Selected quad for the lemma: water_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
water_n	air_n	element_n	fire_n	13,062	5	7.1789	4	true

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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A00429	The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed; Elements. English	Euclid.; Dee, John, 1527-1608.; Candale, François de Foix, comte de, 1502-1594.; Billingsley, Henry, Sir, d. 1606.	1570 (1570)	STC 10560; ESTC S106699	1,020,889	884

Creatures of their Naturall Vertue being most mighty most beneficiall to all elementall Generation Corr●p●ion and the appa●●●nances● and most Harmonious in thei● Monarchie For which thinges being ●nowen and modestly vsed we might highly ●nd continually glorifie God with the princely Prophet saying The Heauens declare the Glorie of God who made the Heauēs in his wisedome who made the Sonne for to haue dominion of the day the Mone and Sterres to haue dominion of the nyght whereby Day to day ●●●●reth tal●●● and night to night declareth knowledge Prayse him all ye St●rr●s and Light. Amen IN order now foloweth of Statike somewhat to say what we meane by ●hat name● and what 〈…〉 doth on 〈◊〉 Art depend Statike is an Arte Mathematicall which demonstra●●th the causes of heauynes and lightnes of all thynges and of motions and properties to heauynes and lightnes belonging● And for asmuch as by the Bilanx or Balance as the chief sensible Instrument Experience of these demonstrations may be had we call this Art Statike that is the Experimentes of the Balance Oh that men wist what proffit all maner of wayes by this Arte might grow to the hable examiner and diligent practiser Thou onely knowest all thinges precisely O God who hast made weight and Balance thy Iudgement who hast created all thinges in Number Waight and Measure and hast wayed the mountaines and hils in a Balance who hast peysed in thy hand both Heauen and earth We therfore warned by the Sacred word to Consider thy Creatures and by that consideration to wynne a glyms as it were or shaddow of perceiuerance that thy wisedome might and goodnes is infinite and vnspeakable in thy Creatures declared And being farder aduertised by thy mercifull goodnes that three principall wayes were of the vsed in Creation of all thy Creatures namely Number Waight and Measure And for as much as of Number and Measure the two Artes auncient famous and to humaine vses most necessary are all ready sufficiently knowen and extant This third key we beseche thee through thy accustomed goodnes that it may come to the nedefull and sufficient knowledge of such thy Seruauntes as in thy workemanship would gladly finde thy true occasions purposely of the vsed whereby we should glorifie thy name and shew forth to the weaklinges in faith thy wondrous wisedome and Goodnes Amen Meruaile nothing at this pang godly frend you Gentle and zelous Student An other day perchaunce you will perceiue what occasion moued me Here as now I will giue you some ground and withall some shew of certaine commodities by this Arte arising And bycause this Arte is rare my wordes and practises might be to darke vnleast you had some light holden before the matter and that best will be in giuing you out of Archimedes demonstrations a few principal Conclusions as foloweth 1. The Superficies of euery Liquor by it selfe consistyng and in quyet is Sphaericall the centre whereof is the same which is the centre of the Earth 2. If Solide Magnitudes being of the same bignes or quātitie that any Liquor is and hauyng also the same Waight be let downe into the same Liquor they will settle downeward so that no parte of them shall be aboue the Superficies of the Liquor and yet neuertheles they will not sinke vtterly downe or drowne 3. If any Solide Magnitude beyng Lighter then a Liquor be let downe into the same Liquor it will settle downe so farre into the same Liquor that so great a quantitie of that Liquor as is the parte of the Solid Magnitude settled dow●e into the same Liquor ● is in Waight aequall to the waight of the whole Solid Magni●ude 4. Any Solide Magnitude Lighter then a Liquor forced downe into the same Liquor will moue vpward with so great a power by how much the Liquor hauyng aequall quantitie to the whole Magnitude is heauyer then the same Magnitude 5. Any Solid Magnitude heauyer then a Liquor beyng let downe into the same Liquor will sinke downe vtterly And wil be in that Liquor Lighter by so much as is the waight or heauynes of the Liquor hauing bygnes or quantitie aequall to the Solid Magnitude 6. If any Solide Magnitude Lighter then a Liquor be let downe into the same Liquor the waight of the same Magnitude will be to the Waight of the Liquor Which is aequall in quantitie to the whole Magnitude in that proportion that the parte of the Magnitude settled downe is to the whole Magnitude BY these verities great Errors may be reformed in Opinion of the Naturall Motion of thinges Light and Heauy Which errors are in Naturall Philosophie almost of all mē allowed to much trusting to Authority and false Suppositions As Of any two bodyes the heauyer to moue downward faster