{"schema_version":1,"access":"public","rights":"Public-domain or compatibly licensed source editions; provenance is retained per record.","work":{"slug":"dee-praeface","title":"The Mathematicall Praeface to the Elements of Geometrie","author":"John Dee","tradition":"european","language":"en-16c","genre":"treatise on the mathematical sciences","period":"1570","source_url":"https://www.gutenberg.org/ebooks/22062","original":{"text":"### To the Lovers of Truth and Students of Noble Sciences\n\nTO THE VNFAINED LOVERS of truthe, and constant Studentes of Noble Sciences, IOHN DEE of London, hartily wisheth grace from heauen, and most prosperous successe in all their honest attemptes and exercises.\n\nDiuine Plato, the great Master of many worthy Philosophers, and the constant auoucher, and pithy perswader of Vnum, Bonum, and Ens: in his Schole and Academie, sundry times (besides his ordinary Scholers) was visited of a certaine kinde of men, allured by the noble fame of Plato, and the great commendation of hys profound and profitable doctrine. But when such Hearers, after long harkening to him, perceaued, that the drift of his discourses issued out, to conclude, this Vnum, Bonum, and Ens, to be Spirituall, Infinite, Æternall, Omnipotent, &c. Nothyng beyng alledged or expressed, How, worldly goods: how, worldly dignitie: how, health, Strẽgth or lustines of body: nor yet the meanes, how a merueilous sensible and bodyly blysse and felicitie hereafter, might be atteyned: Straightway, the fantasies of those hearers, were dampt: their opinion of Plato, was clene chaunged: yea his doctrine was by them despised: and his schole, no more of them visited. Which thing, his Scholer, Aristotle, narrowly cõsidering, founde the cause therof, to be, “For that they had no forwarnyng and information, in generall,” whereto his doctrine tended. For, so, might they haue had occasion, either to haue forborne his schole hauntyng: (if they, then, had misliked his Scope and purpose) or constantly to haue continued therin: to their full satisfaction: if such his finall scope & intent, had ben to their desire. Wherfore, Aristotle, euer, after that, vsed in brief, to forewarne his owne Scholers and hearers, “both of what matter, and also to what ende, he tooke in hand to speake, or teach.” While I consider the diuerse trades of these two excellent Philosophers (and am most sure, both, that Plato right well, otherwise could teach: and that Aristotle mought boldely, with his hearers, haue dealt in like sorte as Plato did) I am in no little pang of perplexitie: Bycause, that, which I mislike, is most easy for me to performe (and to haue Plato for my exãple.) And that, which I know to be most commendable: and (in this first bringyng, into common handling, the Artes Mathematicall) to be most necessary: is full of great difficultie and sundry daungers. Yet, neither do I think it mete, for so straunge matter (as now is ment to be published) and to so straunge an audience, to be bluntly, at first, put forth, without a peculiar Preface: Nor (Imitatyng Aristotle) well can I hope, that accordyng to the amplenes and dignitie of the State Mathematicall, I am able, either playnly to prescribe the materiall boundes: or precisely to expresse the chief purposes, and most wonderfull applications therof. And though I am sure, that such as did shrinke from Plato his schole, after they had perceiued his finall conclusion, would in these thinges haue ben his most diligent hearers (so infinitely mought their desires, in fine and at length, by our Artes Mathematicall be satisfied) yet, by this my Præface & forewarnyng, Aswell all such, may (to their great behofe) the soner, hither be allured: as also the Pythagoricall, and Platonicall perfect scholer, and the constant profound Philosopher, with more ease and spede, may (like the Bee,) gather, hereby, both wax and hony.\n\n(The intent of this Preface.)\n\nWherfore, seyng I finde great occasion (for the causes alleged, and farder, in respect of my Art Mathematike generall) to vse “a certaine forewarnyng and Præface, whose content shalbe, that mighty, most plesaunt, and frutefull Mathematicall Tree, with his chief armes and second (grifted) braunches: Both, what euery one is, and also, what commodity, in generall, is to be looked for, aswell of griff as stocke: And forasmuch as this enterprise is so great, that, to this our tyme, it neuer was (to my knowledge) by any achieued: And also it is most hard, in these our drery dayes, to such rare and straunge Artes, to wyn due and common credit:” Neuertheles, if, for my sincere endeuour to satisfie your honest expectation, you will but lend me your thãkefull mynde a while: and, to such matter as, for this time, my penne (with spede) is hable to deliuer, apply your eye or eare attentifely: perchaunce, at once, and for the first salutyng, this Preface you will finde a lesson long enough. And either you will, for a second (by this) be made much the apter: or shortly become, well hable your selues, of the lyons claw, to coniecture his royall symmetrie, and farder propertie. Now then, gentle, my frendes, and countrey men, Turne your eyes, and bend your myndes to that doctrine, which for our present purpose, my simple talent is hable to yeld you.\n\nAll thinges which are, & haue beyng, are found vnder a triple diuersitie generall. For, either, they are demed Supernaturall, Naturall, or, of a third being. Thinges Supernaturall, are immateriall, simple, indiuisible, incorruptible, & vnchangeable. Things Naturall, are materiall, compounded, diuisible, corruptible, and chaungeable. Thinges Supernaturall, are, of the minde onely, comprehended: Things Naturall, of the sense exterior, ar hable to be perceiued. In thinges Naturall, probabilitie and coniecture hath place: But in things Supernaturall, chief demõstration, & most sure Science is to be had. By which properties & comparasons of these two, more easily may be described, the state, condition, nature and property of those thinges, which, we before termed of a third being: which, by a peculier name also, are called Thynges Mathematicall. For, these, beyng (in a maner) middle, betwene thinges supernaturall and naturall: are not so absolute and excellent, as thinges supernatural: Nor yet so base and grosse, as things naturall: But are thinges immateriall: and neuerthelesse, by materiall things hable somewhat to be signified. And though their particular Images, by Art, are aggregable and diuisible: yet the generall Formes, notwithstandyng, are constant, vnchaungeable, vntrãsformable, and incorruptible. Neither of the sense, can they, at any tyme, be perceiued or iudged. Nor yet, for all that, in the royall mynde of man, first conceiued. But, surmountyng the imperfectiõ of coniecture, weenyng and opinion: and commyng short of high intellectuall cõceptiõ, are the Mercurial fruite of Dianœticall discourse, in perfect imagination subsistyng. A meruaylous newtralitie haue these thinges Mathematicall, and also a straunge participatiõ betwene thinges supernaturall, immortall, intellectual, simple and indiuisible: and thynges naturall, mortall, sensible, compounded and diuisible. Probabilitie and sensible prose, may well serue in thinges naturall: and is commendable: In Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor yet the testimony of sense, any whit credited: But onely a perfect demonstration, of truthes certaine, necessary, and inuincible: vniuersally and necessaryly concluded: is allowed as sufficient for “an Argument exactly and purely Mathematical.”\n\n(Note the worde, Vnit, to expresse the Greke Monas, & not Vnitie: as we haue all, commonly, till now, vsed.)\n\nOf Mathematicall thinges, are two principall kindes: namely, Number, and Magnitude.\n\n(Number.)\n\nNumber, we define, to be, a certayne Mathematicall Sũme, of Vnits. And, an Vnit, is that thing Mathematicall, Indiuisible, by participation of some likenes of whose property, any thing, which is in deede, or is counted One, may resonably be called One. We account an Vnit, a thing Mathematicall, though it be no Number, and also indiuisible: because, of it, materially, Number doth consist: which, principally, is a thing Mathematicall.\n\n(Magnitude.)\n\nMagnitude is a thing Mathematicall, by participation of some likenes of whose nature, any thing is iudged long, broade, or thicke. “A thicke Magnitude we call a Solide, or a Body. What Magnitude so euer, is Solide or Thicke, is also broade, & long. A broade magnitude, we call a Superficies or a Plaine. Euery playne magnitude, hath also length. A long magnitude, we terme a Line. A Line is neither thicke nor broade, but onely long: Euery certayne Line, hath two endes:\n\n(A point.)\n\nThe endes of a line, are Pointes called. A Point, is a thing Mathematicall, indiuisible, which may haue a certayne determined situation.” If a Poynt moue from a determined situation, the way wherein it moued, is also a Line: mathematically produced, whereupon, of the auncient Mathematiciens,\n\n(A Line.)\n\na Line is called the race or course of a Point. A Poynt we define, by the name of a thing Mathematicall: though it be no Magnitude, and indiuisible: because it is the propre ende, and bound of a Line: which is a true Magnitude.\n\n(Magnitude.)\n\nAnd Magnitude we may define to be that thing Mathematicall, which is diuisible for euer, in partes diuisible, long, broade or thicke. Therefore though a Poynt be no Magnitude, yet Terminatiuely, we recken it a thing Mathematicall (as I sayd) by reason it is properly the end, and bound of a line. Neither Number, nor Magnitude, haue any Materialitie.\n\n### Of Number and Magnitude\n\nFirst, we will consider of Number, and of the Science Mathematicall, to it appropriate, called Arithmetike: and afterward of Magnitude, and his Science, called Geometrie. But that name contenteth me not: whereof a word or two hereafter shall be sayd. How Immateriall and free from all matter, Number is, who doth not perceaue? yea, who doth not wonderfully wõder at it? For, neither pure Element, nor Aristoteles, Quinta Essentia, is hable to serue for Number, as his propre matter. Nor yet the puritie and simplenes of Substance Spirituall or Angelicall, will be found propre enough thereto. And therefore the great & godly Philosopher Anitius Boetius, sayd: Omnia quæcun[que] a primæua rerum natura constructa sunt, Numerorum videntur ratione formata. Hoc enim fuit principale in animo Conditoris Exemplar. That is: All thinges (which from the very first originall being of thinges, haue bene framed and made) do appeare to be Formed by the reason of Numbers. For this was the principall example or patterne in the minde of the Creator. O comfortable allurement, O rauishing perswasion, to deale with a Science, whose Subiect, is so Auncient, so pure, so excellent, so surmounting all creatures, so vsed of the Almighty and incomprehensible wisdome of the Creator, in the distinct creation of all creatures: in all their distinct partes, properties, natures, and vertues, by order, and most absolute number, brought, from Nothing, to the Formalitie of their being and state. By Numbers propertie therefore, of vs, by all possible meanes, (to the perfection of the Science) learned, we may both winde and draw our selues into the inward and deepe search and vew, of all creatures distinct vertues, natures, properties, and Formes: And also, farder, arise, clime, ascend, and mount vp (with Speculatiue winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all thinges Numerable: both visible and inuisible, mortall and immortall, Corporall and Spirituall. Part of this profound and diuine Science, had Ioachim the Prophesier atteyned vnto: by Numbers Formall, Naturall, and Rationall, forseyng, concludyng, and forshewyng great particular euents, long before their comming. His bookes yet remainyng, hereof, are good profe: And the noble Earle of Mirandula, (besides that,) a sufficient witnesse: that Ioachim, in his prophesies, proceded by no other way, then by Numbers Formall. And this Earle hym selfe, in Rome,\n\n(Ano. 1488.)\n\n* set vp 900. Conclusions, in all kinde of Sciences, openly to be disputed of: and among the rest, in his Conclusions Mathematicall, (in the eleuenth Conclusion) hath in Latin, this English sentence. By Numbers, a way is had, to the searchyng out, and vnderstandyng of euery thyng, hable to be knowen. For the verifying of which Conclusion, I promise to aunswere to the 74. Questions, vnder written, by the way of Numbers. Which Cõclusions, I omit here to rehearse: aswell auoidyng superfluous prolixitie: as, bycause Ioannes Picus, workes, are commonly had. But, in any case, I would wish that those Conclusions were red diligently, and perceiued of such, as are earnest Obseruers and Considerers of the constant law of nũbers: which is planted in thyngs Naturall and Supernaturall: and is prescribed to all Creatures, inuiolably to be kept. For, so, besides many other thinges, in those Conclusions to be marked, it would apeare, how sincerely, & within my boundes, I disclose the wonderfull mysteries, by numbers, to be atteyned vnto.\n\nOf my former wordes, easy it is to be gathered, that Number hath a treble state: One, in the Creator: an other in euery Creature (in respect of his complete constitution:) and the third, in Spirituall and Angelicall Myndes, and in the Soule of mã. In the first and third state, Number, is termed Number Numbryng. But in all Creatures, otherwise, Number, is termed Nũber Numbred. And in our Soule, Nũber beareth such a swaye, and hath such an affinitie therwith: that some of the old Philosophers taught, Mans Soule, to be a Number mouyng it selfe. And in dede, in vs, though it be a very Accident: yet such an Accident it is, that before all Creatures it had perfect beyng, in the Creator, Sempiternally. Number Numbryng therfore, is the discretion discerning, and distincting of thinges. But in God the Creator, This discretion, in the beginnyng, produced orderly and distinctly all thinges. For his Numbryng, then, was his Creatyng of all thinges. And his Continuall Numbryng, of all thinges, is the Conseruation of them in being: And, where and when he will lacke an Vnit: there and then, that particular thyng shalbe Discreated. Here I stay. But our Seuerallyng, distinctyng, and Numbryng, createth nothyng: but of Multitude considered, maketh certaine and distinct determination. And albeit these thynges be waighty and truthes of great importance, yet (by the infinite goodnes of the Almighty Ternarie,) Artificiall Methods and easy wayes are made, by which the zelous Philosopher, may wyn nere this Riuerish Ida, this Mountayne of Contemplation: and more then Contemplation. And also, though Number, be a thyng so Immateriall, so diuine, and æternall: yet by degrees, by litle and litle, stretchyng forth, and applying some likenes of it, as first, to thinges Spirituall: and then, bryngyng it lower, to thynges sensibly perceiued: as of a momentanye sounde iterated: then to the least thynges that may be seen, numerable: And at length, (most grossely,) to a multitude of any corporall thynges seen, or felt: and so, of these grosse and sensible thynges, we are trayned to learne a certaine Image or likenes of numbers: and to vse Arte in them to our pleasure and proffit. So grosse is our conuersation, and dull is our apprehension: while mortall Sense, in vs, ruleth the common wealth of our litle world. Hereby we say, Three Lyons, are three: or a Ternarie. Three Egles, are three, or a Ternarie.\n\n(☞)\n\nWhich * Ternaries, are eche, the Vnion, knot, and Vniformitie, of three discrete and distinct Vnits. That is, we may in eche Ternarie, thrise, seuerally pointe, and shew a part, One, One, and One. Where, in Numbryng, we say One, two, Three. But how farre, these visible Ones, do differre from our Indiuisible Vnits (in pure Arithmetike, principally considered) no man is ignorant. Yet from these grosse and materiall thynges, may we be led vpward, by degrees, so, informyng our rude Imagination, toward the cõceiuyng of Numbers, absolutely (:Not supposing, nor admixtyng any thyng created, Corporall or Spirituall, to support, conteyne, or represent those Numbers imagined:) that at length, we may be hable, to finde the number of our owne name, gloriously exemplified and registred in the booke of the Trinitie most blessed and æternall.\n\nBut farder vnderstand, that vulgar Practisers, haue Numbers, otherwise, in sundry Considerations: and extend their name farder, then to Numbers, whose least part is an Vnit. For the common Logist, Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse partes: and calleth them Fractions. As of an Vnit, he maketh an halfe, and thus noteth it, ½. and so of other, (infinitely diuerse) partes of an Vnit. Yea and farder, hath, Fractions of Fractions. &c.\n\n### Arithmetic, 1\n\nAnd, forasmuch, as, Addition, Substraction, Multiplication, Diuision and Extraction of Rotes, are the chief, and sufficient partes of Arithmetike: which is, the Science that demonstrateth the properties, of Numbers, and all operatiõs, in numbers to be performed:\n\n“How often, therfore, these fiue sundry sortes of Operations, do, for the most part, of their execution, differre from the fiue operations of like generall property and name, in our Whole numbers practisable, So often, (for a more distinct doctrine) we, vulgarly account and name it, an other kynde of Arithmetike.” And by this reason:\n\n(1.)\n\nthe Consideration, doctrine, and working, in whole numbers onely: where, of an Vnit, is no lesse part to be allowed: is named (as it were) an Arithmetike by it selfe. And so of the Arithmetike of Fractions.\n\n(2.)\n\nIn lyke sorte, the necessary, wonderfull and Secret doctrine of Proportion, and proportionalytie hath purchased vnto it selfe a peculier maner of handlyng and workyng: and so may seme an other forme of Arithmetike.\n\n(3.)\n\nMoreouer, the Astronomers, for spede and more commodious calculation, haue deuised a peculier maner of orderyng nũbers, about theyr circular motions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes &c. which commonly is called the Arithmetike of Astronomical or Phisicall Fractions. That, haue I briefly noted, by the name of Arithmetike Circular. Bycause it is also vsed in circles, not Astronomicall. &c.\n\n(4.)\n\nPractise hath led Numbers farder, and hath framed them, to take vpon them, the shew of Magnitudes propertie: Which is Incommensurabilitie and Irrationalitie. (For in pure Arithmetike, an Vnit, is the common Measure of all Numbers.) And, here, Nũbers are become, as Lynes, Playnes and Solides: some tymes Rationall, some tymes Irrationall. And haue propre and peculier characters, (as ²√. ³√. and so of other. Which is to signifie Rote Square, Rote Cubik: and so forth:) & propre and peculier fashions in the fiue principall partes: Wherfore the practiser, estemeth this, a diuerse Arithmetike from the other. Practise bryngeth in, here, diuerse compoundyng of Numbers: as some tyme, two, three, foure (or more) Radicall nũbers, diuersly knit, by signes, of More & Lesse: as thus ²√12 ³√15. Or thus ⁴√19 ³√12 - ²√2. &c. And some tyme with whole numbers, or fractions of whole Number, amõg\n\nthem: as 20 ²√24. ³√16 33 - ²√10. ⁴√44 12¼ ³√9. And so,\n\ninfinitely, may hap the varietie. After this: Both the one and the other hath fractions incident: and so is this Arithmetike greately enlarged, by diuerse exhibityng and vse of Compositions and mixtynges. Consider how, I (beyng desirous to deliuer the student from error and Cauillation) do giue to this Practise, the name of the Arithmetike of Radicall numbers: Not, of Irrationall or Surd Numbers: which other while, are Rationall: though they haue the Signe of a Rote before them, which, Arithmetike of whole Numbers most vsuall, would say they had no such Roote: and so account them Surd Numbers: which, generally spokẽ, is vntrue: as Euclides tenth booke may teach you. Therfore to call them, generally, Radicall Numbers, (by reason of the signe √. prefixed,) is a sure way: and a sufficient generall distinction from all other ordryng and vsing of Numbers: And yet (beside all this) Consider: the infinite desire of knowledge, and incredible power of mans Search and Capacitye: how, they, ioyntly haue waded farder (by mixtyng of speculation and practise) and haue found out, and atteyned to the very chief perfection (almost) of Numbers Practicall vse. Which thing, is well to be perceiued in that great Arithmeticall Arte of Æquation: commonly called the Rule of Coss. or Algebra. The Latines termed it, Regulam Rei & Census, that is, the Rule of the thyng and his value. With an apt name: comprehendyng the first and last pointes of the worke. And the vulgar names, both in Italian, Frenche and Spanish, depend (in namyng it,) vpon the signification of the Latin word, Res: A thing: vnleast they vse the name of Algebra. And therin (commonly) is a dubble error. The one, of them, which thinke it to be of Geber his inuentyng: the other of such as call it Algebra. For, first, though Geber for his great skill in Numbers, Geometry, Astronomy, and other maruailous Artes, mought haue semed hable to haue first deuised the sayd Rule: and also the name carryeth with it a very nere likenes of Geber his name: yet true it is, that a Greke Philosopher and Mathematicien, named Diophantus, before Geber his tyme, wrote 13. bookes therof (of which, six are yet extant: and I had them to *vse,\n\n(* Anno. 1550.)\n\nof the famous Mathematicien, and my great frende, Petrus Montaureus:) And secondly, the very name, is Algiebar, and not Algebra: as by the Arabien Auicen, may be proued: who hath these precise wordes in Latine, by Andreas Alpagus (most perfect in the Arabik tung) so translated. Scientia faciendi Algiebar & Almachabel. i. Scientia inueniendi numerum ignotum, per additionem Numeri, & diuisionem & æquationem. Which is to say: The Science of workyng Algiebar and Almachabel, that is, the Science of findyng an vnknowen number, by Addyng of a Number, & Diuision & æquation. Here haue you the name: and also the principall partes of the Rule, touched. To name it, The rule, or Art of Æquation, doth signifie the middle part and the State of the Rule. This Rule, hath his peculier Characters:\n\n(5.)\n\nand the principal partes of Arithmetike, to it appertayning, do differre from the other Arithmeticall operations. This Arithmetike, hath Nũbers Simple, Cõpound, Mixt: and Fractions, accordingly. This Rule, and Arithmetike of Algiebar, is so profound, so generall and so (in maner) conteyneth the whole power of Numbers Application practicall: that mans witt, can deale with nothyng, more proffitable about numbers: nor match, with a thyng, more mete for the diuine force of the Soule, (in humane Studies, affaires, or exercises) to be tryed in. Perchaunce you looked for, (long ere now,) to haue had some particular profe, or euident testimony of the vse, proffit and Commodity of Arithmetike vulgar, in the Common lyfe and trade of men. Therto, then, I will now frame my selfe: But herein great care I haue, least length of sundry profes, might make you deme, that either I did misdoute your zelous mynde to vertues schole: or els mistrust your hable witts, by some, to gesse much more. A profe then, foure, fiue, or six, such, will I bryng, as any reasonable man, therwith may be persuaded, to loue & honor, yea learne and exercise the excellent Science of Arithmetike.\n\nAnd first: who, nerer at hand, can be a better witnesse of the frute receiued by Arithmetike, then all kynde of Marchants? Though not all, alike, either nede it, or vse it. How could they forbeare the vse and helpe of the Rule, called the Golden Rule? Simple and Compounde: both forward and backward? How might they misse Arithmeticall helpe in the Rules of Felowshyp: either without tyme, or with tyme? and betwene the Marchant & his Factor? The Rules of Bartering in wares onely: or part in wares, and part in money, would they gladly want? Our Marchant venturers, and Trauaylers ouer Sea, how could they order their doynges iustly and without losse, vnleast certaine and generall Rules for Exchaũge of money, and Rechaunge, were, for their vse, deuised? The Rule of Alligation, in how sundry cases, doth it conclude for them, such precise verities, as neither by naturall witt, nor other experience, they, were hable, els, to know? And (with the Marchant then to make an end) how ample & wonderfull is the Rule of False positions? especially as it is now, by two excellent Mathematiciens (of my familier acquayntance in their life time) enlarged? I meane Gemma Frisius, and Simon Iacob. Who can either in brief conclude, the generall and Capitall Rules? or who can Imagine the Myriades of sundry Cases, and particular examples, in Act and earnest, continually wrought, tried and concluded by the forenamed Rules, onely? How sundry other Arithmeticall practises, are commonly in Marchantes handes, and knowledge: They them selues, can, at large, testifie.\n\nThe Mintmaster, and Goldsmith, in their Mixture of Metals, either of diuerse kindes, or diuerse values: how are they, or may they, exactly be directed, and meruailously pleasured, if Arithmetike be their guide? And the honorable Phisiciãs, will gladly confesse them selues, much beholding to the Science of Arithmetike, and that sundry wayes: But chiefly in their Art of Graduation, and compounde Medicines. And though Galenus, Auerrois, Arnoldus, Lullus, and other haue published their positions, aswell in the quantities of the Degrees aboue Temperament, as in the Rules, concluding the new Forme resulting: yet a more precise, commodious, and easy Method, is extant: by a Countreyman of ours\n\n(aboue 200. yeares ago) inuented. And forasmuch as I am vncertaine, who hath the same: or when that litle Latin treatise, (as the Author writ it,) shall come to be Printed: (Both to declare the desire I haue to pleasure my Countrey, wherin I may: and also, for very good profe of Numbers vse, in this most subtile and frutefull, Philosophicall Conclusion,) I entend in the meane while, most briefly, and with my farder helpe, to communicate the pith therof vnto you.\n\nFirst describe a circle: whose diameter let be an inch. Diuide the Circumference into foure equall partes. Frõ the Center, by those 4. sections, extend 4. right lines: eche of 4. inches and a halfe long: or of as many as you liste, aboue 4. without the circumference of the circle: So that they shall be of 4. inches long (at the least) without the Circle. Make good euident markes, at euery inches end. If you list, you may subdiuide the inches againe into 10. or 12. smaller partes, equall. At the endes of the lines, write the names of the 4. principall elementall Qualities. Hote and Colde, one against the other. And likewise Moyst and Dry, one against the other. And in the Circle write Temperate. Which Temperature hath a good Latitude: as appeareth by the Complexion of man. And therefore we haue allowed vnto it, the foresayd Circle: and not a point Mathematicall or Physicall.\n\n(* Take some part of Lullus counsayle in his booke de Q. Essentia.)\n\nNow, when you haue two thinges Miscible, whose degrees are * truely knowen: Of necessitie, either they are of one Quantitie and waight, or of diuerse. If they be of one Quantitie and waight: whether their formes, be Contrary Qualities, or of one kinde (but of diuerse intentions and degrees) or a Temperate, and a Contrary, The forme resulting of their Mixture, is in the Middle betwene the degrees of the formes mixt. As for example, let A, be Moist in the first degree: and B, Dry in the third degree. Adde 1. and 3. that maketh 4: the halfe or middle of 4. is 2. This 2. is the middle, equally distant from A and B\n\n(* Note.)\n\n(for the * Temperament is counted none. And for it, you must put a Ciphre, if at any time, it be in mixture).\n\n![The four elemental qualities and their degrees](/figures/dee-praeface/mixture-diagram.png)\n\nCounting then from B, 2. degrees, toward A: you finde it to be Dry in the first degree: So is the Forme resulting of the Mixture of A, and B, in our example. I will geue you an other example. Suppose, you haue two thinges, as C, and D: and of C, the Heate to be in the 4. degree: and of D, the Colde, to be remisse, euen vnto the Temperament. Now, for C, you take 4: and for D, you take a Ciphre: which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is 2. Wherefore the Forme resulting of C, and D, is Hote in the second degree: for, 2. degrees, accounted from C, toward D, ende iuste in the 2. degree of heate. Of the third maner, I will geue also an example: which let be this:\n\nI haue a liquid Medicine whose Qualitie of heate is in the 4. degree exalted: as was C, in the example foregoing: and an other liquid Medicine I haue: whose Qualitie, is heate, in the first degree. Of eche of these, I mixt a like quantitie: Subtract here, the lesse frõ the more: and the residue diuide into two equall partes: whereof, the one part, either added to the lesse, or subtracted from the higher degree, doth produce the degree of the Forme resulting, by this mixture of C, and E. As, if from 4. ye abate 1. there resteth 3. the halfe of 3. is 1½: Adde to 1. this 1½: you haue 2½. Or subtract from 4. this 1½: you haue likewise 2½ remayning. Which declareth, the Forme resulting, to be Heate, in the middle of the third degree.\n\n(The Second Rule.)\n\n“But if the Quantities of two thinges Commixt, be diuerse, and the Intensions (of their Formes Miscible) be in diuerse degrees, and heigthes. (Whether those Formes be of one kinde, or of Contrary kindes, or of a Temperate and a Contrary, What proportion is of the lesse quantitie to the greater, the same shall be of the difference, which is betwene the degree of the Forme resulting, and the degree of the greater quantitie of the thing miscible, to the difference, which is betwene the same degree of the Forme resulting, and the degree of the lesse quantitie. As for example. Let two pound of Liquor be geuen, hote in the 4. degree: & one pound of Liquor be geuen, hote in the third degree.” I would gladly know the Forme resulting, in the Mixture of these two Liquors. Set downe your nũbers in order, thus.\n\n£. 2. Hote. 4.\n\n£. 1. Hote. 3.\n\n### Arithmetic, 2\n\nNow by the rule of Algiebar, haue I deuised a very easie, briefe, and generall maner of working in this case. Let vs first, suppose that Middle Forme resulting, to be 1{x}: as that Rule teacheth. And because (by our Rule, here geuen) as the waight of 1. is to 2: So is the difference betwene 4. (the degree of the greater quantitie) and 1{x}: to the difference betwene 1{x} and 3: (the degree of the thing, in lesse quãtitie. And with all, 1{x}, being alwayes in a certaine middell, betwene the two heigthes or degrees). For the first difference, I set 4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1. is to 2. so is 4-1{x} to 1{x}-3. Wherfore, of these foure proportionall numbers, the first and the fourth Multiplied, one by the other, do make as much, as the second and the third Multiplied the one by the other. Let these Multiplications be made accordingly. And of the first and the fourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore, our Æquation is betwene 1{x}-3: and 8-2{x}. Which may be reduced, according to the Arte of Algiebar: as, here, adding 3. to eche part, geueth the Æquation, thus, 1{x}11-2{x}. And yet againe, contracting, or Reducing it: Adde to eche part, 2{x}: Then haue you 3{x} æquall to 11: thus represented 3{x}11. Wherefore, diuiding 11. by 3: the Quotient is 3⅔: the Valew of our 1{x}, Coss, or Thing, first supposed. And that is the heigth, or Intension of the Forme resulting: which is, Heate, in two thirdes of the fourth degree: And here I set the shew of the worke in conclusion, thus. The proufe hereof is easie: by subtracting 3. from 3⅔, resteth ⅔. Subtracte the same heigth of the Forme resulting, (which is 3⅔) frõ 4: then resteth ⅓: You see, that ⅔ is double to ⅓: as 2.£. is double to 1.£. So should it be: by the rule here geuen. Note. As you added to eche part of the Æquation, 3: so if ye first added to eche part 2{x}, it would stand, 3{x}-38. And now adding to eche part 3: you haue (as afore) 3{x}11.\n\n£. 2. Hote. 4. ⅓ The forme\n\n3⅔ resulting.\n\n£. 1. Hote. 3. ⅔\n\nAnd though I, here, speake onely of two thyngs Miscible: and most commonly mo then three, foure, fiue or six, (&c.) are to be Mixed: (and in one Compound to be reduced: & the Forme resultyng of the same, to serue the turne) yet these Rules are sufficient: duely repeated and iterated.\n\nIn procedyng first, with any two: and then, with the Forme Resulting, and an other: & so forth: For, the last worke, concludeth the Forme resultyng of them all: I nede nothing to speake, of the Mixture (here supposed) what it is. Common Philosophie hath defined it, saying, Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio. Euery word in the definition, is of great importance. I nede not also spend any time, to shew, how, the other manner of distributing of degrees, doth agree to these Rules. Neither nede I of the farder vse belonging to the Crosse of Graduation (before described) in this place declare, vnto such as are capable of that, which I haue all ready sayd. Neither yet with examples specifie the Manifold varieties, by the foresayd two generall Rules, to be ordered. The witty and Studious, here, haue sufficient: And they which are not hable to atteine to this, without liuely teaching, and more in particular: would haue larger discoursing, then is mete in this place to be dealt withall: And other (perchaunce) with a proude snuffe will disdaine this litle: and would be vnthankefull for much more. I, therfore conclude: and wish such as haue modest and earnest Philosophicall mindes, to laude God highly for this: and to Meruayle, that the profoundest and subtilest point, concerning Mixture of Formes and Qualities Naturall, is so Matcht and maryed with the most simple, easie, and short way of the noble Rule of Algiebar. Who can remaine, therfore vnpersuaded, to loue, alow, and honor the excellent Science of Arithmetike? For, here, you may perceiue that the litle finger of Arithmetike, is of more might and contriuing, then a hunderd thousand mens wittes, of the middle sorte, are hable to perfourme, or truely to conclude, with out helpe thereof.\n\nNow will we farder, by the wise and valiant Capitaine, be certified, what helpe he hath, by the Rules of Arithmetike: in one of the Artes to him appertaining: And of the Grekes named\n\n(Τακτικὴ.)\n\nΤακτικὴ. “That is, the Skill of Ordring Souldiers in Battell ray after the best maner to all purposes.” This Art so much dependeth vppon Numbers vse, and the Mathematicals, that Ælianus (the best writer therof,) in his worke, to the Emperour Hadrianus, by his perfection, in the Mathematicals, (beyng greater, then other before him had,) thinketh his booke to passe all other the excellent workes, written of that Art, vnto his dayes. For, of it, had written Æneas: Cyneas of Thessaly: Pyrrhus Epirota: and Alexander his sonne: Clearchus: Pausanias: Euangelus: Polybius, familier frende to Scipio: Eupolemus: Iphicrates, Possidonius: and very many other worthy Capitaines, Philosophers and Princes of Immortall fame and memory: Whose fayrest floure of their garland (in this feat) was Arithmetike: and a litle perceiuerance, in Geometricall Figures. But in many other cases doth Arithmetike stand the Capitaine in great stede. As in proportionyng of vittayles, for the Army, either remaining at a stay: or suddenly to be encreased with a certaine number of Souldiers: and for a certain tyme. Or by good Art to diminish his company, to make the victuals, longer to serue the remanent, & for a certaine determined tyme: if nede so require. And so in sundry his other accountes, Reckeninges, Measurynges, and proportionynges, the wise, expert, and Circumspect Capitaine will affirme the Science of Arithmetike, to be one of his chief Counsaylors, directers and aiders. Which thing (by good meanes) was euident to the Noble, the Couragious, the loyall, and Curteous\n\n(☞)\n\nIohn, late Earle of Warwicke. Who was a yong Gentleman, throughly knowne to very few. Albeit his lusty valiantnes, force, and Skill in Chiualrous feates and exercises: his humblenes, and frendelynes to all men, were thinges, openly, of the world perceiued. But what rotes (otherwise,) vertue had fastened in his brest, what Rules of godly and honorable life he had framed to him selfe: what vices, (in some then liuing) notable, he tooke great care to eschew: what manly vertues, in other noble men, (florishing before his eyes,) he Sythingly aspired after: what prowesses he purposed and ment to achieue: with what feats and Artes, he began to furnish and fraught him selfe, for the better seruice of his Kyng and Countrey, both in peace & warre. These (I say) his Heroicall Meditations, forecastinges and determinations, no twayne, (I thinke) beside my selfe, can so perfectly, and truely report. And therfore, in Conscience, I count it my part, for the honor, preferment, & procuring of vertue (thus, briefly) to haue put his Name, in the Register of Fame Immortall.\n\nTo our purpose. This Iohn, by one of his actes (besides many other: both in England and Fraunce, by me, in him noted.) did disclose his harty loue to vertuous Sciences: and his noble intent, to excell in Martiall prowesse: When he, with humble request, and instant Solliciting: got the best Rules (either in time past by Greke or Romaine, or in our time vsed: and new Stratagemes therin deuised) for ordring of all Companies, summes and Numbers of mẽ, (Many, or few) with one kinde of weapon, or mo, appointed: with Artillery, or without: on horsebacke, or on fote: to giue, or take onset: to seem many, being few: to seem few, being many. To marche in battaile or Iornay: with many such feates, to Foughten field, Skarmoush, or Ambushe appartaining:\n\n(This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age: hauing no issue by his wife: Daughter to the Duke of Somerset.)\n\nAnd of all these, liuely designementes (most curiously) to be in velame parchement described: with Notes & peculier markes, as the Arte requireth: and all these Rules, and descriptions Arithmeticall, inclosed in a riche Case of Gold, he vsed to weare about his necke: as his Iuell most precious, and Counsaylour most trusty. Thus, Arithmetike, of him, was shryned in gold: Of Numbers frute, he had good hope. Now, Numbers therfore innumerable, in Numbers prayse, his shryne shall finde.\n\nWhat nede I, (for farder profe to you) of the Scholemasters of Iustice, to require testimony: how nedefull, how frutefull, how skillfull a thing Arithmetike is? I meane, the Lawyers of all sortes. Vndoubtedly, the Ciuilians, can meruaylously declare: how, neither the Auncient Romaine lawes, without good knowledge of Numbers art, can be perceiued: Nor (Iustice in infinite Cases) without due proportion, (narrowly considered,) is hable to be executed. How Iustly, & with great knowledge of Arte, did Papinianus institute a law of partition, and allowance, betwene man and wife after a diuorce? But how Accursius, Baldus, Bartolus, Iason, Alexander, and finally Alciatus, (being otherwise, notably well learned) do iumble, gesse, and erre, from the æquity, art and Intent of the lawmaker: Arithmetike can detect, and conuince: and clerely, make the truth to shine. Good Bartolus, tyred in the examining & proportioning of the matter: and with Accursius Glosse, much cumbred: burst out, and sayd: Nulla est in toto libro, hac glossa difficilior: Cuius computationem nec Scholastici nec Doctores intelligunt. &c. That is: In the whole booke, there is no Glosse harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor the Doctours vnderstand. &c. What can they say of Iulianus law, Si ita Scriptum. &c. Of the Testators will iustly performing, betwene the wife, Sonne and daughter? How can they perceiue the æquitie of Aphricanus, Arithmeticall Reckening, where he treateth of Lex Falcidia? How can they deliuer him, from his Reprouers: and their maintainers: as Ioannes, Accursius Hypolitus and Alciatus? How Iustly and artificially, was Africanus reckening made? Proportionating to the Sommes bequeathed, the Contributions of eche part? Namely, for the hundred presently receiued, 17-1/7. And for the hundred, receiued after ten monethes, 12-6/7: which make the 30: which were to be cõtributed by the legataries to the heire. For, what proportion, 100 hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law, require an expert Arithmeticien, to vnderstand the deepe Iudgemẽt, & Iust determinatiõ of the Auncient Romaine Lawmakers. But much more expert ought he to be, who should be hable, to decide with æquitie, the infinite varietie of Cases, which do, or may happen, vnder euery one of those lawes and ordinances Ciuile. Hereby, easely, ye may now coniecture: that in the Canon law: and in the lawes of the Realme (which with vs, beare the chief Authoritie), Iustice and equity might be greately preferred, and skilfully executed, through due skill of Arithmetike, and proportions appertainyng. The worthy Philosophers, and prudent lawmakers (who haue written many bookes De Republica: How the best state of Common wealthes might be procured and mainteined,) haue very well determined of Iustice: (which, not onely, is the Base and foundacion of Common weales: but also the totall perfection of all our workes, words, and thoughtes:) defining it,\n\n(Iustice.)\n\n“to be that vertue, by which, to euery one, is rendred, that to him appertaineth.” God challengeth this at our handes, to be honored as God: to be loued, as a father: to be feared as a Lord & master. Our neighbours proportiõ, is also prescribed of the Almighty lawmaker: which is, to do to other, euen as we would be done vnto. These proportions, are in Iustice necessary: in duety, commendable: and of Common wealthes, the life, strength, stay and florishing. Aristotle in his Ethikes (to fatch the sede of Iustice, and light of direction, to vse and execute the same) was fayne to fly to the perfection, and power of Numbers: for proportions Arithmeticall and Geometricall. Plato in his booke called Epinomis (which boke, is the Threasury of all his doctrine) where, his purpose is, to seke a Science, which, when a man had it, perfectly: he might seme, and so be, in dede, Wise. He, briefly, of other Sciences discoursing, findeth them, not hable to bring it to passe: But of the Science of Numbers, he sayth. Illa, quæ numerum mortalium generi dedit, id profecto efficiet. Deum autem aliquem, magis quam fortunam, ad salutem nostram, hoc munus nobis arbitror contulisse. &c. Nam ipsum bonorum omnium Authorem, cur non maximi boni, Prudentiæ dico, causam arbitramur? That Science, verely, which hath taught mankynde number, shall be able to bryng it to passe. And, I thinke, a certaine God, rather then fortune, to haue giuen vs this gift, for our blisse. For, why should we not Iudge him, who is the Author of all good things, to be also the cause of the greatest good thyng, namely, Wisedome? There, at length, he proueth Wisedome to be atteyned, by good Skill of Numbers. With which great Testimony, and the manifold profes, and reasons, before expressed, you may be sufficiently and fully persuaded: of the perfect Science of Arithmetike, to make this accounte: That\n\n(☞)\n\nof all Sciences, next to Theologie, it is most diuine, most pure, most ample and generall, most profounde, most subtile, most commodious and most necessary. Whose next Sister, is the Absolute Science of Magnitudes: of which (by the Direction and aide of him, whose Magnitude is Infinite, and of vs Incomprehensible) I now entend, so to write, that both with the Multitude, and also with the Magnitude of Meruaylous and frutefull verities, you (my frendes and Countreymen) may be stird vp, and awaked, to behold what certaine Artes and Sciences, (to our vnspeakable behofe) our heauenly father, hath for vs prepared, and reuealed, by sundry Philosophers and Mathematiciens.\n\n### Arithmetic, 3\n\nBoth, Number and Magnitude, haue a certaine Originall sede, (as it were) of an incredible property: and of man, neuer hable, Fully, to be declared. Of Number, an Vnit, and of Magnitude, a Poynte, doo seeme to be much like Originall causes: But the diuersitie neuerthelesse, is great. We defined an Vnit, to be a thing Mathematicall Indiuisible: A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And farder, that a Point may haue a certaine determined Situation: that is, that we may assigne, and prescribe a Point, to be here, there, yonder. &c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to be tyed to any place, or seat: diuisible or indiuisible. Agayne, by reason, a Point may haue a Situation limited to him: a certaine motion, therfore (to a place, and from a place) is to a Point incident and appertainyng. But an Vnit, can not be imagined to haue any motion. A Point, by his motion, produceth, Mathematically, a line: (as we sayd before) which is the first kinde of Magnitudes, and most simple: An Vnit, can not produce any number. A Line, though it be produced of a Point moued, yet, it doth not consist of pointes: Number, though it be not produced of an Vnit, yet doth it Consist of vnits, as a materiall cause. But formally,\n\n(Number.)\n\nNumber, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting, is the workemanship of our minde: which, of distinct and discrete Vnits, maketh a Number: by vniformitie, resulting of a certaine multitude of Vnits. And so, euery number, may haue his least part, giuen: namely, an Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be giuẽ: by cause, infinitly, diuision therof, may be conceiued. All Magnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine, or Solid, of no Sense, can be perceiued, nor exactly by hãd (any way) represented: nor of Nature produced: But, as (by degrees) Number did come to our perceiuerance: So, by visible formes, we are holpen to imagine, what our Line Mathematicall, is. What our Point, is. So precise, are our Magnitudes, that one Line is no broader then an other: for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we can haue Smaller, then either Arte or Nature can produce any: and Greater also, then all the world can comprehend. Our least Magnitudes, can be diuided into so many partes, as the greatest. As, a Line of an inch long, (with vs) may be diuided into as many partes, as may the diameter of the whole world, from East to West: or any way extended: What priuiledges, aboue all manual Arte, and Natures might, haue our two Sciences Mathematicall? to exhibite, and to deale with thinges of such power, liberty, simplicity, puritie, and perfection? And in them, so certainly, so orderly, so precisely to procede: as, excellent is that workemã Mechanicall Iudged, who nerest can approche to the representing of workes, Mathematically demonstrated?\n\n(☞)\n\nAnd our two Sciences, remaining pure, and absolute, in their proper termes, and in their owne Matter: to haue, and allowe, onely such Demonstrations, as are plaine, certaine, vniuersall, and of an æternall veritye?\n\n### Geometry\n\nThis Science of Magnitude, his properties, conditions, and appertenances: commonly, now is, and from the beginnyng, hath of all Philosophers, ben called Geometrie. But, veryly, with a name to base and scant, for a Science of such dignitie and amplenes. And, perchaunce, that name, by cõmon and secret consent, of all wisemen, hitherto hath ben suffred to remayne: that it might carry with it a perpetuall memorye, of the first and notablest benefite, by that Science, to common people shewed: Which was, when Boundes and meres of land and ground were lost, and confounded (as in Egypt, yearely, with the ouerflowyng of Nilus, the greatest and longest riuer in the world) or, that ground bequeathed, were to be assigned: or, ground sold, were to be layd out: or (when disorder preuailed) that Commõs were distributed into seueralties. For, where, vpon these & such like occasiõs, Some by ignorãce, some by negligẽce, Some by fraude, and some by violence, did wrongfully limite, measure, encroach, or challenge (by pretence of iust content, and measure) those landes and groundes: great losse, disquietnes, murder, and warre did (full oft) ensue: Till, by Gods mercy, and mans Industrie, The perfect Science of Lines, Plaines, and Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. The people then, by this art pleasured, and greatly relieued, in their landes iust measuring: & other Philosophers, writing Rules for land measuring: betwene them both, thus, confirmed the name of Geometria, that is, (according to the very etimologie of the word) Land measuring. Wherin, the people knew no farder, of Magnitudes vse, but in Plaines: and the Philosophers, of thẽ, had no feet hearers, or Scholers: farder to disclose vnto, then of flat, plaine Geometrie. And though, these Philosophers, knew of farder vse, and best vnderstode the etymologye of the worde, yet this name Geometria, was of them applyed generally to all sortes of Magnitudes: vnleast, otherwhile, of Plato, and Pythagoras: When they would precisely declare their owne doctrine. Then, was\n\n(* Plato. 7. de Rep.)\n\n* Geometria, with them, Studium quod circa planum versatur. But, well you may perceiue by Euclides Elementes, that more ample is our Science, then to measure Plaines: and nothyng lesse therin is tought (of purpose) then how to measure Land. An other name, therfore, must nedes be had, for our Mathematicall Science of Magnitudes: which regardeth neither clod, nor turff: neither hill, nor dale: neither earth nor heauen: but is absolute Megethologia: not creping on ground, and dasseling the eye, with pole perche, rod or lyne: but “liftyng the hart aboue the heauens, by inuisible lines, and\n\n(☞)\n\nimmortall beames meteth with the reflexions, of the light incomprehensible: and so procureth Ioye, and perfection vnspeakable.” Of which true vse of our Megethica, or Megethologia, Diuine Plato seemed to haue good taste, and iudgement: and (by the name of Geometrie) so noted it: and warned his Scholers therof: as, in hys seuenth Dialog, of the Common wealth, may euidently be sene. Where (in Latin) thus it is: right well translated: Profecto, nobis hoc non negabunt, Quicun[que] vel paululum quid Geometriæ gustârunt, quin hæc Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa\n\nversantur. In English, thus. Verely (sayth Plato) whosoeuer\n\nhaue, (but euen very litle) tasted of Geometrie, will not denye vnto vs, this: but that this Science, is of an other condicion, quite contrary to that, which they that are exercised in it, do speake of it. And there it followeth, of our Geometrie, Quòd quæritur cognoscendi illius gratia, quod semper est, non & eius quod oritur quando[que] & interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur (ô Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum preparabit cogitationem, vt ad supera conuertamus: quæ, nunc, contra quàm decet, ad inferiora deijcimus. &c. Quàm maximè igitur præcipiendum est, vt qui præclarissimam hanc habitãt Civitatem, nullo modo, Geometriam spernant. Nam & quæ præter ipsius propositum, quodam modo esse videntur, haud exigua sunt. &c. It must nedes be confessed (saith Plato) That [Geometrie] is learned, for the knowyng of that, which is euer: and not of that, which, in tyme, both is bred and is brought to an ende. &c. Geometrie is the knowledge of that which is euerlastyng. It will lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by that meanes, it will prepare the Thought, to the Philosophicall loue of wisdome: that we may turne or conuert, toward heauenly thinges [both mynde and thought] which now, otherwise then becommeth vs, we cast down on base or inferior things. &c. Chiefly, therfore, Commaundement must be giuen, that such as do inhabit this most honorable Citie, by no meanes, despise Geometrie. For euen those thinges [done by it] which, in manner, seame to be, beside the purpose of Geometrie: are of no small importance. &c. And besides the manifold vses of Geometrie, in matters appertainyng to warre, he addeth more, of second vnpurposed frute, and commoditye, arrising by Geometrie: saying: Scimus quin etiam, ad Disciplinas omnes facilius per discendas, interesse omnino, attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam, secundo loco discendam Iuuenibus statuamus. That is. But, also, we know, that for the more easy learnyng of all Artes, it importeth much, whether one haue any knowledge in Geometrie, or no. &c. Let vs therfore make an ordinance or decree, that this Science, of young men shall be learned in the second place. This was Diuine Plato his Iudgement, both of the purposed, chief, and perfect vse of Geometrie: and of his second, dependyng, deriuatiue commodities. And for vs, Christen men, a thousand thousand mo occasions are, to haue nede of the helpe of *\n\n(I. D. * Herein, I would gladly shake of, the earthly name, of Geometrie.)