1 which they do1 ,there1 then1 the ,archemaster steppeth in1 and leadeth forth on1 the ,experiences1 by order of his doctrine ,experimentall1 to the chief and finall power of ,naturall and ,mathematicall ,artes4 ,of two or three men1 in whom1 this ,description of ,archemastry was ,experimentally1 verified1 ,i haue read and hard3 and good record1 is of their such perfection4 ,so that1 this ,art1 is no fantasticall ,imagination3 as some ,sophister1 might1 ,cum suis ,insolubilibus1 make a florish3 and dassell your ,imagination3 and dash your honest desire and ,courage1 from beleuing these thinges1 so vnheard of1 so meruaylous1 of such ,importance4 ,well3 as you will4 ,i haue forewarned you4 ,i haue done the part of a frende3 ,i haue discharged my ,duety toward ,god3 for my small ,talent1 at hys most mercyfull handes receiued4 ,to this ,science1 doth the ,science ,alnirangiat1 great ,seruice4 ,muse nothyng of this name4 ,i chaunge not the name1 so vsed1 and in ,print published by other3 beyng a name1 propre to the ,science4 ,vnder this1 commeth ,ars ,sintrillia1 by ,artephius1 briefly written4 ,but the chief ,science1 of the ,archemaster1 7in this world7 as yet knowen1 is an other 7as it were7 ,,optical ,science3 wherof1 the name shall be told 7,god willyng7 when ,i shall haue some1 7more iust7 occasion1 therof1 to ,discourse4 ,here1 ,i must end1 thus abruptly 7,gentle frende1 and vnfayned louer of honest and necessary verities47 ,for1 they1 who haue 7for your sake1 and vertues cause7 requested me1 7an old forworne ,mathematicien7 to take pen in hand3 7through the confidence they reposed in my long experience3 and tryed sincerity7 for the declaryng and reportyng somewhat1 of the frute and commodity1 by the ,artes ,mathematicall1 to be atteyned vnto3 euen they1 ,sore agaynst their willes1 are forced1 for sundry causes1 to satisfie the workemans request1 in endyng forthwith3 ,he1 so feareth this1 so new an attempt1 so costly3 ,and in matter so slenderly 7hetherto7 among the common ,sorte of ,studentes1 considered or estemed4 ,and where ,i was willed1 somewhat to alledge1 why1 in our vulgare ,speche1 this part of the ,principall ,science of ,geometrie1 called ,euclides ,geometricall ,elementes1 is published1 to your handlyng3 being vnlatined people1 and not ,vniuersitie ,scholers3 ,verily1 ,i thinke it nedelesse4 #a4 ,for1 the ,honour1 and ,estimation of the ,vniuersities1 and ,graduates1 is1 hereby1 nothing diminished4 ,seing1 from1 and by their ,nurse ,children1 you receaue all this ,benefite3 how great soeuer it be4 #b4 ,neither are their ,studies1 hereby1 any whit hindred4 ,no more1 then the ,italian ,vniuersities1 as ,academia ,bononiensis1 ,ferrariensis1 ,florentina1 ,mediolanensis1 ,patauina1 ,papiensis1 ,perusina1 ,pisana1 ,romana1 ,senensis1 or any one of them1 finde them selues1 any deale1 disgraced1 or their ,studies any thing hindred1 by ,frater ,lucas de ,burgo1 or by ,nicolaus ,tartalea1 who in vulgar ,italian language1 haue published1 not onely ,euclides ,geometrie1 but of ,archimedes somewhat3 and in ,arithmetike and ,practicall ,geometrie1 very large volumes1 all in their vulgar speche4 ,nor in ,germany haue the famous ,vniuersities1 any thing bene discontent with ,albertus ,durerus1 his ,geometricall ,institutions in ,dutch3 or with ,gulielmus ,xylander1 his learned translation of the first sixe bookes of ,euclide1 out of the ,greke into the high ,dutch4 ,nor with ,gualterus ,h4 ,riffius1 his ,geometricall ,volume3 very diligently translated into the high ,dutch tounge1 and published4 ,nor yet the ,vniuersities of ,spaine1 or ,portugall1 thinke their reputation to be decayed3 or suppose any their ,studies to be