then the lighter This error is not first by me Noted but by one Iohn Baptist de Benedictis The chief of his propositions is this which seemeth a Paradox If there be two bodyes of one forme and of one kynde aequall in quantitie or vnaequall they will moue by aequall space in aequall tyme So that both theyr mouynges be in ayre or both in water or in any one Middle Hereupon in the feate of Gunnyng certaine good discourses otherwise may receiue great amendement and furderance In the entended purpose also allowing somwhat to the imperfection of Nature not aunswerable to the precisenes of demonstration Moreouer by the foresaid propositions wisely vsed The Ayre the water the Earth the Fire may be nerely knowen how light or heauy they are Naturally in their ●●●gned partes or in the whole And then to thinges Elementall turning your practise you may deale for the proportion of the Elementes in the thinges Compounded Then to the proportions of the Humours in Man their waightes and the waight of his bones and flesh c. Than by waight to haue consideration of the Force of man any maner of way in whole or in part Then may you of Ships water drawing diuersly in the Sea and in fresh water haue pleasant consideration and of waying vp of any thing sonken in Sea or in fresh water c. And to lift vp your head a loft by waight you may as precisely as by any instrument els measure the Diameters of Sonne and Mone c. Frende I pray you way these thinges with the iust Balance of Reason And you will finde Meruailes vpon Meruailes And esteme one Drop of Truth yea in Naturall Philosophie more worth then whole Libraries of Opinions vndemonstrated or not aunswering to Natures Law and your experience Leauing these thinges thus I will giue you two or three light practises to great purpose● and so finish my Annotation Staticall In Mathematicall matters by the Mechaniciens ayde we will behold here the Commodity of waight Make a Cube of any one Vniforme and through like heauy stuffe of the same Stuffe make a Sphaere or Globe precisely of a Diameter aequall to the Radicall side of the Cube Your stuffe may

professions borow or challenge home peculier partes hereof and farder procede as God Nature Reason and Experience shall informe you The Anatomistes will restore to you some part The Physiognomistes some The Chyromantistes some The Metaposcopistes some The excellent Albert Durer a good part the Arte of Perspectiue will somwhat for the Eye helpe forward Pythagoras Hipocrates Plato Galenus Meletius many other in certaine thinges will be Contributaries And farder the Heauen the Earth and all other Creatures will eche shew and offer their Harmonious seruice to fill vp that which wanteth hereof and with your own Experience concluding you may Methodically register the whole for the posteritie Whereby good profe will be had of our Harmonious and Microcosmicall constitution The outward Image and vew hereof to the Art of Zographie and Painting to Sculpture and Architecture for Church House Fort or Ship is most necessary and profitable for that it is the chiefe base and foundation of them Looke in Vitruuius whether I deale sincerely for your behoufe or no. Looke in Albertus Durerus De Symmetria humani Corporis Looke in the 27. and 28. Chapters of the second booke De occulta Philosophia Consider the Arke of Noe. And by that wade farther Remember the Delphicall Oracle NOSCE TEIPSVM Knowe thy selfe so long agoe pronounced of so many a Philosopher repeated and of the Wisest attempted And then you will perceaue how long agoe you haue bene called to the Schole where this Arte might be learned Well I am nothing affrayde of the disdayne of some such as thinke Sciences and Artes to be but Seuen Perhaps those Such may with ignorance and shame enough come short of them Seuen also and yet neuerthelesse they can not prescribe a certaine number of Artes and in eche certaine vnpassable boundes to God Nature and mans Industrie New Artes dayly rise vp and there was no such order taken that All Artes should in one age or in one land or of one man be made knowen to the world Let vs embrace the giftes of God and wayes to wisedome in this time of grace from aboue continually bestowed on them who thankefully will receiue them Et bonis Omnia Cooperabuntur in bonum Trochilike is that Art Mathematicall which demonstrateth the properties of all Circular motions Simple and Compounde And bycause the frute hereof vulgarly receiued is in Wheles it hath the name of Trochilike as a man would say Whele Art. By this art a Whele may be geuen which shall moue ones about in any tyme assigned Two Wheles may be giuen whose turnynges about in one and the same tyme or equall tymes shall haue one to the other any proportion appointed By Wheles may a straight line be described Likewise a Spirall line in plaine Conicall Section lines and other Irregular lines at pleasure may be drawen These and such like are principall Conclusions of this Arte and helpe forward many pleasant and profitable Mechanicall workes As Milles to Saw great and very long Deale bordes no man being by Such haue I seene in Germany and in the Citie of Prage in the