\n\nMegethologicall Contemplations: wherby, to trayne our Imaginations and Myndes, by litle and litle, to forsake and abandon, the grosse and corruptible Obiectes, of our vtward senses: and to apprehend, by sure doctrine demonstratiue, Things Mathematicall. And by them, readily to be holpen and conducted to conceiue, discourse, and conclude of things Intellectual, Spirituall, æternall, and such as concerne our Blisse euerlasting: which, otherwise (without Speciall priuiledge of Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally) is hable to reach vnto, or to Compasse. And, veryly, by my small Talent (from aboue) I am hable to proue and testifie, that the litterall Text, and order of our diuine Law, Oracles, and Mysteries, require more skill in Numbers, and Magnitudes: then (commonly) the expositors haue vttered: but rather onely (at the most) so warned: & shewed their own want therin. (To name any, is nedeles: and to note the places, is, here, no place: But if I be duely asked, my answere is ready.) And without the litterall, Grammaticall, Mathematicall or Naturall verities of such places, by good and certaine Arte, perceiued, no Spirituall sense (propre to those places, by Absolute Theologie) will thereon depend.\n\n(☞)\n\n“No man, therfore, can doute, but toward the atteyning of knowledge incomparable, and Heauenly Wisedome: Mathematicall Speculations, both of Numbers and Magnitudes: are meanes, aydes, and guides: ready, certaine, and necessary.” From henceforth, in this my Preface, will I frame my talke, to Plato his fugitiue Scholers: or, rather, to such, who well can, (and also wil,) vse their vtward senses, to the glory of God, the benefite of their Countrey, and their owne secret contentation, or honest preferment, on this earthly Scaffold. To them, I will orderly recite, describe & declare a great Number of Artes, from our two Mathematicall fountaines, deriued into the fieldes of Nature. Wherby, such Sedes, and Rotes, as lye depe hyd in the groũd of Nature, are refreshed, quickened, and prouoked to grow, shote vp, floure, and giue frute, infinite, and incredible. And these Artes, shalbe such, as vpon Magnitudes properties do depende, more, then vpon Number. And by good reason we may call them Artes, and Artes Mathematicall Deriuatiue: for (at this tyme) I Define\n\n(An Arte.)\n\nAn Arte, to be a Methodicall cõplete Doctrine, hauing abundancy of sufficient, and peculier matter to deale with, by the allowance of the Metaphisicall Philosopher: the knowledge whereof, to humaine state is necessarye. And that I account,\n\n(Art Mathematicall Deriuatiue.)\n\nAn Art Mathematicall deriuatiue, which by Mathematicall demonstratiue Method, in Nũbers, or Magnitudes, ordreth and confirmeth his doctrine, as much & as perfectly, as the matter subiect will admit. And for that, I entend to vse the name and propertie of a\n\n(A Mechanitien.)\n\nMechanicien, otherwise, then (hitherto) it hath ben vsed, I thinke it good, (for distinction sake) to giue you also a brief description, what I meane therby. A Mechanicien, or a Mechanicall workman is he, whose skill is, without knowledge of Mathematicall demonstration, perfectly to worke and finishe any sensible worke, by the Mathematicien principall or deriuatiue, demonstrated or demonstrable. Full well I know, that he which inuenteth, or maketh these demonstrations, is generally called A speculatiue Mechanicien: which differreth nothyng from a Mechanicall Mathematicien. So, in respect of diuerse actions, one man may haue the name of sundry artes: as, some tyme, of a Logicien, some tymes (in the same matter otherwise handled) of a Rethoricien. Of these trifles, I make, (as now, in respect of my Preface,) small account: to fyle thẽ for the fine handlyng of subtile curious disputers. In other places, they may commaunde me, to giue good reason: and yet, here, I will not be vnreasonable.\n\n(1.)\n\nFirst, then, from the puritie, absolutenes, and Immaterialitie of Principall Geometrie, is that kinde of Geometrie deriued, which vulgarly is counted Geometrie: and is the Arte of Measuring sensible magnitudes, their iust quãtities and contentes.\n\n### Common Geometry\n\nThis, teacheth to measure, either at hand: and the practiser, to be by the thing Measured: and so, by due applying of Cumpase, Rule, Squire, Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to the Length, Plaine, or Solide measured,\n\n(1.)\n\n* to be certified, either of the length, perimetry, or distance lineall: and this is called, Mecometrie. Or\n\n(2.)\n\n* to be certified of the content of any plaine Superficies: whether it be in ground Surueyed, Borde, or Glasse measured, or such like thing: which measuring, is named Embadometrie.\n\n(3.)\n\n* Or els to vnderstand the Soliditie, and content of any bodily thing: as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels, small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale vessells, &c, the Measuring, commonly, hath a peculier name: and is called Gaging. And the generall name of these Solide measures, is Stereometrie.\n\n(2.)\n\nOr els, this vulgar Geometrie, hath consideration to teach the practiser, how to measure things, with good distance betwene him and the thing measured: and to vnderstand thereby, either\n\n(1.)\n\n* how Farre, a thing seene (on land or water) is from the measurer: and this may be called Apomecometrie:\n\n(2.)\n\nOr, how High or depe, aboue or vnder the leuel of the measurers stãding, any thing is, which is sene on land or water, called Hypsometrie.\n\n(3.)\n\n* Or, it informeth the measurer, how Broad any thing is, which is in the measurers vew: so it be on Land or Water, situated: and may be called Platometrie. Though I vse here to condition, the thing measured, to be on Land, or Water Situated:\n\nyet, know for certaine, that the sundry heigthe of Cloudes, blasing Starres, and of the Mone, may (by these meanes) haue their distances from the earth: and, of the blasing Starres and Mone, the Soliditie (aswell as distances) to be measured: But because, neither these things are vulgarly taught: nor of a common practiser so ready to be executed: I, rather, let such measures be reckened incident to some of our other Artes, dealing with thinges on high, more purposely, then this vulgar Land measuring Geometrie doth: as in Perspectiue and Astronomie, &c.\n\nOf these Feates (farther applied) is Sprong the Feate of Geodesie, or Land Measuring: more cunningly to measure & Suruey Land, Woods, and Waters, a farre of. More cunningly, I say: But God knoweth (hitherto) in these Realmes of England and Ireland (whether through ignorance or fraude, I can not tell, in euery particular)\n\nhow great wrong and iniurie hath (in my time) bene committed by vntrue measuring and surueying of Land or Woods, any way. And, this I am sure: that the Value of the difference, betwene the truth and such Surueyes, would haue bene hable to haue foũd (for euer) in eche of our two Vniuersities, an excellent Mathematicall Reader: to eche, allowing (yearly) a hundred Markes of lawfull money of this realme: which, in dede, would seme requisit, here, to be had (though by other wayes prouided for) as well, as, the famous Vniuersitie of Paris, hath two Mathematicall Readers: and eche, two hundreth French Crownes yearly, of the French Kinges magnificent liberalitie onely. Now, againe, to our purpose returning: Moreouer, of the former knowledge Geometricall, are growen the Skills of Geographie, Chorographie, Hydrographie, and Stratarithmetrie.\n\n### Geography\n\n“Geographie teacheth wayes, by which, in sũdry formes, (as Sphærike, Plaine or other), the Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, & such other things, vpõ the outface of the earthly Globe (either in the whole, or in some principall mẽber and portion therof cõtayned) may be described and designed, in cõmensurations Analogicall to Nature and veritie: and most aptly to our vew, may be represented.” Of this Arte how great pleasure, and how manifolde commodities do come vnto vs, daily and hourely: of most men, is perceaued. While, some, to beautifie their Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other some, for thinges past, as battels fought, earthquakes, heauenly fyringes, & such occurentes, in histories mentioned: therby liuely, as it were, to vewe the place, the region adioyning, the distance from vs: and such other circumstances. Some other, presently to vewe the large dominion of the Turke: the wide Empire of the Moschouite: and the litle morsell of ground, where Christendome (by profession) is certainly knowen. Litle, I say, in respecte of the rest. &c. Some, either for their owne iorneyes directing into farre landes: or to vnderstand of other mens trauailes. To conclude, some, for one purpose: and some, for an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, & Geographicall Globes. Of whose vse, to speake sufficiently, would require a booke peculier.\n\n### Chorography\n\nChorographie seemeth to be an vnderling, and a twig, of Geographie: and yet neuerthelesse, is in practise manifolde, and in vse very ample. “This teacheth Analogically to describe a small portion or circuite of ground, with the contentes: not regarding what commensuration it hath to the whole, or any parcell, without it, contained. But in the territory or parcell of ground which it taketh in hand to make description of, it leaueth out (or vndescribed) no notable, or odde thing, aboue the ground visible. Yea and sometimes, of thinges vnder ground, geueth some peculier marke: or warning: as of Mettall mines, Cole pittes, Stone quarries. &c.” Thus, a Dukedome, a Shiere, a Lordship, or lesse, may be described distinctly. But marueilous pleasant, and profitable it is, in the exhibiting to our eye, and commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true Symmetry: not approching to any of them: and out of Gunne shot. &c. Hereby, the Architect may furnishe him selfe, with store of what patterns he liketh: to his great instruction: euen in those thinges which outwardly are proportioned: either simply in them selues: or respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also, terme this particular description of places, Topographie.\n\n### Hydrography\n\n“Hydrographie, deliuereth to our knowledge, on Globe or in Plaine, the perfect Analogicall description of the Ocean Sea coastes, through the whole world: or in the chiefe and principall partes thereof:” with the Iles and chiefe particular places of daungers, conteyned within the boundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes, Rockes, Races, Countertides, Whorlepooles. &c. This, dealeth with the Element of the water chiefly: as Geographie did principally take the Element of the Earthes description (with his appertenances) to taske. And besides thys, Hydrographie, requireth a particular Register of certaine Landmarkes (where markes may be had) from the sea, well hable to be skried, in what point of the Seacumpase they appeare, and what apparent forme, Situation, and bignes they haue, in respecte of any daungerous place in the sea, or nere vnto it, assigned: And in all Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes, come and go, the Hydrographer ought to recorde. The Soundinges likewise: and the Chanels wayes: their number, and depthes ordinarily, at ebbe and flud, ought the Hydrographer, by obseruation and diligence of Measuring, to haue certainly knowen. And many other pointes, are belonging to perfecte Hydrographie, and for to make a Rutter, by: of which, I nede not here speake: as of the describing, in any place, vpon Globe or Plaine, the 32. pointes of the Compase, truely: (wherof, scarsly foure, in England, haue right knowledge: bycause, the lines therof, are no straight lines, nor Circles.) Of making due proiection of a Sphere in plaine. Of the Variacion of the Compas, from true Northe: And such like matters (of great importance, all) I leaue to speake of, in this place: bycause, I may seame (al ready) to haue enlarged the boundes, and duety of an Hydographer, much more, then any man (to this day) hath noted, or prescribed. Yet am I well hable to proue, all these thinges, to appertaine, and also to be proper to the Hydrographer. The chief vse and ende of this Art, is the Art of Nauigation: but it hath other diuerse vses: euen by them to be enioyed, that neuer lacke sight of land.\n\n### Stratarithmetrie\n\nStratarithmetrie, is the Skill, (appertainyng to the warre,) by which a man can set in figure, analogicall to any Geometricall figure appointed, any certaine number or summe of men: of such a figure capable: (by reason of the vsuall spaces betwene Souldiers allowed: and for that, of men, can be made no Fractions. Yet, neuertheles, he can order the giuen summe of men, for the greatest such figure, that of them, cã be ordred) and certifie, of the ouerplus: (if any be) and of the next certaine summe, which, with the ouerplus, will admit a figure exactly proportionall to the figure assigned. By which Skill, also, of any army or company of men: (the figure & sides of whose orderly standing, or array, is knowen) he is able to expresse the iust number of men, within that figure conteined: or (orderly) able to be conteined.\n\n(* Note.)\n\n* And this figure, and sides therof, he is hable to know: either beyng by, and at hand: or a farre of. Thus farre, stretcheth the description and property of Stratarithmetrie: sufficient for this tyme and place.\n\n(The difference betwene Stratarithmetrie and Tacticie.)\n\n“It differreth from the Feate Tacticall, De aciebus instruendis. bycause, there, is necessary the wisedome and foresight, to what purpose he so ordreth the men: and Skillfull hability, also, for any occasion, or purpose, to deuise and vse the aptest and most necessary order, array and figure of his Company and Summe of men.” By figure, I meane: as, either of a Perfect Square, Triangle, Circle, Ouale, long square, (of the Grekes it is called Eteromekes) Rhombe, Rhomboïd, Lunular, Ryng, Serpentine, and such other Geometricall figures: Which, in warres, haue ben, and are to be vsed: for commodiousnes, necessity, and auauntage &c. And no small skill ought he to haue, that should make true report, or nere the truth, of the numbers and Summes, of footemen or horsemen, in the Enemyes ordring. A farre of, to make an estimate, betwene nere termes of More and Lesse, is not a thyng very rife, among those that gladly would do it.\n\n(I. D. Frende, you will finde it hard, to performe my description of this Feate. But by Chorographie, you may helpe your selfe some what: where the Figures knowne (in Sides and Angles) are not Regular: And where, Resolution into Triangles can serue. &c. And yet you will finde it strange to deale thus generally with Arithmeticall figures: and, that for Battayle ray. Their contentes, differ so much from like Geometricall Figures.)\n\nGreat pollicy may be vsed of the Capitaines, (at tymes fete, and in places conuenient) as to vse Figures, which make greatest shew, of so many as he hath: and vsing the aduauntage of the three kindes of vsuall spaces: (betwene footemen or horsemen) to take the largest: or when he would seme to haue few, (beyng many:) contrarywise, in Figure, and space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer is carefull to come nere the truth herein, besides the Iudgement of his expert eye, his skill of Ordering Tacticall, the helpe of his Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously framed for cariage and vse) He may wonderfully helpe him selfe, by perspectiue Glasses. In which, (I trust) our posterity will proue more skillfull and expert, and to greater purposes, then in these dayes, can (almost) be credited to be possible.\n\nThus haue I lightly passed ouer the Artificiall Feates, chiefly dependyng vpon vulgar Geometrie: & commonly and generally reckened vnder the name of Geometrie. But there are other (very many) Methodicall Artes, which, declyning from the purity, simplicitie, and Immateriality, of our Principall Science of Magnitudes: do yet neuertheles vse the great ayde, direction, and Method of the sayd principall Science, and haue propre names, and distinct: both from the Science of Geometrie, (from which they are deriued) and one from the other.\n\n### The Mixed Sciences and Perspective\n\nAs Perspectiue, Astronomie, Musike, Cosmographie, Astrologie, Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie, Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture,\n\nNauigation, Thaumaturgike and Archemastrie. I thinke it necessary,\n\norderly, of these to giue some peculier descriptions: and withall, to touch some of their commodious vses, and so to make this Preface, to be a little swete, pleasant Nosegaye for you: to comfort your Spirites, beyng almost out of courage, and in despayre, (through brutish brute) Weenyng that Geometrie, had but serued for buildyng of an house, or a curious bridge, or the roufe of Westminster hall, or some witty pretty deuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That the thing is farre otherwise, then the world, (commonly) to this day, hath demed, by worde and worke, good profe wilbe made.\n\nAmong these Artes, by good reason, Perspectiue ought to be had, ere of Astronomicall Apparences, perfect knowledge can be atteyned. And bycause of the prerogatiue of Light, beyng the first of Gods Creatures: and the eye, the light of our body, and his Sense most mighty, and his organ most Artificiall and Geometricall: At Perspectiue, we will begyn therfore. Perspectiue, is an Art Mathematicall, which demonstrateth the maner, and properties, of all Radiations Direct, Broken, and Reflected. This Description, or Notation, is brief: but it reacheth so farre, as the world is wyde. It concerneth all Creatures, all Actions, and passions, by Emanation of beames perfourmed. Beames, or naturall lines, (here) I meane, not of light onely, or of colour (though they, to eye, giue shew, witnes, and profe, wherby to ground the Arte vpon) but also of other Formes, both Substantiall, and Accidentall, the certaine and determined actiue Radiall emanations. By this Art (omitting to speake of the highest pointes) we may vse our eyes, and the light, with greater pleasure: and perfecter Iudgement: both of things, in light seen, & of other: which by like order of Lightes Radiations, worke and produce their effectes. We may be ashamed to be ignorant of the cause, why so sundry wayes our eye is deceiued, and abused: as, while the eye weeneth a roũd Globe or Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise iudgeth a plaine Square, to be roũd: supposeth walles parallels, to approche, a farre of: rofe and floure parallels, the one to bend downward, the other to rise vpward, at a little distance from you. Againe, of thinges being in like swiftnes of mouing, to thinke the nerer, to moue faster: and the farder, much slower. Nay, of two thinges, wherof the one (incomparably) doth moue swifter then the other, to deme the slower to moue very swift, & the other to stand: what an error is this, of our eye? Of the Raynbow, both of his Colours, of the order of the colours, of the bignes of it, the place and heith of it, (&c) to know the causes demonstratiue, is it not pleasant, is it not necessary? of two or three Sonnes appearing: of Blasing Sterres: and such like thinges: by naturall causes, brought to passe, (and yet neuertheles, of farder matter, Significatiue) is it not commodious for man to know the very true cause, & occasion Naturall? Yea, rather, is it not, greatly, against the Souerainty of Mans nature, to be so ouershot and abused, with thinges (at hand) before his eyes? as with a Pecockes tayle, and a Doues necke: or a whole ore, in water, holden, to seme broken. Thynges, farre of, to seeme nere: and nere, to seme farre of. Small thinges, to seme great: and great, to seme small. One man, to seme an Army. Or a man to be curstly affrayed of his owne shaddow. Yea, so much, to feare, that, if you, being (alone) nere a certaine glasse, and proffer, with dagger or sword, to foyne at the glasse, you shall suddenly be moued to giue backe (in maner) by reason of an Image,\n\n(☞ A marueilous Glasse.)\n\nappearing in the ayre, betwene you & the glasse, with like hand, sword or dagger, & with like quicknes, foyning at your very eye, likewise as you do at the Glasse. Straunge, this is, to heare of: but more meruailous to behold, then these my wordes can signifie. And neuerthelesse by demonstration Opticall, the order and cause therof, is certified: euen so, as the effect is consequent. Yea, thus much more, dare I take vpon me, toward the satisfying of the noble courrage, that longeth ardently for the wisedome of Causes Naturall: as to let him vnderstand, that, in London, he may with his owne eyes, haue profe of that, which I haue sayd herein. A Gentleman, (which, for his good seruice, done to his Countrey, is famous and honorable:\n\nand for skill in the Mathematicall Sciences, and Languages, is the Od man of this land. &c.) euen he, is hable: and (I am sure) will, very willingly, let the Glasse, and profe be sene: and so I (here) request him: for the encrease of wisedome, in the honorable: and for the stopping of the mouthes malicious: and repressing the arrogancy of the ignorant. Ye may easily gesse, what I meane. This Art of Perspectiue, is of that excellency, and may be led, to the certifying, and executing of such thinges, as no man would easily beleue: without Actuall profe perceiued. I speake nothing of Naturall Philosophie, which, without Perspectiue, can not be fully vnderstanded, nor perfectly atteined vnto. Nor, of Astronomie: which, without Perspectiue, can not well be grounded: Nor Astrologie, naturally Verified, and auouched. That part hereof, which dealeth with Glasses (which name, Glasse, is a generall name, in this Arte, for any thing, from which, a Beame reboundeth) is called Catoptrike: and hath so many vses, both merueilous, and proffitable: that, both, it would hold me to long, to note therin the principall conclusions, all ready knowne: And also (perchaunce) some thinges, might lacke due credite with you: And I, therby, to leese my labor: and you, to slip into light Iudgement,\n\n(* ☞)\n\nBefore you haue learned sufficiently the powre of Nature and Arte.\n\n### Astronomy\n\nNow, to procede: Astronomie, is an Arte Mathematicall, which demonstrateth the distance, magnitudes, and all naturall motions, apparences, and passions propre to the Planets and fixed Sterres: for any time past, present and to come: in respect of a certaine Horizon, or without respect of any Horizon. By this Arte we are certified of the distance of the Starry Skye, and of eche Planete from the Centre of the Earth: and of the greatnes of any Fixed starre sene, or Planete, in respect of the Earthes greatnes. As, we are sure (by this Arte) that the Solidity, Massines and Body of the Sonne, conteineth the quantitie of the whole Earth and Sea, a hundred thre score and two times, lesse by ⅛ one eight parte of the earth. But the Body of the whole earthly globe and Sea, is bigger then the body of the Mone, three and forty times lesse by ⅛ of the Mone. Wherfore the Sonne is bigger then the Mone, 7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then the Mone. And yet the vnskillfull man, would iudge them a like bigge. Wherfore, of Necessity, the one is much farder from vs, then the other. The Sonne, when he is fardest from the earth (which, now, in our age, is, when he is in the 8. degree, of Cancer) is, 1179 Semidiameters of the Earth, distante. And the Mone when she is fardest from the earth, is 68 Semidiameters of the earth and ⅓ The nerest, that the Mone commeth to the earth, is Semidiameters 52¼ The distance of the Starry Skye is, frõ vs, in Semidiameters of the earth 20081½ Twenty thousand fourescore, one, and almost a halfe. Subtract from this, the Mones nerest distance, from the Earth: and therof remaineth Semidiameters of the earth 20029¼ Twenty thousand nine and twenty and a quarter.\n\nSo thicke is the heauenly Palace, that the Planetes haue all their exercise in, and most meruailously perfourme the Commaũdement and Charge to them giuen by the omnipotent Maiestie of the king of kings. This is that, which in Genesis is called Ha Rakia. Consider it well. The Semidiameter of the earth, cõteineth of our common miles 3436 4/11 three thousand, foure hundred thirty six and foure eleuenth partes of one myle: Such as the whole earth and Sea, round about, is 21600. One and twenty thousand six hundred of our myles. Allowyng for euery degree of the greatest circle, thre score myles. Now if you way well with your selfe but this litle parcell of frute Astronomicall, as concerning the bignesse, Distances of Sonne, Mone, Sterry Sky, and the huge massines of Ha Rakia, will you not finde your Consciences moued, with the kingly Prophet, to sing the confession of Gods Glory, and say, The Heauens declare the glory of God, and the Firmament [Ha Rakia] sheweth forth the workes of his handes. And so forth, for those fiue first staues, of that kingly Psalme. Well, well, It is time for some to lay hold on wisedome, and to Iudge truly of thinges: and notso to expound the Holy word, all by Allegories: as to Neglect the wisedome, powre and Goodnes of God, in, and by his Creatures, and Creation to be seen and learned. By parables and Analogies of whose natures and properties, the course of the Holy Scripture, also, declareth to vs very many Mysteries. The whole Frame of Gods Creatures, (which is the whole world,) is to vs, a bright glasse: from which, by reflexion, reboundeth to our knowledge and perceiuerance, Beames, and Radiations: representing the Image of his Infinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taught and persuaded to Glorifie our Creator, as God: and be thankefull therfore. Could the Heathenistes finde these vses, of these most pure, beawtifull, and Mighty Corporall Creatures: and shall we, after that the true Sonne of rightwisenesse is risen aboue the Horizon, of our temporall Hemisphærie, and hath so abundantly streamed into our hartes, the direct beames of his goodnes, mercy, and grace: Whose heat All Creatures feele: Spirituall and Corporall: Visible and Inuisible. Shall we (I say) looke vpon the Heauen, Sterres, and Planets, as an Oxe and an Asse doth: no furder carefull or inquisitiue, what they are: why were they Created, How do they execute that they were Created for? Seing, All Creatures, were for our sake created: and both we, and they, Created, chiefly to glorifie the Almighty Creator: and that, by all meanes, to vs possible. Nolite ignorare (saith Plato in Epinomis) Astronomiam, Sapientissimũ quiddam esse. Be ye not ignorant, Astronomie to be a thyng of excellent wisedome. Astronomie, was to vs, from the beginning commended, and in maner commaunded by God him selfe. In asmuch as he made the Sonne, Mone, and Sterres, to be to vs, for Signes, and knowledge of Seasons, and for Distinctions of Dayes, and yeares. Many wordes nede not. But I wish, euery man should way this word, Signes. And besides that, conferre it also with the tenth Chapter of Hieremie. And though Some thinke, that there, they haue found a rod: Yet Modest Reason, will be indifferent Iudge, who ought to be beaten therwith, in respect of our purpose. Leauing that: I pray you vnderstand this: that without great diligence of Obseruation, examination and Calculation, their periods and courses (wherby Distinction of Seasons, yeares, and New Mones might precisely be knowne) could not exactely be certified. Which thing to performe, is that Art, which we here haue Defined to be Astronomie. Wherby, we may haue the distinct Course of Times, dayes, yeares, and Ages: aswell for Consideratiõ of Sacred Prophesies, accomplished in due time, foretold: as for high Mysticall Solemnities holding: And for all other humaine affaires, Conditions, and couenantes, vpon certaine time, betwene man and man: with many other great vses: Wherin, (verely), would be great incertainty, Confusion, vntruth, and brutish Barbarousnes: without the wonderfull diligence and skill of this Arte: continually learning, and determining Times, and periodes of Time, by the Record of the heauenly booke, wherin all times are written: and to be read with an Astronomicall staffe, in stede of a festue.\n\n### Music\n\nMusike, of Motion, hath his Originall cause: Therfore, after the motions most swift, and most Slow, which are in the Firmament, of Nature performed: and vnder the Astronomers Consideration: now I will Speake of an other kinde of Motion, producing sound, audible, and of Man numerable. Musike I call here that Science, which of the Grekes is called Harmonice. Not medling with the Controuersie betwene the auncient Harmonistes, and Canonistes. Musike is a Mathematicall Science, which teacheth, by sense and reason, perfectly to iudge, and order the diuersities of soundes, hye and low. Astronomie and Musike are Sisters, saith Plato. As, for Astronomie, the eyes: So, for Harmonious Motion, the eares were made. But as Astronomie hath a more diuine Contemplation, and cõmodity, then mortall eye can perceiue: So, is Musike to be considered,\n\n(1.)\n\nthat the * Minde may be preferred, before the eare. And from audible sound, we ought to ascende, to the examination: which numbers are Harmonious, and which not. And why, either, the one are: or the other are not. I could at large,\n\n(2.)\n\nin the heauenly * motions and distances, describe a meruailous Harmonie, of Pythagoras Harpe\n\n(3.)\n\nwith eight stringes. Also, somwhat might be sayd of Mercurius * two Harpes,\n\n(4.)\n\neche of foure Stringes Elementall. And very straunge matter, might be alledged of the Harmonie,\n\n(5.)\n\nto our * Spirituall part appropriate. As in Ptolomaus third boke, in the fourth and sixth Chapters may appeare. *\n\n(6.)\n\nAnd what is the cause of the apt bonde, and frendly felowship, of the Intellectuall and Mentall part of vs, with our grosse & corruptible body: but a certaine Meane, and Harmonious Spiritualitie, with both participatyng, & of both (in a maner) resultynge In\n\n(7.)\n\nthe * Tune of Mans voyce, and also\n\n(8.)\n\n* the sound of Instrument, what might be sayd, of Harmonie: No common Musicien would lightly beleue.\n\n(I. D. Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7. chapters. Where you shall haue some occasion farder to thinke of Musike, than commonly is thought.)\n\nBut of the sundry Mixture (as I may terme it) and concurse, diuerse collation, and Application of these Harmonies: as of thre, foure, fiue, or mo: Maruailous haue the effectes ben: and yet may be founde, and produced the like: with some proportionall consideration for our time, and being: in respect of the State, of the thinges then: in which, and by which, the wondrous effectes were wrought. Democritus and Theophrastus affirmed, that, by Musike, griefes and diseases of the Minde, and body might be cured, or inferred. And we finde in Recorde, that Terpander, Arion, Ismenias, Orpheus, Amphion, Dauid, Pythagoras, Empedocles, Asclepiades and Timotheus, by Harmonicall Consonãcy, haue done, and brought to pas, thinges, more then meruailous, to here of. Of them then, making no farder discourse, in this place: Sure I am, that Common Musike, commonly vsed, is found to the Musiciens and Hearers, to be so Commodious and pleasant, That if I would say and dispute, but thus much: That it were to be otherwise vsed, then it is, I should finde more repreeuers, then I could finde priuy, or skilfull of my meaning. In thinges therfore euident, and better knowen, then I can expresse: and so allowed and liked of, (as I would wish, some other thinges, had the like hap) I will spare to enlarge my lines any farder, but consequently follow my purpose.\n\n### Cosmography\n\nOf Cosmographie, I appointed briefly in this place, to geue you some intelligence. Cosmographie, is the whole and perfect description of the heauenly, and also elementall parte of the world, and their homologall application, and mutuall collation necessarie. This Art, requireth Astronomie, Geographie, Hydrographie and Musike. Therfore, it is no small Arte, nor so simple, as in common practise, it is (slightly) considered. This matcheth Heauen, and the Earth, in one frame, and aptly applieth parts Correspõdent: So, as, the Heauenly Globe, may (in practise) be duely described vpon the Geographicall, and Hydrographicall Globe. And there, for vs to consider an Æquonoctiall Circle, an Ecliptike line, Colures, Poles, Sterres in their true Longitudes, Latitudes, Declinations, and Verticalitie: also Climes, and Parallels: and by an Horizon annexed, and reuolution of the earthly Globe (as the Heauen, is, by the Primouant, caried about in 24. æquall Houres) to learne the Risinges and Settinges of Sterres (of Virgill in his Georgikes: of Hesiod: of Hippocrates in his Medicinall Sphære, to Perdicca King of the Macedonians: of Diocles, to King Antigonus, and of other famous Philosophers prescribed) a thing necessary, for due manuring of the earth, for Nauigation, for the Alteration of mans body: being, whole, Sicke, wounded, or brused. By the Reuolution, also, or mouing of the Globe Cosmographicall, the Rising and Setting of the Sonne: the Lengthes, of dayes and nightes: the Houres and times (both night and day) are knowne: with very many other pleasant and necessary vses: Wherof, some are knowne: but better remaine, for such to know and vse:\n\n(☞)\n\nwho of a sparke of true fire, can make a wonderfull bonfire, by applying of due matter, duely.\n\n### Astrology\n\nOf Astrologie, here I make an Arte, seuerall from Astronomie: not by new deuise, but by good reason and authoritie: for, Astrologie, is an Arte Mathematicall, which reasonably demonstrateth the operations and effectes, of the naturall beames, of light, and secrete influence: of the Sterres and Planets: in euery element and elementall body: at all times, in any Horizon assigned. This Arte is furnished with many other great Artes and experiences: As with perfecte Perspectiue, Astronomie, Cosmographie, Naturall Philosophie of the 4. Elementes, the Arte of Graduation, and some good vnderstãding in Musike: and yet moreouer, with an other great Arte, hereafter following, though I, here, set this before, for some considerations me mouing. Sufficient (you see) is the stuffe, to make this rare and secrete Arte, of: and hard enough to frame to the Conclusion Syllogisticall. Yet both the manifolde and continuall trauailes of the most auncient and wise Philosophers, for the atteyning of this Arte: and by examples of effectes, to confirme the same: hath left vnto vs sufficient proufe and witnesse: and we, also, daily may perceaue, That mans body, and all other Elementall bodies, are altered, disposed, ordred, pleasured, and displeasured, by the Influentiall working of the Sunne, Mone, and the other Starres and Planets. And therfore, sayth Aristotle, in the first of his Meteorologicall bookes, in the second Chapter: Est autem necessariò Mundus iste, supernis lationibus ferè continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causà prima putanda omnibus est, vnde motus principium existit. That is: This [Elementall] World is of necessitie, almost, next adioyning, to the heauenly motions: That, from thence, all his vertue or force may be gouerned. For, that is to be thought the first Cause vnto all: from which, the beginning of motion, is. And againe, in the tenth Chapter. Oportet igitur & horum principia sumamus, & causas omnium similiter. Principium igitur vt mouens, præcipuum[que] & omnium primum, Circulus ille est, in quo manifeste Solis latio, &c. And so forth. His Meteorologicall bookes, are full of argumentes, and effectuall demonstrations, of the vertue, operation, and power of the heauenly bodies, in and vpon the fower Elementes, and other bodies, of them (either perfectly, or vnperfectly) composed. And in his second booke, De Generatione & Corruptione, in the tenth Chapter. Quocirca & prima latio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea nam[que] & continua est, & duobus motibus fit: In Englishe, thus. Wherefore the vppermost motion, is not the cause of Generation and Corruption, but the motion of the Zodiake: for, that, both, is continuall, and is caused of two mouinges. And in his second booke, and second Chapter of hys Physikes. Homo nam[que] generat hominem, at[que] Sol. For Man (sayth he) and the Sonne, are cause of mans generation. Authorities may be brought, very many: both of 1000. 2000. yea and 3000. yeares Antiquitie: of great Philosophers, Expert, Wise, and godly men, for that Conclusion: which, daily and hourely, we men, may discerne and perceaue by sense and reason: All beastes do feele, and simply shew, by their actions and passions, outward and inward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally, the Elementes, and all thinges of the Elementes composed, do geue Testimonie (as Aristotle sayd) that theyr Whole Dispositions, vertues, and naturall motions, depend of the Actiuitie of the heauenly motions and Influences. Whereby, beside the specificall order and forme, due to euery seede: and beside the Nature, propre to the Indiuiduall Matrix, of the thing produced: What shall be the heauenly Impression, the perfect and circumspecte Astrologien hath to Conclude. Not onely (by Apotelesmes) τὸ ὁτὶ]. but by Naturall and Mathematicall demonstration τὸ διότι. Whereunto, what Sciences are requisite (without exception) I partly haue here warned: And in my Propædeumes (besides other matter there disclosed) I haue Mathematically furnished vp the whole Method: To this our age, not so carefully handled by any, that euer I saw, or heard of. I was,\n\n(* Anno. 1548 and 1549. in Louayn.)\n\n(for * 21. yeares ago) by certaine earnest disputations, of the Learned Gerardus Mercator, and Antonius Gogaua, (and other,) therto so prouoked: and (by my constant and inuincible zeale to the veritie) in obseruations of Heauenly Influencies (to the Minute of time,) than, so diligent: And chiefly by the Supernaturall influence, from the Starre of Iacob, so directed: That any Modest and Sober Student, carefully and diligently seking for the Truth, will both finde & cõfesse, therin, to be the Veritie, of these my wordes: And also become a Reasonable Reformer, of three Sortes of people: about these Influentiall Operations, greatly erring from the truth.\n\nWherof, the one, is Light Beleuers, the other, Light Despisers, and\n\nthe third Light Practisers. The first, & most cõmon Sort, thinke the Heauen and Sterres, to be answerable to any their doutes or desires:\n\n(1.)\n\nwhich is not so: and, in dede, they, to much, ouer reache. The Second sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare any Sway in Generation\n\n(2.)\n\nand Corruption, in this Elementall world. And to the Sunne, Mone and Sterres (being so many, so pure, so bright, so wonderfull bigge, so farre in distance, so manifold in their motions, so constant in their periodes. &c.) they assigne a sleight, simple office or two, and so allow vnto thẽ (according to their capacities) as much vertue, and power Influentiall, as to the Signe of the Sunne, Mone, and seuen Sterres, hanged vp (for Signes) in London, for distinction of houses, & such grosse helpes, in our worldly affaires: And they vnderstand not (or will not vnderstand) of the other workinges, and vertues of the Heauenly Sunne, Mone, and Sterres: not so much, as the Mariner, or Husband man: no, not so much, as the Elephant doth, as the Cynocephalus, as the Porpentine doth: nor will allow these perfect, and incorruptible mighty bodies, so much vertuall Radiation, & Force, as they see in a litle peece of a Magnes stone: which, at great distance, sheweth his operation. And perchaunce they thinke, the Sea & Riuers (as the Thames) to be some quicke thing, and so to ebbe, and flow, run in and out, of them selues, at their owne fantasies. God helpe, God helpe. Surely, these men, come to short: and either are to dull: or willfully blind: or, perhaps, to malicious. The third man, is the common and vulgare Astrologien, or Practiser: who, being not duely, artificially, and perfectly\n\n(3.)\n\nfurnished: yet, either for vaine glory, or gayne: or like a simple dolt, & blinde Bayard, both in matter and maner, erreth: to the discredit of the Wary, and modest Astrologien: and to the robbing of those most noble corporall Creatures, of their Naturall Vertue: being most mighty: most beneficiall to all elementall Generation, Corruption and the appartenances: and most Harmonious in their Monarchie: For which thinges, being knowen, and modestly vsed: we might highly, and continually glorifie God, with the princely Prophet, saying. The Heauens declare the Glorie of God: who made the Heauẽs in his wisedome: who made the Sonne, for to haue dominion of the day: the Mone and Sterres to haue dominion of the nyght: whereby, Day to day vttereth talke: and night, to night declareth knowledge. Prayse him, all ye Sterres, and Light. Amen.\n\n### Statics, 1\n\nIn order, now foloweth, of Statike, somewhat to say, what we meane by that name: and what commodity, doth, on such Art, depend. Statike, is an Arte Mathematicall, which demonstrateth the causes of heauynes, and lightnes of all thynges: and of motions and properties, to heauynes and lightnes, belonging. And for asmuch as, by the Bilanx, or Balance (as the chief sensible Instrument,) Experience of these demonstrations may be had: we call this Art, Statike: that is, the Experimentes of the Balance. Oh, that men wist, what proffit, (all maner of wayes) by this Arte might grow, to the hable examiner, and diligent practiser. “Thou onely, knowest all thinges precisely (O God) who hast made weight and Balance, thy Iudgement: who hast created all thinges in Number, Waight, and Measure: and hast wayed the mountaines and hils in a Balance: who hast peysed in thy hand, both Heauen and earth. We therfore warned by the Sacred word, to Consider thy Creatures: and by that consideration, to wynne a glyms (as it were,) or shaddow of perceiuerance, that thy wisedome, might, and goodnes is infinite, and vnspeakable, in thy Creatures declared: And being farder aduertised, by thy mercifull goodnes, that, three principall wayes, were, of the, vsed in Creation of all thy Creatures, namely, Number, Waight and Measure, And for as much as, of Number and Measure, the two Artes (auncient, famous, and to humaine vses most necessary,) are, all ready, sufficiently knowen and extant: This third key, we beseche thee (through thy accustomed goodnes,) that it may come to the nedefull and sufficient knowledge, of such thy Seruauntes, as in thy workemanship, would gladly finde, thy true occasions (purposely of the vsed) whereby we should glorifie thy name, and shew forth (to the weaklinges in faith) thy wondrous wisedome and Goodnes. Amen.”\n\nMeruaile 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.\n\n1.\n\nThe Superficies of euery Liquor, by it selfe consistyng, and in quyet, is Sphæricall: the centre whereof, is the same, which is the centre of the Earth.\n\n2.\n\nIf Solide Magnitudes, being of the same bignes, or quãtitie, that any Liquor is, and hauyng also the same Waight: be let downe into the same Liquor, they will settle downeward, so, that no parte of them, shall be aboue the Superficies of the Liquor: and yet neuertheles, they will not sinke vtterly downe, or drowne.\n\n3.\n\nIf any Solide Magnitude beyng Lighter then a Liquor, be let downe into the same Liquor, it will settle downe, so farre into the same Liquor, that so great a quantitie of that Liquor, as is the parte of the Solid Magnitude, settled downe into the same Liquor: is in Waight, æquall, to the waight of the whole Solid Magnitude.\n\n4.\n\nAny Solide Magnitude, Lighter then a Liquor, forced downe into the same Liquor, will moue vpward, with so great a power, by how much, the Liquor hauyng æquall quantitie to the whole Magnitude, is heauyer then the same Magnitude.\n\n5.\n\nAny Solid Magnitude, heauyer then a Liquor, beyng let downe into the same Liquor, will sinke downe vtterly: And wilbe in that Liquor, Lighter by so much, as is the waight or heauynes of the Liquor, hauing bygnes or quantitie, æquall to the Solid Magnitude.\n\n6.\n\n(I. D. The Cutting of a Sphære according to any proportion assigned may by this proposition be done Mechanically by tempering Liquor to a certayne waight in respect of the waight of the Sphære therein Swymming.)\n\nIf any Solide Magnitude, Lighter then a Liquor, be let downe into the same Liquor, the waight of the same Magnitude, will be, to the Waight of the Liquor. (Which is æquall in quantitie to the whole Magnitude,) in that proportion, that the parte, of the Magnitude settled downe, is to the whole Magnitude.\n\nBy these verities, great Errors may be reformed, in Opinion of the Naturall Motion of thinges, Light and Heauy. Which errors, are in Naturall Philosophie (almost) of all mẽ allowed: to much trusting to Authority: and false Suppositions. As, Of any two bodyes, the heauyer, to moue downward faster then the lighter.\n\n(A common error, noted.)\n\nThis 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.\n\nIf there be two bodyes of one forme, and of one kynde, æquall in quantitie or vnæquall,\n\n(A paradox.)\n\nthey will moue by æquall space, in æquall tyme: So that both theyr mouynges be in ayre, or both in water: or in any one Middle.\n\nHereupon, in the feate of Gunnyng,\n\ncertaine good discourses (otherwise) may receiue great amendement, and furderance.\n\n(The wonderfull vse of these Propositions.)\n\nIn the entended purpose, also, allowing somwhat to the imperfection of Nature: not aunswerable to the precisenes of demonstration. Moreouer, by the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth, the Fire, may be nerely, knowen, how light or heauy they are (Naturally) in their assigned partes: or in the whole. And then, to thinges Elementall, turning your practise: you may deale for the proportion of the Elementes, in the thinges Compounded. Then, to the proportions of the Humours in Man: their waightes: and the waight of his bones, and flesh. &c. Than, by waight, to haue consideration of the Force of man, any maner of way: in whole or in part. Then, may you, of Ships water drawing, diuersly, in the Sea and in fresh water, haue pleasant consideration: and of waying vp of any thing, sonken in Sea or in fresh water &c. And (to lift vp your head a loft:) by waight, you may, as precisely, as by any instrument els, measure the Diameters of Sonne and Mone. &c. Frende, I pray you, way these thinges, with the iust Balance of Reason. And you will finde Meruailes vpon Meruailes: And esteme one Drop of Truth (yea in Naturall Philosophie) more worth, then whole Libraries of Opinions, vndemonstrated: or not aunswering to Natures Law, and your experience. Leauing these thinges, thus: I will giue you two or three, light practises, to great purpose: and so finish my Annotation Staticall. In Mathematicall matters, by the Mechaniciens ayde, we will behold, here, the Commodity of waight.\n\n(The practise Staticall, to know the proportion, betwene the Cube, and the Sphære.)\n\nMake a Cube, of any one Vniforme: and through like heauy stuffe: of the same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall to the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne, Lead, Siluer. &c. (being, as I sayd, of like nature, condition, and like waight throughout.) And you may, by Say Balance, haue prepared a great number of the smallest waightes: which, by those Balance can be discerned or tryed: and so, haue proceded to make you a perfect Pyle, company & Number of waightes: to the waight of six, eight, or twelue pound waight: most diligently tryed, all. And of euery one, the Content knowen, in your least waight, that is wayable. [They that can not haue these waightes of precisenes: may, by Sand, Vniforme, and well dusted, make them a number of waightes, somewhat nere precisenes: by halfing euer the Sand: they shall, at length, come to a least common waight. Therein, I leaue the farder matter, to their discretion, whom nede shall pinche.] The Venetians consideration of waight, may seme precise enough: by eight descentes progressionall, * halfing, from a grayne.\n\n(I. D. * For, so, haue you.256. partes of a Graine.)\n\nYour Cube, Sphære, apt Balance, and conuenient waightes, being ready: fall to worke.❉. First, way your Cube. Note the Number of the waight. Way, after that, your Sphære. Note likewise, the Nũber of the waight. If you now find the waight of your Cube, to be to the waight of the Sphære, as 21. is to 11: Then you see, how the Mechanicien and Experimenter, without Geometrie and Demonstration, are (as nerely in effect) tought the proportion of the Cube to the Sphere: as I haue demonstrated it, in the end of the twelfth boke of Euclide. Often, try with the same Cube and Sphære. Then, chaunge, your Sphære and Cube, to an other matter: or to an other bignes: till you haue made a perfect vniuersall Experience of it. Possible it is, that you shall wynne to nerer termes, in the proportion.\n\nWhen you haue found this one certaine Drop of Naturall veritie, procede on, to Inferre, and duely to make assay, of matter depending. As, bycause it is well demonstrated, that a Cylinder, whose heith, and Diameter of his base, is æquall to the Diameter of the Sphære, is Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of the waight of the Sphære, adde halfe so much, as it is: and so haue you the number of the waight of that Cylinder. Which is also Comprehended of our former Cube: So, that the base of that Cylinder, is a Circle described in the Square, which is the base of our Cube. But the Cube and the Cylinder, being both of one heith, haue their Bases in the same proportion, in the which, they are, one to an other, in their Massines or Soliditie. But, before, we haue two numbers, expressing their Massines, Solidities, and Quantities, by waight: wherfore,\n\n(* The proportion of the Square to the Circle inscribed.)\n\nwe haue * the proportion of the Square, to the Circle, inscribed in the same Square. And so are we fallen into the knowledge sensible, and Experimentall of Archimedes great Secret: of him, by great trauaile of minde, sought and found. Wherfore, to any Circle giuen, you can giue a Square æquall:\n\n(* The Squaring of the Circle, Mechanically.)\n\n* as I haue taught, in my Annotation, vpon the first proposition of the twelfth boke, And likewise, to any Square giuen, you may giue a Circle æquall:\n\n(* To any Square geuen, to geue a Circle, equall.)