hindred by the ,excellent ,p4 ,nonnius1 his ,mathematicall workes1 in vulgare speche by him put forth4 ,haue you not1 likewise1 in the ,french tounge1 the whole ,mathematicall ,quadriuie8 and yet neither ,paris1 ,orleance1 or any of the other ,vniuersities of ,fraunce1 at any time1 with the ,translaters1 or ,publishers offended3 or any mans ,studie thereby hindred8 #c4 ,and surely1 the ,common and ,vulgar ,scholer 7much more1 the ,gramarian7 before his comming to the ,vniuersitie1 shall 7or may7 be1 now 7according to ,plato his ,counsell7 sufficiently instructed in ,arithmetike and ,geometrie1 for the better and easier learning of all maner of ,philosophie1 ,academicall1 or ,peripateticall4 ,and by that meanes1 goe more cherefully1 more skilfully1 and spedily forwarde1 in his ,studies1 there to be learned4 ,and1 so1 in lesse time1 profite more1 then 7otherwise7 he should1 or could do4 #d4 ,also many good and pregnant ,englishe wittes1 of young ,gentlemen1 and of other1 who neuer intend to meddle with the profound search and ,studie of ,philosophie 7in the ,vniuersities to be learned7 may neuerthelesse1 now1 with more ease and libertie1 haue good occasion1 vertuously to occupie the sharpnesse of their wittes3 where1 els 7perchance7 otherwise1 they would in fond exercises1 spend 7or rather leese7 their time3 neither seruing ,god3 nor furdering the ,weale1 common or priuate4 #e4 ,and great ,comfort1 with good hope1 may the ,vniuersities haue1 by reason of this ,englishe ,geometrie1 and ,mathematicall ,prface1 that they 7hereafter7 shall be the more regarded1 esteemed1 and resorted vnto4 ,for1 when it shall be knowen and reported1 that of the ,mathematicall ,sciences onely1 such great ,commodities are ensuing 7as ,i haue specified73 and that in dede1 some of you vnlatined ,studentes1 can be good witnesse1 of such rare fruite by you enioyed 7thereby73 as either1 before this1 was not heard of3 or els1 not so fully credited3 8,well1 may all men coniecture1 that farre greater ayde1 and better furniture1 to winne to the ,perfection of all ,philosophie1 ,vniuersities4 may in the ,vniuersities be had3 being the ,storehouses ,threasory of all ,sciences1 and all ,artes1 necessary for the best1 and most noble ,state of ,common ,wealthes40 #f4 ,besides this1 how many a ,common ,artificer1 is there1 in these ,realmes of ,england and ,ireland1 that dealeth with ,numbers1 ,rule1 ,cumpasse3 ,who1 with their owne ,skill and experience1 already had1 will be hable 7by these good helpes and informations7 to finde out1 and deuise1 new workes1 straunge ,engines1 and ,instrumentes3 for sundry purposes in the ,common ,wealth8 or for priuate pleasure8 and for the better maintayning of their owne estate8 ,i will not 7therefore7 fight against myne owne shadowe4 ,for1 no man 7,i am sure7 will open his mouth against this ,enterprise4 ,no m 7,i say7 who either hath ,charitie toward his brother 7and would be glad of his furtherance in vertuous knowledge73 or that hath any care zeale for the bettering of the ,cmon state of this ,realme4 ,neither any1 that make accompt1 what the wiser sort of men 7,sage and ,stayed7 do thinke of them4 ,to none 7therefore7 will ,i make any ,apologie1 for a vertuous acte doing3 and for cmending1 or setting forth1 ,profitable ,artes to ,english men1 in the ,english toung4 8,but1 vnto ,god our ,creator1 let vs all be thankefull3 for that1 ,as he1 of his ,goodnes1 by his ,powre1 and in his wisedome1 hath ,created all thynges1 in ,number1 ,waight1 and ,measure3 ,so1 to vs1 of hys great ,mercy1 he hath reuealed ,meanes1 whereby1 to atteyne the sufficient and necessary knowledge of the foresayd hys three principall ,instrumentes3 ,which ,meanes1 ,i haue abundantly proued vnto you1 to be the ,sciences and ,artes ,mathematicall40 ,and though ,i haue ben pinched with straightnes of tyme3 that1 no way1 ,i could so pen downe the matter 7in my ,mynde7 as ,i determined3 hopyng of conuenient laysure3 ,yet4 if vertuous zeale1 and honest ,intent prouoke and bryng you to the readyng and examinyng of this ,compendious treatise1 ,i do not doute1 but1 as the veritie therof 7accordyng to our purpose7 will be euident vnto you3 ,so the pith and force therof1 will persuade you3 and the wonderfull frute therof1 highly pleasure you4 ,and that you may the easier perceiue1 and better remember1 the principall pointes1 whereof my ,preface treateth1 ,the ,ground platt of this ,prface in a ,table4 ,i will giue you the ,groundplatt of my whole discourse1 in a ,table annexed3 from the first to the last1 somewhat ,methodically contriued4 ,if ,hast1 hath caused my poore pen1 any where1 to stumble3 ,you will1 7,i am sure7 in part of recompence1 7for my earnest and sincere good will to pleasure you71 ,consider the rockish huge mountaines1 and the perilous vnbeaten wayes1 which 7both night and day1 for the while7 it hath toyled and labored through1 to bryng you this good ,newes1 and ,comfortable profe1 of ,vertues frute4 ,so1 ,i ,commit you vnto ,gods ,mercyfull direction1 for the rest3 hartely besechyng hym1 to prosper your ,studyes1 and honest ,intentes3 to his ,glory1 the ,commodity of our ,countrey4 ,amen4 ,written at my poore ,house ,at ,mortlake4 ,anno4 #aegj4 ,february4 #i4 ,decoration ,transcriber's ,note3 ,the 8,groundplat0 was printed in the form of a stemma1 or tree1 on an oversized fold-out page4 ,the layout was impossible to reproduce for this e-text1 so the information has been rearranged in nested-list form4 ,size markings 7see note at beginning of e-text7 are relative within each paragraph4 ,j4 ,,dee ,here haue you 7according to my promisse7 the ,groundplat of my ,,mathematicall ,prface3 annexed to ,euclide 7now first7 published in our ,englishe tounge4 ,an4 #aegj4 ,febr4 #c4 ,sciences1 and ,artes ,mathematicall1 are1 either ,principall1 which are two1 onely1 ,arithmetike4 ,simple1 ,which dealeth with ,numbers onely3 and demonstrateth all their properties and appertenances3 where1 an ,vnit1 is ,indiuisible4 ,mixt1 ,which with aide of ,geometrie principall1 demonstrateth some ,arithmeticall ,conclusion1 or ,purpose4 ,geometrie4 ,simple1 ,which dealeth with ,magnitudes1 onely3 and demonstrateth all their properties1 passions1 and appertenances3 whose ,point1 is ,indiuisible4 ,mixt1 ,which with aide of ,arithmetike principall1 demonstrateth some ,geometricall purpose1 as ,,evclides ,,elementes4 ,the vse whereof1 is either1 ,in thinges ,supernaturall1 ternall1 ,diuine3 ,by ,application1 ,ascending4 ,in thinges ,mathematicall3 without farther ,application4 ,in thinges ,naturall3 both ,substtiall1 ,accidentall1 ,visible1 ,inuisible4 c4 ,by ,application3 ,descending4 ,the like ,vses and ,applications are1 7though in a degree lower7 in the ,artes ,mathematicall ,deriuatiue4 ,deriuatiue fr the ,principalls3 of which1 some haue ,the names of the ,principalls3 as1 ,arithmetike1 vulgar3 which considereth --,arithmetike of most vsuall whole numbers3 ,and of ,fractions to them appertaining4 --,arithmetike of ,proportions4 --,arithmetike ,circular4 --,arithmetike of ,radicall ,nbers3 ,simple1 ,compound1 ,mixt3 ,and of their ,fractions4 --,arithmetike of ,cossike ,nbers3 with their ,fractions3 ,and the great ,arte of ,algiebar4 ,geometrie1 vulgar3 which teacheth ,measuring ,at hand ,all ,lengthes4--,mecometrie4 ,all ,plaines3 ,as1 ,land1 ,borde1 ,glasse1 c4--,embadometrie4 ,all ,solids3 ,as1 ,timber1 ,stone1 ,vessels1 c4--,stereometrie4 ,with distce from the thing ,measured1 as1 ,how farre1 from the ,measurer1 any thing is3 of him sene1 on ,land or ,water3 called ,apomecometrie4 ,how high or deepe1 from the leuell of the ,measurers standing1 any thing is3 ,seene of hym1 on ,land or ,water3 called ,hypsometrie4 ,how broad1 a thing is1 which is in the ,measurers view3 so it be situated on ,land or ,water3 called ,platometrie4 ,of which are growen the ,feates ,artes of ,geodesie3 more cunningly to ,measure and ,suruey ,landes1 ,woods1 ,waters4 c4 ,geographie4 ,chorographie4 ,hydrographie4 ,stratarithmetrie4 ,propre names as1 ,perspectiue1--,which demonstrateth the maners and properties of all ,radiations3 ,directe1 ,broken1 and ,reflected4 ,astronomie1--,which demonstrateth the ,distances1 ,magnitudes1 and all ,naturall motions1 ,apparences1 and ,passions1 proper to the ,planets and fixed ,starres3 for any time1 past1 present1 and to come3 in respecte of a certaine ,horizon1 or without respecte of any ,horizon4 ,musike1--,which demonstrateth by reason1 and teacheth by sense1 perfectly to iudge and order the diuersitie of ,soundes1 hie or low4 ,cosmographie1--,which1 wholy and perfectly maketh description of the ,heauenlym and also ,elementall part of the ,world3 and of these partes1 maketh homologall application1 and mutuall collation necessary4 ,astrologie1--,which reasonably demonstrateth the operations and effectes of the naturall beames of light1 and secrete ,influence of the ,planets1 and fixed ,starres1 in euery ,element and ,elementall body3 at all times1 in any ,horizon assigned4 ,statike1--,which demonstrateth the causes of heauines and lightnes of all thinges3 and of the motions and properties to heauines and lightnes belonging4 ,anthropographie1 ,which describeth the ,nber1 ,measure1 ,waight1 ,figure1 ,situation1 and colour of euery diuers thing contained in the perfecte body of ,,man3 and geueth certaine knowledge of the ,figure1 ,symmetrie1 ,waight1 ,characterization1 due ,locall motion of any percell of the said body assigned3 and of numbers to the said percell appertaining4 ,trochilike1--,which demonstrateth the properties of all ,circular motions3 ,simple and ,compound4 ,helicosophie1--,which demonstrateth the designing of all ,spirall lines3 in ,plaine1 on ,cylinder1 ,cone1 ,sphre1 ,conod1 and ,sphroid3 and their properties4 ,pneumatithmie1--,which demonstrateth by close hollow ,geometricall figures 7,regular and ,irregular7 the straunge properties 7in motion or stay7 of the ,water1 ,ayre1 ,smoke1 and ,fire1 in their ,continuitie1 and as they are ioyned to the ,elementes next them4 ,menadrie1--,which demonstrateth1 how1 aboue ,natures ,vertue1 and power simple3 ,vertue and force1 may be multiplied3 and so to directe1 to lift1 to pull to1 and to put or cast fro1 any multiplied1 or simple determined ,vertue1 ,waight1 or ,force3 naturally1 not1 so1 directible1 or moueable4 ,hypogeiodie1--,which demonstrateth1 how1 vnder the ,sphricall ,superficies of the ,earth1 at any depth1 to any perpendicular line assigned 7whose distance from the perpendicular of the entrance3 and the ,azimuth likewise1 in respecte of the sayd entrance1 is knowen7 certaine way1 may be prescribed and gone1 c4 ,hydragogie1--,which demonstrateth the possible leading of water by ,natures law1 and by artificiall helpe1 from any head 7being ,spring1 standing1 or running water7 to any other place assigned4 ,horometrie1--,which demonstrateth1 how1 at all times appointed1 the precise1 vsuall denomination of time1 may be knowen1 for any place assigned4 ,zographie1--,which demonstrateth and teacheth1 how1 the ,intersection of all visuall ,pyramids1 made by any plaine assigned 7the ,center1 distance1 and lightes being determined7 may be1 by lines1 and proper colours represented4 ,architecture1--,which is a ,science garnished with many doctrines1 and diuers ,instructions3 by whose iudgement1 all workes by other workmen finished1 are iudged4 ,nauigation1--,which demonstrateth1 how1 by the ,shortest good way1 by the aptest direction1 and in the shortest time3 a sufficient ,shippe1 betwene any two places 7in passage nauigable7 assigned1 may be conducted3 and in all stormes and naturall disturbances chauncing1 how to vse the best possible meanes1 to recouer the place first assigned4 ,thaumaturgike1--,which geueth certaine order to make straunge workes1 of the sense to be perceiued3 and of men greatly to be wondred at4 ,archemastrie1--,which teacheth to bring to actuall experience sensible1 all worthy conclusions1 by all the ,artes ,mathematicall purposed3 and by true ,naturall philosophie1 concluded3 ,and both addeth to them a farder ,scope1 in the termes of the same ,artes3 and also1 by his proper ,method1 and in peculiar termes1 procedeth1 with helpe of the forsayd ,artes1 to the performance of complete ,experiences3 which1 of no particular ,arte1 are hable 7,formally7 to be challenged4 ,imprinted by ,iohn ,day4 ,an4 #aegj4 ,feb4 #be4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,errors and ,anomalies3 ,unless otherwise noted1 spelling and punctuation are unchanged4 ,errors are listed below1 with the original form1 if changed1 shown in brackets4 ,unusual words include 8fatch0 7probably used as a variant of 8fetch07 and the mathematical terms 8sexagene0 and 8sexagesme04 ,how1 worldly goods3 how1 worldly dignitie 8o0 in second 8worldly0 invisible his most diligent hearers 7so infinitely mought hearers7 so the boundes1 and duety of an ,hydrographer ,hydographer of the ,grekes it is called ,eteromekes text unchanged3 correct form is 8,heteromekes0 accent unchanged in our worldly affaires wordly fall to worke44 ,some text readers may not display the oversized-asterisk symbol4 ,emptying the first4 ,emptyting 1 1 444 of the suddeyne snddeyne that the right and absolute way may be had he had ,georgic ,i3 ,the quoted segments1 each ending in 8c401 are #dch-dci2 #dea-deg2 #dfc-dfd4 ,additional ,notes3 ,the ,greek letter 7eta7 was consistently printed as if it were the ou-ligature 4 ,the ,latin 8-que0 was written as an abbreviation resembling 8-q204 ,it is shown here as que4 ,mathematical symbols seen in the section accompanying the diagrams could not be reproduced4 ,the following substitutions were made3 --,the curly 8,p0 used for 8,pounds0 is shown as ,p4 --,the 8potestas0 symbol1 used to represent 8x0 7the unknown71 is shown as x4 --,all roots were expressed as the 8root0 sign combined with symbols for the power of #b 7doubled for power of #d1 or fourth root7 and #c4 ,they are shown as 4 ,euclid3 ,the following ,propositions were identified by number4 #f4ab3 7,how7 to find a fourth 7line7 proportional to three given straight lines4 #aa4cd3 ,in equal parallelepipedal solids the bases are reciprocally proportional to the heights2 and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal4 #aa4cf3 ,if three straight lines are proportional1 then the parallelepipedal solid formed out of the three equals the parallelepipedal solid on the mean which is equilateral1 but equiangular with the aforesaid solid4 #ab4a3 ,similar polygons inscribed in circles are to one another as the squares on their diameters4 #ab4b3 ,circles are to one another as the squares on their diameters4 #ab4ah 78last073 ,spheres are to one another in triplicate ratio of their respective diameters4