kingdome of Bohemia Coyning Milles Hand Milles for Corne grinding And all maner of Milles and Whele worke By Winde Smoke Water Waight Spring Man or Beast moued Take in your hand Agricola ●ere Metallica and then shall you in all Mines perceaue how great nede is of Whele worke By Wheles straunge workes and incredible are done as will in other Artes hereafter appeare A wonderfull example of farther possibilitie and present commoditie was sene in my time in a certaine Instrument which by the Inuenter and Artificer before was solde for xx Talentes of Golde and then had by misfortune receaued some iniurie and hurt And one Ianellus of Cremona did mend the same and presented it vnto the Emperour Charles the fifth Hieronymus Cardanus can be my witnesse that therein was one Whele which moued and that in such rate that in 7000. yeares onely his owne periode should be finished A thing almost incredible But how farre I keepe me within my boundes very many men yet aliue can tell Helicosophie is nere Sister to Trochilike and is An Arte Mathematicall which demonstrateth the designing of all Spirall lines in Plaine on Cylinder Cone Sphaere Conoid and Sphaeroid and their properties appertayning The vse hereof in Architecture and diuerse Instrumentes and Engines is most necessary For in many thinges the Skrue worketh the feate which els could not be performed By helpe hereof it is recorded that where all the power of the Citie of Syracusa was not hable to moue a certaine Ship being on ground mightie Archimedes setting to his Skruish Engine caused Hiero the king by him self at ease to remoue her as he would Wherat the King wondring 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 From this day forward said the King Credit ought to be giuen to Archimedes what so●uer he sayth Pneumatithmie demonstrateth by close hollow Geometricall Figures regular and irregular the straunge properties in motion or stay of the Water Ayre Smoke and Fire in theyr cōtinuitie and as they are ioyned to the Elementes next them This Arte to the Naturall Philosopher is very proffitable to proue that Vacuum or Emptines is not in the world And that all Nature abhorreth it so much that contrary to ordinary law the Elementes will moue or stand As Water to ascend rather then betwene him and Ayre Spac● or place should be left more then naturally that quātitie of Ayre requireth or can fill Againe Water to hang and not descend rather then by descending to leaue Emptines at his backe The like is of Fire and Ayre they will descend when either their Cōtinuitie should be dissolued or their next Element forced from them And as they will not be extended to discontinuitie So will they not nor yet of mans force can be prest or pent in space not sufficient and aunswerable to their bodily substance Great force and violence will they vse to enioy their naturall right and libertie Hereupon two or three men together by keping Ayre vnder a great Cauldron and forcyng the same downe orderly may without harme descend to the Sea bottome and continue there a tyme c. Where Note how the thicker Element as the Water giueth place to the thynner as is the ayre and receiueth violence of the thinner in maner c. Pumps and all maner of Bellowes haue their ground of this Art and many other straunge deuises As Hydraulica Organes goyng by water c. Of this Feat called commonly Pneumatica goodly workes are extant both in Greke and Latin. With old and learned Schole men it is called Scientia de pleno vacuo Menadrie is an Arte Mathematicall which demonstrateth how aboue Natures vertue and power simple Vertue and force may be multiplied and so to direct to lift to pull to and to put o● cast fro any multiplied or simple determined Vertue Waight or Force naturally not so directible or

Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both Substātiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Mixt Which with aide of Geometrie principall demonstrateth some Arithmeticall Conclusion or Purpose The vse whereof is either In thinges Supernaturall ●ternall Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both Substātiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Geometrie Simple Which dealeth with Magnitudes onely and demonstrat●th all their properties passions and appertenances whose Point is Indiuisible The vse whereof is either In thinges Supernaturall ●ternall Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both Substātiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Mixt Which with aide of Arithmetike principall demonstrateth some Geometricall purpose as EVCLIDES ELEMENTES The vse whereof is either In thinges Supernaturall ●ternall Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both Substātiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Deriuatiue frō the Principalls o● which some haue The names of the Principalls as Arithmetike vulgar which considereth Arithmetike of most vsuall whole Numbers And of Fractions to them appertaining Arithmetike of Proportions Arithmetike Circular Arithmetike of Radicall Nūbers Simple Compound Mixt And of their Fractions Arithmetike of Cossike Nūbers with their Fractions