\n\n* If you describe a Circle, which shall be in that proportion, to your Circle inscribed, as the Square is to the same Circle: This, you may do, by my Annotations, vpon the second proposition of the twelfth boke of Euclide, in my third Probleme there. Your diligence may come to a proportion, of the Square to the Circle inscribed, nerer the truth, then is the proportion of 14. to 11. And consider, that you may begyn at the Circle and Square, and so come to conclude of the Sphære, & the Cube, what their proportion is: as now, you came from the Sphære to the Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough one hole drawẽ, as the maner is) if you make a Square figure & way it: and then, describing theron, the Circle inscribed: & cut of, & file away, precisely (to the Circle) the ouerplus of the Square: you shall then, waying your Circle, see, whether the waight of the Square, be to your Circle, as 14. to 11. As I haue Noted, in the beginning of Euclides twelfth boke. &c. after this resort to my last proposition, vpon the last of the twelfth. And there, helpe your selfe, to the end. And, here, Note this, by the way.\n\n(Note Squaring of the Circle without knowledge of the proportion betwene Circumference and Diameter.)\n\nThat we may Square the Circle, without hauing knowledge of the proportion, of the Circumference to the Diameter: as you haue here perceiued. And otherwayes also, I can demonstrate it. So that, many haue cumberd them selues superfluously, by trauailing in that point first, which was not of necessitie, first: and also very intricate. And easily, you may, (and that diuersly) come to the knowledge of the Circumference: the Circles Quantitie, being first knowen. Which thing, I leaue to your consideration: making hast to despatch an other Magistrall Probleme: and to bring it, nerer to your knowledge, and readier dealing with, then the world (before this day,) had it for you, that I can tell of. And that is, A Mechanicall Dubblyng of the Cube: &c. Which may, thus, be done:\n\n(To Dubble the Cube redily: by Art Mechanicall: depending vppon Demonstration Mathematicall.)\n\nMake of Copper plates, or Tyn plates, a foursquare vpright Pyramis, or a Cone: perfectly fashioned in the holow, within. Wherin, let great diligence be vsed, to approche (as nere as may be) to the Mathematicall perfection of those figures. At their bases, let them be all open: euery where, els, most close, and iust to. From the vertex, to the Circumference of the base of the Cone: & to the sides of the base of the Pyramis:\n\n(I. D. The 4. sides of this Pyramis must be 4. Isosceles Triangles alike and æquall.)\n\nLet 4. straight lines be drawen, in the inside of the Cone and Pyramis: makyng at their fall, on the perimeters of the bases, equall angles on both sides them selues, with the sayd perimeters. These 4. lines (in the Pyramis: and as many, in the Cone) diuide: one, in 12. æquall partes: and an other, in 24. an other, in 60, and an other, in 100. (reckenyng vp from the vertex.) Or vse other numbers of diuision, as experience shall teach you.\n\n(I. D. * In all workinges with this Pyramis or Cone, Let their Situations be in all Pointes and Conditions, alike, or all one: while you are about one Worke. Els you will erre.)\n\n### Statics, 2\n\nThen, * set your Cone or Pyramis, with the vertex downward, perpendicularly, in respect of the Base. (Though it be otherwayes, it hindreth nothyng.) So let thẽ most stedily be stayed. Now, if there be a Cube, which you wold haue Dubbled. Make you a prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so proportiõ in respect of your Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the waight of them, in water, 3. or 4. times: at the least: what stuff so euer they be made of. Let not your Solid angle, at the vertex, be to sharpe: but that the water may come with ease, to the very vertex, of your hollow Cone or Pyramis. Put one of your Solid Cubes in a Balance apt: take the waight therof exactly in water. Powre that water, (without losse) into the hollow Pyramis or Cone, quietly. Marke in your lines, what numbers the water Cutteth: Take the waight of the same Cube againe: in the same kinde of water, which you had before:\n\n(I. D. * Consider well whan you must put your waters togyther: and whan, you must empty your first water, out of your Pyramis or Cone. Els you will erre.)\n\nput that* also, into the Pyramis or Cone, where you did put the first. Marke now againe, in what number or place of the lines, the water Cutteth them. Two wayes you may conclude your purpose: it is to wete, either by numbers or lines. By numbers: as, if you diuide the side of your Fundamentall Cube into so many æquall partes, as it is capable of, conueniently, with your ease, and precisenes of the diuision. For, as the number of your first and lesse line (in your hollow Pyramis or Cone,) is to the second or greater (both being counted from the vertex) so shall the number of the side of your Fundamentall Cube, be to the nũber belonging to the Radicall side, of the Cube, dubble to your Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it selfe, whether it be dubble or no, to the Cubik number of your Fundamentall Cube. By lines, thus: As your lesse and first line, (in your hollow Pyramis or Cone,) is to the second or greater, so let the Radical side of your Fundamẽtall Cube, be to a fourth proportionall line, by the 12. proposition, of the sixth boke of Euclide. Which fourth line, shall be the Rote Cubik, or Radicall side of the Cube, dubble to your Fundamentall Cube: which is the thing we desired.\n\n(☞ God be thanked for this Inuention, & the fruite ensuing.)\n\nFor this, may I (with ioy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking the holy and glorious Trinity: hauing greater cause therto, then\n\n(* Vitruuius. Lib. 9. Cap. 3.)\n\n* Archimedes had (for finding the fraude vsed in the Kinges Crowne, of Gold): as all men may easily Iudge: by the diuersitie of the frute following of the one, and the other. Where I spake before, of a hollow Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it with water, precisely full, and poure that water into your Pyramis or Cone. And here note the lines cutting in your Pyramis or Cone. Againe, fill your coffen, like as you did before. Put that Water, also, to the first. Marke the second cutting of your lines. Now, as you proceded before, so must you here procede.\n\n(* Note.)\n\n* And if the Cube, which you should Double, be neuer so great: you haue, thus, the proportion (in small) betwene your two litle Cubes: And then, the side, of that great Cube (to be doubled) being the third, will haue the fourth, found, to it proportionall: by the 12. of the sixth of Euclide.\n\n(Note, as concerning the Sphæricall Superficies of the Water.)\n\nNote, that all this while, I forget not my first Proposition Staticall, here rehearsed: that, the Superficies of the water, is Sphæricall. Wherein, vse your discretion: to the first line, adding a small heare breadth, more: and to the second, halfe a heare breadth more, to his length. For, you will easily perceaue, that the difference can be no greater, in any Pyramis or Cone, of you to be handled. Which you shall thus trye. For finding the swelling of the water aboue leuell.\n\n(☞)\n\n“Square the Semidiameter, from the Centre of the earth, to your first Waters Superficies. Square then, halfe the Subtendent of that watry Superficies (which Subtendent must haue the equall partes of his measure, all one, with those of the Semidiameter of the earth to your watry Superficies): Subtracte this square, from the first: Of the residue, take the Rote Square. That Rote, Subtracte from your first Semidiameter of the earth to your watry Superficies: that, which remaineth, is the heith of the water, in the middle, aboue the leuell.” Which, you will finde, to be a thing insensible. And though it were greatly sensible, *\n\n(* Note.)\n\nyet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides twelfth booke, noted: you may reduce all, to a true Leuell. But, farther diligence, of you is to be vsed, against accidentall causes of the waters swelling: as by hauing (somwhat) with a moyst Sponge, before, made moyst your hollow Pyramis or Cone, will preuent an accidentall cause of Swelling, &c. Experience will teach you abundantly: with great ease, pleasure, and cõmoditie.\n\nThus, may you Double the Cube Mechanically, Treble it, and so forth, in any proportion.\n\n(Note this Abridgement of Dubbling the Cube. &c.)\n\nNow will I Abridge your paine, cost, and Care herein. Without all preparing of your Fundamentall Cubes: you may (alike) worke this Conclusion. For, that, was rather a kinde of Experimentall demõstration, then the shortest way: and all, vpon one Mathematicall Demonstration depending. “Take water (as much as conueniently will serue your turne: as I warned before of your Fundamentall Cubes bignes) Way it precisely. Put that water, into your Pyramis or Cone. Of the same kinde of water, then take againe, the same waight you had before: put that likewise into the Pyramis or Cone. For, in eche time, your marking of the lines, how the Water doth cut them, shall geue you the proportion betwen the Radicall sides, of any two Cubes, wherof the one is Double to the other: working as before I haue taught you:\n\n(* ☞ Note.)\n\n* sauing that for you Fundamentall Cube his Radicall side: here, you may take a right line, at pleasure.”\n\nYet farther proceding with our droppe of Naturall truth:\n\n(To giue Cubes one to the other in any proportion, Rationall or Irrationall.)\n\nyou may (now) geue Cubes, one to the other, in any proportiõ geuẽ: Rationall or Irrationall: on this maner. Make a hollow Parallelipipedon of Copper or Tinne: with one Base wãting, or open: as in our Cubike Coffen. Frõ the bottome of that Parallelipipedon, raise vp, many perpendiculars, in euery of his fower sides. Now if any proportion be assigned you, in right lines: Cut one of your perpendiculars (or a line equall to it, or lesse then it) likewise: by the 10. of the sixth of Euclide. And those two partes, set in two sundry lines of those perpendiculars (or you may set them both, in one line) making their beginninges, to be, at the base: and so their lengthes to extend vpward. Now, set your hollow Parallelipipedon, vpright, perpendicularly, steadie. Poure in water, handsomly, to the heith of your shorter line. Poure that water, into the hollow Pyramis or Cone. Marke the place of the rising. Settle your hollow Parallelipipedon againe. Poure water into it: vnto the heith of the second line, exactly.\n\n(* Emptying the first.)\n\nPoure that water * duely into the hollow Pyramis or Cone: Marke now againe, where the water cutteth the same line which you marked before. For, there, as the first marked line, is to the second: So shall the two Radicall sides be, one to the other, of any two Cubes: which, in their Soliditie, shall haue the same proportion, which was at the first assigned: were it Rationall or Irrationall.\n\nThus, in sundry waies you may furnishe your selfe with such straunge and profitable matter: which, long hath bene wished for. And though it be Naturally done and Mechanically: yet hath it a good Demonstration Mathematicall.\n\n(The demonstrations of this Dubbling of the Cube, and of the rest.)\n\nWhich is this: Alwaies, you haue two Like Pyramids: or two Like Cones, in the proportions assigned: and like Pyramids or Cones, are in proportion, one to the other, in the proportion of their Homologall sides (or lines) tripled. Wherefore, if to the first, and second lines, found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in continuall proportion: that fourth line, shall be to the first, as the greater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of Euclide. If Pyramis to Pyramis, or Cone to Cone, be double,\n\n(I. D. * Hereby, helpe your self to become a præcise practiser. And so consider, how, nothing at all, you are hindred (sensibly) by the Conuexitie of the water.)\n\nthen shall * Line to Line, be also double, &c. But, as our first line, is to the second, so is the Radicall side of our Fundamentall Cube, to the Radicall side of the Cube to be made, or to be doubled: and therefore, to those twaine also, a third and a fourth line, in continuall proportion, ioyned: will geue the fourth line in that proportion to the first, as our fourth Pyramidall, or Conike line, was to his first: but that was double, or treble, &c. as the Pyramids or Cones were, one to an other (as we haue proued) therfore, this fourth, shalbe also double or treble to the first, as the Pyramids or Cones were one to an other: But our made Cube, is described of the second in proportion, of the fower proportionall lines:\n\n(* By the 33. of the eleuenth booke of Euclide.)\n\ntherfore * as the fourth line, is to the first, so is that Cube, to the first Cube: and we haue proued the fourth line, to be to the first, as the Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is to the Cube, as Pyramis is to Pyramis, or Cone is to Cone.\n\n(I. D. * And your diligence in practise, can so (in waight of water) performe it: Therefore, now, you are able to geue good reason of your whole doing.)\n\nBut we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double or treble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was to be demonstrated. And of the Parallelipipedõ, it is euidẽt, that the water Solide Parallelipipedons, are one to the other, as their heithes are, seing they haue one base. Wherfore the Pyramids or Cones, made of those water Parallelipipedons, are one to the other, as the lines are (one to the other) betwene which, our proportion was assigned. But the Cubes made of lines, after the proportiõ of the Pyramidal or Conik homologall lines, are one to the other, as the Pyramides or Cones are, one to the other (as we before did proue) therfore, the Cubes made, shalbe one to the other, as the lines assigned, are one to the other: Which was to be demonstrated. Note.\n\n(* Note this Corollary.)\n\n* This, my Demonstratiõ is more generall, then onely in Square Pyramis or Cone: Consider well. Thus, haue I, both Mathematically and Mechanically, ben very long in wordes: yet (I trust) nothing tedious to them, who, to these thinges, are well affected. And verily I am forced (auoiding prolixitie) to omit sundry such things, easie to be practised: which to the Mathematicien, would be a great Threasure: and to the Mechanicien, no small gaine.\n\n(* The great Commodities following of these new Inuentions.)\n\n* Now may you, Betwene two lines giuen, finde two middle proportionals, in Continuall proportion: by the hollow Parallelipipedon, and the hollow Pyramis, or Cone. Now, any Parallelipipedon rectangle being giuen: thre right lines may be found, proportionall in any proportion assigned, of which, shal be produced a Parallelipipedon, æquall to the Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36. proposition, of the 11. boke of Euclide. Now, all those thinges, which Vitruuius in his Architecture, specified hable to be done, by dubbling of the Cube: Or, by finding of two middle proportionall lines, betwene two lines giuen, may easely be performed. Now, that Probleme, which I noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke of Euclide, is proued possible. Now, may any regular body, be Transformed into an other, &c. Now, any regular body: any Sphere, yea any Mixt Solid: and (that more is) Irregular Solides, may be made (in any proportiõ assigned) like vnto the body, first giuen. Thus, of a Manneken, (as the Dutch Painters terme it) in the same Symmetrie, may a Giant be made: and that, with any gesture, by the Manneken vsed: and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make one, of the same Mould (in any assigned proportion) bigger or lesser.\n\n(* ☞)\n\nNow, may you, of any * Gunne, or little peece of ordinaũce, make an other, with the same Symmetrie (in all pointes) as great, and as little, as you will. Marke that: and thinke on it. Infinitely, may you apply this, so long sought for, and now so easily concluded: and withall, so willingly and frankly communicated to such, as faithfully deale with vertuous studies.\n\n(Such is the Fruite of the Mathematicall Sciences and Artes.)\n\n### Statics, 3\n\nThus, can the Mathematicall minde, deale Speculatiuely in his own Arte: and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he can, by order, Descend, to frame Naturall thinges, to wonderfull vses: and when he list, retire home into his owne Centre: and there, prepare more Meanes, to Ascend or Descend by: and, all, to the glory of God, and our honest delectation in earth.\n\nAlthough, the Printer, hath looked for this Præface, a day or two, yet could I not bring my pen from the paper, before I had giuen you comfortable warning, and brief instructions, of some of the Commodities, by Statike, hable to be reaped: In the rest, I will therfore, be as brief, as it is possible: and with all, describing them, somwhat accordingly. And that, you shall perceiue, by this, which in order commeth next. For, wheras, it is so ample and wonderfull, that, an whole yeare long, one might finde fruitfull matter therin, to speake of: and also in practise, is a Threasure endeles: yet will I glanse ouer it, with wordes very few.\n\n### Anthropography\n\nThis do I call Anthropographie. Which is an Art restored, and of my preferment to your Seruice. I pray you, thinke of it, as of one of the chief pointes, of Humane knowledge. Although it be, but now, first Cõfirmed, with this new name: yet the matter, hath from the beginning, ben in consideration of all perfect Philosophers. Anthropographie, is the description of the Number, Measure, Waight, figure, Situation, and colour of euery diuerse thing, conteyned in the perfect body of MAN: with certain knowledge of the Symmetrie, figure, waight, Characterization, and due locall motion, of any parcell of the sayd body, assigned: and of Nũbers, to the sayd parcell appertainyng. This, is the one part of the Definition, mete for this place: Sufficient to notifie, the particularitie, and excellency of the Arte: and why it is, here, ascribed to the Mathematicals. Yf the description of the heauenly part of the world, had a peculier Art, called Astronomie: If the description of the earthly Globe, hath his peculier arte, called Geographie. If the Matching of both, hath his peculier Arte, called Cosmographie: Which is the Descriptiõ of the whole, and vniuersall frame of the world: Why should not the description of\n\n(MAN is the Lesse World.)\n\nhim, who is the Lesse world: and, frõ the beginning, called Microcosmus (that is. The Lesse World.) And for whose sake, and seruice, all bodily creatures els, were created: Who, also, participateth with Spirites, and Angels: and is made to the Image and similitude of God: haue his peculier Art? and be called the Arte of Artes: rather, then, either to want a name, or to haue to base and impropre a name? You must of sundry professions, borow or challenge home, peculier partes hereof: and farder procede: as, God, Nature, Reason and Experience shall informe you. The Anatomistes will restore to you, some part: The Physiognomistes, some: The Chyromantistes some. The Metaposcopistes, some: The excellent, Albert Durer, a good part: the Arte of Perspectiue, will somwhat, for the Eye, helpe forward: Pythagoras, Hipocrates, Plato, Galenus, Meletius, & many other (in certaine thinges) will be Contributaries. And farder, the Heauen, the Earth, and all other Creatures, will eche shew, and offer their Harmonious seruice, to fill vp, that, which wanteth hereof: and with your own Experience, concluding: you may Methodically register the whole, for the posteritie: Whereby, good profe will be had, of our Harmonious, and\n\n(Micro Cosmus.)\n\nMicrocosmicall constitution.\n\n(* ☞)\n\nThe 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.\n\n(* Lib. 3. Cap. 1.)\n\nLooke 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,\n\n(☞)\n\nAll 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.\n\n### Trochilike\n\nTrochilike, 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:\n\n(Saw Milles.)\n\nAs Milles, to Saw great and very long Deale bordes, no man being by. Such haue I seene in Germany: and in the Citie of Prage: in the kingdome of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight, Spring, Man or Beast, moued. Take in your hand, Agricola De re Metallica: and then shall you (in all Mines) perceaue, how great nede is, of Whele worke. By Wheles, straunge workes and incredible, are done: as will, in other Artes hereafter, appeare. A wonderfull example of farther possibilitie, and present commoditie, was sene in my time, in a certaine Instrument: which by the Inuenter and Artificer (before) was solde for xx. Talentes of Golde: and then had (by misfortune) receaued some iniurie and hurt: And one Ianellus of Cremona did mend the same, and presented it vnto the Emperour Charles the fifth. Hieronymus Cardanus, can be my witnesse, that therein, was one Whele, which moued, and that, in such rate, that, in 7000. yeares onely, his owne periode should be finished. A thing almost incredible: But how farre, I keepe me within my boundes: very many men (yet aliue) can tell.\n\n### Helicosophy\n\nHelicosophie, is nere Sister to Trochilike: and is, An Arte\n\nMathematicall, which demonstrateth the designing of all Spirall lines in Plaine, on Cylinder, Cone, Sphære, Conoid, and Sphæroid, and their properties appertayning. The vse hereof, in Architecture, and diuerse Instrumentes and Engines, is most necessary. For, in many thinges, the Skrue worketh the feate, which, els, could not be performed. By helpe hereof,\n\n(* Atheneus Lib. 5. cap. 8.)\n\nit 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.\n\n(Proclus. Pag. 18.)\n\nWherat, the King wondring: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει λέγοντι πιϛευτεόν. From this day, forward (said the King) Credit ought to be giuen to Archimedes, what soeuer he sayth.\n\n### Pneumatithmie\n\nPneumatithmie demonstrateth by close hollow Geometricall Figures, (regular and irregular) the straunge properties (in motion or stay) of the Water, Ayre, Smoke, and Fire, in theyr cõtinuitie, and as they are ioyned to the Elementes next them. This Arte, to the Naturall Philosopher, is very proffitable: to proue, that Vacuum, or Emptines is not in the world. And that, all Nature, abhorreth it so much: that, contrary to ordinary law, the Elementes will moue or stand. As, Water to ascend: rather then betwene him and Ayre, Space or place should be left, more then (naturally) that quãtitie of Ayre requireth, or can fill. Againe, Water to hang, and not descend: rather then by descending, to leaue Emptines at his backe. The like, is of Fire and Ayre: they will descend: when, either, their Cõtinuitie should be dissolued: or their next Element forced from them. And as they will not be extended, to discontinuitie: So, will they not, nor yet of mans force, can be prest or pent, in space, not sufficient and aunswerable to their bodily substance. Great force and violence will they vse, to enioy their naturall right and libertie.\n\n(To go to the bottom of the Sea without daunger.)\n\nHereupon, 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.\n\n### Menadrie\n\nMenadrie, is an Arte Mathematicall, which demonstrateth, how, aboue Natures vertue and power simple: Vertue and force may be multiplied: and so, to direct, to lift, to pull to, and to put or cast fro, any multiplied or simple, determined Vertue, Waight or Force: naturally, not, so, directible or moueable. Very much is this Art furdred by other Artes: as, in some pointes, by Perspectiue: in some, by Statike: in some, by Trochilike: and in other, by Helicosophie: and Pneumatithmie. By this Art, all Cranes, Gybbettes, & Ingines to lift vp, or to force any thing, any maner way, are ordred: and the certaine cause of their force, is knowne: As, the force which one man hath with the Duche waghen Racke: therwith, to set vp agayne, a mighty waghen laden, being ouerthrowne. The force of the Crossebow Racke, is certainly, here, demonstrated. The reason, why one mã, doth with a leauer, lift that, which Sixe men, with their handes onely, could not, so easily do. By this Arte, in our common Cranes in London, where powre is to Crane vp, the waight of 2000. pound: by two Wheles more (by good order added) Arte concludeth, that there may be Craned vp 200000. pound waight &c. So well knew Archimedes this Arte: that he alone, with his deuises and engynes, (twise or thrise) spoyled and discomfited the whole Army and Hoste of the Romaines, besieging Syracusa,\n\n(Plutarchus in Marco Marcello.)\n\nMarcus Marcellus the Consul, being their Generall Capitaine.\n\n(Synesius in Epistolis.)\n\nSuch huge Stones, so many, with such force, and so farre, did he with his engynes hayle among them, out of the Citie.\n\n(Polybius.)\n\n(Plinius.)\n\n(Quintilianus.)\n\n(T. Liuius.)\n\nAnd by Sea likewise: though their Ships might come to the walls of Syracusa, yet hee vtterly confounded the Romaine Nauye. What with his mighty Stones hurlyng:\n\n(* Athenæus.)\n\nwhat with Pikes of * 18 fote long, made like shaftes: which he forced almost a quarter of a myle. What, with his catchyng hold of their Shyps, and hoysing them vp aboue the water, and suddenly letting them fall into the Sea againe:\n\n(* Galenus.)\n\n(Anthemius.)\n\nwhat with his * Burning Glasses: by which he fired their other Shippes a far-of: what, with his other pollicies, deuises, and engines, he so manfully acquit him selfe: that all the Force, courage, and pollicie of the Romaines (for a great season) could nothing preuaile, for the winning of Syracusa. Wherupon, the Romanes named Archimedes, Briareus, and Centimanus. Zonaras maketh mention of one Proclus, who so well had perceiued Archimedes Arte of Menadrie, and had so well inuented of his owne, that with his Burning Glasses,\n\n(Burning Glasses.)\n\nbeing placed vpon the walles of Bysance, he multiplied so the heate of the Sunne, and directed the beames of the same against his enemies Nauie with such force, and so sodeinly (like lightening) that he burned and destroyed both man and ship. And Dionspecifieth of Priscus, a Geometricien in Bysance, who inuented and vsed sondry Engins, of Force multiplied: Which was cause, that the Emperour Seuerus pardoned him, his life, after he had wonne Bysance: Bycause he honored the Arte, wytt, and rare industrie of Priscus. But nothing inferior to the inuention of these engines of Force, was the inuention of Gunnes.\n\n(Gunnes.)\n\nWhich, from an English man, had the occasion and order of first inuenting: though in an other land, and by other men, it was first executed. And they that should see the record, where the occasion and order generall, of Gunning, is first discoursed of, would thinke: that, “small thinges, slight, and cõmon: comming to wise mens consideration, and industrious mens handling, may grow to be of force incredible.”\n\n### Hypogeiodie\n\nHypogeiodie, is an Arte Mathematicall, demonstratyng, how, vnder the Sphæricall Superficies of the earth, at any depth, to any perpendicular line assigned (whose distance from the perpendicular of the entrance: and the Azimuth, likewise, in respect of the said entrance, is knowen) certaine way may be præscribed and gone: And how, any way aboue the Superficies of the earth designed, may vnder earth, at any depth limited, be kept: goyng alwayes, perpendicularly, vnder the way, on earth designed: And, contrarywise, Any way, (straight or croked,) vnder the earth, beyng giuen: vppon the vtface, or Superficies of the earth, to Lyne out the same: So, as, from the Centre of the earth, perpendiculars drawen to the Sphæricall Superficies of the earth, shall precisely fall in the Correspondent pointes of those two wayes. This, with all other Cases and circumstances herein, and appertenances, this Arte demonstrateth. This Arte, is very ample in varietie of Conclusions: and very profitable sundry wayes to the Common Wealth. The occasion of my Inuenting this Arte, was at the request of two Gentlemen, who had a certaine worke (of gaine) vnder ground: and their groundes did ioyne ouer the worke: and by reason of the crokednes, diuers depthes, and heithes of the way vnder ground, they were in doubt, and at controuersie, vnder whose ground, as then, the worke was. The name onely (before this) was of me published, De Itinere Subterraneo: The rest, be at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole, and for secrete passages vnder ground, betwene place and place (as this land hath diuerse) and for other purposes, any man may easily perceaue, both the great fruite of this Arte, and also in this Arte, the great aide of Geometrie.\n\n### Hydragogie\n\nHydragogie, demonstrateth the possible leading of Water, by Natures lawe, and by artificiall helpe, from any head (being a Spring, standing, or running Water) to any other place assigned. Long, hath this Arte bene in vse: and much thereof written: and very marueilous workes therein, performed: as may yet appeare, in Italy: by the Ruynes remaining of the Aqueductes. In other places, of Riuers leading through the Maine land, Nauigable many a Mile. And in other places, of the marueilous forcinges of Water to Ascend. which all, declare the great skill, to be required of him, who should in this Arte be perfecte, for all occasions of waters possible leading. To speake of the allowance of the Fall, for euery hundred foote: or of the Ventills (if the waters labour be farre, and great) I neede not: Seing, at hand (about vs) many expert men can sufficiently testifie, in effecte, the order: though the Demonstration of the Necessitie thereof, they know not: Nor yet, if they should be led, vp and downe, and about Mountaines, from the head of the Spring: and then, a place being assigned: and of them, to be demaunded, how low or high, that last place is, in respecte of the head, from which (so crokedly, and vp and downe) they be come: Perhaps, they would not, or could not, very redily, or nerely assoyle that question. Geometrie therefore, is necessary to Hydragogie. Of the sundry wayes to force water to ascend, eyther by Tympane, Kettell mills, Skrue, Ctesibike, or such like: in Vitruuius, Agricola, (and other,) fully, the maner may appeare. And so, thereby, also be most euident, how the Artes, of Pneumatithmie, Helicosophie, Statike, Trochilike, and Menadrie, come to the furniture of this, in Speculation, and to the Commoditie of the Common Wealth, in practise.\n\n### Horometry\n\nHorometrie, is an Arte Mathematicall, which demõstrateth, how, at all times appointed, the precise vsuall denominatiõ of time, may be knowen, for any place assigned. These wordes, are smoth and plaine easie Englishe, but the reach of their meaning, is farther, then you woulde lightly imagine. Some part of this Arte, was called in olde time, Gnomonice: and of late, Horologiographia: and in Englishe, may be termed, Dialling. Auncient is the vse, and more auncient, is the Inuention. The vse, doth well appeare to haue bene (at the least) aboue two thousand and three hundred yeare agoe:\n\n(4. Reg. 20.)\n\nin * King Achaz Diall, then, by the Sunne, shewing the distinction of time. By Sunne, Mone, and Sterres, this Dialling may be performed, and the precise Time of day or night knowen. But the demonstratiue delineation of these Dialls, of all sortes, requireth good skill, both of Astronomie, and Geometrie Elementall, Sphæricall, Phænomenall, and Conikall. Then, to vse the groundes of the Arte, for any regular Superficies, in any place offred: and (in any possible apt position therof) theron, to describe (all maner of wayes) how, vsuall howers, may be (by the Sunnes shadow) truely determined: will be found no sleight Painters worke. So to Paint, and prescribe the Sunnes Motion, to the breadth of a heare. In this Feate (in my youth) I Inuented a way, How in any Horizontall, Murall, or Æquinoctiall Diall, &c. At all howers (the Sunne shining) the Signe and Degree ascendent, may be knowen. Which is a thing very necessary for the Rising of those fixed Sterres: whose Operation in the Ayre, is of great might, euidently. I speake no further, of the vse hereof. Bur forasmuch as, Mans affaires require knowledge of Times & Momentes, when, neither Sunne, Mone, or Sterre, can be sene: Therefore, by Industrie Mechanicall, was inuented, first, how, by Water, running orderly, the Time and howers might be knowen: whereof, the famous Ctesibius, was Inuentor: a man, of Vitruuius, to the Skie (iustly) extolled. Then, after that, by Sand running, were howers measured: Then, by Trochilike with waight: And of late time, by Trochilike with Spring: without waight. All these, by Sunne or Sterres direction (in certaine time) require ouersight and reformation, according to the heauenly Æquinoctiall Motion: besides the inæqualitie of their owne Operation. There remayneth (without parabolicall meaning herein) among the Philosophers,\n\n(A perpetuall Motion.)\n\na more excellent, more commodious, and more marueilous way, then all these: of hauing the motion of the Primouant (or first æquinoctiall motion,) by Nature and Arte, Imitated: which you shall (by furder search in waightier studyes) hereafter, vnderstand more of. And so, it is tyme to finish this Annotation, of Tymes distinction, vsed in our common, and priuate affaires: The commoditie wherof, no man would want, that can tell, how to bestow his tyme.\n\n### Zography\n\nZographie, is an Arte Mathematicall, which teacheth and demonstrateth, how, the Intersection of all visuall Pyramides, made by any playne assigned, (the Centre, distance, and lightes, beyng determined) may be, by lynes, and due propre colours, represented. A notable Arte, is this: and would require a whole Volume, to declare the property thereof: and the Commodities ensuyng. Great skill of Geometrie, Arithmetike, Perspectiue, and Anthropographie, with many other particular Artes, hath the Zographer, nede of, for his perfection. For, the most excellent Painter, (who is but the propre Mechanicien, & Imitator sensible, of the Zographer) hath atteined to such perfection, that Sense of Man and beast, haue iudged thinges painted, to be things naturall, and not artificiall: aliue, and not dead. This Mechanicall Zographer (commonly called the Painter) is meruailous in his skill: and seemeth to haue a certaine diuine power: As, of frendes absent, to make a frendly, present comfort: yea, and of frendes dead, to giue a continuall, silent presence: not onely with vs, but with our posteritie, for many Ages. And so procedyng, Consider, How, in Winter, he can shew you, the liuely vew of Sommers Ioy, and riches: and in Sommer, exhibite the countenance of Winters dolefull State, and nakednes. Cities, Townes, Fortes, Woodes, Armyes, yea whole Kingdomes (be they neuer so farre, or greate) can he, with ease, bring with him, home (to any mans Iudgement) as Paternes liuely, of the thinges rehearsed. In one little house, can he, enclose (with great pleasure of the beholders,) the portrayture liuely, of all visible Creatures, either on earth, or in the earth, liuing: or in the waters lying, Creping, slyding, or swimming: or of any foule, or fly, in the ayre flying. Nay, in respect of the Starres, the Skie, the Cloudes: yea, in the shew of the very light it selfe (that Diuine Creature) can he match our eyes Iudgement, most nerely. What a thing is this? thinges not yet being, he can represent so, as, at their being, the Picture shall seame (in maner) to haue Created them. To what Artificer, is not Picture, a great pleasure and Commoditie? Which of them all, will refuse the Direction and ayde of Picture? The Architect, the Goldsmith, and the Arras Weauer: of Picture, make great account. Our liuely Herbals, our portraitures of birdes, beastes, and fishes: and our curious Anatomies, which way, are they most perfectly made, or with most pleasure, of vs beholden? Is it not, by Picture onely? And if Picture, by the Industry of the Painter, be thus commodious and meruailous: what shall be thought of Zographie, the Scholemaster of Picture, and chief gouernor? Though I mencion not Sculpture, in my Table of Artes Mathematicall: yet may all men perceiue, How, that Picture and Sculpture, are Sisters germaine: and both, right profitable, in a Commõ wealth. and of Sculpture, aswell as of Picture, excellent Artificers haue written great bokes in commendation. Witnesse I take, of Georgio Vasari, Pittore Aretino: of Pomponius Gauricus: and other. To these two Artes, (with other,) is a certaine od Arte, called Althalmasat, much beholdyng: more, then the common Sculptor, Entayler, Keruer, Cutter, Grauer, Founder, or Paynter (&c) know their Arte, to be commodious.\n\n### Architecture\n\n(An objection.)\n\nArchitecture, to many may seme not worthy, or not mete, to be reckned among the Artes Mathematicall. To whom, I thinke good, to giue some account of my so doyng. Not worthy, (will they say,) bycause it is but for building, of a house, Pallace, Church, Forte, or such like, grosse workes. And you, also, defined the Artes Mathematicall, to be such, as dealed with no Materiall or corruptible thing: and also did demonstratiuely procede in their faculty, by Number or Magnitude. First,\n\n(The Answer.)\n\nyou see, that I count, here, Architecture, among those Artes Mathematicall, which are Deriued from the Principals: and you know, that such, may deale with Naturall thinges, and sensible matter. Of which, “some draw nerer, to the Simple and absolute Mathematicall Speculation, then other do.\n\n(☞)\n\nAnd though, the Architect procureth, enformeth, & directeth, the Mechanicien, to handworke, & the building actuall, of house, Castell, or Pallace, and is chief Iudge of the same: yet, with him selfe (as chief Master and Architect,) remaineth the Demonstratiue reason and cause, of the Mechaniciens worke: in Lyne, plaine, and Solid: by Geometricall, Arithmeticall, Opticall, Musicall, Astronomicall, Cosmographicall” (& to be brief) by all the former Deriued Artes Mathematicall, and other Naturall Artes, hable to be confirmed and stablished. If this be so: then, may you thinke, that Architecture, hath good and due allowance, in this honest Company of Artes Mathematicall Deriuatiue. I will, herein, craue Iudgement of two most perfect Architectes: the one, being Vitruuius, the Romaine: who did write ten bookes thereof, to the Emperour Augustus (in whose daies our Heauenly Archemaster, was borne): and the other, Leo Baptista Albertus, a Florentine: who also published ten bookes therof. Architectura (sayth Vitruuius) est Scientia pluribus disciplinis & varijs eruditionibus ornata: cuius Iudicio probantur omnia, quæ ab cæteris Artificibus perficiuntur opera. That is. Architecture, is a Science garnished with many doctrines & diuerse instructions: by whose Iudgement, all workes, by other workmen finished, are Iudged. It followeth. Ea nascitur ex Fabrica, & Ratiocinatione. &c. Ratiocinatio autem est, quæ, res fabricatas, Solertia ac ratione proportionis, demonstrare at[que] explicare potest. Architecture, groweth of Framing, and Reasoning. &c. Reasoning, is that, which of thinges framed, with forecast, and proportion: can make demonstration, and manifest declaration. Againe. Cùm, in omnibus enim rebus, tùm maximè etiam in Architectura, hæc duo insunt: quod significatur, & quod significat. Significatur proposita res, de qua dicitur: hanc autem Significat Demonstratio, rationibus doctrinarum explicata. Forasmuch as, in all thinges: therefore chiefly in Architecture, these two thinges are: the thing signified: and that which signifieth. The thing propounded, whereof we speake, is the thing Signified. But Demonstration, expressed with the reasons of diuerse doctrines, doth signifie the same thing. After that. Vt literatus sit, peritus Graphidos, eruditus Geometriæ, & Optices non ignarus: instructus Arithmetica: historias complures nouerit, Philosophos diligenter audiuerit: Musicam sciuerit: Medicinæ non sit ignarus, responsa Iurisperitorũ nouerit: Astrologiam, Cæli[que]\n\nrationes cognitas habeat. An Architect (sayth he) ought to\n\nvnderstand Languages, to be skilfull of Painting, well instructed in Geometrie, not ignorant of Perspectiue, furnished with Arithmetike, haue knowledge of many histories, and diligently haue heard Philosophers, haue skill of Musike, not ignorant of Physike, know the aunsweres of Lawyers, and haue Astronomie, and the courses Cælestiall, in good knowledge. He geueth reason, orderly, wherefore all these Artes, Doctrines, and Instructions, are requisite in an excellent Architect. And (for breuitie) omitting the Latin text, thus he hath. Secondly, it is behofefull for an Architect to haue the knowledge of Painting: that he may the more easilie fashion out, in patternes painted, the forme of what worke he liketh. And Geometrie, geueth to Architecture many helpes: and first teacheth the Vse of the Rule, and the Cumpasse: wherby (chiefly and easilie) the descriptions of Buildinges, are despatched in Groundplats: and the directions of Squires, Leuells, and Lines. Likewise, by Perspectiue, the Lightes of the heauen, are well led, in the buildinges: from certaine quarters of the world. By Arithmetike, the charges of Buildinges are summed together: the measures are expressed, and the hard questions of Symmetries, are by Geometricall Meanes and Methods discoursed on. &c. Besides this, of the Nature of thinges (which in Greke is called φυσιολογία) Philosophie doth make declaration. Which, it is necessary, for an Architect, with diligence to haue learned: because it hath many and diuers naturall questions: as specially, in Aqueductes. For in their courses, leadinges about, in the leuell ground, and in the mountinges, the naturall Spirites or breathes are ingendred diuers wayes: The hindrances, which they cause, no man can helpe, but he, which out of Philosophie, hath learned the originall causes of thinges. Likewise, who soeuer shall read Ctesibius, or Archimedes bookes, (and of others, who haue written such Rules) can not thinke, as they do: vnlesse he shall haue receaued of Philosophers, instructions in these thinges. And Musike he must nedes know: that he may haue vnderstanding, both of Regular and Mathematicall Musike: that he may temper well his Balistes, Catapultes, and Scorpions. &c. Moreouer, the Brasen Vessels, which in Theatres, are placed by Mathematicall order, in ambries, vnder the steppes: and the diuersities of the soundes (which y^e Grecians call ηχεῖα) are ordred according to Musicall Symphonies & Harmonies: being distributed in y^e Circuites, by Diatessaron, Diapente, and Diapason. That the conuenient voyce, of the players sound, whẽ it came to these preparations, made in order, there being increased: with y^t increasing, might come more cleare & pleasant, to y^e eares of the lokers on. &c. And of Astronomie, is knowẽ y^e East, West, South, and North. The fashion of the heauen, the Æquinox, the Solsticie, and the course of the sterres. Which thinges, vnleast one know: he can not perceiue, any thyng at all, the reason of Horologies. Seyng therfore this ample Science, is garnished, beautified and stored, with so many and sundry skils and knowledges: I thinke, that none can iustly account them selues Architectes, of the suddeyne. But they onely, who from their childes yeares, ascendyng by these degrees of knowledges, beyng fostered vp with the atteynyng of many Languages and Artes, haue wonne to the high Tabernacle of Architecture. &c. And to whom Nature hath giuen such quicke Circumspection, sharpnes of witt, and Memorie, that they may be very absolutely skillfull in Geometrie, Astronomie, Musike, and the rest of the Artes Mathematicall: Such, surmount and passe the callyng, and state, of Architectes:\n\n(A Mathematicien.)\n\nand are become Mathematiciens. &c. And they are found, seldome. As, in tymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes: Apollonius Pergęus: Eratosthenes Cyreneus: Archimedes, and Scopas, Syracusians. Who also, left to theyr posteritie, many Engines and Gnomonicall workes: by numbers and naturall meanes, inuented and declared.\n\nThus much, and the same wordes (in sense) in one onely Chapter of this Incõparable Architect Vitruuius, shall you finde. And if you should, but take his boke in your hand, and slightly loke thorough it, you would say straight way:\n\n(Vitruuius.)\n\nThis is Geometrie, Arithmetike, Astronomie, Musike, Anthropographie, Hydragogie, Horometrie. &c. and (to cõclude) the Storehouse of all workmãship. Now, let vs listen to our other Iudge, our Florentine, Leo Baptista: and narrowly consider, how he doth determine of Architecture. Sed ante[que] vltra progrediar. &c. But before I procede any further (sayth he) I thinke, that I ought to expresse, what man I would haue to bee allowed an Architect. For, I will not bryng in place a Carpenter: as though you might Compare him to the Chief Masters of other Artes. For the hand of the Carpenter, is the Architectes Instrument.\n\n(VVho is an Architect.)\n\nBut I will appoint the Architect to be “that man, who hath the skill, (by a certaine and meruailous meanes and way,) both in minde and Imagination to determine and also in worke to finish: what workes so euer, by motion of waight, and cuppling and framyng together of bodyes, may most aptly be Commodious for the worthiest Vses of Man.” And that he may be able to performe these thinges, he hath nede of atteynyng and knowledge of the best, and most worthy thynges. &c. The whole Feate of Architecture in buildyng, consisteth in Lineamentes, and in Framyng. And the whole power and skill of Lineamentes, tendeth to this: that the right and absolute way may be had, of Coaptyng and ioyning Lines and angles: by which, the face of the buildyng or frame, may be comprehended and concluded. And it is the property of Lineamentes, to prescribe vnto buildynges, and euery part of them, an apt place, & certaine nũber: a worthy maner, and a semely order: that, so, y^e whole forme and figure of the buildyng, may rest in the very Lineamentes. &c. And we may prescribe in mynde and imagination the whole formes, *\n\n(* The Immaterialitie of perfect Architecture.)\n\nall material stuffe beyng secluded. Which point we shall atteyne, by Notyng and forepointyng the angles, and lines, by a sure and certaine direction and connexion. Seyng then, these thinges, are thus:\n\n(What, Lineament is.)\n\nLineamente, shalbe the certaine and constant prescribyng, conceiued in mynde: made in lines and angles: and finished with a learned minde and wyt. “We thanke you Master Baptist, that you haue so aptly brought your Arte, and phrase therof, to haue some Mathematicall perfection:\n\nby certaine order, nũber, forme, figure, and Symmetrie mentall:” all naturall & sensible stuffe set a part. Now, then, it is euident, (Gentle reader) how aptely and worthely, I haue preferred Architecture, to be bred and fostered vp in the Dominion of the pereles Princesse, Mathematica: and to be a naturall Subiect of hers. And the name of Architecture, is of the principalitie, which this Science hath, aboue all other Artes. And Plato affirmeth, the Architect to be Master ouer all, that make any worke. Wherupon, he is neither Smith, nor Builder: nor, separately, any Artificer: but the Hed, the Prouost, the Directer, and Iudge of all Artificiall workes, and all Artificers. For, the true Architect, is hable to teach, Demonstrate, distribute, describe, and Iudge all workes wrought. And he, onely, searcheth out the causes and reasons of all Artificiall thynges. Thus excellent, is Architecture: though few (in our dayes) atteyne thereto: yet may not the Arte, be otherwise thought on, then in very dede it is worthy. Nor we may not, of auncient Artes, make new and imperfect Definitions in our dayes: for scarsitie of Artificers: No more, than we may pynche in, the Definitions of Wisedome, or Honestie, or of Frendeshyp or of Iustice. No more will I consent, to Diminish any whit, of the perfection and dignitie, (by iust cause) allowed to absolute Architecture. Vnder the Direction of this Arte, are thre principall, necessary Mechanicall Artes. Namely, Howsing, Fortification, and Naupegie. Howsing, I vnderstand, both for Diuine Seruice, and Mans common vsage: publike, and priuate. Of Fortification and Naupegie, straunge matter might be told you: But perchaunce, some will be tyred, with this Bederoll, all ready rehearsed: and other some, will nycely nip my grosse and homely discoursing with you: made in post hast: for feare you should wante this true and frendly warnyng, and tast giuyng, of the Power Mathematicall. Lyfe is short, and vncertaine: Tymes are perilouse: &c. And still the Printer awayting, for my pen staying: All these thinges, with farder matter of Ingratefulnes, giue me occasion to passe away, to the other Artes remainyng, with all spede possible.\n\n### Navigation\n\nThe Arte of Nauigation, demonstrateth how, by the shortest good way, by the aptest Directiõ, & in the shortest time, a sufficient Ship, betwene any two places (in passage Nauigable,) assigned: may be cõducted: and in all stormes, & naturall disturbances chauncyng, how, to vse the best possible meanes, whereby to recouer the place first assigned. What nede, the Master Pilote, hath of other Artes, here before recited, it is easie to know: as, of Hydrographie, Astronomie, Astrologie, and Horometrie. Presupposing continually, the common Base, and foundacion of all: namely Arithmetike and Geometrie. So that, he be hable to vnderstand, and Iudge his own necessary Instrumentes, and furniture Necessary: Whether they be perfectly made or no: and also can, (if nede be) make them, hym selfe. As Quadrantes, The Astronomers Ryng, The Astronomers staffe, The Astrolabe vniuersall. An Hydrographicall Globe. Charts Hydrographicall, true, (not with parallell Meridians). The Common Sea Compas: The Compas of variacion: The Proportionall, and Paradoxall Compasses\n\n(Anno. 1559.)\n\n(of me Inuented, for our two Moscouy Master Pilotes, at the request of the Company) Clockes with spryng: houre, halfe houre, and three houre Sandglasses: & sundry other Instrumẽtes: And also, be hable, on Globe, or Playne to describe the Paradoxall Compasse: and duely to vse the same, to all maner of purposes, whereto it was inuented. And also, be hable to Calculate the Planetes places for all tymes.\n\nMoreouer, with Sonne Mone or Sterre (or without) be hable to define the Longitude & Latitude of the place, which he is in: So that, the Longitude & Latitude of the place, from which he sayled, be giuen: or by him, be knowne. whereto, appertayneth expert meanes, to be certified euer, of the Ships way. &c. And by foreseing the Rising, Settyng, Nonestedyng, or Midnightyng of certaine tempestuous fixed Sterres: or their Coniunctions, and Anglynges with the Planetes, &c. he ought to haue expert coniecture of Stormes, Tempestes, and Spoutes: and such lyke Meteorologicall effectes, daungerous on Sea. For (as Plato sayth,) Mutationes, opportunitates[que] temporum presentire, non minus rei militari, quàm Agriculturæ, Nauigationi[que] conuenit. To foresee the alterations and opportunities of tymes, is conuenient, no lesse to the Art of Warre, then to Husbandry and Nauigation. And besides such cunnyng meanes, more euident tokens in Sonne and Mone, ought of hym to be knowen: such as (the Philosophicall Poëte) Virgilius teacheth, in hys Georgikes. Where he sayth,\n\n(Sidenote: Georgic. 1.)\n\nSol quo[que] & exoriens & quum se condet in vndas, Signa dabit, Solem certissima signa sequuntur. &c. -------- Nam sæpe videmus, Ipsius in vultu varios errare colores. Cæruleus, pluuiam denunciat, igneus Euros. Sin maculæ incipient rutilo immiscerier igni, Omnia tum pariter vento, nimbis[que] videbis Feruere: non illa quisquam me nocte per altum Ire, ne[que] a terra moueat conuellere funem. &c. Sol tibi signa dabit. Solem quis dicere falsum Audeat? -------- &c.