And the great Arte of Algiebar Geometrie vulgar which teacheth Measuring At hand All Lengthes Mecometrie All Plaines As Land Borde Glasse c. Embadometrie All Solids As Timber Stone Vessels c. Stereometrie With distāce from the thing Measured as How farre from the Measurer any thing is of him sene on Land or Water called Apomecometrie Of which are growen the Feates Artes of Geodesie more cunningly to Measure and Suruey Landes Woods Waters c. Geographie Chorographie Hydrographie Stratarithmetrie How high or deepe from the leuell of the Measurers standing any thing is Seene of hym on Land or Water called Hypsometrie Of which are growen the Feates Artes of Geodesie more cunningly to Measure and Suruey Landes Woods Waters c. Geographie Chorographie Hydrographie Stratarithmetrie How broad a thing is which is in the Measurers vew so it be situated on Land or Water called Platometrie Of which are growen the Feates Artes of Geodesie more cunningly to Measure and Suruey Landes Woods Waters c. Geographie Chorographie Hydrographie Stratarithmetrie Propre names as Perspectiue Which demonstrateth the maners and properties of all Radiations Directe Broken and Reflected Astronomie Which demonstrateth the Distances Magnitudes and all Naturall motions Apparences and Passions proper to the Planets and fixed Starres f●r any time past pr●sent and to come in respecte of a certaine Horizon or without respecte of any Horizon Musike Which demonstrateth by reason and teacheth by sense perfectly to iudge and order the diuersitie of Soundes hi● or l●w Cosmographie Which wholy and perfectly maketh description of the Heauenly and also Elementall part of the World and of these partes maketh h●m●l●gall application and mutuall collation necessary Astrologie Which reasonably demonstrateth the operations and effectes of the naturall bea●es of light and 〈◊〉 In●luence of the Planets and fixed Starres 〈◊〉 euery Element and Elementall body at all times in any Horiz●n assigned Statike Which demonstrateth the causes of heauines and lightnes of all thinges and of the motions and properties to heauines and lightnes belonging Anthropographie Which describeth the Nūber Measure Waight Figure Situation and colour of euery diuers thing contained in the perfect● body of ●● AN and geueth certaine knowledge of the Figure Symmetri● Waight Characterization due Locall motion of any p●rcell of the sayd body assigned and of numbers to the said p●rcell appertaining Trochilike Which demonstrateth the properties of all Circular motions Simple and Compound Helicosophie Which demonstrateth the designing of all Spirall lines in Plaine on Cylinder Co●● Sph●re C●n●id and Spharo●d and their properties Pneumatithmie Which demonstrateth by close hollow Geometricall figures Regular and Irregular the straunge properties in motion or stay of the Water Ayre Smoke and Fire in their Continuiti● and as they are ioyned to the Elementes next them Menadrie Which demonstrateth how about Natures Vertue and power simple Vertue and force may be multiplied and so to directe to lif● to pull to a●d to put or cast fro any multiplied or simple determined Vertue Waight or Force naturally not so directible or moueable Hypogeiodie Which demonstrateth how vnder the Spharicall Superficie● of the E●rth at ●ny depth to any perpendicular line assigned whose distance from the perpendicular of the entrance and the Azi●uth likewise 〈◊〉 respe●●e of the sayd entrance is knowen certaine way may be prescribed and g●ne c. Hydragogie Which demonstr●teth the possible leading of water by Natures l●● and by artificiall helpe fr●● any head being Spring standing or running water to any other place assigned Horometrie Which demonstrateth how at all times appointed the precise vsuall denomination of time ●●y ●e know●n for any place assigned Zographie Which demonstrateth and teacheth how the Intersection of all vsuall 〈…〉 assigned the Center distanc● and lightes b●ing determined may be by lines and proper col●urs repre●●●● Architecture Which is a Sci●●●● gar●ished with many doctrines and 〈…〉 are iudged Nauigation Thaumaturgike Archemastrie ¶ The first booke of Euclides Elementes IN THIS FIRST BOOKE is intreated of the most simple easie and first matters and groundes of Geometry as namely of Lynes Angles Triangles Parallels Squares and Parallelogrammes First of theyr definitions showyng what they are After that it teach●th how to draw Parallel lynes and how to forme diuersly figures of three sides foure sides according to the varietie of their sides and Angles

be termed Dialling Auncient is the vse and more auncient is the Inuention The vse doth well appeare to haue bene at the least aboue two thousand and three hundred yeare agoe in King Acha● Diall then by the Sunne shewing the distinction of time By Sunne Mone and Sterres this Dialling may be performed and the precise Time of day or night knowen But the demonstratiue delineation of these Dialls of all sortes requireth good skill both of Astronomie and Geometrie Elementall Sphaericall Phaenomenall and Conikall Then to vse the groundes of the Arte for any regular Superficies in any place offred and in any possible apt position therof th●ron to describe all maner of wayes how vsuall howers may be by the Sunnes shadow truely determined will be found no sleight Painters worke So to Paint and prescribe the Sunnes Motion to the breadth of a heare In this Feate in my youth I Inuented a way How in any Horizontall Murall or AEquinoctiall Diall c. At all howers the Sunne shining the Signe and Degree ascendent may be knowen Which is a thing very necessary for the Rising of those fixed Sterres whose Operation in the Ayre is of great might euidently I speake no further of the vse hereof But forasmuch as Mans affaires require knowledge of Times Momentes when neither Sunne Mone or Sterre can be sene Therefore by Industrie Mechanicall was inuented first how by Water running orderly the Time and howers might be knowen whereof the famous Ctesibius was Inuentor a man of Vitruuius to the Skie iustly extolled Then after that by Sand running were howers measured Then by Trochilike with waight And of late time by Trochilike with Spring without waight All these by Sunne or Sterres direction in certaine time require ouersight and reformation according to the heauenly AEquinoctiall Motion besides the inaequalitie of their owne Operation There remayneth without parabolicall meaning herein among the Philosophers a more excellent more commodious and more marueilous way then all these of hauing the motion of the Primouant or first ●quinoctiall motion by Nature and Arte● Imitated which you shall by furder search in waightier studyes hereafter vnderstand more of And so it is tyme to finish this Annotation of Tymes distinction vsed in our common and priuate affaires The commoditie wherof no man would want that can tell how to bestow his tyme. Zographie is an Arte Mathematicall which teacheth and demonstrateth how the Intersection of all visuall Pyramides made by any playne assigned the Centre distance and lightes beyng determined may be by lynes and due propre colours represented A notable Arte is this and would require a whole Volume to declare the property thereof and the Commodities ensuyng Great skill of Geometrie Arithmetike Perspectiue and Anthropographie with many other particular Art●s hath the Zographer nede of for his perfection For the most excellent Painter who is but the propre Mechanicien Imitator sensible of the Zographer hath atteined to such perfection that Sense of Man and beast haue iudged thinges painted to be things naturall and not artificiall aliue and not dead This Mechanicall Zographer commonly called the Painter is meruailous in his skill and seemeth to haue a certaine diuine power As of frendes absent to make a frendly present comfort yea and of frendes dead to giue a continuall silent presence not onely with vs but with our posteritie for many Ages And so procedyng Consider How in Winter he can shew you the liuely vew of Sommers Ioy and riches and in Sommer exhibite the countenance of Winters dolefull State and nakednes Cities Townes Fortes Woodes Armyes yea whole Kingdomes be they neuer so farre or greate can he with ease bring with him home to any mans Iudgement as Paternes liuely of the thinges rehearsed In one little house can he enclose with great pleasure of the beholders the portrayture liuely of all visible Creatures either on earth or in the earth liuing or in the waters lying Creping slyding or swimming or of any ●oule or fly in the ayre flying Nay in respect of the Starres the Skie the Cloudes yea in the shew of the very light it selfe that Diuine Creature can he match our eyes Iudgement most nerely What a thing is this thinges not yet being he can represent so as at their being the Picture shall seame in maner to haue Created them To what Artificer is not Picture a great pleasure and Commoditie● Which of them all will refuse the Direction and ayde of Picture The Architect the Goldsmith and the Arras Weauer of Picture make great account Our liuely Herbals our portraitures of birdes beastes and fishes and our curious Anatomies which way are they most perfectly made or with most pleasure of vs beholden Is it not by Picture onely And if Picture by the Industry of the Painter be thus commodious and meruailous what shall be thought of Zographie the Scholemaster of Picture and chief gouernor Though I mencion not Sculpture in my Table of Artes Mathematicall yet may all men perceiue How that Picture and Sculpture are Sisters germaine and both right profitable in a Commō wealth and of Sculpture aswell as of Picture excellent Artificers haue written great bokes in commendation Witnesse I take of Georgio Vasari Pittore Aretino of Pomponius Gauricus ● and other To these two Artes with other is a certaine od Arte called Althalmasat much beholdyng more then the common Sculptor Entayler Keruer Cut●er Grauer Founder or Paynter c know their Arte to be commodious Architecture to many may seme not worthy or not mete to be reckned among the Artes Mathematicall ● To whom I thinke good to giue some account of my so doyng Not worthy will they say bycause it is but for building of a house Pallace Church Forte or such like grosse workes And you also defined