\n\nAnd so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and Beastes, and of many thynges els, a certaine Sympathicall forewarnyng may be had: sometymes to great pleasure and proffit, both on Sea and Land. Sufficiently, for my present purpose, it doth appeare, by the premisses, how Mathematicall, the Arte of Nauigation, is: and how it nedeth and also vseth other Mathematicall Artes: And now, if I would go about to speake of the manifold Commodities, commyng to this Land, and others, by Shypps and Nauigation, you might thinke, that I catch at occasions, to vse many wordes, where no nede is.\n\nYet, this one thyng may I, (iustly) say. In Nauigation, none ought to haue greater care, to be skillfull, then our English Pylotes. And perchaunce, Some, would more attempt: And other Some, more willingly would be aydyng, it they wist certainely, What Priuiledge, God had endued this Iland with, by reason of Situation, most commodious for Nauigation, to Places most Famous & Riche. And though,\n\n(* Anno. 1567 S. H. G.)\n\n(of * Late) a young Gentleman, a Courragious Capitaine, was in a great readynes, with good hope, and great causes of persuasion, to haue ventured, for a Discouerye, (either Westerly, by Cape de Paramantia: or Esterly, aboue Noua Zemla, and the Cyremisses) and was, at the very nere tyme of Attemptyng, called and employed otherwise (both then, and since,) in great good seruice to his Countrey, as the Irish Rebels haue * tasted:\n\n(* Anno. 1569)\n\nYet, I say, (though the same Gentleman, doo not hereafter, deale therewith) Some one, or other, should listen to the Matter: and by good aduise, and discrete Circumspection, by little, and little, wynne to the\n\nsufficient knowledge of that Trade and Voyage: Which, now, I would\n\nbe sory, (through Carelesnesse, want of Skill, and Courrage,) should remayne Vnknowne and vnheard of. Seyng, also, we are herein, halfe Challenged, by the learned, by halfe request, published. Therof, verely, might grow Commoditye, to this Land chiefly, and to the rest of the Christen Common wealth, farre passing all riches and worldly Threasure.\n\n### Thaumaturgike, 1\n\nThaumaturgike, is that Art Mathematicall, which giueth certaine order to make straunge workes, of the sense to be perceiued, and of men greatly to be wondred at. By sundry meanes, this Wonder-worke is wrought. Some, by Pneumatithmie. As the workes of Ctesibius and Hero, Some by waight. wherof Timæus speaketh. Some, by Stringes strayned, or Springs, therwith Imitating liuely Motions. Some, by other meanes, as the Images of Mercurie: and the brasen hed, made by Albertus Magnus, which dyd seme to speake. Boethius was excellent in these feates. To whom, Cassiodorus writyng, sayth. Your purpose is to know profound thynges: and to shew meruayles. By the disposition of your Arte, Metals do low: Diomedes of brasse, doth blow a Trumpet loude: a brasen Serpent hisseth: byrdes made, sing swetely. Small thynges we rehearse of you, who can Imitate the heauen. &c. Of the straunge Selfmouyng, which, at Saint Denys, by Paris,\n\n(* Anno. 1551)\n\n* I saw, ones or twise (Orontius beyng then with me, in Company) it were to straunge to tell. But some haue written it. And yet, (I hope) it is there, of other to be sene. And by Perspectiue also straunge thinges, are done. As partly (before) I gaue you to vnderstand in Perspectiue. As, to see in the Ayre, a loft, the lyuely Image of an other man, either walkyng to and fro: or standyng still. Likewise, to come into an house, and there to see the liuely shew of Gold, Siluer or precious stones: and commyng to take them in your hand, to finde nought but Ayre. Hereby, haue some men (in all other matters counted wise) fouly ouershot thẽ selues: misdeaming of the meanes. Therfore sayd Claudius Cælestinus.\n\n(De his quæ Mundo mirabiliter eueniunt. cap. 8.)\n\nHodie magnæ literaturæ viros & magna reputationis videmus, opera quedam quasi miranda, supra Naturã putare: de quibus in Perspectiua doctus causam faciliter reddidisset. That is. Now a dayes, we see some men, yea of great learnyng and reputation, to Iudge certain workes as meruaylous, aboue the power of Nature: Of which workes, one that were skillfull in Perspectiue might easely haue giuen the Cause. Of Archimedes Sphære, Cicero witnesseth.\n\n(Tusc. 1.)\n\nWhich is very straunge to thinke on. For when Archimedes (sayth he) did fasten in a Sphære, the mouynges of the Sonne, Mone, and of the fiue other Planets, he did, as the God, which (in Timæus of Plato) did make the world. That, one turnyng, should rule motions most vnlike in slownes, and swiftnes. But a greater cause of meruayling we haue by Claudianus report hereof. Who affirmeth this Archimedes worke, to haue ben of Glasse. And discourseth of it more at large: which I omit. The Doue of wood, which the Mathematicien Archytas did make to flye, is by Agellius spoken of. Of Dædalus straunge Images, Plato reporteth. Homere of Vulcans Selfmouers, (by secret wheles) leaueth in writyng. Aristotle, in hys Politikes, of both, maketh mention. Meruaylous was the workemanshyp, of late dayes, performed by good skill of Trochilike. &c. For in Noremberge, A flye of Iern, beyng let out of the Artificers hand, did (as it were) fly about by the gestes, at the table, and at length, as though it were weary, retourne to his masters hand agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of the same Towne, a mighty way, and that a loft in the Ayre, toward the Emperour comming thether: and followed hym, beyng come to the gate of the towne. *\n\n(* ☞)\n\nThus, you see, what, Arte Mathematicall can performe, when Skill, will, Industry, and Hability, are duely applyed to profe.\n\n(A Digression.)\n\nAnd for these, and such like marueilous Actes and Feates, Naturally, Mathematically, and Mechanically, wrought and contriued:\n\n(Apologeticall.)\n\nought any honest Student, and Modest Christian Philosopher, be counted, & called a Coniurer? Shall the folly of Idiotes, and the Mallice of the Scornfull, so much preuaile, that He, who seeketh no worldly gaine or glory at their handes: But onely, of God, the threasor of heauenly wisedome, & knowledge of pure veritie: Shall he (I say) in the meane space, be robbed and spoiled of his honest name and fame? He that seketh (by S. Paules aduertisement) in the Creatures Properties, and wonderfull vertues, to finde iuste cause, to glorifie the Æternall, and Almightie Creator by: Shall that man, be (in hugger mugger) condemned, as a Companion of the Helhoundes, and a Caller, and Coniurer of wicked and damned Spirites? He that bewaileth his great want of time, sufficient (to his contentation) for learning of Godly wisdome, and Godly Verities in: and onely therin setteth all his delight: Will that mã leese and abuse his time, in dealing with the Chiefe enemie of Christ our Redemer: the deadly foe of all mankinde: the subtile and impudent peruerter of Godly Veritie: the Hypocriticall Crocodile: the Enuious Basiliske, continually desirous, in the twinke of an eye, to destroy all Mankinde, both in Body and Soule, æternally? Surely (for my part, somewhat to say herein) I haue not learned to make so brutish, and so wicked a Bargaine. Should I, for my xx. or xxv. yeares Studie: for two or three thousand Markes spending: seuen or eight thousand Miles going and trauailing, onely for good learninges sake: And that, in all maner of wethers: in all maner of waies and passages: both early and late: in daunger of violence by man: in daunger of destruction by wilde beastes: in hunger: in thirst: in perilous heates by day, with toyle on foote: in daungerous dampes of colde, by night, almost bereuing life: (as God knoweth): with lodginges, oft times, to small ease: and somtime to lesse securitie. And for much more (then all this) done & suffred, for Learning and attaining of Wisedome: Should I (I pray you) for all this, no otherwise, nor more warily: or (by Gods mercifulnes) no more luckily, haue fished, with so large, and costly, a Nette, so long time in drawing (and that with the helpe and aduise of Lady Philosophie, & Queene Theologie): but at length, to haue catched, and drawen vp, * a Frog?\n\n(* A prouerb. Fayre fisht, and caught a Frog.)\n\nNay, a Deuill? For, so, doth the Common peuish Pratler Imagine and Iangle: And, so, doth the Malicious skorner, secretly wishe, & brauely and boldly face down, behinde my backe. Ah, what a miserable thing, is this kinde of Men? How great is the blindnes & boldnes, of the Multitude, in thinges aboue their Capacitie? What a Land: what a People: what Maners: what Times are these? Are they become Deuils, them selues: and, by false witnesse bearing against their Neighbour, would they also, become Murderers? Doth God, so long geue them respite, to reclaime them selues in, from this horrible slaundering of the giltlesse: contrary to their owne Consciences: and yet will they not cease? Doth the Innocent, forbeare the calling of them, Iuridically to aunswere him, according to the rigour of the Lawes: and will they despise his Charitable pacience? As they, against him, by name, do forge, fable, rage, and raise slaunder, by Worde & Print: Will they prouoke him, by worde and Print, likewise, to Note their Names to the World: with their particular deuises, fables, beastly Imaginations, and vnchristen-like slaunders? Well: Well. O (you such) my vnkinde Countrey men. O vnnaturall Countrey men. O vnthankfull Countrey men. O Brainsicke, Rashe, Spitefull, and Disdainfull Countrey men. Why oppresse you me, thus violently, with your slaundering of me: Contrary to Veritie: and contrary to your owne Consciences? And I, to this hower, neither by worde, deede, or thought, haue bene, any way, hurtfull, damageable, or iniurious to you, or yours? Haue I, so long, so dearly, so farre, so carefully, so painfully, so daungerously sought & trauailed for the learning of Wisedome, & atteyning of Vertue: And in the end (in your iudgemẽt) am I become, worse, then when I begã? Worse, thẽ a Mad man? A dangerous Member in the Common Wealth: and no Member of the Church of Christ? Call you this, to be Learned? Call you this, to be a Philosopher? and a louer of Wisedome? To forsake the straight heauenly way: and to wallow in the broad way of damnation? To forsake the light of heauenly Wisedome: and to lurke in the dungeon of the Prince of darkenesse? To forsake the Veritie of God, & his Creatures: and to fawne vpon the Impudent, Craftie, Obstinate Lier, and continuall disgracer of Gods Veritie, to the vttermost of his power? To forsake the Life & Blisse Æternall: and to cleaue vnto the Author of Death euerlasting? that Murderous Tyrant, most gredily awaiting the Pray of Mans Soule? Well: I thanke God and our Lorde Iesus Christ, for the Comfort which I haue by the Examples of other men, before my time: To whom, neither in godlines of life, nor in perfection of learning, I am worthy to be compared: and yet, they sustained the very like Iniuries, that I do: or rather, greater. Pacient Socrates, his Apologie will testifie: Apuleius his Apologies, will declare the Brutishnesse of the Multitude. Ioannes Picus, Earle of Mirandula, his Apologie will teach you, of the Raging slaunder of the Malicious Ignorant against him. Ioannes Trithemius, his Apologie will specifie, how he had occasion to make publike Protestation: as well by reason of the Rude Simple: as also, in respect of such, as were counted to be of the wisest sort of men. “Many could I recite: But I deferre the precise and determined handling of this matter: being loth to detect the Folly & Mallice of my Natiue Countrey men. *\n\n(* ☞)\n\nWho, so hardly, can disgest or like any extraordinary course of Philosophicall Studies: not falling within the Cumpasse of their Capacitie: or where they are not made priuie of the true and secrete cause, of such wonderfull Philosophicall Feates.” These men, are of fower sortes, chiefly. The first, I may name, Vaine pratling busie bodies: The second, Fond Frendes: The third, Imperfectly zelous: and the fourth, Malicious Ignorant. To eche of these (briefly, and in charitie) I will say a word or two, and so returne to my Præface.\n\n(1.)\n\nVaine pratling busie bodies, vse your idle assemblies, and conferences, otherwise, then in talke of matter, either aboue your Capacities, for hardnesse: or contrary to your Consciences, in Veritie.\n\n(2.)\n\nFonde Frendes, leaue of, so to commend your vnacquainted frend, vpon blinde affection: As, because he knoweth more, then the common Student: that, therfore, he must needes be skilfull, and a doer, in such matter and maner, as you terme Coniuring. Weening, thereby, you aduaunce his fame: and that you make other men, great marueilers of your hap, to haue such a learned frend. Cease to ascribe Impietie, where you pretend Amitie. For, if your tounges were true, then were that your frend, Vntrue, both to God, and his Soueraigne. Such Frendes and Fondlinges, I shake of, and renounce you: Shake you of, your Folly.\n\n(3.)\n\nImperfectly zelous, to you, do I say: that (perhaps) well, do you Meane: But farre you misse the Marke: If a Lambe you will kill, to feede the flocke with his bloud. Sheepe, with Lambes bloud, haue no naturall sustenaunce: No more, is Christes flocke, with horrible slaunders, duely ædified. Nor your faire pretense, by such rashe ragged Rhetorike, any whit, well graced. But such, as so vse me, will finde a fowle Cracke in their Credite. Speake that you know: And know, as you ought: Know not, by Heare say, when life lieth in daunger. Search to the quicke, & let Charitie be your guide.\n\n(4.)\n\nMalicious Ignorant, what shall I say to thee? Prohibe linguam tuam a malo. A detractione parcite linguæ. Cause thy toung to refraine frõ euill. Refraine your toung from slaunder. Though your tounges be sharpned, Serpent like, & Adders poyson lye in your lippes:\n\n(Psal. 140.)\n\nyet take heede, and thinke, betimes, with your selfe, Vir linguosus non stabilietur in terra. Virum violentum venabitur malum, donec præcipitetur. For, sure I am, Quia faciet Dominus Iudicium afflicti: & vindictam pauperum.\n\nThus, I require you, my assured frendes, and Countrey men (you Mathematiciens, Mechaniciens, and Philosophers, Charitable and discrete) to deale in my behalf, with the light & vntrue tounged, my enuious Aduersaries, or Fond frends. And farther, I would wishe, that at leysor, you would consider, how Basilius Magnus, layeth Moses and Daniel, before the eyes of those, which count all such Studies Philosophicall (as mine hath bene) to be vngodly, or vnprofitable. Waye well S. Stephen his witnesse of Moses.\n\n(Act. 7. C.)\n\nEruditus est Moses omni Sapientia Ægyptiorũ: & erat potens in verbis & operibus suis. Moses was instructed in all maner of wisedome of the Ægyptians: and he was of power both in his wordes, and workes. You see this Philosophicall Power & Wisedome, which Moses had, to be nothing misliked of the Holy Ghost. Yet Plinius hath recorded, Moses to be a wicked Magicien. And that (of force) must be, either for this Philosophicall wisedome, learned, before his calling to the leading of the Children of Israel: or for those his wonders, wrought before King Pharao, after he had the conducting of the Israelites. As concerning the first, you perceaue, how S. Stephen, at his Martyrdome (being full of the Holy Ghost) in his Recapitulation of the olde Testament, hath made mention of Moses Philosophie: with good liking of it: And Basilius Magnus also, auoucheth it, to haue bene to Moses profitable (and therefore, I say, to the Church of God, necessary). But as cõcerning Moses wonders, done before King Pharao: God, him selfe, sayd: Vide vt omnia ostenta, quæ posui in manu tua, facias coram Pharaone. See that thou do all those wonders before Pharao, which I haue put in thy hand. Thus, you euidently perceaue, how rashly, Plinius hath slaundered Moses,\n\n(Lib. 30. Cap. 1.)\n\nof vayne fraudulent Magike, saying: Est & alia Magices Factio, a Mose, Iamne, & Iotape, Iudæis pendens: sed multis millibus annorum post Zoroastrem. &c.\n\n(1.)\n\nLet all such, therefore, who, in Iudgement and Skill of Philosophie, are farre Inferior to Plinie, “take good heede, least they ouershoote them selues rashly,” in\n\n(☞)\n\nIudging of Philosophers straunge Actes: and the Meanes, how they are done.\n\n### Thaumaturgike, 2\n\n(2.)\n\nBut, much more, ought they to beware of forging, deuising, and imagining monstrous feates, and wonderfull workes, when and where, no such were done: no, not any sparke or likelihode, of such, as they, without all shame, do report.\n\n(3.)\n\nAnd (to conclude) most of all, let them be ashamed of Man, and afraide of the dreadfull and Iuste Iudge: both Folishly or Maliciously to deuise: and then, deuilishly to father their new fond Monsters on me: Innocent, in hand and hart: for trespacing either against the lawe of God, or Man, in any my Studies or Exercises, Philosophicall, or Mathematicall: As in due time, I hope, will be more manifest.\n\n### Archemastrie\n\nNow end I, with Archemastrie. Which name, is not so new, as this Arte is rare. For an other Arte, vnder this, a degree (for skill and power) hath bene indued with this English name before. And yet, this, may serue for our purpose, sufficiently, at this present. This Arte, teacheth to bryng to actuall experience sensible, all worthy conclusions by all the Artes Mathematicall purposed, & by true Naturall Philosophie concluded: & both addeth to them a farder scope, in the termes of the same Artes, & also by hys propre Method, and in peculier termes, procedeth, with helpe of the foresayd Artes, to the performance of complet Experiẽces, which of no particular Art, are hable (Formally) to be challenged. If you remember, how we considered Architecture, in respect of all common handworkes: some light may you haue, therby, to vnderstand the Souerainty and propertie of this Science. Science I may call it, rather, then an Arte: for the excellency and Mastershyp it hath, ouer so many, and so mighty Artes and Sciences. And bycause it procedeth by Experiences, and searcheth forth the causes of Conclusions, by Experiences: and also putteth the Conclusions them selues, in Experience, it is named of some, Scientia Experimentalis. The Experimentall Science. Nicolaus Cusanus termeth it so, in hys Experimentes Statikall, And an other Philosopher,\n\nof this land Natiue (the floure of whose worthy fame, can neuer dye nor wither) did write therof largely, at the request of Clement the sixt. The Arte carrieth with it, a wonderfull Credit: By reason, it certefieth, sensibly, fully, and completely to the vtmost power of Nature, and Arte. This Arte, certifieth by Experience complete and absolute: and other Artes, with their Argumentes, and Demonstrations, persuade: and in wordes, proue very well their Conclusions. *\n\n(☞)\n\nBut wordes, and Argumentes, are no sensible certifying: nor the full and finall frute of Sciences practisable. And though some Artes, haue in them, Experiences, yet they are not complete, and brought to the vttermost, they may be stretched vnto, and applyed sensibly. As for example: the Naturall Philosopher disputeth and maketh goodly shew of reason: And the Astronomer, and the Opticall Mechanicien, put some thynges in Experience: but neither, all, that they may: nor yet sufficiently, and to the vtmost, those, which they do, There, then, the Archemaster steppeth in, and leadeth forth on, the Experiences, by order of his doctrine Experimentall, to the chief and finall power of Naturall and Mathematicall Artes. Of two or three men, in whom, this Description of Archemastry was Experimentally, verified, I haue read and hard: and good record, is of their such perfection. So that, this Art, is no fantasticall Imagination: as some Sophister, might, Cum suis Insolubilibus, make a florish: and dassell your Imagination: and dash your honest desire and Courage, from beleuing these thinges, so vnheard of, so meruaylous, & of such Importance. Well: as you will. I haue forewarned you. I haue done the part of a frende: I haue discharged my Duety toward God: for my small Talent, at hys most mercyfull handes receiued. To this Science, doth the Science Alnirangiat, great Seruice. Muse nothyng of this name. I chaunge not the name, so vsed, and in Print published by other: beyng a name, propre to the Science. Vnder this, commeth Ars Sintrillia, by Artephius, briefly written. But the chief Science, of the Archemaster, (in this world) as yet knowen, is an other (as it were) OPTICAL Science: wherof, the name shall be told (God willyng) when I shall haue some, (more iust) occasion, therof, to Discourse.\n\nHere, I must end, thus abruptly (Gentle frende, and vnfayned louer of honest and necessary verities.) For, they, who haue (for your sake, and vertues cause) requested me, (an old forworne Mathematicien) to take pen in hand: (through the confidence they reposed in my long experience: and tryed sincerity) for the declaryng and reportyng somewhat, of the frute and commodity, by the Artes Mathematicall, to be atteyned vnto: euen they, Sore agaynst their willes, are forced, for sundry causes, to satisfie the workemans request, in endyng forthwith: He, so feareth this, so new an attempt, & so costly: And in matter so slenderly (hetherto) among the common Sorte of Studentes, considered or estemed.\n\nAnd where I was willed, somewhat to alledge, why, in our vulgare Speche, this part of the Principall Science of Geometrie, called Euclides Geometricall Elementes, is published, to your handlyng: being vnlatined people, and not Vniuersitie Scholers: Verily, I thinke it nedelesse.\n\n(1.)\n\nFor, the Honour, and Estimation of the Vniuersities, and Graduates, is, hereby, nothing diminished. Seing, from, and by their Nurse Children, you receaue all this Benefite: how great soeuer it be.\n\n(2.)\n\nNeither are their Studies, hereby, any whit hindred. No more, then the Italian Vniuersities, as Academia Bononiensis, Ferrariensis, Florentina, Mediolanensis, Patauina, Papiensis, Perusina, Pisana, Romana, Senensis, or any one of them, finde them selues, any deale, disgraced, or their Studies any thing hindred, by Frater Lucas de Burgo, or by Nicolaus Tartalea, who in vulgar Italian language, haue published, not onely Euclides Geometrie, but of Archimedes somewhat: and in Arithmetike and Practicall Geometrie, very large volumes, all in their vulgar speche. Nor in Germany haue the famous Vniuersities, any thing bene discontent with Albertus Durerus, his Geometricall Institutions in Dutch: or with Gulielmus Xylander, his learned translation of the first sixe bookes of Euclide, out of the Greke into the high Dutch. Nor with Gualterus H. Riffius, his Geometricall Volume: very diligently translated into the high Dutch tounge, and published. Nor yet the Vniuersities of Spaine, or Portugall, thinke their reputation to be decayed: or suppose any their Studies to be hindred by the Excellent P. Nonnius, his Mathematicall workes, in vulgare speche by him put forth. Haue you not, likewise, in the French tounge, the whole Mathematicall Quadriuie? and yet neither Paris, Orleance, or any of the other Vniuersities of Fraunce, at any time, with the Translaters, or Publishers offended: or any mans Studie thereby hindred?\n\n(3.)\n\nAnd surely, the Common and Vulgar Scholer (much more, the Gramarian) before his comming to the Vniuersitie, shall (or may) be, now (according to Plato his Counsell) sufficiently instructed in Arithmetike and Geometrie, for the better and easier learning of all maner of Philosophie, Academicall, or Peripateticall. And by that meanes, goe more cherefully, more skilfully, and spedily forwarde, in his Studies, there to be learned. And, so, in lesse time, profite more, then (otherwise) he should, or could do.\n\n(4.)\n\nAlso many good and pregnant Englishe wittes, of young Gentlemen, and of other, who neuer intend to meddle with the profound search and Studie of Philosophie (in the Vniuersities to be learned) may neuerthelesse, now, with more ease and libertie, haue good occasion, vertuously to occupie the sharpnesse of their wittes: where, els (perchance) otherwise, they would in fond exercises, spend (or rather leese) their time: neither seruing God: nor furdering the Weale, common or priuate.\n\n(5.)\n\nAnd great Comfort, with good hope, may the Vniuersities haue, by reason of this Englishe Geometrie, and Mathematicall Præface, that they (hereafter) shall be the more regarded, esteemed, and resorted vnto. For, when it shall be knowen and reported, that of the Mathematicall Sciences onely, such great Commodities are ensuing (as I haue specified): and that in dede, some of you vnlatined Studentes, can be good witnesse, of such rare fruite by you enioyed (thereby): as either, before this, was not heard of: or els, not so fully credited: “Well, may all men coniecture, that farre greater ayde, and better furniture, to winne to the Perfection of all Philosophie,\n\n(Vniuersities.)\n\nmay in the Vniuersities be had: being the Storehouses & Threasory of all Sciences,\n\n(☞)\n\nand all Artes, necessary for the best, and most noble State of Common Wealthes.”\n\n(6.)\n\nBesides this, how many a Common Artificer, is there, in these Realmes of England and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who, with their owne Skill and experience, already had, will be hable (by these good helpes and informations) to finde out, and deuise, new workes, straunge Engines, and Instrumentes: for sundry purposes in the Common Wealth? or for priuate pleasure? and for the better maintayning of their owne estate? I will not (therefore) fight against myne owne shadowe. For, no man (I am sure) will open his mouth against this Enterprise. No mã (I say) who either hath Charitie toward his brother (and would be glad of his furtherance in vertuous knowledge): or that hath any care & zeale for the bettering of the Cõmon state of this Realme. Neither any, that make accompt, what the wiser sort of men (Sage and Stayed) do thinke of them. To none (therefore) will I make any Apologie, for a vertuous acte doing: and for cõmending, or setting forth, Profitable Artes to English men, in the English toung. “But, vnto God our Creator, let vs all be thankefull: for that, As he, of his Goodnes, by his Powre, and in his wisedome,\n\n(☞)\n\nhath Created all thynges, in Number, Waight, and Measure: So, to vs, of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne the sufficient and necessary knowledge of the foresayd hys three principall Instrumentes: Which Meanes, I haue abundantly proued vnto you, to be the Sciences and Artes Mathematicall.”\n\nAnd though I haue ben pinched with straightnes of tyme: that, no way, I could so pen downe the matter (in my Mynde) as I determined: hopyng of conuenient laysure: Yet. if vertuous zeale, and honest Intent prouoke and bryng you to the readyng and examinyng of this Compendious treatise, I do not doute, but, as the veritie therof (accordyng to our purpose) will be euident vnto you: So the pith and force therof, will persuade you: and the wonderfull frute therof, highly pleasure you. And that you may the easier perceiue, and better remember, the principall pointes, whereof my Preface treateth,\n\n(The Ground platt of this Præface in a Table.)\n\nI will giue you the Groundplatt of my whole discourse, in a Table annexed: from the first to the last, somewhat Methodically contriued.\n\nIf Hast, hath caused my poore pen, any where, to stumble: You will, (I am sure) in part of recompence, (for my earnest and sincere good will to pleasure you), Consider the rockish huge mountaines, and the perilous vnbeaten wayes, which (both night and day, for the while) it hath toyled and labored through, to bryng you this good Newes, and Comfortable profe, of Vertues frute.\n\nSo, I Commit you vnto Gods Mercyfull direction, for the rest: hartely besechyng hym, to prosper your Studyes, and honest Intentes: to his Glory, & the Commodity of our Countrey. Amen.\n\nWritten at my poore House At Mortlake.\n\nAnno. 1570. February. 9.\n\nJ. DEE\n\n### The Groundplat of the Mathematical Praeface\n\nHere haue you (according to my promisse) the Groundplat of my MATHEMATICALL Præface: annexed to Euclide (now first) published in our Englishe tounge. An. 1570. Febr. 3.\n\nSciences, and Artes Mathematicall, are, either\n\nPrincipall, which are two, onely,\n\nArithmetike.\n\nSimple, Which dealeth with Numbers onely: and demonstrateth all their properties and appertenances: where, an Vnit, is Indiuisible. Mixt, Which with aide of Geometrie principall, demonstrateth some Arithmeticall Conclusion, or Purpose.\n\nGeometrie.\n\nSimple, Which dealeth with Magnitudes, onely: and demonstrateth all their properties, passions, and appertenances: whose Point, is Indiuisible. Mixt, Which with aide of Arithmetike principall, demonstrateth some Geometricall purpose, as EVCLIDES ELEMENTES.\n\nThe vse whereof, is either,\n\nIn thinges Supernaturall, æternall, & Diuine: By Application, Ascending. In thinges Mathematicall: without farther Application. In thinges Naturall: both Substãtiall, & Accidentall, Visible, & Inuisible. &c. By Application: Descending.\n\nThe like Vses and Applications are, (though in a degree lower) in the Artes Mathematicall Deriuatiue.\n\nDeriuatiue frõ the Principalls: of which, some haue\n\nThe names of the Principalls: as,\n\nArithmetike, vulgar: which considereth\n\n--Arithmetike of most vsuall whole numbers: And of Fractions to them appertaining. --Arithmetike of Proportions. --Arithmetike Circular. --Arithmetike of Radicall Nũbers: Simple, Compound, Mixt: And of their Fractions. --Arithmetike of Cossike Nũbers: with their Fractions: And the great Arte of Algiebar.\n\nGeometrie, vulgar: which teacheth Measuring\n\nAt hand\n\nAll Lengthes.--Mecometrie. All Plaines: As, Land, Borde, Glasse, &c.--Embadometrie. All Solids: As, Timber, Stone, Vessels, &c.--Stereometrie.\n\nWith distãce from the thing Measured, as,\n\nHow farre, from the Measurer, any thing is: of him sene, on Land or Water: called Apomecometrie. How high or deepe, from the leuell of the Measurers standing, any thing is: Seene of hym, on Land or Water: called Hypsometrie. How broad, a thing is, which is in the Measurers view: so it be situated on Land or Water: called Platometrie.\n\nOf which are growen the Feates & Artes of\n\nGeodesie: more cunningly to Measure and Suruey Landes, Woods, Waters. &c. Geographie. Chorographie. Hydrographie. Stratarithmetrie.\n\nPropre names as,\n\nPerspectiue,--Which demonstrateth the maners and properties of all Radiations: Directe, Broken, and Reflected.\n\nAstronomie,--Which demonstrateth the Distances, Magnitudes, and all Naturall motions, Apparences, and Passions, proper to the Planets and fixed Starres: for any time, past, present, and to come: in respecte of a certaine Horizon, or without respecte of any Horizon.\n\nMusike,--Which demonstrateth by reason, and teacheth by sense, perfectly to iudge and order the diuersitie of Soundes, hie or low.\n\nCosmographie,--Which, wholy and perfectly maketh description of the Heauenlym and also Elementall part of the World: and of these partes, maketh homologall application, and mutuall collation necessary.\n\nAstrologie,--Which reasonably demonstrateth the operations and effectes of the naturall beames of light, and secrete Influence of the Planets, and fixed Starres, in euery Element and Elementall body: at all times, in any Horizon assigned.\n\nStatike,--Which demonstrateth the causes of heauines and lightnes of all thinges: and of the motions and properties to heauines and lightnes belonging.\n\nAnthropographie, Which describeth the Nũber, Measure, Waight, Figure, Situation, and colour of euery diuers thing contained in the perfecte body of MAN: and geueth certaine knowledge of the Figure, Symmetrie, Waight, Characterization, & due Locall motion of any percell of the said body assigned: and of numbers to the said percell appertaining.\n\nTrochilike,--Which demonstrateth the properties of all Circular motions: Simple and Compound.\n\nHelicosophie,--Which demonstrateth the designing of all Spirall lines: in Plaine, on Cylinder, Cone, Sphære, Conoïd, and Sphæroid: and their properties.\n\nPneumatithmie,--Which demonstrateth by close hollow Geometricall figures (Regular and Irregular) the straunge properties (in motion or stay) of the Water, Ayre, Smoke, and Fire, in their Continuitie, and as they are ioyned to the Elementes next them.\n\nMenadrie,--Which demonstrateth, how, aboue Natures Vertue, and power simple: Vertue and force, may be multiplied: and so to directe, to lift, to pull to, and to put or cast fro, any multiplied, or simple determined Vertue, Waight, or Force: naturally, not, so, directible, or moueable.\n\nHypogeiodie,--Which demonstrateth, how, vnder the Sphæricall Superficies of the Earth, at any depth, to any perpendicular line assigned (whose distance from the perpendicular of the entrance: and the Azimuth likewise, in respecte of the sayd entrance, is knowen) certaine way, may be prescribed and gone, &c.\n\nHydragogie,--Which demonstrateth the possible leading of water by Natures law, and by artificiall helpe, from any head (being Spring, standing, or running water) to any other place assigned.\n\nHorometrie,--Which demonstrateth, how, at all times appointed, the precise, vsuall denomination of time, may be knowen, for any place assigned.\n\nZographie,--Which demonstrateth and teacheth, how, the Intersection of all visuall Pyramids, made by any plaine assigned (the Center, distance, and lightes being determined) may be, by lines, and proper colours represented.\n\nArchitecture,--Which is a Science garnished with many doctrines, and diuers Instructions: by whose iudgement, all workes by other workmen finished, are iudged.\n\nNauigation,--Which demonstrateth, how, by the Shortest good way, by the aptest direction, and in the shortest time: a sufficient Shippe, betwene any two places (in passage nauigable) assigned, may be conducted: and in all stormes and naturall disturbances chauncing, how to vse the best possible meanes, to recouer the place first assigned.\n\nThaumaturgike,--Which geueth certaine order to make straunge workes, of the sense to be perceiued: and of men greatly to be wondred at.\n\nArchemastrie,--Which teacheth to bring to actuall experience sensible, all worthy conclusions, by all the Artes Mathematicall purposed: and by true Naturall philosophie, concluded: And both addeth to them a farder Scope, in the termes of the same Artes: and also, by his proper Method, and in peculiar termes, procedeth, with helpe of the forsayd Artes, to the performance of complete Experiences: which, of no particular Arte, are hable (Formally) to be challenged.\n\n¶ Imprinted by Iohn Day.\n\nAn. 1570. Feb. 25.","download_url":"https://mouseia.org/api/works/dee-praeface/original.txt"},"translations":[{"id":"trn_dee_praeface_mouseia","translator":"Mouseia","kind":"curated","note":"Modern machine-assisted Musean edition by Mouseia, from Dee's 1570 Praeface. Latin quotations and titles are translated into English, using Dee's own English glosses where provided. Printer's type-size markup is normalized; marginal notes are kept as asides. Billingsley's separate address is omitted. Uncertain readings are marked.","text":"### To the Lovers of Truth and Students of Noble Sciences\n\nTo the unfeigned lovers of truth, and the constant students of the noble sciences, John Dee of London heartily wishes grace from heaven, and the most prosperous success in all their honorable attempts and exercises.\n\nDivine Plato, that great master of many worthy philosophers, the steady affirmer and vigorous persuader of the One, the Good, and Being, was visited in his school and academy at various times — besides his ordinary scholars — by a certain kind of men, allured by the noble fame of Plato and the great commendation of his profound and profitable doctrine. But when such hearers, after long listening to him, perceived that the drift of his discourses came out to conclude that the One, the Good, and Being are spiritual, infinite, eternal, omnipotent, etc., nothing being alleged or expressed as to how worldly goods, how worldly dignity, how health, strength, or vigor of body might be attained, nor yet by what means a marvelous sensible and bodily bliss and felicity hereafter might be reached — at once the fancies of those hearers were dampened; their opinion of Plato was completely changed; indeed his doctrine was despised by them, and his school visited by them no more. Which thing his scholar Aristotle, narrowly considering, found the cause of it to be \"that they had no forewarning and general information as to where his doctrine tended.\" For so they might have had occasion either to refrain from haunting his school — if they then disliked his aim and purpose — or to continue in it constantly, to their full satisfaction, if such his final scope and intent had been to their desire. For this reason Aristotle, ever after, used briefly to forewarn his own scholars and hearers \"both of what matter, and also to what end, he undertook to speak or teach.\"\n\nWhile I consider the diverse methods of these two excellent philosophers (and am most sure both that Plato could very well have taught otherwise, and that Aristotle might have dealt boldly with his hearers in the same way as Plato did), I am in no small pang of perplexity: because that which I dislike is easiest for me to perform (and to have Plato for my example), and that which I know to be most commendable, and (in this first bringing into common handling the mathematical arts) most necessary, is full of great difficulty and sundry dangers. Yet neither do I think it fitting, for so strange a matter as is now meant to be published, and to so strange an audience, to be bluntly at first put forth without a special preface; nor (imitating Aristotle) can I well hope that, according to the breadth and dignity of the mathematical discipline, I am able either plainly to set out the bounds of its subject matter, or precisely to express its chief purposes and most wonderful applications. And though I am sure that such as shrank from Plato's school, after they had perceived his final conclusion, would in these things have been his most diligent hearers — so infinitely might their desires in the end and at length be satisfied by our mathematical arts — yet by this my preface and forewarning, all such may, to their great benefit, be the sooner lured here; and also the Pythagorean and Platonic perfect scholar, and the constant profound philosopher, with more ease and speed may, like the bee, gather from this both wax and honey.\n\n(The intent of this preface.)\n\nTherefore, since I find great occasion — for the causes alleged, and further in respect of my art of mathematics in general — to use \"a certain forewarning and preface, whose content shall be that mighty, most pleasant, and fruitful mathematical tree, with its main arms and second grafted branches: what each one of them is, and also what benefit in general is to be looked for, both from graft and from stock.\" And since this enterprise is so great that, to our time, it has never (to my knowledge) been achieved by anyone; and also it is most hard in these dreary days to win due and common credit for such rare and strange arts — nevertheless, if for my sincere endeavor to satisfy your honest expectation you will lend me your thankful mind a while, and apply your eye or ear attentively to such matter as my pen is able to deliver this time with speed, perhaps at once, and for the first saluting, you will find this preface a long enough lesson. And either you will by this be made much the apter for a second, or shortly become well able yourselves, from the lion's claw, to conjecture his royal symmetry and further property. Now then, gentle friends and countrymen, turn your eyes and bend your minds to that doctrine which for our present purpose my simple talent is able to yield you.\n\nAll things that are, and have being, are found under a triple general diversity. For either they are deemed supernatural, natural, or of a third being. Supernatural things are immaterial, simple, indivisible, incorruptible, and unchangeable. Natural things are material, compounded, divisible, corruptible, and changeable. Supernatural things are comprehended by the mind alone; natural things are able to be perceived by the outward sense. In natural things probability and conjecture have a place; but in supernatural things chief demonstration and most sure knowledge is to be had. By which properties and comparisons of these two, the state, condition, nature, and property of those things which we earlier called \"of a third being\" may more easily be described; which also, by a special name, are called Mathematical Things. For these, being in a manner middle between supernatural and natural things, are not so absolute and excellent as supernatural things, nor yet so base and gross as natural things; but they are immaterial, and nevertheless by material things able to be somewhat signified. And though their particular images by art are aggregable and divisible, yet the general forms notwithstanding remain constant, unchangeable, untransformable, and incorruptible. Neither can they at any time be perceived or judged by the sense. Nor yet for all that are they first conceived in the royal mind of man. But surmounting the imperfection of conjecture, supposition, and opinion, and falling short of high intellectual conception, they are the mercurial fruit of discursive reasoning, subsisting in perfect imagination. A marvelous neutrality have these mathematical things, and also a strange participation between supernatural, immortal, intellectual, simple and indivisible things, and natural, mortal, sensible, compounded and divisible things. Probability and sensible prose may well serve in natural things, and is commendable; in mathematical reasonings a probable argument is nothing regarded, nor yet the testimony of sense at all credited; but only a perfect demonstration of truths certain, necessary, and invincible, universally and necessarily concluded, is allowed as sufficient for \"an argument exactly and purely mathematical.\"\n\n(Note the word \"unit\" to express the Greek Monas, and not \"unity,\" as we have all commonly used until now.)\n\nOf mathematical things there are two principal kinds: namely, Number and Magnitude.\n\n(Number.)\n\nNumber, we define, to be a certain mathematical sum of units. And a unit is that mathematical thing, indivisible, by sharing in some likeness of whose property any thing which in deed or in count is One may reasonably be called One. We account a unit a mathematical thing, though it is no number and also indivisible, because number materially consists of it, which principally is a mathematical thing.\n\n(Magnitude.)\n\nMagnitude is a mathematical thing, by sharing in some likeness of whose nature any thing is judged long, broad, or thick. \"A thick magnitude we call a solid, or a body. What magnitude soever is solid or thick is also broad and long. A broad magnitude we call a surface or a plane. Every plane magnitude has also length. A long magnitude we call a line. A line is neither thick nor broad, but only long; every certain line has two ends:\n\n(A point.)\n\nThe ends of a line are called points. A point is a mathematical thing, indivisible, which may have a certain determined position.\" If a point moves from a determined position, the way in which it moved is also a line, mathematically produced; after which, of the ancient mathematicians,\n\n(A line.)\n\na line is called the race or course of a point. A point we define by the name of a mathematical thing, though it is no magnitude and is indivisible, because it is the proper end and bound of a line, which is a true magnitude.\n\n(Magnitude.)\n\nAnd magnitude we may define to be that mathematical thing which is divisible forever into divisible parts — long, broad, or thick. Therefore though a point is no magnitude, yet terminatively we reckon it a mathematical thing (as I said) by reason it is properly the end and bound of a line. Neither number nor magnitude has any materiality.\n\n### Of Number and Magnitude\n\nFirst we will consider of number, and of the science mathematical appropriate to it, called Arithmetic: and afterward of magnitude, and its science, called geometry. But that name does not content me; of which a word or two shall be said hereafter. How immaterial and free from all matter Number is, who does not perceive? Indeed, who does not wonder at it greatly? For neither pure Element, nor Aristotle's Fifth Essence, is able to serve as the proper matter of Number. Nor yet will the purity and simplicity of Spiritual or Angelic Substance be found proper enough for it. And therefore the great and godly philosopher Anitius Boetius said: All things (which from the very first original being of things have been framed and made) do appear to be formed by the reason of Numbers. For this was the principal example or pattern in the mind of the Creator. O comfortable allurement, O ravishing persuasion, to deal with a Science whose Subject is so Ancient, so pure, so excellent, so surpassing all creatures, so used by the Almighty and incomprehensible wisdom of the Creator in the distinct creation of all creatures: in all their distinct parts, properties, natures, and virtues, by order and most absolute number, brought from Nothing to the formality of their being and state. By the property of numbers, therefore, learned by us by all possible means (to the perfection of the Science), we may both wind and draw ourselves into the inward and deep search and view of all creatures' distinct virtues, natures, properties, and forms; and also arise further, climb, ascend, and mount up (with speculative wings) in spirit to see in the Glass of Creation the Form of Forms, the Exemplar Number of all things numerable, both visible and invisible, mortal and immortal, corporeal and spiritual. Part of this profound and divine Science had Joachim the prophet attained: by Formal, Natural, and Rational Numbers, foreseeing, concluding, and foretelling great particular events long before their coming. His books yet remaining are good proof of this; and the noble Earl of Mirandula, besides that, a sufficient witness that Joachim in his prophecies proceeded by no other way than by Formal Numbers. And this Earl himself, in Rome,\n\n(A.D. 1488.)\n\n* set up 900 Conclusions, in all kinds of sciences, openly to be disputed; and among the rest, in his Mathematical Conclusions (in the eleventh Conclusion) he has in Latin this English sentence: By Numbers, a way is had to the searching out and understanding of every thing able to be known. For the verifying of which Conclusion, I promise to answer the 74 questions written below by the way of Numbers. Which Conclusions I omit here to rehearse: as well avoiding superfluous prolixity, as because the works of Joannes Picus are commonly available. But in any case I would wish that those Conclusions were read diligently and understood by such as are earnest observers and considerers of the constant law of numbers, which is planted in natural and supernatural things, and is prescribed to all creatures inviolably to be kept. For so, besides many other things in those Conclusions to be marked, it would appear how sincerely, and within my bounds, I disclose the wonderful mysteries to be attained by numbers.\n\nOf my former words it is easy to gather that Number has a triple state: one in the Creator; another in every creature (in respect of its complete constitution); and the third in Spiritual and Angelic Minds, and in the Soul of man. In the first and third state Number is termed Number Numbering. But in all creatures otherwise, Number is termed Number Numbered. And in our Soul Number bears such sway, and has such affinity with it, that some of the old philosophers taught the human Soul to be a self-moving Number. And indeed in us, though it is a mere accident, yet such an accident is it that before all creatures it had perfect being in the Creator, eternally. Number Numbering, therefore, is the discretion that discerns and distinguishes things. But in God the Creator this discretion in the beginning produced all things orderly and distinctly. For his Numbering then was his Creating of all things. And his continual Numbering of all things is the conservation of them in being; and where and when he will lack a Unit, there and then that particular thing shall be discreated. Here I stop. But our separating, distinguishing, and Numbering creates nothing; of Multitude considered it makes a certain and distinct determination. And although these things are weighty and truths of great importance, yet (by the infinite goodness of the Almighty Ternary) artificial methods and easy ways are made by which the zealous philosopher may come near this Ida of many rivers, this Mountain of Contemplation, and more than Contemplation. And also, though Number be a thing so immaterial, so divine, and eternal, yet by degrees, little by little, stretching forth and applying some likeness of it — first to spiritual things, then bringing it lower to things sensibly perceived, as of a momentary sound repeated, then to the least things that may be seen, numerable, and at length (most grossly) to a multitude of any corporeal things seen or felt — and so from these gross and sensible things we are trained to learn a certain image or likeness of numbers, and to use art in them to our pleasure and profit. So gross is our conversation, and dull our apprehension, while mortal Sense in us rules the commonwealth of our little world. By this we say, Three lions are three, or a ternary. Three eagles are three, or a ternary.\n\n(☞)\n\nWhich * ternaries are each the union, knot, and uniformity of three discrete and distinct units. That is, we may in each ternary thrice separately point out and show a part: One, One, and One. Where in numbering we say One, two, Three. But how far these visible Ones differ from our indivisible Units (principally considered in pure arithmetic) no one is ignorant. Yet from these gross and material things may we be led upward by degrees, so informing our rude imagination toward the conceiving of Numbers absolutely — not supposing, nor admixing any created thing, corporeal or spiritual, to support, contain, or represent those Numbers imagined — that at length we may be able to find the number of our own name gloriously exemplified and registered in the book of the Trinity most blessed and eternal.\n\nBut further understand that vulgar practitioners have Numbers otherwise in sundry considerations, and extend their name further than to Numbers whose least part is a Unit. For the common accountant, reckoning master, or arithmetician, in his using of Numbers, imagines lesser parts of a Unit, and calls them Fractions. As of a Unit he makes a half, and thus notes it, ½; and so of other (infinitely diverse) parts of a Unit. Indeed he further has fractions of fractions, etc.\n\n### Arithmetic, 1\n\nAnd since Addition, Subtraction, Multiplication, Division, and Extraction of Roots are the chief and sufficient parts of arithmetic: which is the science that demonstrates the properties of numbers, and all operations to be performed in numbers:\n\n\"How often, therefore, these five sundry sorts of operations differ, for the most part in their execution, from the five operations of like general property and name practicable in our whole numbers, so often (for a more distinct doctrine) we commonly account and name it another kind of arithmetic.\" And by this reason:\n\n(1.)\n\nthe consideration, doctrine, and working in whole numbers only, where of a Unit no lesser part is to be allowed, is named (as it were) an arithmetic by itself. And so of the arithmetic of fractions.\n\n(2.)