the Artes Mathematicall to be such as dealed with no Materiall or corruptible thing and al●o did demonstrat●uely procede in their faculty by Number or Magnitude First you see that I count here Architecture among those Artes Mathematicall which are Deriued from the Principals and you know that such may deale with Naturall thinges and sensib●●●a●●er Of which some draw nerer to the Simple and absolute Mathematicall Speculation then other do And though the Architect procureth enformeth directeth the Mechanicien to handworke the building actuall of house Castell or Pallace and is chief Iudge of the same yet with him selfe as chief Master and Architect remaineth the Demonstratiue reason and cause of the Mechaniciens worke in Lyne plaine and Solid by Geometricall Arithmeticall Opticall Musi●all Astronomicall Cosmographicall to be brief by all the former Deriued Artes Mathematicall and other Naturall Artes hable to be confirmed and stablished If this be so●then may you thinke that Architecture hath good and due allowance in this honest Company of Artes Mathematicall Deriuatiue I will herein craue Iudgement of two most perfect Architect●s the one being Vitruuius the Romaine who did write ten

but not euery Pyramis a Tetrahedron And in dede Psellus in numbring of these fiue solides or bodies calleth a Tetrahedron a Pyramis in manifest wordes This I say might make Flussas others as I thinke it did to omitte the definition of a Tetrahedron in this place as sufficiently comprehended within the definition of a Pyramis geuen before But why then did he not count that de●inition of a Pyramis faultie for that it extendeth it selfe to large and comprehendeth vnder it a Tetrahedron which differeth from a Pyramis by that it is contayned of equall triangles as he not so aduisedly did before the definition of a Prisme 23 An Octohedron is a solide or bodily figure cōtained vnder eight equall and equilater triangles As a Cube is a solide figure contayned vnder sixe superficiall figures of foure sides or squares which are equilater equiangle and equall the one to the other so is an Octohedron a solide figure contained vnder eight triangles which are equilater and equall the one to the other As ye may in these two figures here set beholde Whereof the first is drawen according as this solide is commonly described vpon a plaine superficies The second is drawen as it is described by arte vpon a plaine to shewe bodilike And in deede although the second appeare to the eye more bodilike yet as I before noted in a Cube for the vnderstanding of diuers Propositions in these fiue bookes following is the first description of more vse yea of necessitie For without it ye can not cōceaue the draught of lines and sections in any one of the eight sides which are sometimes in the descriptions of some of those Propositions required Wherefore to the consideration of this first description imagine first that vppon the vpper face of the superficies of the parallelogramme ABCD be described a Pyramis hauing his fower triangles AFB AFC CFD and DFB equilater and equiangle and concurring in the point F. Thē cōceaue that on the lower face of the super●icies of the former parallelogramme be described an other Pyramis hauing his fower triangles AEB AEC CED DEB equilater and equiangle and concurring in the point E. For so although somewhat grosly by reason the triangles can not be described equilater you may in a plaine perceaue the forme of this solide and by that meanes conceaue any lines or sections required to be drawen in any of the sayd eight triangles which are the sides of that body 24 A Dodecahedron is a solide or bodily figure cōtained vnder twelue equall equilater and equiangle Pentagons As a Cube a Tetrahedron and an Octohedron are contayned vnder equall plaine figures a Cube vnder squares the other two vnder triangles so is this solide figure contained vnder twelue equilater equiangle and equall Pentagons or figures of fiue sides As in these two figures here set you may perceaue Of which the first which thinge also was before noted of a Cube a Tetrahedron and an Octohedron is the common description of it in a plaine the other is the description of it by arte vppon a plaine to make it to appeare somwhat bodilike The first description in deede is very obscure to conceaue but yet of necessitie it must so neyther can it otherwise be in a plaine described to vnderstād those Propositions of Euclide in these fiue bokes a following which concerne the same For in it although rudely may you see all the twelue Pentagons which should in deede be all equall equilater and equiangle And now how you may somewhat conceaue the first figure described in the plaine to be a body Imagine first the Pentagon ABCDE ●o be vpon a ground plaine superficies then imagine the Pentagon FGHKL to be on high opposite vnto the Pentagon ABCDE And betwene those two Pentagons there will be ten Pentagons pulled vp fiue frō the fiue sides of the ground Pentagon namely from the side AB the Pentagon ABONM from the side BC the Pentagon BCQPO from the side CD the Pentagon CDSRQ from the side DE the Pentagon DEVTS from the side EA the Pentagon EAMXV the other fiue Pentagons haue eche one of their sides common with one of the sides of the Pentagon FGHKL which is opposite vnto the Pentagon in the ground superficies namely these are the other fiue Pentagons FGNMX GHPON HKRQP KLRST LFXVT So here you may behold twelue Pentagons which if you imagine to be equall equilater equiangle and to be lifted vp ye shall although somewhat rudely conceaue the bodily forme of a Pentagon And some light it will geue to the vnderstanding of certaine Propositions of the fiue bookes following concerning the same 25 An Icosahedron is a solide or bodily figure contained vnder twentie equall and equilater triangles These ●iue solides now last defined namely a Cube a Tetrahedrō an Octohedron a Dodecahedron and an Icosahedrō are called regular bodies As in plaine superficieces those are called regular figures whose sides and angles are equal as are equilater triangles equilater pentagons hexagons such lyke so in solides such only are counted and called regular which are cōprehēded vnder equal playne superficieces which haue equal sides and equal angles as all these fiue foresayd haue as manifestly appeareth by their definitions which were all geuen by this proprietie of equalitie of their superficieces which haue also their sides and angles equall And in all the course of nature there are no other bodies of this condition and perfection but onely these fiue Wherfore they haue euer of the auncient Philosophers bene had in great estimation and admiration and haue bene thought worthy of much contemplacion about which they haue bestowed most diligent study and endeuour to searche out the natures properties of them They are as it were the ende and perfection of all Geometry for whose sake is written whatsoeuer is written in Geometry They were as men say first inuented by the most witty Pithagoras then afterward set forth by the diuine Plato and last of all meruelously taught and declared by the most excellent Philosopher Euclide in these bookes following and euer since wonderfully embraced of all learned Philosophers The knowledge of them containeth infinite secretes of nature Pithag●ras Timeus and Plato by them searched out the cōposition of the world with the harmony and preseruation therof and applied these ●iue solides to the simple partes therof the Pyramis or Tetrahedrō they ascribed to the ●ire for that it ascendeth vpward according to the figure of the Pyramis To the ayre they ascribed the Octohedron for that through the subtle moisture which it hath it extendeth it selfe euery way to the one side and to the other accordyng as that figure doth Vnto the water they assigned the Ikosahedron for that it is continually flowing and mouing and as it were makyng angle● 〈…〉 ●ide according to that figure And to the earth they attributed a Cube as to a thing stable● 〈◊〉 and sure as the figure

Second part of the first case The second case First part of the secōd case Second part of the secōd case Construction Two cases in this Proposition The first case The first part of the first case 〈◊〉 second 〈◊〉 of the 〈◊〉 case The second case A Corollary The first Senary by substraction Demonstration An other demonstration after Campane Diffinition of the eight irrationall line Diffinition of 〈◊〉 ●inth irrationall line An other demonstratiō after Campane Construction Demonstration Diffinition of the tenth ir●ationall line Diffinition of the eleuēth irrationall line ●●●●i●ition of the twelueth irra●ionall line Diffinition of the thirtenth and last irrationall line An Assumpt of Campane I. Dee Though Campanes lemma be true ye● the maner of demonstrating it narrowly considered is not artificiall Second Senary Demonstration leading to an impossibilitie Demonstration leading to an absurditie Construction Demonstration leading to an absurditie Demonstration leading to an absurditie Demonstratiō leading to an impossibilitie Construction Demonstration 〈◊〉 an abjurd●t●●● Sixe kindes of re●iduall lines First diffinition Second diffinition Third diffinition Fourth diffinition Fifth diffinition Sixth diffinition Third Senary Construction Demonstratio● Construction Demo●strati●● Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstratio● An other more redie way to finde out the sixe residuall lines Fourth Senary The ●irst par● of the Construction The first part of the demonstration Note AI and FK concluded rational parallelogramme Note DH and FK parallelogrammes mediall Second part of the construction Second part of the demonstration LN is the onely li●e ●hat we sought consider First part of the construction The first part of th● demonstration AI and FK concluded parallelogrāmes mediall DH EK rationall The second part of the construction The second part of the demonstration * Analytically the pro●e hereof followeth amōg● other thinges The line LN found which is the principall drift of all the former discourse The first part of the Construction The fi●st part of the demonstration Note AI and FK mediall Note DH and EK mediall Note AI incommensurable to EK Second part of the Constructiō The principall line LN foūde * Because the lines AF