\n\nIn like manner the necessary, wonderful, and secret doctrine of Proportion and proportionality has won for itself a special manner of handling and working, and so may seem another form of arithmetic.\n\n(3.)\n\nMoreover, astronomers, for speed and more convenient calculation, have devised a special manner of ordering numbers about their circular motions by sixties and sixtieths — by Signs, Degrees, and Minutes, etc. — which commonly is called the arithmetic of astronomical or physical fractions. That have I briefly noted by the name of Circular Arithmetic, because it is also used in circles that are not astronomical, etc.\n\n(4.)\n\nPractice has led Numbers further, and has framed them to take on the show of magnitude's property, which is incommensurability and irrationality. (For in pure arithmetic a Unit is the common measure of all numbers.) And here numbers are become like lines, planes, and solids: sometimes rational, sometimes irrational. And they have proper and peculiar characters (as ²√, ³√, and so of others, which is to signify square root, cubic root, and so forth), and proper and peculiar fashions in the five principal parts; therefore the practitioner esteems this a diverse arithmetic from the other. Practice brings in here diverse compounding of numbers: as sometimes two, three, four (or more) radical numbers diversely knit by signs of More and Less, as thus ²√12 ³√15; or thus ⁴√19 ³√12 - ²√2, etc.; and sometimes with whole numbers, or fractions of whole numbers, among them: as 20 ²√24, ³√16, 33 - ²√10, ⁴√44, 12¼, ³√9. And so infinitely may the variety happen. After this, both the one and the other have fractions incident, and so is this arithmetic greatly enlarged by the diverse exhibiting and use of compositions and mixtures. Consider how I, being desirous to deliver the student from error and cavillation, give to this practice the name of the arithmetic of radical numbers — not of irrational or surd numbers, which are at times rational, though they have the sign of a root before them, which the arithmetic of whole numbers, most usual, would say they had no such root of, and so account them surd numbers; which generally spoken is untrue, as Euclid's tenth book may teach you. Therefore to call them generally radical numbers (by reason of the sign √ prefixed) is a sure way, and a sufficient general distinction from all other ordering and use of numbers. And yet besides all this, consider the infinite desire of knowledge and incredible power of man's search and capacity: how they jointly have waded further (by mixing speculation and practice), and have found out and attained the very chief perfection (almost) of the practical use of numbers. Which thing is well to be perceived in that great arithmetic art of equation, commonly called the Rule of Coss, or Algebra. The Latins termed it the Rule of the Thing and its Value; an apt name, comprehending the first and last points of the work. And the common names, both in Italian, French, and Spanish, depend (in naming it) upon the meaning of the Latin word for \"a thing\" — unless they use the name Algebra. And in this (commonly) is a double error: the one of them that think it to be of Geber's inventing; the other of such as call it Algebra. For first, though Geber, for his great skill in numbers, geometry, astronomy, and other marvelous arts, might have seemed able to have first devised the said rule, and also the name carries with it a very near likeness to Geber's name, yet it is true that a Greek philosopher and mathematician named Diophantus, before Geber's time, wrote 13 books of it (of which six are yet extant; and I had them for use,\n\n(* A.D. 1550.)\n\nof the famous mathematician, and my great friend, Petrus Montaureus:) And secondly, the very name is Algiebar, and not Algebra: as by the Arabian Avicen may be proved, who has these precise words, translated into Latin by Andreas Alpagus (most perfect in the Arabic tongue): The Science of working Algiebar and Almachabel, that is, the Science of finding an unknown number, by adding a number, and by division and equation. Here have you the name, and also the principal parts of the rule, touched. To name it The Rule or Art of Equation signifies the middle part and the state of the rule. This Rule has its peculiar characters:\n\n(5.)\n\nand the principal parts of arithmetic pertaining to it do differ from the other arithmetic operations. This arithmetic has numbers simple, compound, mixed, and fractions accordingly. This Rule and arithmetic of Algiebar is so profound, so general, and so (in a manner) contains the whole power of the practical application of numbers, that man's wit can deal with nothing more profitable about numbers, nor match with a thing more fit for the divine force of the Soul (in human studies, affairs, or exercises) to be tried in. Perhaps you looked for (long before now) to have had some particular proof or evident testimony of the use, profit, and commodity of vulgar arithmetic in the common life and trade of men. To that, then, I will now frame myself; but in this I have great care, for fear that the length of sundry proofs might make you think that either I doubted your zealous mind in the school of virtue, or mistrusted your able wits to guess much more by themselves. A proof, then, four, five, or six such will I bring as any reasonable man may by that be persuaded to love and honor, yes even to learn and exercise, the excellent Science of Arithmetic.\n\nAnd first: who, nearer at hand, can be a better witness of the fruit received by arithmetic than all kinds of merchants? Though not all alike either need it or use it. How could they forgo the use and help of the rule called the Golden Rule — simple and compound, both forward and backward? How might they miss arithmetic help in the Rules of Fellowship, either without time or with time, and between the merchant and his factor? The Rules of Bartering, in wares only, or part in wares and part in money, would they gladly do without? Our merchant venturers and travelers overseas, how could they order their dealings justly and without loss, unless certain and general rules for exchange of money and re-exchange were devised for their use? The Rule of Alligation, in how sundry cases does it conclude for them such precise truths as neither by natural wit nor by other experience were they able otherwise to know? And (with the merchant then to make an end) how ample and wonderful is the Rule of False Positions, especially as it is now enlarged by two excellent mathematicians of my familiar acquaintance in their lifetime — I mean Gemma Frisius and Simon Jacob? Who can either in brief conclude the general and capital rules? Or who can imagine the myriads of sundry cases and particular examples, in act and earnest, continually wrought, tried, and concluded by the forenamed rules alone? How many other arithmetic practices are commonly in merchants' hands and knowledge, they themselves can at large testify.\n\nThe mintmaster and goldsmith, in their mixture of metals, either of diverse kinds or diverse values, how are they, or may they, exactly be directed and wonderfully pleased, if arithmetic be their guide? And the honorable physicians will gladly confess themselves much beholden to the Science of Arithmetic, and that in sundry ways; but chiefly in their Art of Graduation and compound medicines. And though Galen, Averroes, Arnoldus, and Lullus have published their positions, as well in the quantities of the Degrees above temperament as in the rules concluding the new Form resulting, yet a more precise, convenient, and easy Method is extant, invented by a countryman of ours\n\n(Invented above 200 years ago.) And since I am uncertain who has the same, or when that little Latin treatise (as the author wrote it) shall come to be printed — both to declare the desire I have to pleasure my country as far as I may, and also for very good proof of the use of numbers in this most subtle and fruitful philosophical conclusion — I intend in the meanwhile, most briefly and with my further help, to communicate its pith to you.\n\nFirst draw a circle whose diameter is an inch. Divide the circumference into four equal parts. From the center, by those 4 sections, extend 4 straight lines: each 4½ inches long, or as many as you please, above 4, outside the circumference of the circle; so that they shall be 4 inches long (at the least) outside the circle. Make clear and evident marks at the end of every inch. If you wish, you may subdivide the inches again into 10 or 12 smaller equal parts. At the ends of the lines write the names of the 4 principal elemental qualities: Hot and Cold, one opposite the other; and likewise Moist and Dry, one opposite the other. And in the circle write Temperate. Which temperature has a good latitude: as appears by the complexion of man. And therefore we have allowed to it the said circle, and not a mathematical or physical point.\n\n(* Take some of Lullus's counsel in his book On the Fifth Essence.)\n\nNow, when you have two miscible things whose degrees are * truly known: of necessity either they are of one quantity and weight, or of diverse. If they are of one quantity and weight: whether their forms are contrary qualities, or of one kind (but of diverse intentions and degrees), or a temperate and a contrary, the form resulting from their mixture is in the middle between the degrees of the forms mixed. As for example, let A be moist in the first degree, and B dry in the third degree. Add 1 and 3, that makes 4; the half or middle of 4 is 2. This 2 is the middle, equally distant from A and B\n\n(* Note.)\n\n(for the * temperate quality is counted as none, and for it you must put a zero, if at any time it is in a mixture).\n\n![The four elemental qualities and their degrees](/figures/dee-praeface/mixture-diagram.png)\n\nCounting then from B, 2 degrees toward A, you find it to be dry in the first degree; so is the form resulting from the mixture of A and B in our example. I will give you another example. Suppose you have two things, as C and D, and of C the heat to be in the 4th degree, and of D the cold to be relaxed even down to the temperament. Now for C you take 4, and for D you take a zero, which, added to 4, yields only 4. The middle, or half, of which is 2. Therefore the form resulting from C and D is hot in the second degree; for 2 degrees counted from C toward D end exactly in the second degree of heat. Of the third manner I will give also an example; which let be this:\n\nI have a liquid medicine whose quality of heat is exalted in the 4th degree, as was C in the example foregoing; and another liquid medicine I have whose quality is heat in the first degree. Of each of these I mixed a like quantity. Subtract here the less from the more, and divide the residue into two equal parts; of which the one part, either added to the less or subtracted from the higher degree, produces the degree of the form resulting from this mixture of C and E. As, if from 4 you subtract 1, there remains 3; the half of 3 is 1½: add to 1 this 1½, you have 2½; or subtract from 4 this 1½, you have likewise 2½ remaining. Which declares the form resulting to be heat in the middle of the third degree.\n\n(The Second Rule.)\n\n\"But if the quantities of two commixed things are diverse, and the intensities (of their miscible forms) are in diverse degrees and heights (whether those forms be of one kind, or of contrary kinds, or of a temperate and a contrary), what proportion the lesser quantity has to the greater, the same shall be between the difference which is from the degree of the form resulting to the degree of the greater quantity of the miscible thing, and the difference which is from the same degree of the form resulting to the degree of the lesser quantity. As for example. Let two pounds of liquor be given, hot in the 4th degree; and one pound of liquor be given, hot in the third degree.\" I would gladly know the form resulting in the mixture of these two liquors. Set down your numbers in order, thus.\n\n£. 2.    Hot. 4.\n£. 1.    Hot. 3.\n\n### Arithmetic, 2\n\nBy the rule of algebra I have devised a very easy, brief, and general way of working in this case. Let us first suppose that the middle Form resulting is x, as that Rule teaches. And because, by our Rule here given, as the weight of 1 is to 2, so is the difference between 4 (the degree of the greater quantity) and x, to the difference between x and 3 (the degree of the thing in lesser quantity; and all the while, x always lies in a certain middle between the two heights or degrees). For the first difference I set 4 - x, and for the second I set x - 3. And now again I say, as 1 is to 2, so is 4 - x to x - 3. Therefore, of these four proportional numbers, the first and the fourth multiplied, one by the other, make as much as the second and the third multiplied, the one by the other. Let these multiplications be made accordingly. And of the first and the fourth we have x - 3, and of the second and the third, 8 - 2x. Therefore our equation lies between x - 3 and 8 - 2x. Which may be reduced according to the art of algebra: for here, adding 3 to each part gives the equation thus: x = 11 - 2x. And again, contracting or reducing it: add 2x to each part; then you have 3x equal to 11, thus represented: 3x = 11. Therefore, dividing 11 by 3, the quotient is 3⅔: the value of our x, the Coss, or Thing, first supposed. And that is the height, or intension, of the Form resulting: which is heat in two thirds of the fourth degree. And here I set out the work, in conclusion, thus. The proof of this is easy: subtracting 3 from 3⅔ leaves ⅔. Subtract the same height of the Form resulting, which is 3⅔, from 4, and ⅓ remains: you see that ⅔ is double ⅓, as 2 pounds is double 1 pound. So it should be, by the Rule here given. Note: as you added 3 to each part of the equation, so if you had first added 2x to each part, it would stand: 3x - 3 = 8. And now, adding 3 to each part, you have, as before, 3x = 11.\n\n2 pounds, hot in the 4th degree: ⅓; the Form resulting: 3⅔; 1 pound, hot in the 3rd degree: ⅔.\n\nAnd though I speak here only of two miscible things, when most commonly three, four, five, or six, etc., are to be mixed (and reduced into one compound, with the Form resulting from the same to serve the turn), yet these Rules are sufficient, duly repeated and iterated.\n\nIn proceeding first with any two, and then with the Form Resulting and another, and so forth: for the last work concludes the Form Resulting of them all. I need say nothing of what the Mixture here supposed is. Common Philosophy has defined it, saying: Mixture is the union of miscible things that have been altered and joined through their smallest parts. Every word in the definition is of great importance. I need spend no time either to show how the other manner of distributing degrees agrees with these Rules. Neither need I declare here the further use belonging to the Cross of Graduation, before described, to those who are capable of what I have already said. Nor need I specify with examples the manifold varieties to be ordered by the two general Rules aforesaid. The witty and studious have enough here; and those who are not able to attain to this without lively teaching, and in more detail, would need a fuller discussion than is fitting to be dealt with in this place. And others, perhaps, with a proud disdain will scorn this little, and would be unthankful for much more. I therefore conclude, and wish such as have modest and earnest philosophical minds to praise God highly for this, and to wonder that the profoundest and subtlest point concerning the mixture of Forms and natural Qualities is so matched and married with the simplest, easiest, and shortest way of the noble Rule of Algebra. Who can remain, therefore, unpersuaded to love, allow, and honor the excellent science of arithmetic? For here you may perceive that the little finger of arithmetic has more might and contrivance than a hundred thousand men of the middle sort are able to perform, or truly to conclude, without the help of it.\n\nNow we shall be further certified, by the wise and valiant Captain, what help he has from the Rules of arithmetic in one of the arts belonging to him, and which the Greeks name:\n\nΤακτικὴ.\n\nΤακτικὴ. \"That is, the skill of ordering soldiers in battle array in the best manner for all purposes.\" This art depends so much upon the use of numbers and the mathematics, that Aelianus, the best writer of it, in his work to the Emperor Hadrian, by his perfection in the mathematics, being greater than others before him had, thinks his book surpasses all other excellent works written of that art up to his days. For of it have written Aeneas, Cyneas of Thessaly, Pyrrhus Epirota and his son Alexander, Clearchus, Pausanias, Euangelus, Polybius, familiar friend to Scipio, Eupolemus, Iphicrates, Posidonius, and very many other worthy captains, philosophers, and princes of immortal fame and memory: whose fairest flower in their garland (in this feat) was arithmetic, and a little perception of geometric figures. But in many other cases arithmetic stands the captain in great stead. As in proportioning the victuals for the army, whether it is remaining at a stand, or is suddenly to be increased with a certain number of soldiers, and for a certain time. Or by good art to diminish his company, so that the victuals serve the remainder longer, and for a certain determined time, if need so requires. And so, in his sundry other accounts, reckonings, measurings, and proportionings, the wise, expert, and circumspect captain will affirm the science of arithmetic to be one of his chief counselors, directors, and aiders. Which thing (by good means) was evident to the noble, the courageous, the loyal, and courteous\n\n(☞)\n\nJohn, late Earl of Warwick. Who was a young gentleman thoroughly known to very few. Although his vigorous valor, force, and skill in chivalrous feats and exercises, his humility, and friendliness to all men, were things openly perceived by the world. But what roots virtue had otherwise fastened in his breast, what rules of godly and honorable life he had framed for himself, what notable vices (in some then living) he took great care to avoid, what manly virtues in other noblemen (flourishing before his eyes) he unceasingly aspired after, what prowess he purposed and meant to achieve, with what feats and arts he began to furnish and provide himself for the better service of his King and country, both in peace and war: these, I say, his heroic meditations, forecastings, and determinations, no two, I think, besides myself, can report so perfectly and truly. And therefore, in conscience, I count it my part, for the honor, advancement, and procuring of virtue, thus briefly to have put his Name in the Register of Immortal Fame.\n\nTo our purpose. This John, by one of his acts (besides many others, both in England and France, noted by me in him) disclosed his hearty love of virtuous sciences, and his noble intent to excel in martial prowess: when, with humble request and constant soliciting, he obtained the best rules (whether used in times past by Greeks or Romans, or in our own time, with new stratagems devised tin this) for ordering all companies, sums, and numbers of men, many or few, appointed with one kind of weapon or more, with artillery or without, on horseback or on foot, to give or take the onset, to seem many when few, and few when many; to march in battle or on a journey, and to many such feats belonging to a pitched field, skirmish, or ambush:\n\nThis noble Earl died in the year 1554, scarce 24 years of age, having no issue by his wife, daughter to the Duke of Somerset.\n\nAnd all these lifelike designs (most carefully) to be drawn on vellum parchment, with notes and peculiar marks as the art requires; and all these rules and arithmetic descriptions enclosed in a rich case of gold, he used to wear about his neck, as his most precious jewel and most trustworthy counselor. Thus was arithmetic enshrined in gold by him; of the fruit of Numbers he had good hope. Now, therefore, innumerable Numbers shall find his shrine in the praise of Numbers.\n\nWhat need have I (for further proof to you) of the schoolmasters of Justice, to require testimony, how needful, how fruitful, how skillful a thing arithmetic is? I mean lawyers of all sorts. Undoubtedly the civilians can wonderfully declare how neither the ancient Roman laws, without good knowledge of the art of numbers, can be perceived, nor (Justice in infinite cases) without due proportion, narrowly considered, be executed. How justly, and with great knowledge of the art, did Papinian institute a law of partition and allowance between man and wife after a divorce? But how Accursius, Baldus, Bartolus, Jason, Alexander, and finally Alciatus (otherwise notably well learned) jumble, guess, and err from the equity, art, and intent of the lawmaker, arithmetic can detect and convict, and clearly make the truth shine. Good Bartolus, tired in examining and proportioning the matter, and much encumbered with Accursius's Gloss, burst out and said: In the whole book there is no gloss harder than this, whose account or reckoning neither the scholars nor the doctors understand. etc. What can they say of Julianus's law beginning \"If it was thus written,\" etc., of the testator's will justly carried out between the wife, son, and daughter? How can they perceive the equity of Aphricanus's arithmetic reckoning, where he treats of the Falcidian Law? How can they clear him from his reprovers and those who maintain them, such as Joannes, Accursius Hypolitus, and Alciatus? How justly and skillfully was Africanus's reckoning made, proportioning to the sums bequeathed the contributions of each part? Namely, for the hundred presently received, 17-1/7; and for the hundred received after ten months, 12-6/7: which make the 30 which were to be contributed by the legatees to the heir. For what proportion 100 has to 75, the same has 17-1/7 to 12-6/7: which is Sesquitertia, that is, as 4 to 3, which make 7. Wonderfully many places in the civil law require an expert arithmetician to understand the deep judgment and just determination of the ancient Roman lawmakers. But much more expert ought he to be who would be able to decide with equity the infinite variety of cases which do or may happen under every one of those civil laws and ordinances. By this you may now easily conjecture that in the canon law, and in the laws of the realm (which with us bear the chief authority), justice and equity might be greatly preferred, and skillfully executed, through due skill of arithmetic and the proportions belonging to it. The worthy philosophers and prudent lawmakers (who have written many books On the Commonwealth, how the best state of commonwealths might be procured and maintained) have very well determined of justice (which not only is the base and foundation of commonwealths, but also the total perfection of all our works, words, and thoughts), defining it,\n\nJustice.\n\n\"to be that virtue by which, to everyone, is rendered that to him appertains.\" God demands this at our hands: to be honored as God, to be loved as a father, to be feared as Lord and master. Our neighbor's proportion is also prescribed by the Almighty lawmaker: which is, to do to others as we would have them do to us. These proportions are necessary in justice, commendable in duty, and are the life, strength, support, and flourishing of commonwealths. Aristotle in his Ethics (to fetch the seed of justice and the light of direction, for using and executing the same) had to fly to the perfection and power of numbers: for arithmetic and geometric proportions. Plato, in his book called Epinomis (which book is the treasury of all his doctrine), where his purpose is to seek a science which, when a man had it perfectly, he might seem and so be, in deed, wise. He, briefly discoursing of other sciences, finds them unable to bring it to pass; but of the science of numbers he says: That science, truly, which has taught mankind number, shall be able to bring it to pass. And I think that a certain God, rather than fortune, has given us this gift for our bliss. For why should we not judge him who is the Author of all good things to be also the cause of the greatest good thing, namely, Wisdom? There at length he proves Wisdom to be attained by good skill of numbers. With which great testimony, and the manifold proofs and reasons before expressed, you may be sufficiently and fully persuaded of the perfect science of arithmetic, to make this account: that\n\n(☞)\n\nof all sciences, next to theology, it is most divine, most pure, most ample and general, most profound, most subtle, most commodious and most necessary. Whose next sister is the absolute science of magnitudes: of which (by the direction and aid of him whose Magnitude is Infinite, and of us Incomprehensible) I now intend so to write, that both with the multitude and also with the magnitude of marvelous and fruitful truths, you (my friends and countrymen) may be stirred up and awakened to see what certain arts and sciences (to our unspeakable benefit) our heavenly father has prepared and revealed for us, by sundry philosophers and mathematicians.\n\n### Arithmetic, 3\n\nBoth Number and Magnitude have a certain original seed (as it were) of incredible property, which man is never able fully to declare. Of Number, a Unit, and of Magnitude, a Point, seem to be much alike as original causes: but the difference, nevertheless, is great. We defined a Unit to be a mathematically indivisible thing; a Point, likewise, we said to be a mathematically indivisible thing. And further, that a Point may have a certain determined situation: that is, that we may assign and prescribe a Point to be here, there, or over there, etc. In this, behold, our Unit is free, and can abide no bondage, nor be tied to any place or seat, divisible or indivisible. Again, because a Point may have a situation limited to it, a certain motion (to a place and from a place) is incident and belonging to a Point. But a Unit cannot be imagined to have any motion. A Point, by its motion, produces mathematically a line (as we said before), which is the first kind of magnitudes and most simple: a Unit cannot produce any number. A Line, though it is produced by a Point moved, yet does not consist of points: Number, though it is not produced by a Unit, yet does consist of units as a material cause. But formally,\n\nNumber.\n\nNumber is the union and unity of units. Which uniting and knitting is the workmanship of our mind, which, from distinct and discrete units, makes a Number, by a uniformity resulting from a certain multitude of units. And so every number may have its least part given, namely a unit: but of a Magnitude (no, not even of a Line) can the least part be given, because an infinite division of it may be conceived. All Magnitude is either a Line, a Plane, or a Solid. Which Line, Plane, or Solid can be perceived by no sense, nor exactly represented by hand in any way, nor produced by Nature: but as Number (by degrees) came to our perception, so by visible forms we are helped to imagine what our mathematical Line is, what our Point is. So precise are our magnitudes that one Line is no broader than another, for they have no breadth; nor our planes any thickness; nor yet our bodies any weight, be they ever so large of dimension. Our bodies we can have smaller than either Art or Nature can produce any, and greater also than all the world can contain. Our least magnitudes can be divided into as many parts as the greatest. As a Line one inch long (with us) may be divided into as many parts as may the diameter of the whole world from East to West, or extended any way: what privileges, above all manual Art and the might of Nature, have our two mathematical sciences, to exhibit and to deal with things of such power, liberty, simplicity, purity, and perfection? And in them to proceed so certainly, so orderly, so precisely: as excellent is that mechanical workman judged who can most nearly approach the representing of works mathematically demonstrated?\n\n(☞)\n\nAnd our two sciences, remaining pure and absolute in their proper terms and in their own matter, to have and allow only such demonstrations as are plain, certain, universal, and of an eternal truth?\n\n### Geometry\n\nThis Science of Magnitude, its properties, conditions, and appurtenances, has commonly been called geometry by all philosophers, now and from the beginning. But, truly, with a name too base and scant for a science of such dignity and amplitude. And perhaps that name, by common and secret consent of all wise men, has hitherto been allowed to remain, so that it might carry with it a perpetual memory of the first and most notable benefit shown to common people by that science: which was, when boundaries and meres of land and ground were lost and confounded (as in Egypt, yearly, with the overflowing of the Nile, the greatest and longest river in the world), or ground bequeathed was to be assigned, or ground sold was to be laid out, or (when disorder prevailed) commons were distributed into severalties. For where, upon these and such like occasions, some by ignorance, some by negligence, some by fraud, and some by violence, did wrongfully limit, measure, encroach upon, or claim (by pretense of just content and measure) those lands and grounds: great loss, disquiet, murder, and war did full often ensue: till, by God's mercy and men's industry, the perfect Science of Lines, Planes, and Solids (like a divine justiciar) gave to every man his own. The people then, pleased by this art and greatly relieved in the just measuring of their lands, and other philosophers writing rules for land measuring, between them both thus confirmed the name Geometria, that is (according to the very etymology of the word) land measuring. In it, the people knew no further of the use of magnitudes but in planes; and the philosophers had no fit hearers or scholars among them, to disclose further to than of flat, plain geometry. And though these philosophers knew of further use, and best understood the etymology of the word, yet this name Geometria was by them applied generally to all sorts of magnitudes: except at other times by Plato and Pythagoras, when they would precisely declare their own doctrine. Then it was\n\nPlato, Republic 7.\n\nGeometria, with them, the study concerned with the plane. But well you may perceive by Euclid's Elements that our Science is more ample than to measure planes, and nothing less is taught in it (on purpose) than how to measure land. Another name, therefore, must be had for our mathematical Science of Magnitudes: which regards neither clod nor turf, neither hill nor dale, neither earth nor heaven, but is absolute Megethologia: not creeping on ground and dazzling the eye with pole, perch, rod, or line, but \"lifting the heart above the heavens by invisible lines, and\n\n(☞)\n\nimmortal beams meeting with the reflections of the incomprehensible light, and so procuring joy and unspeakable perfection.\" Of which true use of our Megethica, or Megethologia, Divine Plato seemed to have good taste and judgment, and (by the name of geometry) so noted it, and warned his scholars of it, as may clearly be seen in his seventh Dialogue of the Commonwealth. Where it is thus, rightly translated: Truly, says Plato, whoever has (even very little) tasted of geometry will not deny us this: that this Science is of another condition, quite contrary to that which they who are exercised in it speak of it. And there it follows, of our geometry: It must be confessed, says Plato, that [geometry] is learned for the knowing of that which is ever, and not of that which in time both is bred and is brought to an end. etc. Geometry is the knowledge of that which is everlasting. It will lift up, therefore (O Gentle Sir), our mind to the Truth, and by that means it will prepare the Thought to the philosophical love of wisdom: that we may turn or convert toward heavenly things [both mind and thought] which now, otherwise than becomes us, we cast down on base or inferior things. etc. Chiefly, therefore, commandment must be given that such as inhabit this most honorable City by no means despise geometry. For even those things [done by it] which in manner seem to be beside the purpose of geometry are of no small importance. etc. And besides the manifold uses of geometry in matters pertaining to war, he adds more, of second unpurposed fruit and commodity arising from geometry, saying: But also we know that for the more easy learning of all arts it much imports whether one has any knowledge in geometry or no. etc. Let us therefore make an ordinance or decree that this Science shall be learned by young men in the second place. This was Divine Plato's judgment, both of the purposed, chief, and perfect use of geometry, and of his second, dependent, derivative commodities. And for us, Christian men, a thousand thousand more occasions are, to have need of the help of\n\nI. D. In this, I would gladly shake off the earthly name of geometry.\n\nmetaphysical contemplations, by which to train our imaginations and minds little by little to forsake and abandon the gross and corruptible objects of our outward senses, and to apprehend, by sure demonstrative doctrine, Mathematical Things. And by them, readily to be helped and conducted to conceive, discourse, and conclude of things Intellectual, Spiritual, eternal, and such as concern our everlasting Bliss: which otherwise (without special privilege of Illumination or Revelation from heaven) no mortal man's wit (naturally) is able to reach or to compass. And truly, by my small Talent (from above) I am able to prove and testify that the literal Text and order of our divine Law, Oracles, and Mysteries require more skill in Numbers and Magnitudes than (commonly) the expositors have uttered: but rather only (at most) so warned, and showed their own want tin this. (To name any is needless; and to note the places is here no place; but if I am duly asked, my answer is ready.) And without the literal, grammatical, mathematical, or natural truths of such places, perceived by good and certain Art, no spiritual sense (proper to those places by Absolute Theology) will depend thereon.\n\n(☞)\n\n\"No man, therefore, can doubt but toward the attaining of knowledge incomparable, and Heavenly Wisdom: Mathematical Speculations, both of Numbers and Magnitudes: are means, aids, and guides: ready, certain, and necessary.\" From from now on, in this my Preface, will I frame my talk to Plato's fugitive Scholars; or rather to such as well can (and also will) use their outward senses, to the glory of God, the benefit of their Country, and their own secret contentment, or honest advancement, on this earthly Scaffold. To them I will orderly recite, describe, and declare a great Number of arts, from our two mathematical fountains, derived into the fields of Nature. By which such Seeds and Roots as lie deep hidden in the ground of Nature are refreshed, quickened, and provoked to grow, shoot up, flower, and give fruit, infinite and incredible. And these arts shall be such as depend upon the properties of Magnitudes more than upon Number. And by good reason we may call them arts, and derived mathematical arts: for (at this time) I define\n\nAn art.\n\nAn art to be a methodical complete doctrine, having abundance of sufficient and peculiar matter to deal with, by the allowance of the metaphysical philosopher: the knowledge of which is necessary to the human estate. And that I count,\n\nDerived mathematical art.\n\nA derived mathematical art, which by a mathematical demonstrative method, in Numbers or Magnitudes, orders and confirms its doctrine, as much and as perfectly as the subject matter will admit. And for that, I intend to use the name and property of a\n\nA mechanician.\n\nmechanician, otherwise than (hitherto) it has been used, I think it good (for distinction's sake) to give you also a brief description of what I mean by that. A mechanician, or a mechanical workman, is he whose skill is, without knowledge of mathematical demonstration, perfectly to work and finish any sensible work, by the mathematician, principal or derivative, demonstrated or demonstrable. Full well I know that he who invents or makes these demonstrations is generally called a speculative mechanician, which differs nothing from a mechanical mathematician. So, in respect of diverse actions, one man may have the names of sundry arts: as sometimes a logician, sometimes (in the same matter otherwise handled) a rhetorician. Of these trifles I make (as now, in respect of my Preface) small account: to file them for the fine handling of subtle curious disputers. In other places they may command me to give good reason: and yet here I will not be unreasonable.\n\n(1.)\n\nFirst, then, from the purity, absoluteness, and immateriality of Principal Geometry is derived that kind of geometry which vulgarly is counted geometry: and is the Art of measuring sensible magnitudes, their just quantities and contents.\n\n### Common Geometry\n\nThis teaches measuring either at hand, with the practitioner standing by the thing measured: and so, by duly applying the compass, rule, square, yard, ell, perch, pole, line, gaging rod (or the like instrument) to the length, plane, or solid being measured,\n\n(1.)\n\nTo be certified of the length, perimeter, or linear distance: and this is called Mecometry. Or\n\n(2.)\n\nTo be certified of the content of any plane surface: whether it be surveyed ground, a measured board or pane of glass, or the like thing: which measuring is named Embadometry.\n\n(3.)\n\nOr else to understand the solidity and capacity of any bodily thing: such as timber and stone, or the capacity of pits, ponds, wells, vessels small and great of all kinds. Where the measuring of wine, oil, beer, or ale vessels, etc., commonly has a peculiar name: and is called gaging. And the general name of these solid measures is stereometry.\n\n(2.)\n\nOr else this common geometry also undertakes to teach the practitioner how to measure things while standing at a good distance from the thing measured: and by that to understand either\n\n(1.)\n\nHow far a thing seen (on land or water) is from the measurer: and this may be called Apomecometry:\n\n(2.)\n\nOr, how high or deep, above or below the level of the measurer's standing, anything is that is seen on land or water, called Hypsometry.\n\n(3.)\n\nOr, it informs the measurer how broad anything is that lies in the measurer's view, provided it is situated on land or water: and may be called Platometry. Though I am in the habit of stipulating that the thing measured be situated on land or water:\n\nYet know for certain that the various heights of clouds, comets, and the moon may by these means have their distances from the earth; and that of comets and the moon, the solidity as well as the distances may be measured. But because neither these things are taught in the common way, nor are so ready for the ordinary practitioner to carry out, I prefer to let such measures be reckoned as incidental to some of our other arts, which deal more purposely with things on high than this common land-measuring geometry does: as in perspective and astronomy, etc.\n\nFrom these feats, further applied, has sprung the feat of geodesy, or land surveying: to measure and survey land, woods, and waters more skillfully from afar. More skillfully, I say: but God knows, up to now in these realms of England and Ireland (whether through ignorance or fraud, I cannot tell in every particular)\n\nHow great a wrong and injury has been done in my time by untrue measuring and surveying of land or woods in every way. And of this I am sure: that the value of the difference between the truth and such surveys would have been able to fund forever, in each of our two universities, an excellent mathematics reader: each allowed yearly a hundred marks of lawful money of this realm; which indeed would seem to be required here (though provided for in other ways), just as the famous University of Paris has two mathematics readers, each with two hundred French crowns yearly, from the French king's magnificent liberality alone.\n\nNow, returning again to our purpose: moreover, from that earlier geometrical knowledge have grown the skills of geography, chorography, hydrography, and Stratarithmetrie.\n\n### Geography\n\n\"Geography teaches the ways by which, in various forms (as spherical, planar, or other), the situation of cities, towns, villages, forts, castles, mountains, woods, harbors, rivers, creeks, and such other things upon the surface of the earthly globe (either in the whole, or in some principal member and portion of it contained) may be described and drawn in proportions analogous to nature and truth, and represented most aptly to our view.\" Of this art how great pleasure and how manifold the commodities that come to us daily and hourly, most men perceive. While some, to beautify their halls, parlors, chambers, galleries, studies, or libraries; others, for things past, as battles fought, earthquakes, heavenly fires, and such occurrences mentioned in histories: by that livelike, as it were, to view the place, the neighboring region, the distance from us, and such other circumstances. Others, presently to view the wide dominion of the Turk, the vast empire of the Muscovite, and the little morsel of ground where Christendom (by profession) is certainly known. Little, I say, in respect of the rest, etc. Some, either to direct their own journeys into far countries, or to learn of other men's travels. To conclude, some for one purpose and some for another like, love, get, and use maps, charts, and geographic globes. Of whose use to speak adequately would require a book of its own.\n\n### Chorography\n\nChorography seems to be an underling and a twig of geography: and yet nevertheless is manifold in practice and very ample in use. \"This teaches, analogically, to describe a small portion or circuit of ground with its contents: without regard to what commensuration it has to the whole, or any parcel contained outside it. But in the territory or parcel of ground it takes in hand to describe, it leaves out nothing notable or odd standing above ground and visible. And sometimes it gives some peculiar mark or warning of things underground: as of metal mines, coal pits, stone quarries, etc.\" Thus a duchy, a shire, a lordship, or less may be described distinctly. But it is marvelous pleasant and profitable to set before our eye, and in commensuration, the plan of a city, town, fort, or palace in true symmetry: without approaching any of them, and out of gunshot, etc. By that the architect can furnish himself with a store of whatever patterns he likes: to his great instruction, even in those things that are outwardly proportioned: either simply in themselves, or respectively to the hills, rivers, harbors, and woods adjoining. Some also call this particular description of places topography.\n\n### Hydrography\n\n\"Hydrography gives us, on globe or in plane, the perfect analogical description of the ocean sea coasts throughout the whole world, or in its chief and principal parts:\" with the islands and chief particular places of danger contained within the bounds and sea coasts described: such as quicksands, banks, pits, rocks, races, counter-tides, whirlpools, etc. This deals chiefly with the element of water: just as geography did principally take in hand the description of the element of earth (with its appurtenances). And besides this, hydrography requires a particular register of certain landmarks (where marks can be had) well able to be spied from the sea: at what point of the sea compass they appear, and what apparent form, situation, and bigness they have in respect of any dangerous place in the sea, or near to it, assigned. And on all coasts, what moon makes full sea, and in what way the tides and ebbs come and go, the hydrographer ought to record. The soundings likewise, and the channel ways: their number and depths ordinarily at ebb and flood, the hydrographer ought by observation and diligent measuring to have certainly known. And many other points belong to perfect hydrography, and to making sailing directions: of which I need not speak here: such as truly describing, in any place, on globe or plane, the 32 points of the compass (of which scarcely four in England have right knowledge, because the lines of it are neither straight lines nor circles). Of making due projection of a sphere in plane. Of the variation of the compass from true north. And such like matters (all of great importance) I leave to speak of here: because I may seem already to have widened the bounds and duty of a hydrographer far more than any man (to this day) has noted or prescribed. Yet I am well able to prove that all these things pertain, and are also proper, to the hydrographer. The chief use and end of this art is the art of navigation: but it has other diverse uses: even to be enjoyed by those who never lack sight of land.\n\n### Stratarithmetrie\n\nStratarithmetrie is the skill (appertaining to war) by which a man can set in a figure, analogous to any geometric figure appointed, any certain number or sum of men: capable of such a figure (by reason of the usual spaces allowed between soldiers, and because no fractions can be made of men. Yet nevertheless he can order the given sum of men into the greatest such figure that can be ordered of them) and certify the surplus (if any) and the next certain sum which, with the surplus, will admit a figure exactly proportional to the assigned figure. By which skill also, of any army or company of men (the figure and sides of whose orderly standing or array is known), he is able to express the exact number of men contained within that figure: or (in order) able to be contained.\n\nNote.\n\nAnd this figure and its sides he is able to know: either being by and at hand, or far off. Thus far stretches the description and property of Stratarithmetrie: sufficient for this time and place.\n\nThe difference between Stratarithmetrie and Tacticie.\n\n\"It differs from the tactical feat of drawing up battle lines, because there it takes wisdom and foresight to what purpose he orders the men; and skillful ability also, for any occasion or purpose, to devise and use the fittest and most necessary order, array, and figure of his company and sum of men.\" By figure I mean such as a perfect square, triangle, circle, oval, long square (which the Greeks call Eteromekes), rhomb, rhomboid, crescent, ring, serpentine, and such other geometric figures: which in wars have been and are to be used for convenience, necessity, and advantage, etc. And he ought to have no small skill who can give a true report, or near the truth, of the numbers and sums of footmen or horsemen in the enemy's ordering. To make an estimate from afar, between close terms of more and less, is not a thing very common among those who would gladly do it.\n\nI. D., friend, you will find it hard to carry out my description of this feat. But by chorography you can help yourself somewhat: where the figures known (in sides and angles) are not regular, and where a resolution into triangles can serve, etc. And yet you will find it strange to deal so generally with arithmetic figures, and that for battle array: their contents differ so much from like geometric figures.\n\nGreat policy may be used by the captains (at fitting times and in convenient places): as to use figures that make the greatest show of as many men as he has; and using the advantage of the three kinds of usual spaces (between footmen or horsemen) to take the largest; or when he would seem to have few, being many: and the contrary, in figure and space. The herald, pursuivant, sergeant royal, captain, or whoever is careful to come near the truth in this, besides the judgment of his expert eye, his skill of tactical ordering, and the help of his geometric instrument, ring or astronomical staff (conveniently framed for carrying and use), he may wonderfully help himself with perspective glasses. In which (I trust) our posterity will prove more skillful and expert, and for greater purposes, than in these days can almost be credited to be possible.\n\nThus I have lightly passed over the artificial feats depending chiefly upon common geometry, and commonly and generally reckoned under the name of geometry.\n\n### The Mixed Sciences and Perspective\n\nBut there are other (very many) methodical arts, which, declining from the purity, simplicity, and immateriality of our principal science of magnitudes, do nevertheless use the great aid, direction, and method of the said principal science, and have proper names, distinct both from the science of geometry (from which they are derived) and from one another: such as perspective, astronomy, music, Cosmography, astrology, statics, anthropography, Trochilike, Helicosophy, Pneumatithmie, Menadrie, Hypogeiodie, Hydragogie, horometry, painting, architecture, navigation, Thaumaturgike and Archemastrie. I think it necessary to give, in order, some particular description of these, and also to touch on some of their useful applications, and so to make this preface a little sweet, pleasant nosegay for you: to comfort your spirits when they are almost out of courage and in despair (through a brutish turn of mind), supposing that geometry has served only for the building of a house, or an ornate bridge, or the roof of Westminster Hall, or some clever little device or engine suited to a carpenter or a joiner, etc. That things are far otherwise than the world has commonly judged them, to this day, will be well proved both in word and in deed.\n\nAmong these arts, by good reason, perspective ought to be had before perfect knowledge of astronomical appearances can be attained. And because light holds the prerogative as the first of God's creatures, and the eye is the light of our body, whose sense is the mightiest and whose organ is the most skillful and geometric, we will begin, therefore, with perspective. Perspective is a mathematical art which demonstrates the manner and properties of all radiations—direct, broken, and reflected. This description, or statement, is brief, but it reaches as far as the world is wide. It concerns all creatures, all actions and passions brought about by the emanation of rays. By rays, or natural lines, I mean here not only those of light or color (though these give the eye appearance, witness, and proof on which to ground the art) but also the certain and determined active radial emanations of other forms, both substantial and accidental. Through this art (setting aside its highest points) we may use our eyes and the light with greater pleasure and more perfect judgment, both of things seen in light and of other things, which work and produce their effects by the same order of the radiation of lights. We should be ashamed to be ignorant of the cause of the many ways our eye is deceived and misled: as when the eye takes a round globe or sphere, far off, to be a flat plain circle, and likewise judges a flat square to be round; supposes parallel walls to draw together at a distance; and takes roof and floor, parallel to each other, the one to bend downward and the other to rise up, at a little distance from you. Again, of things moving at equal speed, to think the nearer moves faster and the farther much slower. And of two things, one of which moves incomparably faster than the other, to judge the slower to move very swiftly and the other to stand still—what an error is this of our eye! Of the rainbow: of its colors, the order of the colors, its size, its place and height, etc.—to know the demonstrative causes, is it not pleasant, is it not necessary? Of two or three suns appearing, of blazing stars, and such things, brought to pass by natural causes (and yet, nevertheless, significant of matters beyond): is it not useful for humankind to know the very true cause and the natural occasion? Is it not, rather, greatly beneath the dignity of human nature to be so misled and deceived by things at hand, before one's eyes: as with a peacock's tail and a dove's neck, or a whole oar held in water seeming broken? Distant things seeming near, and near things seeming distant. Small things seeming great, and great things seeming small. One man seeming an army. Or a person being terribly afraid of his own shadow. Yes, to fear so much that, if you are alone near a certain mirror and make as if to thrust at it with a dagger or sword, you will suddenly be moved to draw back, as it were, by reason of an image,\n\nA marvelous mirror.\n\nappearing in the air between you and the mirror, with the same hand, sword or dagger, and with equal quickness, lunging at your very eye just as you do at the mirror. Strange this is to hear of, but more marvelous to see than these words of mine can express. And yet by optical demonstration the order and cause of it are made certain, even as its effect follows from it. Yes, this much more I dare undertake, toward satisfying the noble spirit that longs ardently for the wisdom of natural causes: to let him understand that in London he may with his own eyes have proof of what I have said here. A gentleman (who, for the good service he has done his country, is famous and honorable:\n\nand for skill in the mathematical sciences and languages is the odd man of this land, etc.) Even he is able, and I am sure will very willingly, let the mirror and the proof be seen; and so I here request it of him: for the increase of wisdom among the honorable, for the stopping of malicious mouths, and for the repressing of the arrogance of the ignorant. You may easily guess what I mean. This art of perspective is of such excellence, and may be led to the establishing and carrying out of such things as no one would readily believe without actual proof perceived. I say nothing of natural philosophy, which, without perspective, cannot be fully understood nor perfectly attained. Nor of astronomy, which, without perspective, cannot well be grounded; nor of astrology, verified and confirmed on natural grounds. That part of it which deals with mirrors (the word mirror being, in this art, a general name for anything from which a ray bounces back) is called catoptrics, and has so many uses, both marvelous and profitable, that it would keep me too long to note the principal conclusions already known in it; and also some things might perhaps lack due credit with you, and I would by that lose my labor, and you would slip into a light judgment,\n\nBefore you have learned sufficiently the power of nature and art.\n\n### Astronomy\n\nNow, to proceed: Astronomy is a mathematical art which demonstrates the distances, magnitudes, and all natural motions, appearances, and affections proper to the planets and fixed stars: for any time past, present, or to come: with respect to a given horizon, or without respect to any horizon. By this art we are assured of the distance of the starry sky and of each planet from the center of the Earth, and of the size of any fixed star seen, or of any planet, in proportion to the Earth's size. Thus we are sure (by this art) that the solidity, bulk, and body of the Sun contain the quantity of the whole Earth and Sea a hundred and sixty-two times, less by one eighth part of the Earth. But the body of the whole earthly globe and Sea is bigger than the body of the Moon forty-three times, less by one eighth of the Moon. Therefore the Sun is bigger than the Moon 7000 times, less by 59 39/64—that is, precisely 6940 25/64 times bigger than the Moon. And yet an unskillful person would judge them alike in size. Therefore, necessarily, one is much farther from us than the other. The Sun, when it is farthest from the Earth (which now, in our age, is when it is in the 8th degree of Cancer), is 1179 semidiameters of the Earth distant. And the Moon, when she is farthest from the Earth, is 68 semidiameters of the Earth and ⅓. The nearest the Moon comes to the Earth is 52¼ semidiameters. The distance of the starry sky from us, in semidiameters of the Earth, is 20081½—twenty thousand, eighty-one and a half. Subtract from this the Moon's nearest distance from the Earth, and there remain 20029¼ semidiameters of the Earth—twenty thousand twenty-nine and a quarter.\n\nSo thick is the heavenly palace in which the planets have all their exercise, and so marvelously do they perform the command and charge given to them by the omnipotent majesty of the King of kings. This is that which in Genesis is called Ha Rakia. Consider it well. The semidiameter of the Earth contains, in our common miles, 3436 4/11—that is, three thousand four hundred thirty-six and four elevenths of one mile; so that the whole Earth and Sea round about is 21600, twenty-one thousand six hundred of our miles, allowing sixty miles for every degree of the greatest circle. Now if you weigh well within yourself but this little parcel of astronomical fruit, concerning the size and distances of the Sun, Moon, and starry sky, and the huge mass of Ha Rakia, will you not find your conscience moved, with the kingly prophet, to sing the confession of God's glory, and say, \"The heavens declare the glory of God, and the firmament [Ha Rakia] shows forth the works of his hands.\" And so forth for those first five staves of that kingly psalm. Well, well, it is time for some to lay hold of wisdom and to judge things truly, and not so to expound the Holy Word all by allegories, as to neglect the wisdom, power, and goodness of God to be seen and learned in and through his creatures and creation. Through parables and analogies of their natures and properties, the course of Holy Scripture also declares to us very many mysteries. The whole frame of God's creatures (which is the whole world) is to us a bright mirror, from which, by reflection, beams and radiations bounce back to our knowledge and perception, representing the image of his infinite goodness, omnipotence, and wisdom. And we are by that taught and persuaded to glorify our Creator as God, and to be thankful for it. Could the heathens find these uses in these most pure, beautiful, and mighty corporeal creatures, and shall we—after the true Sun of righteousness has risen above the horizon of our temporal hemisphere, and has so abundantly streamed into our hearts the direct beams of his goodness, mercy, and grace, whose heat all creatures feel, spiritual and corporeal, visible and invisible? Shall we, I say, look upon the heaven, stars, and planets as an ox and a donkey do, no further caring or inquiring what they are, why they were created, or how they carry out the purpose for which they were created? Seeing that all creatures were created for our sake, and that both we and they were created chiefly to glorify the Almighty Creator, and that by every means possible to us. Do not be ignorant (says Plato in Epinomis) that astronomy is a thing of excellent wisdom. Astronomy was from the beginning commended to us, and in a manner commanded by God himself, since he made the sun, moon, and stars to be for us signs, and knowledge of seasons, and for the distinction of days and years. Many words are not needed. But I wish every person would weigh this word, signs. And besides that, compare it also with the tenth chapter of Jeremiah. And though some think that there they have found a rod, yet modest reason will be an impartial judge of who ought to be beaten with it, in respect of our purpose. Leaving that, I ask you to understand this: that without great diligence of observation, examination, and calculation, their periods and courses (by which the distinction of seasons, years, and new moons might be precisely known) could not be exactly established. To accomplish this is the art which we have here defined as astronomy. By it we may have the distinct course of times, days, years, and ages: both for the consideration of sacred prophecies foretold and accomplished in due time, and for the holding of high mystical solemnities, and for all other human affairs, conditions, and covenants made at a fixed time between person and person, with many other great uses: in which, truly, there would be great uncertainty, confusion, untruth, and brutish barbarism without the wonderful diligence and skill of this art, continually learning and determining times and periods of time by the record of the heavenly book in which all times are written, to be read with an astronomical staff instead of a pointer.\n\n### Music\n\nMusic has its original cause in motion: therefore, after the motions most swift and most slow, which are in the firmament, performed by nature and under the astronomer's consideration, now I will speak of another kind of motion, producing sound, audible and countable by man. By music I mean, here, that science which the Greeks call Harmonice. I do not meddle with the controversy between the ancient Harmonists and Canonists. Music is a mathematical science which teaches, by sense and reason, perfectly to judge and order the diversities of sounds, high and low. Astronomy and music are sisters, says Plato. As, for astronomy, the eyes were made; so, for harmonious motion, the ears. But as astronomy has a more divine contemplation and benefit than mortal eye can perceive, so music is to be considered,\n\n(1.)\n\nthat the mind may be preferred to the ear. And from audible sound we ought to rise to the examination: which numbers are harmonious and which are not. And why the one are, and the other are not. I could at length,\n\n(2.)\n\nin the heavenly motions and distances, describe a marvelous harmony, of Pythagoras' harp\n\n(3.)\n\nwith eight strings. Also, something might be said of Mercury's two harps,\n\n(4.)\n\neach of four elemental strings. And very strange matter might be alleged of the harmony,\n\n(5.)\n\nappropriate to our spiritual part. As may appear in Ptolemy's third book, in the fourth and sixth chapters.\n\n(6.)\n\nAnd what is the cause of the fitting bond and friendly fellowship of the intellectual and mental part of us with our gross and corruptible body, but a certain mean and harmonious spirituality, participating in both and (in a manner) resulting in\n\n(7.)\n\nthe tune of the human voice, and also\n\n(8.)\n\nthe sound of instruments: what might be said of harmony, no common musician would readily believe.\n\nI. D. Read in Aristotle's 8th book of Politics, the 5th, 6th, and 7th chapters. Where you will have further occasion to think about music than is commonly thought.\n\nBut of the sundry mixture (as I may call it) and concourse, the diverse comparison and application of these harmonies, as of three, four, five, or more: marvelous have the effects been; and yet the like may be found and produced, with some proportional consideration for our time and being, in respect of the state of the things then, in which and by which the wondrous effects were wrought. Democritus and Theophrastus affirmed that, by music, griefs and diseases of the mind and body might be cured, or inferred. And we find in record that Terpander, Arion, Ismenias, Orpheus, Amphion, David, Pythagoras, Empedocles, Asclepiades, and Timotheus, by harmonic consonance, have done and brought to pass things more than marvelous to hear of. Of them, then, making no further discourse here: I am sure that common music, commonly used, is found by musicians and hearers to be so beneficial and pleasant that if I were to say and argue only this much—that it ought to be used otherwise than it is—I should find more critics than I could find people privy to, or skilled in, my meaning. In things, therefore, evident and better known than I can express, and so allowed and approved of (as I would wish some other things had the like luck), I will spare to enlarge my lines any further, but consequently follow my purpose.\n\n### Cosmography\n\nOf cosmography, I have briefly set out here to give you some information. Cosmography is the whole and perfect description of the heavenly and also the elemental part of the world, and their corresponding application and necessary mutual comparison. This art requires astronomy, geography, hydrography, and music. Therefore it is no small art, nor so simple as it is (slightly) considered in common practice. This matches heaven and earth in one frame, and fittingly applies corresponding parts, so that the heavenly globe may (in practice) be duly described upon the geographic and hydrographic globe. And there, for us to consider an equinoctial circle, an ecliptic line, colures, poles, stars in their true longitudes, latitudes, declinations, and verticality; also climes and parallels: and by an annexed horizon, and the revolution of the earthly globe (as the heaven is carried about by the primum mobile, the outermost moving sphere, in 24 equal hours), to learn the risings and settings of stars (of Virgil in his Georgics: of Hesiod: of Hippocrates in his medical sphere, to Perdicca, king of the Macedonians: of Diocles, to King Antigonus, and of other famous philosophers prescribed), a thing necessary for the due tillage of the earth, for navigation, for the alteration of a man's body, being whole, sick, wounded, or bruised. By the revolution, also, or motion of the cosmographic globe, the rising and setting of the sun, the lengths of days and nights, and the hours and times (both night and day) are known: with very many other pleasant and necessary uses: of which some are known, but the better remain, for such as know and use them:\n\n(☞)\n\nwho, from a spark of true fire, can make a wonderful bonfire, by duly applying due matter.\n\n### Astrology\n\nOf astrology, here I make an art, separate from astronomy: not by new device, but by good reason and authority: for astrology is a mathematical art which reasonably demonstrates the operations and effects of the natural beams of light and the secret influence of the stars and planets: in every element and elemental body: at all times, in any given horizon. This art is furnished with many other great arts and experiences: as with perfect perspective, astronomy, cosmography, natural philosophy of the 4 elements, the art of graduation, and some good understanding in music: and yet moreover with another great art, following hereafter, though I set this one before, for some considerations that move me. Sufficient (you see) is the stuff for making this rare and secret art; and hard enough to shape into a syllogistic conclusion. Yet both the manifold and continual travels of the most ancient and wise philosophers, for the attaining of this art, and, by examples of effects, to confirm it, have left us sufficient proof and witness; and we also may perceive daily that a man's body, and all other elemental bodies, are altered, disposed, ordered, pleased, and displeased by the influential working of the sun, the moon, and the other stars and planets. And therefore, says Aristotle, in the first of his Meteorological books, in the second chapter: This [elemental] world is of necessity almost adjoining the heavenly motions: that from there all its virtue or force may be governed. For that is to be thought the first cause of all: from which the beginning of motion is. And again, in the tenth chapter: We must therefore also take up the principles of these things, and the causes of all alike. The principle, then, as that which moves, chief and first of all, is that circle in which the sun's motion is plainly seen, etc. And so forth. His Meteorological books are full of arguments and effective demonstrations of the virtue, operation, and power of the heavenly bodies in and upon the four elements, and other bodies composed of them (either perfectly or imperfectly). And in his second book, On Generation and Corruption, in the tenth chapter: Therefore the uppermost motion is not the cause of generation and corruption, but the motion of the zodiac: for that both is continual and is caused by two motions. And in his second book, and second chapter of his Physics. For man (says he) and the sun are the cause of man's generation. Authorities may be brought, very many: of 1000, 2000, yes and 3000 years of antiquity: from great philosophers, expert, wise, and godly men, for that conclusion: which daily and hourly we men may discern and perceive by sense and reason: all beasts feel, and simply show, by their actions and passions, outward and inward: all plants, herbs, trees, flowers, and fruits. And finally, the elements, and all things composed of the elements, give testimony (as Aristotle said) that their whole dispositions, virtues, and natural motions depend on the activity of the heavenly motions and influences. By which, besides the specific order and form due to every seed: and besides the nature proper to the individual matrix of the thing produced: what the heavenly impression shall be, the perfect and circumspect astrologer has to conclude. Not only (by apotelesms) τὸ ὁτὶ, but by natural and mathematical demonstration τὸ διότι. As to which, what sciences are requisite (without exception) I have partly warned here: and in my Propædeumes (besides other matter disclosed there) I have mathematically furnished up the whole method: which in this age of ours has not been so carefully handled by anyone I ever saw or heard of. I was,\n\nYears 1548 and 1549, in Louvain.\n\n(for 21 years ago) by certain earnest disputations of the learned Gerardus Mercator and Antonius Gogaua (and others), so provoked thereto: and (by my constant and invincible zeal to the truth) in observations of heavenly influences (to the minute of time), then so diligent: and chiefly by the supernatural influence from the star of Jacob, so directed: that any modest and sober student, carefully and diligently seeking the truth, will both find and confess tin this the truth of these my words: and also become a reasonable reformer of three sorts of people, greatly erring from the truth about these influential operations.\n\nOf these, the one is light believers, the other light despisers, and the third light practitioners. The first and most common sort think the heaven and stars answerable to any of their doubts or desires:\n\n(1.)\n\nwhich is not so: and indeed they overreach too much. The second sort think no influential virtue (from the heavenly bodies) bears any sway in generation\n\n(2.)\n\nand corruption, in this elemental world. And to the sun, moon, and stars (being so many, so pure, so bright, so wonderfully big, so far in distance, so manifold in their motions, so constant in their periods, &c.) they assign a slight, simple office or two, and so allow them (according to their capacities) as much virtue and influential power as they allow to the sign of the sun, moon, and seven stars, hung up (for signs) in London to distinguish houses, and such gross helps in our worldly affairs: and they do not understand (or will not understand) the other workings and virtues of the heavenly sun, moon, and stars: not so much as the mariner or husbandman: no, not so much as the elephant does, as the Cynocephalus, as the porcupine does; nor will they allow these perfect and incorruptible mighty bodies so much virtual radiation and force as they see in a little piece of magnet stone, which at a great distance shows its operation. And perhaps they think the sea and rivers (as the Thames) to be some living thing, and so to ebb and flow, run in and out, of themselves at their own fantasies. God help, God help. Surely these men come short: and either are too dull, or willfully blind, or perhaps too malicious. The third man is the common and vulgar astrologer, or practitioner: who, not being duly, skillfully, and perfectly\n\n(3.)\n\nfurnished: yet either for vain glory or gain, or like a simple dolt and blind Bayard, errs both in matter and manner: to the discredit of the wary and modest astrologer, and to the robbing of those most noble corporeal creatures of their natural virtue: being most mighty, most beneficial to all elemental generation, corruption, and their appurtenances: and most harmonious in their monarchy: for which things, being known and modestly used, we might highly and continually glorify God, with the princely prophet, saying: The heavens declare the glory of God: who made the heavens in his wisdom: who made the sun to have dominion of the day: the moon and stars to have dominion of the night: by which day to day utters speech, and night to night declares knowledge. Praise him, all you stars, and light. Amen.\n\n### Statics, 1\n\nIn order, now follows, of Statike: to say something of what we mean by that name, and what benefit depends on such an art. Statike is a mathematical art, which demonstrates the causes of heaviness and lightness in all things, and of the motions and properties belonging to heaviness and lightness. And since 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 experiments of the balance. Oh, that men knew what profit, in every way, might grow from this art to the able examiner and diligent practitioner. \"You alone know all things precisely (O God), who have made weight and balance your judgment; who have created all things in number, weight, and measure; who have weighed the mountains and hills in a balance, and poised in your hand both heaven and earth. We therefore, warned by the sacred word to consider your creatures, and by that consideration to win a glimpse, as it were, or a shadow of perception, that your wisdom, might, and goodness are infinite and unspeakable, declared in your creatures; and being further advised by your merciful goodness that three principal means were used in the creation of all your creatures, namely number, weight, and measure; and since the two arts of number and measure (ancient, famous, and to human uses most necessary) are already sufficiently known and available — this third key we beseech of you (through your accustomed goodness) that it may come to the needed and sufficient knowledge of such of your servants as would gladly find in your workmanship your true occasions (purposely designed) by which we should glorify your name, and show forth (to those weak in faith) your wondrous wisdom and goodness. Amen.\"\n\nDo not marvel at this outburst, godly friend, you gentle and zealous student. Another day, perhaps, you will perceive what moved me to it. Here, for now, I will give you some foundation, and with it some show of the benefits arising from this art. And because this art is rare, my words and practices might be too dark, unless some light were held before the matter; and that will best be done by giving you, out of Archimedes' demonstrations, a few principal conclusions, as follows.\n\n1.\n\nThe surface of every liquid, standing by itself and at rest, is spherical; its center is the same as the center of the Earth.\n\n2.\n\nIf solid magnitudes of the same size or quantity as any liquid, and having the same weight, are lowered into that liquid, they will settle downward so that no part of them lies above the surface of the liquid; and yet for all that they will not sink utterly down, or drown.\n\n3.\n\nIf any solid magnitude lighter than a liquid is lowered into that same liquid, it will settle down into it so far that the quantity of that liquid equal to the part of the solid magnitude which has settled down into it is, in weight, equal to the weight of the whole solid magnitude.\n\n4.\n\nAny solid magnitude lighter than a liquid, forced down into that same liquid, will move upward with a power equal to the amount by which the liquid of equal quantity to the whole magnitude is heavier than that same magnitude.\n\n5.\n\nAny solid magnitude heavier than a liquid, lowered into that same liquid, will sink down utterly; and in that liquid it will be lighter by exactly the weight or heaviness of the liquid having size or quantity equal to the solid magnitude.\n\n6.\n\nI. D. The cutting of a sphere according to any proportion assigned may by this proposition be done mechanically, by tempering a liquid to a certain weight in respect of the weight of the sphere swimming in it.\n\nIf any solid magnitude lighter than a liquid is lowered into that same liquid, the weight of that magnitude is to the weight of the liquid (which is equal in quantity to the whole magnitude) in the same proportion that the part of the magnitude settled down is to the whole magnitude.\n\nBy these truths, great errors may be corrected in the opinion of the natural motion of things, light and heavy. Which errors are allowed in natural philosophy by almost all men, from too much trust in authority and false assumptions; as, that of any two bodies, the heavier moves downward faster than the lighter.\n\nA common error, noted.\n\nThis error was not first noted by me, but by one John Baptist de Benedictis. The chief of his propositions is this, which seems a paradox:\n\nA paradox.\n\nIf there are two bodies of one form and of one kind, equal in quantity or unequal,\n\nthey will move through equal spaces in equal times: provided both their motions are in air, or both in water, or in any one medium.\n\nHereupon, in the practice of gunnery, certain good discourses may otherwise receive great amendment and furtherance.\n\nThe wonderful use of these propositions.\n\nIn the intended purpose also, allowing something to the imperfection of nature, which does not answer to the precision of demonstration. Moreover, by the foresaid propositions (wisely used) air, water, earth, and fire may be nearly known: how light or heavy they are (naturally) in their assigned parts, or in the whole. And then, turning your practice to elemental things, you may deal for the proportion of the elements in compounded things. Then, for the proportions of the humors in man: their weights, and the weight of his bones and flesh, etc. Then, by weight, to consider the force of a man in any manner, whole or in part. Then you may have pleasant consideration of the water-draft of ships, differing in the sea and in fresh water; and of weighing up anything sunk in sea or fresh water, etc. And (to lift your head aloft) by weight you may, as precisely as by any other instrument, measure the diameters of sun and moon, etc.\n\nFriend, please weigh these things with the just balance of reason, and you will find wonders upon wonders; and esteem one drop of truth (yes, in natural philosophy) more worth than whole libraries of opinions undemonstrated, or not answering to nature's law and your experience. Leaving these things thus, I will give you two or three light practices to great purpose, and so finish my statical annotation. In mathematical matters, by the mechanician's aid, we will here see the benefit of weight.\n\nThe statical practice, to know the proportion between the cube and the sphere.\n\nMake a cube of one uniform material throughout, of like heavy stuff; and of the same stuff make a sphere or globe, precisely, of a diameter equal to the radical side of the cube. Your stuff may be wood, copper, tin, lead, silver, etc. (being, as I said, of like nature, condition, and like weight throughout). And you may, by assay balance, have prepared a great number of the smallest weights, which by that balance can be discerned or tried; and so have proceeded to make a perfect pile, company and number of weights, up to the weight of six, eight, or twelve pounds: all most diligently tried. And of every one, the content known, in your least weight that is weighable.\n\n[Those who cannot have weights of precision may, with uniform and well-sifted sand, make themselves a number of weights somewhat near precision, by always halving the sand; they will at length come to a least common weight. There I leave the further matter to the discretion of those whom necessity shall press.]\n\nThe Venetian reckoning of weight may seem precise enough: by eight successive descents, halving from a grain.\n\nI. D. For so you have 256 parts of a grain.\n\nYour cube, sphere, suitable balance, and convenient weights being ready, fall to work. First, weigh your cube. Note the number of the weights. Then weigh your sphere, and likewise note the number of the weights. If you now find the weight of your cube to be to the weight of the sphere as 21 is to 11, then you see how the mechanician and experimenter, without geometry and demonstration, are (as nearly in effect) taught the proportion of the cube to the sphere, as I have demonstrated it at the end of the twelfth book of Euclid. Often try with the same cube and sphere. Then change your sphere and cube to another material, or to another size, until you have made a perfect universal experience of it. It is possible that you may win to nearer terms in the proportion.\n\nWhen you have found this one certain drop of natural truth, proceed on to infer, and duly to make trial of, the matter depending. For, since it is well demonstrated that a cylinder whose height and the diameter of its base are equal to the diameter of the sphere is sesquialter to that same sphere (that is, as 3 is to 2), to the number of the weight of the sphere add half as much again, and so you have the number of the weight of that cylinder. Which is also contained in our former cube: so that the base of that cylinder is a circle described in the square which is the base of our cube. But the cube and the cylinder, being both of one height, have their bases in the same proportion in which they are, one to another, in their bulk or solidity. But, beforehand, we have two numbers expressing their bulk, solidities, and quantities by weight; therefore,\n\nThe proportion of the square to the circle inscribed.\n\nwe have the proportion of the square to the circle inscribed in the same square. And so we have fallen into the sensible and experimental knowledge of Archimedes' great secret, sought and found by him through great travail of mind. Therefore, to any circle given you can give a square equal:\n\nThe squaring of the circle, mechanically.\n\nas I have taught in my annotation upon the first proposition of the twelfth book; and likewise to any square given you may give a circle equal:\n\nTo any square given, to give a circle equal.\n\nIf you describe a circle which shall be in that proportion to your inscribed circle as the square is to the same circle, this you may do by my annotations upon the second proposition of the twelfth book of Euclid, in my third problem there. Your diligence may come to a proportion of the square to the circle inscribed nearer the truth than is the proportion of 14 to 11. And consider that you may begin at the circle and square, and so come to conclude of the sphere and the cube what their proportion is, as now you came from the sphere to the circle. For, of silver or gold or brass sheets or plates (drawn through one hole, as is the manner) if you make a square figure and weigh it; and then, describing thereon the inscribed circle, cut off and file away precisely (to the circle) the surplus of the square, you shall then, weighing your circle, see whether the weight of the square be to your circle as 14 to 11. As I have noted at the beginning of Euclid's twelfth book, etc. After this, resort to my last proposition upon the last of the twelfth; and there help yourself to the end. And here note this by the way.\n\nNote: squaring the circle without knowledge of the proportion between circumference and diameter.\n\nThat we may square the circle without having knowledge of the proportion of the circumference to the diameter, as you have here perceived. And otherwise also, I can demonstrate it. So that many have encumbered themselves superfluously, by laboring in that point first which was not of necessity first, and also very intricate. And easily you may (and that variously) come to the knowledge of the circumference, the circle's quantity being first known. Which thing I leave to your consideration, making haste to dispatch another masterly problem, and to bring it nearer to your knowledge and readier dealing with than the world (before this day) had it for you, that I can tell of. And that is, a mechanical doubling of the cube, etc. Which may thus be done:\n\nTo double the cube readily, by mechanical art, depending upon mathematical demonstration.\n\nMake of copper plates or tin plates a four-square upright pyramid, or a cone, perfectly fashioned hollow within. In this let great diligence be used to approach (as near as may be) to the mathematical perfection of those figures. At their bases let them be all open; everywhere else most close and just. From the vertex to the circumference of the base of the cone, and to the sides of the base of the pyramid:\n\nI. D. The 4 sides of this pyramid must be 4 isosceles triangles alike and equal.\n\nLet 4 straight lines be drawn in the inside of the cone and pyramid, making at their fall upon the perimeters of the bases equal angles on both sides of themselves with those perimeters. These 4 lines (in the pyramid, and as many in the cone) divide: one into 12 equal parts, another into 24, another into 60, and another into 100 (counting up from the vertex). Or use other numbers of division, as experience shall teach you.\n\nI. D. In all workings with this pyramid or cone, let their situations be in all points and conditions alike, or all the same, while you are about one work. Else you will err.\n\n### Statics, 2\n\nThen set your cone or pyramid, with the vertex downward, perpendicularly with respect to the base. (Though it is placed otherwise, it hinders nothing.) So let them be most steadily supported. Now, if there is a cube which you would have doubled, make a neat cube of copper, silver, lead, tin, wood, stone, or bone; or else make a hollow cube, or cubic coffer, of copper, silver, tin, or wood, etc. You may so proportion these with respect to your pyramid or cone that the pyramid or cone will be able to contain their weight in water 3 or 4 times over, at the least, whatever stuff they are made of. Let not your solid angle at the vertex be too sharp, but let the water come with ease to the very vertex of your hollow cone or pyramid. Put one of your solid cubes in a suitable balance; take its weight exactly in water. Pour that water (without loss) into the hollow pyramid or cone, quietly. Mark in your lines what numbers the water cuts at. Take the weight of the same cube again, in the same kind of water which you had before:\n\nI. D. Consider well when you must put your waters together, and when you must empty your first water out of your pyramid or cone. Else you will err.\n\nput that also into the pyramid or cone where you put the first. Mark now again in what number or place of the lines the water cuts them. Two ways you may reach your conclusion: that is, either by numbers or by lines. By numbers: as, if you divide the side of your fundamental cube into as many equal parts as it is capable of, conveniently, with your ease and the precision of the division. For, as the number of your first and lesser line (in your hollow pyramid or cone) is to the second or greater (both being counted from the vertex), so shall the number of the side of your fundamental cube be to the number belonging to the radical side of the cube double to your fundamental cube; which, being multiplied cubically, will soon show whether it is double the cubic number of your fundamental cube or not. By lines, thus: as your lesser and first line (in your hollow pyramid or cone) is to the second or greater, so let the radical side of your fundamental cube be to a fourth proportional line, by the 12th proposition of the sixth book of Euclid. Which fourth line shall be the cube root, or radical side, of the cube double to your fundamental cube; which is the thing we desired.\n\nGod be thanked for this invention and the fruit ensuing.\n\nFor this may I (with joy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, thanking the holy and glorious Trinity, having greater cause thereto than\n\nVitruvius, Book 9, Chapter 3.\n\nArchimedes had, for finding the fraud used in the king's crown of gold; as all men may easily judge by the diversity of the fruit following from the one and the other.\n\nWhere I spoke before of a hollow cubic coffer, the like use is of it, and without weight. Thus: fill it with water precisely full, and pour that water into your pyramid or cone. And here note where the lines are cut in your pyramid or cone. Again fill your coffer, just as you did before. Put that water also to the first. Mark the second cutting of your lines. Now, as you proceeded before, so must you proceed here.\n\nNote.\n\nAnd if the cube which you should double be ever so great, you have thus the proportion (in small) between your two little cubes; and then the side of that great cube (to be doubled), being the third, will have the fourth found to it proportional, by the 12th of the sixth book of Euclid.\n\nNote, concerning the spherical surface of the water.\n\nNote that all this while I do not forget my first statical proposition, here rehearsed: that the surface of the water is spherical. In this use your discretion: to the first line adding a small hair's breadth more, and to the second half a hair's breadth more to its length. For you will easily perceive that the difference can be no greater in any pyramid or cone you are to handle. Which you shall thus try. For finding the swell of the water above level.\n\n\"Square the semidiameter from the center of the earth to the surface of your first water. Then square half the subtendent of that watery surface (which subtendent must have the equal parts of its measure all the same as those of the semidiameter of the earth to your watery surface). Subtract this square from the first; of the residue take the square root. Subtract that root from your first semidiameter of the earth to your watery surface: what remains is the height of the water in the middle above the level.\"\n\nWhich you will find to be a thing imperceptible. And though it were greatly perceptible,\n\nNote.\n\nyet, by help of my sixth theorem upon the last proposition of Euclid's twelfth book, you may reduce all to a true level. But further diligence is to be used by you against accidental causes of the water's swelling: as by having beforehand made moist your hollow pyramid or cone somewhat with a moist sponge, which will prevent an accidental cause of swelling, etc. Experience will teach you abundantly, with great ease, pleasure, and convenience.\n\nThus may you double the cube mechanically, treble it, and so forth, in any proportion.\n\nNote this abridgment of doubling the cube, etc.\n\nNow will I abridge your pain, cost, and care in this. Without all the preparing of your fundamental cubes, you may (alike) work this conclusion. For that was rather a kind of experimental demonstration than the shortest way, and all depending upon one mathematical demonstration. \"Take water (as much as will conveniently serve your turn, as I warned before of your fundamental cube's size). Weigh it precisely. Put that water into your pyramid or cone. Of the same kind of water then take again the same weight you had before, and put that likewise into the pyramid or cone. For each time, your marking of the lines, how the water cuts them, shall give you the proportion between the radical sides of any two cubes, of which the one is double to the other; working as before I have taught you:\n\nNote.\n\nsaving that, for your fundamental cube's radical side, here you may take a straight line at pleasure.\"\n\nYet further proceeding with our drop of natural truth:\n\nTo give cubes one to another in any proportion, rational or irrational.\n\nyou may (now) give cubes one to another in any proportion given, rational or irrational, in this manner. Make a hollow parallelepiped of copper or tin, with one base wanting, or open, as in our cubic coffer. From the bottom of that parallelepiped raise up many perpendiculars in each of its four sides. Now if any proportion be assigned you in straight lines, cut one of your perpendiculars (or a line equal to it, or less than it) likewise, by the 10th of the sixth book of Euclid. And set those two parts in two separate lines of those perpendiculars (or you may set them both in one line), making their beginnings to be at the base, and so their lengths to extend upward. Now set your hollow parallelepiped upright, perpendicularly, steady. Pour in water to the height of your shorter line. Pour that water into the hollow pyramid or cone. Mark the place of the rise. Settle your hollow parallelepiped again. Pour water into it, to the height of the second line, exactly.\n\nEmptying the first.\n\nPour that water duly into the hollow pyramid or cone; mark now again where the water cuts the same line which you marked before. For there, as the first marked line is to the second, so shall the two radical sides be, one to the other, of any two cubes which in their solidity shall have the same proportion which was at the first assigned, whether it be rational or irrational.\n\nThus, in sundry ways you may furnish yourself with such strange and profitable matter, which long has been wished for. And though it be done naturally and mechanically, yet it has a good mathematical demonstration.\n\nThe demonstrations of this doubling of the cube, and of the rest.\n\nWhich is this: always you have two like pyramids, or two like cones, in the proportions assigned; and like pyramids or cones are in proportion, one to another, in the proportion of their homologous sides (or lines) tripled. Therefore, if to the first and second lines found in your hollow pyramid or cone you join a third and a fourth in continual proportion, that fourth line shall be to the first as the greater pyramid or cone is to the lesser, by the 33rd of the eleventh book of Euclid. If pyramid to pyramid, or cone to cone, be double,\n\nI. D. Hereby help yourself to become a precise practitioner. And so consider how you are hindered not at all (sensibly) by the convexity of the water.\n\nthen shall line to line be also double, etc. But, as our first line is to the second, so is the radical side of our fundamental cube to the radical side of the cube to be made or to be doubled; and therefore to those two also a third and a fourth line in continual proportion joined will give the fourth line in that proportion to the first as our fourth pyramidal or conic line was to its first; but that was double or triple, etc., as the pyramids or cones were one to another (as we have proved); therefore this fourth shall be also double or triple to the first, as the pyramids or cones were one to another. But our made cube is described of the second in proportion, of the four proportional lines:\n\nBy the 33rd of the eleventh book of Euclid.\n\ntherefore, as the fourth line is to the first, so is that cube to the first cube; and we have proved the fourth line to be to the first as the pyramid or cone is to the pyramid or cone. Therefore the cube is to the cube as pyramid is to pyramid, or cone is to cone.\n\nI. D. And your diligence in practice can so (in weight of water) perform it; therefore, now, you are able to give good reason for your whole doing.\n\nBut we suppose pyramid to pyramid, or cone to cone, to be double or triple, etc. Therefore cube is to cube double, or triple, etc. Which was to be demonstrated. And of the parallelepiped it is evident that the solid water parallelepipeds are one to another as their heights are, seeing they have one base. Therefore the pyramids or cones made of those water parallelepipeds are one to another as the lines are (one to another) between which our proportion was assigned. But the cubes made of lines, after the proportion of the pyramidal or conic homologous lines, are one to another as the pyramids or cones are one to another (as we before did prove); therefore the cubes made shall be one to another as the lines assigned are one to another. Which was to be demonstrated. Note.\n\nNote this corollary.\n\nThis my demonstration is more general than only in a square pyramid or cone; consider well. Thus have I, both mathematically and mechanically, been very long in words; yet (I trust) nothing tedious to those who are well affected toward these things. And truly I am forced (avoiding prolixity) to omit sundry such things easy to be practiced, which to the mathematician would be a great treasure, and to the mechanician no small gain.\n\nThe great benefits following from these new inventions.\n\nNow may you, between two lines given, find two middle proportionals in continual proportion, by the hollow parallelepiped and the hollow pyramid or cone. Now, any rectangular parallelepiped being given, three straight lines may be found proportional in any proportion assigned, of which shall be produced a parallelepiped equal to the parallelepiped given. Hereof I noted somewhat upon the 36th proposition of the 11th book of Euclid. Now, all those things which Vitruvius in his Architecture specified as able to be done by doubling of the cube, or by finding of two middle proportionals between two lines given, may easily be performed. Now, that problem which I noted to you at the end of my addition upon the 34th of the 11th book of Euclid is proved possible. Now may any regular body be transformed into another, etc. Now, any regular body: any sphere, yes, any mixed solid; and (what is more) irregular solids may be made (in any proportion assigned) like the body first given. Thus, of a mannequin (as the Dutch painters term it) in the same symmetry may a giant be made, and that with any gesture used by the mannequin, and conversely. Now may you, of any mold or model of a ship, make one of the same mold (in any assigned proportion) bigger or lesser.\n\nNow may you, of any gun, or little piece of ordnance, make another, with the same symmetry (in all points) as great and as little as you will. Mark that, and think on it. Infinitely may you apply this, so long sought for and now so easily concluded, and withal so willingly and frankly communicated to such as faithfully deal with virtuous studies.\n\nSuch is the fruit of the mathematical sciences and arts.\n\n### Statics, 3\n\nThus the mathematical mind can deal speculatively in its own art, and by good means mount above the clouds and stars; and thirdly, it can, in order, descend, to frame natural things to wonderful uses, and, when it pleases, retire home into its own center, and there prepare more means to ascend or descend by, all to the glory of God and our honorable delight on earth.\n\nAlthough the printer has looked for this preface a day or two, yet I could not bring my pen from the paper before I had given you encouraging notice and brief instructions about some of the benefits that may be reaped by statics. In the rest I will therefore be as brief as I possibly can, and, at the same time, describe them somewhat accordingly. And you will perceive as much from what comes next in order. For, whereas it is so ample and wonderful that one might find in it fruitful matter to speak of for a whole year, and in practice is a treasure without end, yet I will glance over it with very few words.\n\n### Anthropography\n\nThis I call Anthropography: an art restored, and an advancement of mine in your service. I ask you to think of it as one of the chief points of human knowledge. Although it has only now first been confirmed with this new name, yet the matter has, from the beginning, been under consideration by all perfect philosophers. Anthropography is the description of the number, measure, weight, figure, situation, and color of every diverse thing contained in the perfect body of man, with certain knowledge of the symmetry, figure, weight, characterization, and due local motion of any assigned part of the said body, and of the numbers belonging to that part. This is the one part of the definition suited to this place, sufficient to show the particularity and excellency of the art, and why it is here ascribed to the mathematical sciences. If the description of the heavenly part of the world has a peculiar art called astronomy; if the description of the earthly globe has its peculiar art called geography; if the matching of both has its peculiar art called cosmography, which is the description of the whole and universal frame of the world; why should not the description of\n\nMan is the lesser world.\n\nhim who is the lesser world, and from the beginning called Microcosmus (that is, the lesser world), and for whose sake and service all other bodily creatures were created, who also partakes with spirits and angels and is made in the image and similitude of God, have its peculiar art, and be called the art of arts, rather than either lack a name or have too base and improper a one? You must borrow or claim back for yourself the peculiar parts of this from sundry professions, and proceed farther, as God, nature, reason, and experience shall inform you. The anatomists will restore to you some part; the physiognomists, some; the chiromantists, some; the metaposcopists, some; the excellent Albert Durer, a good part; the art of perspective will help the eye forward somewhat; Pythagoras, Hippocrates, Plato, Galen, Meletius, and many others, in certain things, will be contributors. And farther, the heavens, the earth, and all other creatures will each show and offer their harmonious service to fill up whatever is lacking hereof; and, concluding with your own experience, you may methodically register the whole for posterity, by which good proof will be had of our harmonious and\n\nThe microcosm.\n\nmicrocosmic constitution.\n\nThe outward image and view of this is most necessary and profitable to the art of drawing and painting, to sculpture, and to architecture (for church, house, fort, or ship), for it is the chief base and foundation of them.\n\nBook 3, chapter 1.\n\nLook in Vitruvius, whether I deal sincerely for your benefit or not. Look in Albertus Durerus, On the Symmetry of the Human Body. Look in chapters 27 and 28 of the second book, On Occult Philosophy. Consider the Ark of Noah. And from that, wade farther. Remember the Delphic oracle \"Know yourself\", pronounced so long ago, repeated by so many a philosopher, and attempted by the wisest; and then you will perceive how long ago you have been called to the school where this art might be learned. Well, I am not at all afraid of the disdain of some who think the sciences and arts to be only seven. Perhaps such people may, with ignorance and shame enough, come short of those seven as well; and yet nevertheless they cannot set down a certain number of arts, nor in each art certain impassable bounds for God, nature, and human industry. New arts rise up daily; and no such order was taken that all the arts should be made known to the world in one age, or in one land, or by one man. Let us embrace the gifts of God and the ways to wisdom, in this time of grace, continually bestowed from above on those who will receive them thankfully: and to the good, all things shall work together for good.\n\n### Trochilike\n\nTrochilike is that mathematical art which demonstrates the properties of all circular motions, simple and compound. And because the fruit of it, commonly received, is in wheels, it has the name Trochilike, as one might say, Wheel Art. By this art a wheel may be given which shall turn once about in any assigned time. Two wheels may be given whose turnings about in one and the same time (or in equal times) shall have to one another any appointed proportion. By wheels a straight line may be described; likewise a spiral line in a plane, conic section lines, and other irregular lines may be drawn at pleasure. These and such like are principal conclusions of this art, and help forward many pleasant and profitable mechanical works:\n\nSawmills.