and ●G are proued commensurable in length * By the first o● the sixth and tenth of the tenth The first part of the construction The first part of the demonstration Note AK rational Note DK mediall AI and FK incommensurable The second part of the construction The second part of the demonstration LN the chiefe line of this theoreme founde Demonstration The line LN Demonstration The fiueth Senary These sixe propositions following are the conu●rses of the sixe former propositions Construction Demonstration * By the 20. of the tenth ** By the 21. of the tenth * By the 22. of the tenth ●F cōcluded a residual line Construction Demonstration CF concl●ded a residuall line Construction Demonstration CF concluded a residual line Construction Demonstration CF proued a residuall line CF proued a residuall line Construction Demonstration CF ●roued ● residuall The sixt Senary Construction Demonstration CD cōcluded a residuall line Note Construction Demonstration CD proued a mediall Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Seuenth Senary Constraction Demonstration Construction Demonstration Demo●stratio● Construction Demonstration on leading to an impossibilitie A Corollary The determination hath sundry partes orderly to be proued Construction Demonstration This is an Assumpt problematicall artificially vsed and demonstrated * Therfore those three lines are in continuall proportion FE concluded a residuall li●● which is s●●what prep●●icro●sly in respect o● the ●●der propounded both in the propositiō and also in the determinatiō Construction Demonstration Construction Demons●ration Here are the ●ower partes of the propositi● more orderly h●dled the● in the former demöstration Construction Demons●ration An Assumpt An other demonstratiō after Flussas Construction Demons●ration This is in a maner the conuerse of both the former propositions ioyntly Construction Demonstration Construction Demonstration Demonstration An other demonstration Demonstratiō leading to an impossibili●ie An other demonstration leading to an impossibili●ie The argument of the eleuenth booke A point the beginning of all quantitie continuall The methode vsed by Euclide in the ten ●●●mer boo●es ●irst bo●●e Second ●●o●e Third boo●e ●ourth b●o●e ●iueth bo●●e Sixth boo●e Seuenth book● ●ight boo●● Ninth booke Tenth boo●e What is entrea●ea of in the fi●e boo●es foll●wi●● 〈◊〉 ●●●ular bodies● the ●●all ende 〈…〉 o● I u●●●●es ●eome●●●all ●●ementes Co●●a●is●n ●● the 〈◊〉 ●●o●e and 〈◊〉 booke 〈◊〉 First dif●inition A solide the most perfectest quantitie No science of thinges infinite Second diffinition Third diffinition Two dif●initions included in this di●●inition Declaratiō of the first part Declaration of the second part Fourth diffinition Fifth diffinition Sixth diffinition Seuenth def●inition Eighth di●finition Ninth di●●i●ition Tenth diffinition Eleuenth diffinition An other diffinition of a prisme which is a speciall diffinition of a prisme as it is commonly called and vsed This bodie called Figura Serratilis Psellus Twelueth diffinition What is to be ta●●n heede of in the diffinition of a sphere geuen by Iohannes de Sacro Busco Theodosiu● di●●inition of a sphere The circumference of a sphere Galens diffinition 〈◊〉 a sph●r● The dig●itie of a s●here A sphere called a Globe Thirtenth diffinition Theodosius diffinition of the axe of a sphere Fourtenth diffinition Theodosius diffinition of the center of a sphere Flussas diffinition of the center of a sphere Fiuetenth diffinition Difference betwene the diameter axe of a sphere Seuententh diffinition First kinde of Cones A Cone called of Campane a ro●●de Piramis Seuententh diffinition A conicall superficies Eightenth diffinition Ninetenth diffinition A cillindricall superficies Corollary A round● Columne or sphere A Corollary added by Campane Twenty diffinition Twenty one diffinitio● Twenty two diffinition A Tetrahedron one of the fiue regular bodyes Di●●erence betwene a Tetrahedron and a Piramis Psellus calleth a Tetrahedron a Piramis Twenty three definition Twēty ●o●er definition Twenty fiue diffinition Fiue regular bodies The dignity of these bodies A Tetrahedron ascribed vnto the fire An octohedron ascribed vnto the ayre An Ikosahedron assigned vnto the water A cube assigned vnto the earth A dodecahedron assigned to heauen Diffinition of a parallelipipedon A D●d●●●●edron An Icosa●edron Demonstration leading to an impossibilitie An other demonstration after Flussas Construction Demonstration leading to an impossibilitie Demonstration leading to an impossibilitie Construction Demonstration Demonstration leading to an impossibilitie Construction * An Assumpt as M. Dee pr●ueth it Demonstration Demonstration leading to an impossibilitie This proposition is as it were the conuerse of the sixth Construction Demonstration Construction Demonstration Construction Demonstration Construction Two cases in this proposition The first case Iohn Dee * This requireth the imagination of a plaine superficies passing by the pointe A and the straight line BC. And so helpe your selfe in the lyke cases either Mathematically imagining or Mechanically practising Second