\n\nMills that saw great and very long deal boards, with no man standing by. Such I have seen in Germany, and in the city of Prague, in the kingdom of Bohemia; coining mills, hand mills for grinding corn; and all manner of mills and wheelwork, moved by wind, smoke, water, weight, spring, man or beast. Take up in your hand Agricola's On the Nature of Metals, and then you shall perceive, in all mines, how great need there is of wheelwork. By wheels strange and incredible works are done, as will appear hereafter in other arts. A wonderful example of further possibility and of present benefit was seen in my time in a certain instrument: which its inventor and artificer had formerly sold for twenty talents of gold, and which had then, by misfortune, suffered some injury and hurt. One Ianellus of Cremona mended it, and presented it to Emperor Charles the Fifth. Hieronymus Cardanus can be my witness that in it there was one wheel which moved, and that at such a rate that in 7000 years only its own period would be completed. A thing almost incredible; but how far I keep within my bounds, very many men yet alive can tell.\n\n### Helicosophy\n\nHelicosophy is near sister to Trochilike, and is a mathematical art which demonstrates the drawing of all spiral lines, in a plane, on a cylinder, cone, sphere, conoid, and spheroid, and the properties belonging to them. Its use in architecture and in diverse instruments and engines is most necessary. For in many things the screw performs the feat which otherwise could not be accomplished. By the help of this,\n\nAthenaeus, book 5, chapter 8.\n\nit is recorded that, whereas all the power of the city of Syracuse could not move a certain ship (being on the ground), mighty Archimedes, applying his screw engine, caused Hiero the king, by himself and at ease, to move her just as he pleased.\n\nProclus, page 18.\n\nAt this the king, wondering, said: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει λέγοντι πιϛευτεόν. \"From this day forward (said the king), credit ought to be given to Archimedes, whatever he says.\"\n\n### Pneumatithmie\n\nPneumatithmie demonstrates, by closed hollow geometric figures (regular and irregular), the strange properties (in motion or rest) of water, air, smoke, and fire in their continuity, and as they are joined to the elements next them. This art is very profitable to the natural philosopher, to prove that a vacuum, or emptiness, is not in the world; and that all nature abhors it so much that, contrary to ordinary law, the elements will move or stand. Thus water will ascend, rather than leave between it and the air a space or place more than that quantity of air naturally requires or can fill. Again, water will hang and not descend, rather than leave emptiness behind it by descending. The same is true of fire and air: they will descend rather than have their continuity dissolved, or their next element forced away from them. And as they will not be extended to discontinuity, so they will not, nor can they by any human force, be pressed or pent into a space not sufficient and answerable to their bodily substance. Great force and violence will they use to enjoy their natural right and liberty.\n\nTo go to the bottom of the sea without danger.\n\nHereupon, two or three men together, by keeping air under a great cauldron and forcing it down in order, may descend without harm to the bottom of the sea, and stay there for a time, etc. Where note how the thicker element, as water, gives place to the thinner, as is the air, and receives violence from the thinner in like manner, etc. Pumps and all manner of bellows have their ground in this art, and many other strange devices; as hydraulica, organs worked by water, etc. Of this feat, commonly called Pneumatica, goodly works are extant, both in Greek and Latin. With old and learned schoolmen it is called the Science of the Full and the Empty.\n\n### Menadrie\n\nMenadrie is a mathematical art which demonstrates how, above nature's simple virtue and power, virtue and force may be multiplied, and so direct, lift, pull toward, and push or cast from, any multiplied or simple determined virtue, weight, or force that is not naturally so easy to direct or move. This art is furthered very much by other arts: in some points by perspective; in some by statics; in some by Trochilike; and in others by Helicosophy and Pneumatithmie. By this art all cranes, gibbets, and engines to lift up or to force anything, in any manner of way, are ordered, and the certain cause of their force is known: as the force that one man has with the Dutch wagon rack, with which to set up again a mighty laden wagon that has been overturned. The force of the crossbow rack is certainly here demonstrated. The reason why one man with a lever lifts that which six men with their hands only could not so easily do. By this art, in our common cranes in London, where the power is to crane up a weight of 2000 pounds, by two more wheels (added by good ordering) the art concludes that there may be craned up a weight of 200000 pounds, etc. So well did Archimedes know this art that he alone, with his devices and engines, twice or thrice spoiled and discomfited the whole army and host of the Romans besieging Syracuse,\n\nPlutarch in Marco Marcello.\n\nMarcus Marcellus the Consul, being their commander-in-chief.\n\nSynesius, in his Letters.\n\nSuch huge stones, so many, with such force, and so far, did he with his engines hail among them out of the city.\n\nPolybius.\n\nPliny.\n\nQuintilian.\n\nT. Livius.\n\nAnd by sea likewise, though their ships might come to the walls of Syracuse, yet he utterly confounded the Roman navy. What with his mighty stones hurling:\n\nAthenaeus.\n\nwhat with pikes 18 feet long, made like shafts, which he forced almost a quarter of a mile; what with his catching hold of their ships, hoisting them up above the water, and suddenly letting them fall into the sea again;\n\nGalen.\n\nAnthemius.\n\nwhat with his burning glasses, by which he fired their other ships afar off; what with his other policies, devices, and engines, he so manfully acquitted himself that all the force, courage, and policy of the Romans (for a great season) could nothing prevail for the winning of Syracuse. After which the Romans named Archimedes Briareus and Centimanus. Zonaras makes mention of one Proclus, who so well had perceived Archimedes' art of Menadrie, and had so well invented on his own, that with his burning glasses,\n\nBurning glasses.\n\nbeing placed upon the walls of Byzantium, he multiplied the heat of the sun so, and directed its beams against his enemies' navy with such force and so suddenly (like lightning), that he burned and destroyed both man and ship. And Dion specifies of Priscus, a geometrician in Byzantium, who invented and used sundry engines of force multiplied; which was the cause that Emperor Severus pardoned him his life after he had won Byzantium, because he honored the art, wit, and rare industry of Priscus. But nothing inferior to the invention of these engines of force was the invention of guns.\n\nGuns.\n\nWhich took its occasion and order of first invention from an Englishman, though in another land and by other men it was first carried out. And those who should see the record where the occasion and general order of gunning is first discussed would think that \"small things, slight and common, coming to wise men's consideration and industrious men's handling, may grow to be of force incredible.\"\n\n### Hypogeiodie\n\nHypogeiodie is a mathematical art demonstrating how, under the spherical surface of the earth, at any depth, to any assigned perpendicular line (whose distance from the perpendicular of the entrance, and the azimuth likewise with respect to the said entrance, are known), a certain way may be prescribed and traveled; and how any way designed above the surface of the earth may be kept under ground at any limited depth, always going perpendicularly beneath the way designed on the earth; and, conversely, given any way (straight or crooked) under the earth, to lay it out upon the outer face or surface of the earth, so that perpendiculars drawn from the center of the earth to the spherical surface of the earth shall precisely fall in the corresponding points of those two ways. This, with all other cases and circumstances in this, and appurtenances, this art demonstrates. This art is very ample in variety of conclusions, and very profitable in sundry ways to the commonwealth. The occasion of my inventing this art was at the request of two gentlemen who had a certain work (of gain) under ground, and their grounds joined over the work; and by reason of the crookedness, diverse depths, and heights of the way under ground, they were in doubt, and at controversy, over under whose ground the work then was. The name only (before this) was published by me, On the Subterranean Journey; the rest is at God's will. For pioneers, miners, diggers for metals, stone, and coal, and for secret passages under ground between place and place (as this land has diverse), and for other purposes, any man may easily perceive both the great fruit of this art, and also in this art the great aid of geometry.\n\n### Hydragogie\n\nHydragogie demonstrates the possible leading of water, by nature's law and by artificial help, from any head (being a spring, standing, or running water) to any other place assigned. Long has this art been in use, and much of it written, and very marvelous works performed in it, as may still appear in Italy by the ruins remaining of the aqueducts; in other places, of rivers led through the mainland, navigable many a mile; and in other places, of the marvelous forcings of water to ascend. All these declare the great skill that must be required of one who would be perfect in this art for all possible occasions of leading water. I need not speak of the allowance of the fall for every hundred feet, or of the vents (if the water's labor be far and great), seeing that at hand (about us) many expert men can sufficiently testify to the effect of the order, though they do not know the demonstration of its necessity; nor yet, if they were led up and down and about mountains from the head of the spring, and then a place being assigned, and them being asked how low or high that last place is with respect to the head from which (so crookedly and up and down) they have come, perhaps they would not or could not very readily or precisely answer that question. Geometry is therefore necessary to Hydragogie. Of the sundry ways to force water to ascend, whether by tympanum, kettle mills, screw, Ctesibike, or such like, the manner may fully appear in Vitruvius, Agricola, and others; and so by that also may be most evident how the arts of Pneumatithmie, Helicosophy, statics, Trochilike, and Menadrie come to the furnishing of this in speculation, and to the benefit of the commonwealth in practice.\n\n### Horometry\n\nHorometry is a mathematical art that demonstrates how, at any appointed time, the precise customary reckoning of time may be known for any given place. These words are smooth, plain, easy English, but their meaning reaches farther than you would readily imagine. Some part of this art was called in old times Gnomonice, and more recently Horologiographia, and in English it may be called dialing. The use is ancient, and the invention is older still. The use plainly appears to have been in existence, at least, more than two thousand three hundred years ago:\n\n(4. Reg. 20.)\n\nIn King Achaz's dial, then, by the sun, showing the distinction of time. By sun, moon, and stars this dialing may be carried out, and the precise time of day or night known. But the demonstrative drawing of dials of every sort requires good skill in astronomy and in elemental, spherical, phenomenal, and conic geometry. Then, to apply the grounds of the art to any regular surface offered in any place, and there to describe, in every manner of way, how the usual hours may truly be determined by the sun's shadow, will be found no slight work for a draftsman: so to draw and prescribe the sun's motion to the breadth of a hair. In this feat (in my youth) I invented a way how, in any horizontal, mural, or equinoctial dial, etc., at all hours (the sun shining) the ascending sign and degree may be known. Which is a thing very necessary for the rising of those fixed stars whose operation in the air is plainly of great power. I say no more of its use here. But since human affairs require knowledge of times and moments when neither sun, moon, nor star can be seen, mechanical industry first invented how, by water running in orderly course, the time and hours might be known, of which the famous Ctesibius was the inventor, a man whom Vitruvius justly extolled to the sky. After that, hours were measured by running sand; then by Trochilike driven by weight; and in recent times by Trochilike driven by a spring, without weight. All of these, directed by sun or stars at certain times, require oversight and correction according to the heavenly equinoctial motion, besides the inequality of their own working. There remains (without any parabolic meaning here) among the philosophers,\n\n(A perpetual motion.)\n\na more excellent, more useful, and more marvelous way than all these: to have the motion of the Primouant (the first equinoctial motion) imitated by nature and art, which you shall hereafter understand more of through further search in weightier studies. And so it is time to finish this account of the division of time used in our common and private affairs, the benefit of which no man would want who knows how to spend his time.\n\n### Zography\n\nDrawing is a mathematical art which teaches and demonstrates how the intersection of all visual pyramids, made by any assigned plane (the center, distance, and lights being determined), may be represented by lines and proper colors. A notable art this is, and it would require a whole volume to declare its nature and the benefits ensuing from it. Great skill of geometry, arithmetic, perspective, and anthropography, with many other particular arts, the draftsman needs for his perfection. For the most excellent painter (who is but the proper mechanic and the sensible imitator of the draftsman) has attained to such perfection that the senses of men and beasts have judged painted things to be natural and not artificial, alive and not dead. This mechanical draftsman (commonly called the painter) is marvelous in his skill, and seems to have a certain divine power: to make of absent friends a friendly, present comfort; yes, and of dead friends to give a continual, silent presence, not only with us but with our posterity for many ages.\n\nAnd so proceeding, consider: in winter he can show you the lively view of summer's joy and riches, and in summer present the face of winter's doleful state and nakedness. Cities, towns, forts, woods, armies, yes whole kingdoms (however distant or great) he can easily bring home with him (into anyone's judgment) as living patterns of the things rehearsed. In one little house he can enclose (to the great pleasure of the beholders) the lively portrait of all visible creatures: those on earth or living under the earth; or in the waters, lying, creeping, sliding, or swimming; or any bird or fly flying in the air. Even set against the stars, the sky, the clouds, yes, in the very appearance of light itself (that divine creature), can he match the judgment of our eyes most nearly. What a thing this is! Things that do not yet exist he can represent so that, when they do exist, the picture seems (in a manner) to have created them. To what craftsman is painting not a great pleasure and benefit? Which of them all would refuse the direction and aid of painting? The architect, the goldsmith, and the tapestry weaver hold painting in high regard. Our vivid herbals, our portraits of birds, beasts, and fishes, and our elaborate anatomies, in what way are they most perfectly made, or beheld by us with most pleasure? Is it not by painting alone? And if painting, by the painter's industry, is so beneficial and marvelous, what shall be thought of drawing, the schoolmaster of painting and its chief governor? Though I do not mention sculpture in my table of mathematical arts, yet all men may perceive how painting and sculpture are full sisters, and both very profitable in a commonwealth. And of sculpture, as much as of painting, excellent craftsmen have written large books in its praise. I take as witnesses Georgio Vasari, Pittore Aretino; of Pomponius Gauricus; and others. To these two arts (and others) a certain odd art called Althalmasat owes much, more than the ordinary sculptor, carver, cutter, engraver, founder, or painter, etc., knows his art to owe.\n\n### Architecture\n\n(An objection.)\n\nArchitecture seems to many not worthy, or not fit, to be reckoned among the mathematical arts. To them I think it good to give some account of my doing so. Not worthy, they will say, because it is only for the building of a house, palace, church, fort, or such like gross works. And you also defined the mathematical arts to be such as dealt with no material or corruptible thing, and also did demonstratively proceed in their faculty by number or magnitude. First,\n\n(The answer.)\n\nYou see that I count Architecture here among those mathematical arts which are derived from the principal arts; and you know that such may deal with natural things and sensible matter. Of which, \"some draw nearer to simple and absolute mathematical speculation than others do.\n\nAnd though the architect procures, informs, and directs the mechanic in the handwork and the actual building of house, castle, or palace, and is chief judge of the same, yet with himself (as chief master and architect) remains the demonstrative reason and cause of the mechanic's work: in line, plane, and solid; by geometric, arithmetic, optic, musical, astronomical, cosmographic\" (to be brief) by all the former derived mathematical arts and other natural arts, able to be confirmed and established. If this be so, then you may think that Architecture has good and due allowance in this honest company of derived mathematical arts. I will here crave the judgment of two most perfect architects: the one, Vitruvius the Roman, who wrote ten books of it to the Emperor Augustus (in whose days our heavenly Archemaster was born); and the other, Leo Baptista Albertus, a Florentine, who also published ten books of it. Architecture (says Vitruvius) is a science garnished with many doctrines and diverse instructions, by whose judgment all works finished by other workmen are judged. It follows: Architecture grows of framing and reasoning, etc. Reasoning is that which, of things framed with forecast and proportion, can make demonstration and manifest declaration. Again: In all things, therefore chiefly in Architecture, these two things are: the thing signified, and that which signifies. The thing propounded, of which we speak, is the thing signified. But demonstration, expressed with the reasons of diverse doctrines, signifies the same thing. After that: An architect (says he) ought to understand languages, to be skilled in painting, well instructed in geometry, not ignorant of perspective, furnished with arithmetic, to have knowledge of many histories, and diligently to have heard philosophers, to have skill in music, not ignorant of medicine, to know the answers of lawyers, and to have astronomy and the courses of the heavens in good knowledge. He gives orderly reason why all these arts, doctrines, and instructions are requisite in an excellent architect. And (for brevity) omitting the Latin text, thus he has it. Secondly, it is helpful for an architect to have the knowledge of painting: that he may the more easily fashion out, in painted patterns, the form of whatever work he likes. And geometry gives to architecture many helps: and first teaches the use of the rule and the compass, by which (chiefly and easily) the descriptions of buildings are dispatched in ground plans, and the directions of squares, levels, and lines. Likewise, by perspective, the lights of the sky are well led in buildings from certain quarters of the world. By arithmetic the charges of buildings are summed together, the measures are expressed, and the hard questions of symmetries are discoursed on by geometric means and methods, etc. Besides this, of the nature of things (which in Greek is called φυσιολογία) philosophy makes declaration. Which it is necessary for an architect diligently to have learned, because it has many and diverse natural questions, as especially in aqueducts. For in their courses, their leadings about in the level ground and in the mountings, the natural spirits or breaths are engendered diverse ways. The hindrances which they cause no man can help, but he who out of philosophy has learned the original causes of things. Likewise, whoever shall read Ctesibius or Archimedes' books (and others who have written such rules) cannot think as they do, unless he shall have received from philosophers instruction in these things. And he must know music too, that he may have understanding both of regular and mathematical music: that he may tune well his ballistas, catapults, and scorpions, etc. Moreover, the bronze vessels which in theaters are placed by mathematical order in niches beneath the steps, and the diversities of the sounds (which the Greeks call ηχεῖα) are ordered according to musical symphonies and harmonies, being distributed through the compass by Diatessaron, Diapente, and Diapason: so that the fitting voice of the players' sound, coming to these preparations made in order and there being increased, might with that increase come more clear and pleasant to the ears of the listeners, etc. And by astronomy are known the east, west, south, and north; the fashion of the sky, the equinox, the solstice, and the course of the stars. Which things, unless one knows, he cannot perceive anything at all of the reason of horologies. Seeing therefore this ample science is garnished, beautified, and stored with so many and sundry skills and knowledges, I think that none can justly account themselves architects all at once. But only those who, from their childhood years, ascending by these degrees of knowledge, being fostered up with the attainment of many languages and arts, have won their way to the high tabernacle of architecture, etc. And to whom nature has given such quick discernment, sharpness of wit, and memory that they may be thoroughly skilled in geometry, astronomy, music, and the rest of the mathematical arts: such surpass and pass the calling and state of architects:\n\n(A mathematician.)\n\nand have become mathematicians, etc. And they are found seldom. As in times past was Aristarchus Samius; Philolaus and Archytas, Tarentines; Apollonius Pergaeus; Eratosthenes Cyreneus; Archimedes and Scopas, Syracusians. Who also left to their posterity many engines and gnomonical works, invented and declared by numbers and natural means.\n\nThus much, and the same words (in sense) in one single chapter of this incomparable architect Vitruvius shall you find. And if you should but take his book in hand and look through it ever so slightly, you would say at once:\n\n(Vitruvius.)\n\nThis is geometry, arithmetic, astronomy, music, anthropography, Hydragogie, horometry, etc. And (to conclude) the storehouse of all workmanship. Now let us listen to our other judge, our Florentine Leo Baptista, and narrowly consider how he determines of architecture. But before I proceed any further (says he) I think that I ought to express what man I would have allowed an architect. For I will not bring in place a carpenter, as though you might compare him to the chief masters of other arts. For the hand of the carpenter is the architect's instrument.\n\n(Who is an architect.)\n\nBut I will appoint the architect to be \"that man who has the skill (by a certain and marvelous means and way) both in mind and imagination to determine, and also in work to finish, whatever works, by motion of weight and the coupling and framing together of bodies, may most aptly be made useful for the worthiest uses of man.\" And that he may be able to perform these things he needs attainment and knowledge of the best and most worthy things, etc. The whole practice of architecture in building consists in lineaments and in framing. And the whole power and skill of lineaments tends to this: that the right and absolute way may be had of adapting and joining lines and angles, by which the face of the building or frame may be comprehended and concluded. And it is the property of lineaments to prescribe to buildings and every part of them a fitting place and certain number: a worthy manner and a seemly order: that so the whole form and figure of the building may rest in the very lineaments, etc. And we may prescribe in mind and imagination the entire forms,\n\n(The immateriality of perfect architecture.)\n\nall material stuff being excluded. Which point we shall attain by noting and forepointing the angles and lines by a sure and certain direction and connection. Seeing then that these things are thus:\n\n(What lineament is.)\n\nLineament shall be the certain and constant prescribing, conceived in mind: made in lines and angles: and finished with a learned mind and wit. \"We thank you, Master Baptist, that you have so aptly brought your art and the phrase of it to a certain mathematical perfection:\n\nby certain order, number, form, figure, and mental symmetry:\" all natural and sensible stuff set aside. Now, then, it is evident, gentle reader, how aptly and worthily I have preferred Architecture to be bred and fostered in the dominion of the peerless princess Mathematics, and to be a natural subject of hers. And the name of Architecture comes from the principality which this science has above all other arts. And Plato affirms the architect to be master over all who make any work. So he is neither smith nor builder, nor separately any craftsman, but the head, provost, director, and judge of all artificial works and all craftsmen. For the true architect is able to teach, demonstrate, distribute, describe, and judge all works wrought. And he alone searches out the causes and reasons of all artificial things. Thus excellent is architecture, though few in our days attain to it; yet the art must not be thought of as anything less than it truly deserves. Nor may we make new and imperfect definitions of ancient arts in our days, for scarcity of craftsmen: no more than we may cut short the definitions of wisdom, or honesty, or friendship, or justice. Neither will I consent to diminish any whit of the perfection and dignity (by just cause) allowed to absolute architecture. Under the direction of this art are three principal, necessary mechanical arts. Namely housing, fortification, and shipbuilding. By housing I understand buildings both for divine service and human common use, public and private. Of fortification and shipbuilding strange matter might be told you; but perhaps some will be tired with this long list already rehearsed, and others will finely pick apart my coarse, homely discourse written in haste, for fear you should lack this true and friendly warning, and this foretaste of mathematical power. Life is short and uncertain; times are perilous, etc. And with the printer still waiting while my pen lingers: all these things, with further matter of ingratitude, give me occasion to move on to the other arts remaining, with all speed possible.\n\n### Navigation\n\nThe art of Navigation demonstrates how a sufficient ship, between any two places assigned (in navigable passage), may be conducted by the shortest good way, by the aptest direction, and in the shortest time; and in all storms and natural disturbances that may happen, how to use the best possible means to recover the place first assigned. What need the master pilot has of the other arts recited before is easy to know: of hydrography, astronomy, astrology, and horometry; always presupposing the common base and foundation of all, namely arithmetic and geometry. So that he is able to understand and judge his own necessary instruments and necessary equipment, whether they are perfectly made or not, and also can, if need be, make them himself: quadrants, the astronomer's ring, the astronomer's staff, the universal astrolabe, a hydrographical globe, true hydrographical charts (not with parallel meridians), the common sea compass, the compass of variation, the proportional and paradoxical compasses.\n\n(Year 1559.)\n\n(Invented by me for our two Moscouy master pilots, at the request of the Company) clocks with springs: hour, half-hour, and three-hour sand glasses, and sundry other instruments; and also to be able, on globe or plane, to describe the paradoxical compass, and duly to use the same to all manner of purposes for which it was invented; and also to be able to calculate the planets' places for all times.\n\nMoreover, with sun, moon, or star (or without them) he is able to define the longitude and latitude of the place which he is in; so that the longitude and latitude of the place from which he sailed be given, or known to him. To which belong expert means of always being certain of the ship's course, and so on. And by foreseeing the rising, setting, culmination at noon, or culmination at midnight of certain tempestuous fixed stars, or their conjunctions and angles with the planets, and so on, he ought to have expert conjecture of storms, tempests, and waterspouts, and such like meteorological effects, dangerous at sea. For, as Plato says, To foresee the alterations and opportunities of times is fitting no less to the art of war than to husbandry and navigation. And besides such skillful means, more evident tokens in sun and moon ought to be known to him: such as the philosophical poet Virgilius teaches in his Georgics, where he says:\n\n(Georgic. 1.)\n\nThe sun also, both at his rising and when he sinks into the waves, will give signs; the surest signs follow the sun, etc. … For often we see that on his very face various colors wander. Blue, he announces rain; fiery, the east winds. But if spots begin to mingle with the ruddy fire, then you will see everything alike seething with wind and storm clouds: on such a night let no one urge me to go out over the deep, or to pull up the cable from the land, etc. The sun will give you signs. Who would dare call the sun false? … etc.\n\nAnd so of moon, stars, water, air, fire, wood, stones, birds, and beasts, and of many things else, a certain sympathetic forewarning may be had, sometimes to great pleasure and profit, both on sea and land. Enough for my present purpose it appears from the premises how mathematical the art of navigation is, and how it needs and also uses other mathematical arts. And now, if I should go about to speak of the many benefits coming to this land and others by ships and navigation, you might think that I catch at occasions to use many words where none are needed.\n\nYet this one thing may I justly say: in navigation, none ought to take greater care to be skillful than our English pilots. And perhaps some would attempt more, and others more willingly would help, if they knew for certain what privilege God had endowed this island with, by reason of a situation most convenient for navigation to places most famous and rich. And though,\n\n(Year 1567 S. H. G.)\n\n(Of late) a young gentleman, a courageous captain, was in great readiness, with good hope and strong grounds of confidence, to have ventured on a discovery (either westward, by Cape de Paramantia; or eastward, above Nova Zemla and the Cyremisses), and was, at the very near time of attempting it, called and employed otherwise, both then and since, in great good service to his country, as the Irish rebels have experienced:\n\n(Year 1569)\n\nYet I say, though the same gentleman does not hereafter deal with the matter, someone or other should attend to it; and by good advice and careful discretion, little by little, gain sufficient knowledge of that trade and voyage; which now I would be sorry, through carelessness, lack of skill and courage, should remain unknown and unheard of. Seeing also that, in this matter, we are half challenged by the learned and half requested to publish. From this, truly, might grow benefit, to this land chiefly, and to the rest of the Christian commonwealth, far surpassing all riches and worldly treasure.\n\n### Thaumaturgike, 1\n\nThaumaturgike is that mathematical art which gives a certain order to make strange works, to be perceived by the sense, and greatly to be wondered at by men. By sundry means this wonder-work is wrought: some by Pneumatithmie, as the works of Ctesibius and Hero; some by weight, of which Timæus speaks; some by strings strained or springs, by that imitating lively motions; some by other means, as the images of Mercurie, and the brazen head made by Albertus Magnus which seemed to speak. Boethius was excellent in these feats. To whom Cassiodorus, writing, says, \"Your purpose is to know profound things and to show marvels. By the disposition of your art, metals bellow: Diomedes of brass blows a trumpet loud: a brazen serpent hisses: made birds sing sweetly. Small things we rehearse of you, who can imitate heaven, and so on.\" Of the strange self-moving which, at Saint Denys by Paris,\n\n(Year 1551)\n\nI saw, once or twice (Orontius being then with me, in company); it would be too strange to tell. But some have written it; and yet, I hope, it is there to be seen by others. And by perspective also strange things are done, as partly before I gave you to understand in perspective: as, to see in the air aloft the lively image of another man, either walking to and fro or standing still; likewise, to come into a house and there see the lively show of gold, silver, or precious stones, and coming to take them in your hand, find nothing but air. By this have some men (in all other matters counted wise) badly overshot themselves, misjudging the means. Therefore said Claudius Cælestinus:\n\n(On those things that happen marvelously in the world. Chapter 8.)\n\nNowadays we see some men, yes of great learning and reputation, judge certain works to be marvelous, above the power of nature; of which works one skilled in perspective might easily have given the cause. Of Archimedes' sphere, Cicero bears witness.\n\n(Tusculan Disputations 1.)\n\nWhich is very strange to think on. For when Archimedes, says he, fastened in a sphere the movements of the sun, moon, and of the five other planets, he did as the god who (in Timæus of Plato) made the world: that, with one turning, he should rule motions most unlike in slowness and swiftness. But a greater cause for wondering we have from Claudianus' report of this, who affirms this work of Archimedes to have been of glass, and discourses of it more at length, which I omit. The dove of wood, which the mathematician Archytas made to fly, is spoken of by Agellius. Of Dædalus' strange images Plato reports. Homere, of Vulcan's self-movers (by secret wheels), leaves in writing. Aristotle, in his Politics, makes mention of both. Marvelous was the workmanship, in late days, performed by good skill of Trochilike, and so on. For in Noremberge a fly of iron, let out of the artisan's hand, did, as it were, fly about among the guests at the table, and at length, as though it were weary, return to its master's hand again. Moreover, an artificial eagle was ordered to fly out of the same town a mighty way, and that aloft in the air, toward the emperor coming there, and followed him when he had come to the gate of the town.\n\n(☞)\n\nThus you see what the mathematical art can perform, when skill, will, industry, and ability are duly applied to proof.\n\n(A Digression.)\n\nAnd for these, and such like marvelous acts and feats, naturally, mathematically, and mechanically wrought and contrived:\n\n(Apologeticall.)\n\nought any honest student and modest Christian philosopher be counted and called a conjurer? Shall the folly of idiots and the malice of the scornful so much prevail, that he who seeks no worldly gain or glory at their hands, but only from God the treasure of heavenly wisdom and knowledge of pure truth, shall he, I say, in the meantime be robbed and despoiled of his honest name and fame? He who seeks (by S. Paul's admonition) in the creatures' properties and wonderful virtues to find just cause to glorify the eternal and almighty Creator by: shall that man be (in secret) condemned as a companion of the hellhounds, and a caller and conjurer of wicked and damned spirits? He who bewails his great want of time, enough (to his satisfaction) to learn godly wisdom and godly truths in, and sets all his delight only in that: will that man lose and abuse his time in dealing with the chief enemy of Christ our Redeemer: the deadly foe of all humankind, the subtle and impudent perverter of godly truth, the hypocritical crocodile, the envious basilisk, continually desirous, in the twinkling of an eye, to destroy all humankind, both body and soul, eternally? Surely (for my part, somewhat to say in this) I have not learned to make so brutish and so wicked a bargain. Should I, for my twenty or twenty-five years of study: for two or three thousand marks spent: for seven or eight thousand miles gone and traveled, only for good learning's sake; and that in all manner of weather: in all manner of ways and passages: both early and late: in danger of violence by man: in danger of destruction by wild beasts: in hunger: in thirst: in perilous heat by day, with toil on foot: in dangerous damp colds by night, almost depriving life (as God knows): with lodgings often to little ease, and sometimes to less security; and for much more than all this done and suffered, for learning and the attainment of wisdom: should I, I ask you, for all this, no otherwise nor more warily (or by God's mercy no more luckily), have fished with so large and costly a net, so long a time in drawing (and that with the help and advice of Lady Philosophy and Queen Theology), but at length to have caught and drawn up a frog?\n\n(A proverb. Fair fished, and caught a frog.)\n\nNo, a devil? For, so, does the common peevish prattler imagine and jangle; and so does the malicious scorner secretly wish, and bravely and boldly face down behind my back. Ah, what a miserable thing is this kind of men! How great is the blindness and boldness of the multitude in things above their capacity? What a land: what a people: what manners: what times are these? Have they become devils themselves, and, by false witness bearing against their neighbor, would they also become murderers? Does God so long give them respite, to reclaim themselves from this horrible slandering of the guiltless, contrary to their own consciences, and yet will they not cease? Does the innocent forbear calling them to answer him juridically, according to the rigor of the laws, and will they despise his charitable patience? As they, against him, by name, forge, fable, rage, and raise slander by word and print, will they provoke him, by word and print likewise, to note their names to the world, with their particular devices, fables, beastly imaginations, and unchristian-like slanders? Well, well. O you who are such, my unkind countrymen: O unnatural countrymen: O unthankful countrymen: O brainsick, rash, spiteful, and disdainful countrymen. Why do you oppress me so violently with your slandering of me, contrary to truth and contrary to your own consciences? And I, to this hour, neither by word, deed, or thought, have been in any way harmful, damaging, or injurious to you or yours? Have I so long, so dearly, so far, so carefully, so painfully, so dangerously sought and traveled for the learning of wisdom, and the attaining of virtue; and in the end (in your judgment) have I become worse than when I began? Worse than a madman? A dangerous member of the commonwealth, and no member of the church of Christ? Is this what you call being learned? Is this what you call being a philosopher, and a lover of wisdom? To forsake the straight heavenly way, and to wallow in the broad way of damnation? To forsake the light of heavenly wisdom, and to lurk in the dungeon of the prince of darkness? To forsake the truth of God and his creatures, and to fawn upon the impudent, crafty, obstinate liar, and continual disgracer of God's truth, to the utmost of his power? To forsake the life and bliss eternal, and to cleave to the author of everlasting death: that murderous tyrant most greedily awaiting the prey of a human soul? Well, I thank God and our Lord Jesus Christ, for the comfort which I have by the examples of other men before my time: to whom, neither in godliness of life nor in perfection of learning, I am worthy to be compared; and yet they sustained the very same injuries that I do, or rather greater. Patient Socrates, his Apology, will testify: Apuleius, his Apologies, will declare the brutishness of the multitude. Ioannes Picus, Earl of Mirandula, his Apology will teach you of the raging slander of the malicious ignorant against him. Ioannes Trithemius, his Apology will specify how he had occasion to make public protestation, as much by reason of the rude and simple, as also in respect of those who were counted to be of the wisest sort of men. \"Many could I recite: but I defer the precise and determined handling of this matter, being loth to detect the folly and malice of my native countrymen.\n\n(☞)\n\nWho so hardly can digest or like any extraordinary course of philosophical studies: not falling within the compass of their capacity: or where they are not made privy of the true and secret cause of such wonderful philosophical feats.\" These men are of four sorts, chiefly: the first I may name, vain prattling busybodies; the second, fond friends; the third, imperfectly zealous; and the fourth, malicious ignorant. To each of these (briefly and in charity) I will say a word or two, and so return to my Preface.\n\n(1.)\n\nVain prattling busybodies: use your idle assemblies and conferences otherwise than in talk of matter either above your capacities for hardness, or contrary to your consciences in truth.\n\n(2.)\n\nFond friends, leave off so to commend your unacquainted friend upon blind affection: as, because he knows more than the common student, that therefore he must necessarily be skillful and a doer in such matter and manner as you term conjuring. You think by that that you advance his fame, and that you make other men marvel greatly at your luck, to have such a learned friend. Cease to ascribe impiety where you pretend amity. For, if your tongues were true, then would that friend of yours be untrue both to God and his sovereign. Such friends and fondlings, I shake off and renounce you: shake off your folly.\n\n(3.)\n\nImperfectly zealous, to you I say: that perhaps well you mean, but far you miss the mark: if a lamb you will kill to feed the flock with his blood. Sheep, with lamb's blood, have no natural sustenance: no more is Christ's flock duly edified with horrible slanders. Nor is your fair pretense any whit well graced by such rash ragged rhetoric. But such as so use me will find a foul crack in their credit. Speak what you know: and know as you ought: do not know by hearsay when life lies in danger. Search to the quick, and let charity be your guide.\n\n(4.)\n\nMalicious ignorant, what shall I say to you? Keep your tongue from evil. Refrain your tongue from slander. Though your tongues be sharpened serpent-like, and adders' poison lie in your lips:\n\n(Psalm 140.)\n\nyet take heed, and in time think to yourself: A man of loose tongue shall not be established in the land. Evil shall hunt the violent man until he is cast down. For I am sure that the Lord will do justice for the afflicted, and vengeance for the poor.\n\nThus I require you, my assured friends and countrymen (you mathematicians, mechanicians, and philosophers, charitable and discrete), to deal in my behalf with the frivolous and untrue-tongued, my envious adversaries or fond friends. And farther, I would wish that at leisure you would consider how Basilius Magnus lays Moses and Daniel before the eyes of those who count all such philosophical studies (as mine has been) ungodly or unprofitable. Weigh well S. Stephen's witness of Moses.\n\n(Act. 7. C.)\n\nMoses was instructed in all manner of wisdom of the Egyptians: and he was of power both in his words and works. You see this philosophical power and wisdom, which Moses had, to be in no way disapproved by the Holy Ghost. Yet Plinius has recorded Moses to be a wicked magician. And that of necessity must be either for this philosophical wisdom, learned before his calling to the leading of the children of Israel, or for those his wonders, wrought before King Pharao, after he had the conducting of the Israelites. As concerning the first, you perceive how S. Stephen, at his martyrdom (being full of the Holy Ghost), in his recapitulation of the old testament, made mention of Moses' philosophy, with good liking of it: and Basilius Magnus also avouches it to have been profitable to Moses (and therefore, I say, to the church of God, necessary). But as concerning Moses' wonders, done before King Pharao: God himself said: See that you do all those wonders before Pharao, which I have put in your hand. Thus you evidently perceive how rashly Plinius has slandered Moses,\n\n(Book 30, chapter 1.)\n\nof vain fraudulent magic, saying: There is also another sect of magic, depending on Moses, Jannes, and Iotape, and the Jews, but many thousands of years after Zoroaster. etc.\n\n(1.)\n\nLet all such, therefore, who, in judgment and skill of philosophy, are far inferior to Plinie, \"take good heed, for fear that they overshoot themselves rashly,\" in\n\n(☞)\n\njudging of philosophers' strange acts, and the means how they are done.\n\n### Thaumaturgike, 2\n\n(2.)\n\nBut much more ought they to beware of forging, devising, and imagining monstrous feats and wonderful works when and where none such were done: not even a spark or likelihood of such as they report without any shame.\n\n(3.)\n\nAnd, to conclude, most of all let them be ashamed before men and afraid of the dreadful and just Judge: both for foolishly or maliciously devising such things, and then devilishly fathering their newly contrived monsters on me, though I am innocent in hand and heart, as though I had trespassed against the law of God or of man in any of my studies or exercises, philosophical or mathematical—which I hope will in due time be more manifest.\n\n### Archemastrie\n\nNow I end, with Archemastrie. The name is not so new as the art is rare; for another art, a degree below this in skill and power, had already been given this English name. And yet this will serve our purpose well enough for now. This art teaches how to bring to actual sensible experience all worthy conclusions proposed by all the mathematical arts and concluded by true natural philosophy; and it both adds to them a further scope in the terms of the same arts, and also by its own method and in peculiar terms proceeds, with the help of the aforesaid arts, to the performance of complete experiences which of no particular art can formally be challenged. If you remember how we considered architecture in respect of all common handiworks, some light you may have from that toward understanding the sovereignty and property of this science. I may call it a science rather than an art, for the excellence and mastery it has over so many and so mighty arts and sciences. And because it proceeds by experiences, and searches out the causes of conclusions by experiences, and also puts the conclusions themselves into experience, it is named by some the Experimental Science. Nicolaus Cusanus calls it so in his Statical Experiments, and another philosopher, native to this land (the flower of whose worthy fame can never die nor wither), wrote at length on it at the request of Clement the Sixth. The art carries with it wonderful credit, for it certifies sensibly, fully, and completely, to the utmost power of nature and art. This art certifies by complete and absolute experience; other arts, with their arguments and demonstrations, persuade, and in words prove their conclusions very well.\n\nBut words and arguments are no sensible certification, nor the full and final fruit of the sciences as they are practiced. And though some arts have experiences in them, yet they are not complete, nor brought to the utmost they may be stretched to and applied sensibly. As for example: the natural philosopher disputes and makes a goodly show of reason; and the astronomer and the optical mechanician put some things into experience—but neither all that they may, nor yet those they do, sufficiently and to the utmost. There then the Archemaster steps in, and leads the experiences on, by order of his experimental doctrine, to the chief and final power of natural and mathematical arts. Of two or three men in whom this description of Archemastry was experimentally verified, I have read and heard, and there is good record of such perfection in them. So that this art is no fantastic imagination—as some sophister, with his Insolubles, might flourish, and dazzle your imagination, and dash your honest desire and courage from believing these things so unheard-of, so marvelous, and of such importance. Well: as you will. I have warned you. I have done the part of a friend: I have discharged my duty toward God, for my small talent received at his most merciful hands. To this science does the science Alnirangiat render great service. Do not wonder at this name: I do not change a name so used, and published in print by others, being a name proper to the science. Under this comes Ars Sintrillia [title unclear in the source], briefly written by Artephius. But the chief science of the Archemaster (in this world) as yet known is another (as it were) optical science: of which the name shall be told (God willing) when I shall have some more just occasion to discourse of it.\n\nHere I must end, thus abruptly (gentle friend, and unfeigned lover of honest and necessary verities). For they who have, for your sake and virtue's cause, requested me (an old, worn-out mathematician) to take pen in hand (through the confidence they placed in my long experience and tried sincerity), for the declaring and reporting somewhat of the fruit and commodity to be attained by the mathematical arts: even they, sore against their wills, are forced, for sundry causes, to satisfy the workman's request by ending without delay; he so fears this so new an attempt, and so costly, and in a matter so slenderly hitherto considered or esteemed among the common sort of students.\n\nAnd since I was urged to set forth some reason why, in our vulgar speech, this part of the principal science of geometry, called Euclid's Geometrical Elements, is published for your handling—you who have no Latin, and are not university scholars—truly, I think it needless.\n\n(1.)\n\nFor the honor and estimation of the universities and graduates is hereby nothing diminished, seeing that from them, and by those they have reared, you receive all this benefit, however great it be.\n\n(2.)\n\nNeither are their studies in the least hindered: no more than the Italian universities, as the academies of Bologna, Ferrara, Florence, Milan, Padua, Pavia, Perugia, Pisa, Rome, and Siena, or any one of them, find themselves at all disgraced, or their studies anything hindered, by Friar Lucas de Burgo or by Nicolaus Tartalea, who in the vulgar Italian language have published not only Euclid's Geometry, but somewhat of Archimedes: and in arithmetic and practical geometry very large volumes, all in their vulgar speech. Nor in Germany have the famous universities been anything discontent with Albertus Durerus, his geometrical institutions in German; or with Gulielmus Xylander, his learned translation of the first six books of Euclid out of the Greek into high German; nor with Gualterus H. Riffius, his geometrical volume, very diligently translated into the German tongue and published. Nor yet do the universities of Spain or Portugal think their reputation to be decayed, or suppose any of their studies hindered by the excellent P. Nonnius, his mathematical works put forth in vulgar speech. Have you not, likewise, in the French tongue the whole mathematical quadrivium? and yet have neither Paris, Orleans, nor any of the other universities of France ever been offended with the translators or publishers, or any man's study by that hindered?\n\n(3.)\n\nAnd surely the common and ordinary scholar (much more the grammarian) before coming to the university shall (or may) now be sufficiently instructed, according to Plato's counsel, in arithmetic and geometry, for the better and easier learning of all manner of philosophy, academic or peripatetic; and by that means go forward more cheerfully, more skillfully, and more speedily in his studies there to be learned; and so, in less time, profit more than otherwise he should or could do.\n\n(4.)\n\nAlso many good and pregnant English wits, of young gentlemen and of others who never intend to meddle with the profound search and study of philosophy (to be learned in the universities), may nevertheless now, with more ease and liberty, have good occasion to occupy the sharpness of their wits virtuously: where otherwise, perhaps, they would spend their time—or rather lose it—in foolish exercises, neither serving God nor furthering the common or private good.\n\n(5.)\n\nAnd great comfort, with good hope, may the universities have, by reason of this English geometry and mathematical preface, that they hereafter shall be the more regarded, esteemed, and resorted to. For when it shall be known and reported that of the mathematical sciences only such great commodities are ensuing as I have specified, and that indeed some of you students with no Latin can be good witnesses of such rare fruit enjoyed by you by that, as either was not heard of before this, or else not so fully credited: \"Well may all men conjecture that far greater aid and better furniture, to win to the perfection of all philosophy,\n\n(Universities.)\n\nmay in the universities be had: being the storehouses and treasury of all sciences, and all arts necessary for the best and most noble state of commonwealths.\"\n\n(6.)\n\nBesides this, how many a common artisan is there in these realms of England and Ireland who deals with numbers, rule, and compass: who, with their own skill and experience already had, will be able (by these good helps and information) to find out and devise new works, strange engines, and instruments, for sundry purposes in the commonwealth? or for private pleasure? and for the better maintaining of their own estate? I will not therefore fight against my own shadow. For no man (I am sure) will open his mouth against this enterprise. No man (I say) who either has charity toward his brother (and would be glad of his furtherance in virtuous knowledge), or who has any care and zeal for the bettering of the common state of this realm. Neither any who make account of what the wiser sort of men (sage and steady) think of them. To none, therefore, will I make any apology for doing a virtuous act, and for commending or setting forth profitable arts to English men in the English tongue. \"But to God our Creator let us all be thankful: for that, as he, of his goodness, by his power, and in his wisdom, has created all things in number, weight, and measure: so, to us, of his great mercy, he has revealed means by which to attain the sufficient and necessary knowledge of his three principal instruments aforesaid: which means I have abundantly proved to you to be the sciences and arts mathematical.\"\n\nAnd though I have been pinched with straightness of time, that in no way I could pen down the matter (in my mind) as I determined, hoping for convenient leisure: yet if virtuous zeal and honest intent provoke and bring you to the reading and examining of this compendious treatise, I do not doubt but as the truth of it (according to our purpose) will be evident to you, so the pith and force of it will persuade you, and the wonderful fruit of it highly please you. And that you may the easier perceive, and better remember, the principal points of which my preface treats,\n\n(The ground plan of this preface in a table.)\n\nI will give you the ground plan of my whole discourse, in a table annexed, from the first to the last, somewhat methodically contrived.\n\nIf haste has caused my poor pen anywhere to stumble, you will (I am sure), in partial recompense for my earnest and sincere good will in pleasing you, consider the huge rocky mountains and the perilous unbeaten ways through which it (both night and day the while) has toiled and labored to bring you this good news and comfortable proof of virtue's fruit.\n\nSo I commit you to God's merciful direction for the rest: heartily beseeching him to prosper your studies and honest intentions, to his glory and the benefit of our country. Amen.\n\nWritten at my poor house at Mortlake.\n\nIn the year 1570. February 9.\n\nJ. Dee\n\n### The Groundplat of the Mathematical Praeface\n\nHere you have (as I promised) the ground plan of my Mathematical Preface, annexed to Euclid and now first published in our English tongue. An. 1570. Febr. 3.\n\nSciences and arts mathematical are either\n\nPrincipal, which are two only,\n\nArithmetic.\n\nSimple, which deals with numbers only, and demonstrates all their properties and appurtenances, where a unit is indivisible. Mixed, which, with the help of principal geometry, demonstrates some arithmetical conclusion or purpose.\n\nGeometry.\n\nSimple, which deals with magnitudes only, and demonstrates all their properties, affections, and appurtenances, whose point is indivisible. Mixed, which, with the help of principal arithmetic, demonstrates some geometrical purpose, as Euclid's Elements.\n\nThe use of these is either:\n\nIn supernatural, eternal, and divine things: by application, ascending. In mathematical things: without further application. In natural things, both substantial and accidental, visible and invisible, etc.: by application, descending.\n\nThe like uses and applications are found, though a degree lower, in the derivative mathematical arts.\n\nDerivative from the principal ones: of which some have\n\nThe names of the principal ones, as:\n\nArithmetic, vulgar, which considers\n\n—Arithmetic of the most usual whole numbers, and of the fractions belonging to them.—Arithmetic of proportions.—Circular arithmetic.—Arithmetic of radical numbers: simple, compound, and mixed; and of their fractions.—Arithmetic of cossic numbers, with their fractions; and the great art of algebra.\n\nGeometry, vulgar, which teaches measuring\n\nAt hand\n\nAll lengths—Mecometrie. All planes, such as land, board, glass, etc.—Embadometrie. All solids, such as timber, stone, vessels, etc.—Stereometrie.\n\nWith distance from the thing measured, as:\n\nHow far any thing is from the measurer, seen by him on land or water, is called Apomecometrie. How high or deep any thing is from the level of the measurer's standing, seen by him on land or water, is called Hypsometrie. How broad a thing is that lies in the measurer's view, so it be situated on land or water, is called Platometrie.\n\nFrom which have grown the feats and arts of\n\nGeodesy, more skillfully to measure and survey lands, woods, waters, etc. Geography. Chorography. Hydrography. Stratarithmetrie.\n\nProper names, as:\n\nPerspective—which demonstrates the modes and properties of all radiations: direct, broken, and reflected.\n\nAstronomy—which demonstrates the distances, magnitudes, and all natural motions, appearances, and affections proper to the planets and fixed stars: for any time, past, present, and to come: in respect of a certain horizon, or without respect of any horizon.\n\nMusic—which demonstrates by reason, and teaches by sense, perfectly to judge and order the diversity of sounds, high or low.\n\nCosmography—which wholly and perfectly makes description of the heavenly and also the elemental part of the world: and of these parts makes homologous application and mutual collation necessary.\n\nAstrology—which reasonably demonstrates the operations and effects of the natural beams of light and the secret influence of the planets and fixed stars in every element and elemental body: at all times, in any horizon assigned.\n\nStatics—which demonstrates the causes of heaviness and lightness of all things: and of the motions and properties belonging to heaviness and lightness.\n\nAnthropography—which describes the number, measure, weight, figure, situation, and color of every diverse thing contained in the perfect body of man: and gives certain knowledge of the figure, symmetry, weight, characterization, and due local motion of any designated part of the said body, and of the numbers belonging to that part.\n\nTrochilike—which demonstrates the properties of all circular motions: simple and compound.\n\nHelicosophy—which demonstrates the designing of all spiral lines in a plane, on a cylinder, cone, sphere, conoid, and spheroid, and their properties.\n\nPneumatithmie—which demonstrates by closed hollow geometric figures (regular and irregular) the strange properties (in motion or rest) of water, air, smoke, and fire in their continuity, and as they are joined to the elements next to them.\n\nMenadrie—which demonstrates how virtue and force may be multiplied above nature's simple virtue and power: and so to direct, to lift, to pull toward, and to put or cast from, any determined virtue, weight, or force, simple or multiplied, which naturally cannot be so directed or moved.\n\nHypogeiodie—which demonstrates how, under the spherical surface of the earth, at any depth, to any assigned perpendicular line (whose distance from the perpendicular of the entrance, and the azimuth likewise in respect of the said entrance, being known), a certain course may be prescribed and traveled, etc.\n\nHydragogie—which demonstrates the possible leading of water by nature's law and by artificial help from any head (being spring, standing, or running water) to any other place assigned.\n\nHorometry—which demonstrates how, at all times appointed, the precise customary designation of time may be known for any place assigned.\n\nPainting—which demonstrates and teaches how the intersection of all visual pyramids made by any assigned plane (the center, distance, and lights being determined) may be represented by lines and proper colors.\n\nArchitecture—which is a science adorned with many doctrines and diverse instructions: by whose judgment all works finished by other workmen are judged.\n\nNavigation—which demonstrates how, by the shortest good way, by the aptest direction, and in the shortest time, a sufficient ship between any two places (in a navigable passage) assigned may be conducted: and in all storms and natural disturbances happening, how to use the best possible means to recover the place first assigned.\n\nThaumaturgike—which gives certain order to make strange works, to be perceived by the senses, and greatly to be wondered at by men.\n\nArchemastrie—which teaches how to bring to actual sensible experience all worthy conclusions proposed by all the mathematical arts and concluded by true natural philosophy: and both adds to them a further scope in the terms of the same arts, and also by its proper method and in peculiar terms proceeds, with the help of the aforesaid arts, to the performance of complete experiences which of no particular art can formally be challenged.\n\nImprinted by John Day.\n\nAn. 1570. Feb. 25.","download_url":"https://mouseia.org/api/works/dee-praeface/translations/trn_dee_praeface_mouseia.txt"},{"id":"trn_dee_praeface_plain","translator":"Mouseia","kind":"plain","note":"Modern machine-assisted Plain English edition by Mouseia, from Dee's 1570 Praeface. Latin quotations and titles are translated into English, using Dee's own English glosses where provided. Printer's type-size markup is normalized; marginal notes are kept as asides. Billingsley's separate address is omitted. Uncertain readings are marked.","text":"### To the Lovers of Truth and Students of Noble Sciences\n\nTo all who love truth without pretense and who steadily study the noble sciences, John Dee of London warmly wishes grace from heaven and the greatest success in all their honorable work and exercises.\n\nDivine Plato, the great teacher of many worthy philosophers, and the constant affirmer and powerful persuader of the One, the Good, and Being, was visited many times in his school and academy, besides his ordinary students, by a certain kind of men, allured by the noble fame of Plato and the great praise of his deep and profitable teaching. But when such hearers, after listening to him a long time, perceived that the point of his discourses came to this conclusion, that the One, the Good, and Being are spiritual, infinite, eternal, omnipotent, etc., with nothing alleged or explained about worldly goods, about worldly dignity, about health, strength or vigor of body, nor about the means by which a wonderful, sensible and bodily bliss and happiness hereafter might be attained—immediately those hearers were discouraged, their opinion of Plato completely changed, his doctrine was despised by them, and his school was visited by them no more. His student Aristotle, considering this closely, found the cause to be \"that they had no general forewarning and information about where his teaching tended.\" For so they could have had reason either to stay away from his school (if they then disliked his scope and purpose) or to continue in it steadily, to their full satisfaction, if such a final scope and intent had been what they wanted. So Aristotle, from then on, always used to warn his own students and hearers briefly, \"both of what subject matter, and also to what end, he took in hand to speak or teach.\" While I consider the different approaches of these two excellent philosophers (and I am most sure both that Plato could very well have taught otherwise, and that Aristotle could boldly have dealt with his hearers in the same way Plato did), I am in no small anguish of perplexity: because what I dislike is easiest for me to perform (and to have Plato as my example), while what I know to be most commendable, and (in this first bringing of the mathematical arts into common handling) most necessary, is full of great difficulty and various dangers. Yet I do not think it fitting, for so strange a matter as is now meant to be published, and to so strange an audience, to put it out bluntly at first with no special preface; nor, imitating Aristotle, can I well hope that, according to the fullness and dignity of the mathematical state, I am able either to set out plainly the material bounds, or to state precisely the chief purposes and most wonderful applications of it. And though I am sure that those who shrank from Plato's school, after they had perceived his final conclusion, would in these things have been his most diligent hearers (so infinitely might their desires be satisfied in the end and at length by our mathematical arts), yet by this preface and forewarning all such may be drawn here sooner, to their great benefit; and also the fully trained Pythagorean and Platonian student, and the steadfast profound philosopher, may with more ease and speed gather from it both wax and honey, like the bee.\n\nThe intent of this preface.\n\nSeeing that I have good reason (for the reasons given, and further in respect of my general mathematical art) to use \"a certain forewarning and preface, whose content shall be this: the mighty, most pleasant and fruitful Mathematical Tree, with its main arms and its second grafted branches—what each one of them is, and also what benefit in general may be looked for, from graft as much as from stock. And since this undertaking is so great that, up to our time, it has never (to my knowledge) been achieved by anyone; and since it is also very hard in these gloomy days to win proper and general credit for such rare and strange arts,\" still, if you will lend me your thankful mind a while, for my sincere effort to meet your honorable expectation, and apply your eye or ear attentively to whatever my pen can deliver quickly this time, perhaps at once, as a first greeting, you will find this preface a lesson long enough. And either you will be made much readier by it for a second one, or shortly become quite able yourselves, from the lion's claw, to judge its royal symmetry and further properties. So now, my good friends and countrymen, turn your eyes and bend your minds to that doctrine which, for our present purpose, my modest talent is able to yield you.\n\nAll things that are, and have being, fall under a general threefold difference. For either they are judged supernatural, natural, or of a third being. Supernatural things are immaterial, simple, indivisible, incorruptible, and unchangeable. Natural things are material, compound, divisible, corruptible, and changeable. Supernatural things are grasped by the mind alone; natural things can be perceived by the outward senses. In natural things probability and conjecture have a place; but in supernatural things the highest demonstration and the most certain knowledge is to be had. By these properties and comparisons of the two, the state, condition, nature and property of those things we earlier called of a third being can be more easily described, which by a special name are also called Mathematical Things. For these, being in a manner middle between supernatural and natural things, are not so complete and excellent as supernatural things, nor so base and coarse as natural things; but they are immaterial things, and yet can be signified to some degree by material things. And though their particular images can be assembled and divided by art, yet the general forms remain constant, unchangeable, untransformable, and incorruptible. They can never be perceived or judged by the senses. Nor yet for all that are they first conceived in the royal mind of man. But rising above the imperfection of conjecture, guesswork and opinion, and falling short of high intellectual understanding, they are the mediating fruit of Dianoetic reasoning, resting in perfect imagination. These mathematical things have a wonderful neutrality and a strange participation between supernatural, immortal, intellectual, simple and indivisible things, and natural, mortal, sensible, compound and divisible things. Probability and proof from the senses may well serve in natural things, and that is commendable; in mathematical reasoning a probable argument is not regarded at all, nor is the testimony of the senses credited at all; but only a perfect demonstration of certain, necessary and invincible truths, concluded universally and necessarily, is allowed as sufficient for \"an argument exactly and purely mathematical.\"\n\nNote that the word unit is used here to express the Greek monas, and not unity, as we have all commonly used until now.\n\nMathematical things fall into two main kinds: number and magnitude.\n\nNumber.\n\nWe define number as a certain mathematical sum of units. A unit is an indivisible mathematical thing; by partaking in some likeness of its property, anything that exists, or is counted as one, can reasonably be called one. We count a unit to be a mathematical thing, though it is no number and is also indivisible, because number consists of units materially, and number is itself chiefly a mathematical thing.\n\nMagnitude.\n\nMagnitude is a mathematical thing; by partaking in some likeness of its nature, anything is judged long, broad, or thick. \"A thick magnitude we call a solid, or a body. Any magnitude that is solid or thick is also broad and long. A broad magnitude we call a surface or a plane. Every plane magnitude also has length. A long magnitude we call a line. A line is neither thick nor broad, but only long. Every definite line has two ends:\n\nA point.\n\nThe ends of a line are called points. A point is an indivisible mathematical thing that may have a certain definite position.\" If a point moves from a definite position, the way in which it moved is also a line, produced mathematically; after which, according to the ancient mathematicians,\n\nA line.\n\nA line is called the race or course of a point. We define a point as a mathematical thing, though it is no magnitude and is indivisible, because it is the proper end and bound of a line, which is a true magnitude.\n\nMagnitude.\n\nAnd we may define magnitude as that mathematical thing which is forever divisible into divisible parts, long, broad or thick. So though a point is no magnitude, yet in a limiting sense we count it a mathematical thing (as I said), because it is properly the end and bound of a line. Neither number nor magnitude has any materiality.\n\n### Of Number and Magnitude\n\nFirst we will consider number, and the mathematical science proper to it, called Arithmetic; and afterward of magnitude, and its science, called geometry. But that name does not satisfy me; a word or two about that will come later. How immaterial and free from all matter number is, who does not perceive it? Yes, who does not marvel at it? For neither pure element, nor Aristotle's quintessence, can serve as the proper matter of number. Nor will the purity and simplicity of a Spiritual or Angelic Substance be found proper enough for it. And therefore the great and godly philosopher Anitius Boetius said: All things (which from the very first original being of things have been framed and made) appear to be formed by the reason of Numbers. For this was the principal example or pattern in the mind of the Creator. What a comfort, what a rapture, to deal with a Science whose Subject is so ancient, so pure, so excellent, so far above all creatures, and so used by the almighty and incomprehensible wisdom of the Creator in the distinct creation of all creatures: in all their distinct parts, properties, natures, and virtues, brought by order and the most absolute number, from Nothing, to the form of their being and state. Therefore, learning the property of Numbers by every possible means (to the perfection of the Science), we may both wind and draw ourselves into the inward and deep search and view of all creatures' distinct virtues, natures, properties, and Forms; and also further arise, climb, ascend and mount up (with speculative wings) in spirit, to see in the mirror of creation the Form of Forms, the model Number of all things numerable: both visible and invisible, mortal and immortal, bodily and spiritual. Part of this profound and divine Science the prophet Joachim had attained: by Formal, Natural and Rational Numbers, foreseeing, concluding and foretelling great particular events long before they came. His books still remaining are good proof of this; and the noble Earl of Mirandula (besides that) is a sufficient witness that Joachim, in his prophecies, proceeded by no other way than by Formal Numbers. And this Earl himself, in Rome,\n\nIn the year 1488.\n\nHe set up 900 conclusions in all kinds of sciences, openly to be disputed, and among the rest, in his Mathematical Conclusions (in the eleventh Conclusion) he has in Latin this English sentence: \"By Numbers, a way is had to the searching out and understanding of every thing able to be known.\" To verify which Conclusion, I promise to answer the 74 questions written below by the way of Numbers. Which Conclusions I omit here to repeat, both to avoid needless length and because the works of Ioannes Picus are commonly available. But in any case I would wish that those Conclusions were read carefully and understood by such as are earnest observers and considerers of the constant law of numbers, which is planted in natural and supernatural things, and is prescribed to all creatures to be kept inviolably. For so, besides many other things to be noted in those Conclusions, it would appear how sincerely, and within my bounds, I disclose the wonderful mysteries to be attained by numbers.\n\nFrom my earlier words it is easy to gather that Number has a threefold state: one in the Creator, another in every Creature (in respect of its complete constitution), and the third in Spiritual and Angelic Minds and in the Soul of man. In the first and third state, Number is called Number Numbering. But in all Creatures, otherwise, Number is called Number Numbered. And in our Soul, Number holds such sway and has such an affinity with it that some of the old Philosophers taught that the human Soul is a Number moving itself. And indeed, in us, though it is only an accident, yet it is such an accident that before all Creatures it had perfect being in the Creator, eternally. Number Numbering, then, is the discretion that discerns and distinguishes things. But in God the Creator this discretion, in the beginning, produced all things orderly and distinctly. For his Numbering then was his Creating of all things. And his continual Numbering of all things is their conservation in being; and wherever and whenever he will lack a Unit, there and then that particular thing will be uncreated. Here I stop. But our separating, distinguishing, and Numbering creates nothing; of Multitude considered it makes a certain and distinct determination. And although these things are weighty truths of great importance, yet (by the infinite goodness of the Almighty Ternary) Artificial Methods and easy ways are made by which the zealous Philosopher may draw near to this river-rich Ida, this Mountain of Contemplation, and more than Contemplation. And also, though Number is a thing so immaterial, so divine, and eternal, yet by degrees, little by little, stretching forth and applying some likeness of it—first to Spiritual things; then bringing it lower to things sensibly perceived, such as a momentary sound repeated; then to the smallest things that can be seen and numbered; and at length (most coarsely) to a multitude of any bodily things seen or felt—so from these coarse and sensible things we are trained to learn a certain Image or likeness of numbers, and to use Art in them for our pleasure and profit. So coarse is our dealing and dull our apprehension, while mortal Sense rules the commonwealth of our little world. By this we say, three lions are three, or a Ternary. Three eagles are three, or a Ternary.\n\nNote this.\n\nSuch Ternaries are each the Union, knot, and Uniformity of three separate and distinct Units. That is, in each Ternary we can three times separately point out and show a part: One, One, and One. Whereas in Numbering we say One, two, Three. But how far these visible Ones differ from our Indivisible Units (principally considered in pure arithmetic) no one is ignorant. Yet from these coarse and material things we may be led upward, degree by degree, training our crude imagination toward the conceiving of Numbers absolutely (supposing no created thing, bodily or spiritual, to support, contain, or represent those Numbers imagined), that at length we may be able to find the number of our own name, gloriously exemplified and registered in the book of the most blessed and eternal Trinity.\n\nBut further understand that ordinary practitioners consider Numbers in various other ways, and extend the name further than to Numbers whose least part is a Unit. For the common reckoner, master of accounts, or arithmetician, in his use of Numbers, imagines parts smaller than a Unit and calls them Fractions. From a Unit he makes a half, and notes it thus, ½, and so of other (infinitely various) parts of a Unit. And further, he has Fractions of Fractions, etc.\n\n### Arithmetic, 1\n\nAnd, since Addition, Subtraction, Multiplication, Division and Extraction of Roots are the chief and sufficient parts of arithmetic: which is the science that demonstrates the properties of numbers, and all the operations to be performed in numbers:\n\n\"How often, therefore, these five different sorts of operations differ, for the most part in their execution, from the five operations of the same general property and name that are practicable in our whole numbers, so often (for a clearer account) we commonly count and name it another kind of arithmetic.\" And by this reason:\n\n1.\n\nThe study, teaching, and working of whole numbers only, where no part smaller than a unit is allowed, is named (as it were) an arithmetic by itself. And so with the arithmetic of fractions.\n\n2.\n\nIn the same way, the necessary, wonderful, and secret doctrine of proportion and proportionality has gained for itself its own particular way of handling and working. So it may seem another form of arithmetic.\n\n3.\n\nAlso, astronomers, for speed and more convenient calculation, have devised a particular way of ordering numbers for their circular motions: by sixties and sixtieths, by signs, degrees, and minutes, etc. This is commonly called the arithmetic of astronomical or physical fractions. I have briefly noted this by the name Circular Arithmetic, because it is also used in circles that are not astronomical, etc.\n\n4.\n\nPractice has led numbers further, and shaped them to take on the property of magnitudes, which is incommensurability and irrationality. (For in pure arithmetic, a unit is the common measure of all numbers.) Here numbers have become like lines, planes, and solids: sometimes rational, sometimes irrational. They have their own proper characters (as ²√, ³√, and so on for others, which is to signify square root, cube root, and so forth) and their own proper forms in the five principal parts. For that reason, the practitioner counts this a different arithmetic from the others. Practice brings in here various combinations of numbers: sometimes two, three, four (or more) radical numbers, joined in different ways by the signs of more and less, as thus ²√12 ³√15, or thus ⁴√19 ³√12 - ²√2, etc. And sometimes with whole numbers, or fractions of whole numbers, among them, as 20 ²√24. ³√16 33 - ²√10. ⁴√44 12¼ ³√9. And so the variety may be infinite. After this, both the one and the other have fractions attached, and so this arithmetic is greatly enlarged by the different showing and use of compositions and mixtures. Consider how I, being eager to free the student from error and quibbling, give this practice the name arithmetic of radical numbers, not of irrational or surd numbers. Those are at times rational, even though they have the sign of a root before them. The arithmetic of whole numbers, being the most usual, would say they have no such root, and so would count them as surd numbers. But that, spoken generally, is untrue, as Euclid's tenth book can teach you. So to call them, generally, radical numbers, because of the sign √ prefixed, is a sure way and a sufficient general distinction from every other ordering and use of numbers. And yet, beyond all this, consider the infinite desire for knowledge and the incredible power of human search and capacity: how together they have gone further (by mixing theory and practice) and have found and reached the very highest perfection, almost, of the practical use of numbers. This can be well seen in the great arithmetical art of equation, commonly called the Rule of Coss, or algebra. The Latins called it the rule of the thing and its value. A fitting name, covering the first and last points of the work. And the common names in Italian, French, and Spanish depend, in naming it, on the meaning of the Latin word Res, a thing, unless they use the name algebra. And in this there are commonly two errors. One is the error of those who think it was Geber's invention; the other is the error of those who call it algebra. First, though Geber, for his great skill in numbers, geometry, astronomy, and other marvelous arts, might have seemed able to have first devised the said rule, and though the name carries with it a very close likeness to Geber's name, yet it is true that a Greek philosopher and mathematician named Diophantus, before Geber's time, wrote 13 books about it (of which six are still in existence, and I had them for use,\n\nIn 1550.\n\nfrom the famous mathematician and my great friend Petrus Montaureus.) And secondly, the very name is Algiebar, not algebra, as the Arab Auicen can prove, who has these exact words, translated into Latin by Andreas Alpagus, most skilled in the Arabic tongue: the science of working Algiebar and Almachabel, that is, the science of finding an unknown number by adding a number, and by division, and equation. Here you have the name, and also the principal parts of the rule, touched on. To call it the rule or art of equation signifies the middle part and the state of the rule. This rule has its own particular characters:\n\n5.\n\nand the principal parts of arithmetic belonging to it differ from the other arithmetical operations. This arithmetic has simple, compound, and mixed numbers, and fractions accordingly. This rule and the arithmetic of Algiebar are so profound and so general, and so (in a manner) contain the whole power of the practical application of numbers, that the human mind can deal with nothing more profitable in numbers, nor find anything better suited to the soul's divine power (in human studies, business, or exercises) to be tested in. Perhaps you have been expecting, long before now, some particular proof or clear testimony of the use, profit, and benefit of ordinary arithmetic in the common life and trade of people. To that, then, I will now turn. But I take great care here, for fear that the length of several proofs make you think that I either doubted your zealous mind in the school of virtue, or distrusted your able wits to guess much more. A proof, then: four, five, or six such proofs I will bring, as any reasonable person may be persuaded by them to love and honor, yes, learn and practice the excellent science of arithmetic.\n\nAnd first, who closer at hand could be a better witness of the fruit received from arithmetic than all kinds of merchants? Though not all of them equally need it or use it. How could they do without the use and help of the rule called the Golden Rule, simple and compound, both forward and backward? How could they do without arithmetic's help in the rules of partnership, either without time or with time, and between the merchant and his agent? Would they gladly do without the rules of bartering in goods alone, or part in goods and part in money? Our merchant venturers and travelers overseas, how could they order their dealings justly and without loss, unless certain and general rules for the exchange of money and re-exchange were devised for their use? In how many different cases does the rule of alligation conclude for them such precise truths as they could not otherwise know, either by natural wit or by other experience? And, to make an end with the merchant, how wide-ranging and wonderful is the rule of false positions, especially as it has now been enlarged by two excellent mathematicians (of my close acquaintance in their lifetimes)? I mean Gemma Frisius and Simon Iacob. Who can briefly sum up the general and capital rules? And who can imagine the myriads of different cases and particular examples, in real earnest, continually worked out, tried, and settled by the forenamed rules alone? How many other arithmetical practices are commonly in merchants' hands and knowledge, they themselves can testify at length.\n\nThe mintmaster and the goldsmith, in their mixing of metals, whether of different kinds or different values, how exactly are they, or may they be, directed and wonderfully served, if arithmetic is their guide? And the honorable physicians will gladly confess that they owe much to the science of arithmetic, and in several ways. But chiefly in their art of graduation and in compound medicines. And though Galenus, Auerrois, Arnoldus, Lullus, and others have published their positions, both in the quantities of the degrees above temperament and in the rules that conclude the newly resulting form, yet a more precise, convenient, and easy method exists, invented by a countryman of ours\n\n(over 200 years ago) invented. And since I am uncertain who has it, or when that little Latin treatise (as the author wrote it) will come to be printed: (both to declare the desire I have to please my country, where I can, and also as very good proof of the use of numbers in this most subtle and fruitful philosophical conclusion,) I intend in the meantime, very briefly and with my further help, to communicate its essence to you.\n\nFirst draw a circle whose diameter is one inch. Divide the circumference into four equal parts. From the center, through those 4 points, extend 4 straight lines, each 4½ inches long, or as long as you like, more than 4, outside the circumference of the circle: so that they are 4 inches long (at least) outside the circle. Make clear marks at the end of every inch. If you like, you may subdivide the inches again into 10 or 12 smaller equal parts. At the ends of the lines, write the names of the 4 principal elemental qualities: hot and cold, one opposite the other, and likewise moist and dry, one opposite the other. And in the circle write Temperate. This temperature has a good range, as appears in a person's complexion. And therefore we have allowed it the said circle, and not a mathematical or physical point.\n\nTake some of Lullus's counsel in his book On the Fifth Essence.\n\nNow, when you have two mixable things whose degrees are truly known, they must be either of one quantity and weight, or of different ones. If they are of one quantity and weight, whether their forms are contrary qualities, or of one kind (but of different intentions and degrees), or a temperate and a contrary, the form resulting from their mixture is in the middle between the degrees of the mixed forms. For example, let A be moist in the first degree, and B dry in the third degree. Add 1 and 3, which makes 4. The half or middle of 4 is 2. This 2 is the middle, equally distant from A and B.\n\nNote.\n\nFor the temperament counts as none. For it you must put a zero, if at any time it is in the mixture.\n\n![The four elemental qualities and their degrees](/figures/dee-praeface/mixture-diagram.png)\n\nCounting from B, 2 degrees toward A, you find it to be dry in the first degree. So is the form resulting from the mixture of A and B in our example. I will give you another example. Suppose you have two things, C and D: and the heat of C in the 4th degree, and the cold of D so weak that it reaches the temperament. For C you take 4, and for D you take a zero. Added to 4, that yields only 4. The middle, or half, of it is 2. Therefore the form resulting from C and D is hot in the second degree, because 2 degrees, counted from C toward D, end exactly in the 2nd degree of heat. I will also give an example of the third manner, which is this:\n\nI have a liquid medicine whose quality of heat is raised to the 4th degree, as C was in the example above. And I have another liquid medicine whose quality is heat in the first degree. Of each of these I mixed an equal quantity. Subtract the lesser from the greater here, and divide the remainder into two equal parts. One of those two parts, either added to the lesser or subtracted from the higher degree, produces the degree of the form resulting from this mixture of C and E. Thus, if from 4 you take away 1, 3 remains. The half of 3 is 1½. Add this 1½ to 1, and you have 2½. Or subtract this 1½ from 4, and you likewise have 2½ remaining. This shows the resulting form to be heat in the middle of the third degree.\n\nThe second rule.\n\n\"But if the quantities of two things mixed together are different, and the intensities (of their mixable forms) are in different degrees and heights. (Whether those forms are of one kind, or of contrary kinds, or a temperate and a contrary, what proportion the lesser quantity has to the greater, the same proportion shall the difference between the degree of the resulting form and the degree of the greater quantity of the mixable thing have to the difference between that same degree of the resulting form and the degree of the lesser quantity.) As for example. Let two pounds of liquor be given, hot in the 4th degree, and one pound of liquor be given, hot in the third degree.\" I would gladly know the form resulting in the mixture of these two liquors. Set down your numbers in order, like this.\n\n£. 2.    Hot. 4.\n\n£. 1.    Hot. 3.\n\n### Arithmetic, 2\n\nBy the rule of algebra I have devised a very easy, brief and general way of working in this case. First, let us suppose that the middle resulting form is 1x, as that rule teaches. For, by the rule given here: as the weight of 1 is to 2, so is the difference between 4 (the degree of the greater quantity) and 1x to the difference between 1x and 3 (the degree of the thing in lesser quantity; and all along, 1x always sits at a certain middle between the two heights or degrees). For the first difference I set 4 - 1x; for the second, I set 1x - 3. And now again I say: as 1 is to 2, so is 4 - 1x to 1x - 3. Of these four proportional numbers, then, the first and the fourth multiplied together make as much as the second and the third multiplied together. Let the multiplications be made accordingly. From the first and the fourth we have 1x - 3, and from the second and the third, 8 - 2x. So our equation lies between 1x - 3 and 8 - 2x. It can be reduced by the art of algebra: here, adding 3 to each side gives the equation, thus, 1x = 11 - 2x. And to reduce it again: add 2x to each side. Then you have 3x equal to 11, written 3x = 11. Dividing 11 by 3, the quotient is 3⅔: the value of our 1x, the Coss, or Thing, first supposed. That is the height, or intensity, of the resulting form: it is heat at two thirds of the fourth degree. And here I set out the work itself in conclusion, like this. The proof is easy. Subtract 3 from 3⅔ and ⅔ remains. Subtract the same height of the resulting form (which is 3⅔) from 4, and ⅓ remains. You see that ⅔ is double ⅓, as 2 parts are double 1 part. So it should be, by the rule given here. Note: as you added 3 to each side of the equation, so if you had first added 2x to each side, it would stand as 3x - 3 = 8. And now, adding 3 to each side, you have (as before) 3x = 11.\n\nTwo parts, heat 4, ⅓; the form resulting, 3⅔; one part, heat 3, ⅔.\n\nAlthough I speak here of only two mixable things, and most commonly more than three, four, five or six (etc.) are to be mixed (and reduced into one compound, with the resulting form of them doing the job), yet these rules are enough, if they are repeated and applied in turn.\n\nStart with any two; then take the resulting form and another; and so on. For the last operation gives the resulting form of all of them. I need say nothing here about what the mixture (supposed here) is. Common philosophy has defined it, saying: Mixture is the union of miscible things that have been altered and joined through their smallest parts. Every word in the definition is of great importance. Nor do I need to spend any time showing how the other way of distributing degrees agrees with these rules. Neither do I need to explain here the further use of the Cross of Graduation (described before) to those who can take in what I have already said. Nor do I need to specify with examples the many varieties that the two general rules above can order. The clever and the studious have enough here. Those who cannot reach this without live teaching, and in more detail, would need a longer discussion than belongs in this place. And others (perhaps) will turn up their noses at this little, and would be ungrateful for much more. So I conclude, and I wish those with modest and earnest philosophical minds to praise God highly for this, and to marvel that the deepest and subtlest point concerning the mixing of forms and natural qualities is so matched and married to the simplest, easiest and shortest way of the noble rule of algebra. Who, then, can stay unpersuaded from loving, approving and honoring the excellent science of arithmetic? For here you can see that the little finger of arithmetic has more power and contrivance than the wits of a hundred thousand ordinary people can perform or truly conclude without its help.\n\nNow we will go further, and by the wise and brave captain learn what help he has from the rules of arithmetic in one of the arts belonging to him, which the Greeks call\n\nΤακτικὴ.\n\nΤακτικὴ. \"That is, the skill of ordering soldiers in battle array in the best way for all purposes.\" This art depends so much on the use of numbers and on mathematics that Ælianus (the best writer on it), in his work for the Emperor Hadrianus, thinks his book, by his perfection in mathematics (which was greater than others before him had), surpasses all other excellent works on that art written up to his day. For on it had written Æneas; Cyneas of Thessaly; Pyrrhus Epirota and Alexander his son; Clearchus; Pausanias; Euangelus; Polybius, familiar friend of Scipio; Eupolemus; Iphicrates; Possidonius; and very many other worthy captains, philosophers and princes of immortal fame and memory, whose fairest flower in their garland (in this feat) was arithmetic, and a little knowledge of geometric figures. But in many other cases arithmetic stands the captain in good stead: in apportioning provisions for the army, whether it stays at its current size, or is suddenly increased by a certain number of soldiers, for a certain time; or, by good skill, in reducing his company, so that the provisions last longer for those who remain, for a fixed period, if need requires. And so, in his other accounts, reckonings, measurements and proportions, the wise, expert and careful captain will affirm that the science of arithmetic is one of his chief counselors, directors and helpers. This was clear, by good means, to the noble, courageous, loyal and courteous John, late Earl of Warwick. He was a young gentleman thoroughly known by very few. Yet his lively valor, strength and skill in chivalrous feats and exercises, and his humility and friendliness to all men, were things the world saw openly. But what roots virtue had otherwise fastened in his breast; what rules of godly and honorable life he had set for himself; what notable vices (in some then living) he took great care to avoid; what manly virtues (in other noblemen flourishing before his eyes) he constantly aspired to; what prowess he intended and meant to achieve; and with what feats and arts he began to furnish and load himself for better service to his king and country, in peace and war: these, I say, his heroic meditations, forecasts and decisions, no two people besides myself, I think, can report so perfectly and truly. And so, in conscience, I count it my part, for the honor, advancement and promotion of virtue, briefly to have put his name in the Register of Immortal Fame.\n\nTo our point. This John, by one of his acts (besides many others, both in England and France, that I noted in him), revealed his heartfelt love of the virtuous sciences and his noble aim to excel in martial prowess, when he obtained, by humble request and constant urging, the best rules (used in times past by Greeks or Romans, or used in our own time, with new stratagems devised in them) for ordering all companies, bodies and numbers of men (many or few) appointed with one kind of weapon or more, with artillery or without, on horseback or on foot; to give or take the onset; to seem many when few, and few when many; to march in battle or on a journey; and many such feats belonging to a pitched battle, skirmish or ambush:\n\nThis noble earl died in the year 1554, scarcely 24 years of age, having no children by his wife, daughter to the Duke of Somerset.\n\nAnd all these lively designs, most carefully, were to be drawn on vellum parchment, with notes and special marks as the art requires; and all these rules and arithmetic descriptions, enclosed in a rich case of gold, he used to wear around his neck as his most precious jewel and his most trusted counselor. Thus arithmetic was enshrined in him in gold; of the fruit of numbers he had good hope. Now, therefore, innumerable numbers shall find, in the praise of numbers, his shrine.\n\nWhy need I (for further proof to you) ask the schoolmasters of justice for testimony of how necessary, how fruitful and how skillful a thing arithmetic is? I mean the lawyers of all sorts. Undoubtedly the civil lawyers can wonderfully show how neither the ancient Roman laws can be understood without good knowledge of the art of numbers, nor can justice (in infinite cases) be carried out without due proportion, closely considered. How justly, and with great knowledge of the art, did Papinianus lay down a law of division and allowance between man and wife after a divorce? But how Accursius, Baldus, Bartolus, Iason, Alexander, and finally Alciatus (otherwise notably well learned) jumble, guess and err from the equity, art and intent of the lawmaker, arithmetic can detect and convict, and can clearly make the truth shine. Good Bartolus, worn out from examining and apportioning the matter, and much troubled by Accursius's gloss, burst out and said: In the whole book there is no gloss harder than this, whose account or reckoning neither the scholars nor the doctors understand, etc. What can they say of the law of Julianus beginning \"If it was thus written,\" etc., of the testator's will justly performed between the wife, son and daughter? How can they perceive the equity of Aphricanus's arithmetic reckoning, where he treats of the Falcidian Law? How can they clear him from his reprovers and their maintainers, such as Ioannes, Accursius Hypolitus and Alciatus? How justly and skillfully was Africanus's reckoning made, apportioning to the sums bequeathed the contributions of each part? Namely, for the hundred received right away, 17-1/7; and for the hundred received after ten months, 12-6/7; which make the 30 that the legatees were to contribute to the heir. For what proportion 100 has to 75, the same 17-1/7 has to 12-6/7; which is sesquitertia, that is, as 4 to 3, which make 7. Very many places in the civil law require an expert arithmetician to understand the deep judgment and just determination of the ancient Roman lawmakers. But he ought to be far more expert who can decide with equity the infinite variety of cases that do or may happen under every one of those civil laws and ordinances. From this you may easily conjecture that in canon law, and in the laws of the realm (which with us carry the chief authority), justice and equity might be greatly preferred and skillfully executed, through due skill of arithmetic and the proportions belonging to it. The worthy philosophers and prudent lawmakers (who have written many books On the Commonwealth: how the best state of commonwealths might be procured and maintained) have judged very well of justice (which is not only the base and foundation of commonwealths, but also the total perfection of all our works, words and thoughts), defining it,\n\nJustice.\n\n\"to be that virtue by which to everyone is rendered what belongs to him.\" God demands this of us: to be honored as God, to be loved as a father, to be feared as a Lord and master. Our neighbor's proportion is also laid down by the Almighty lawmaker: to do to others as we would have done to us. These proportions are necessary in justice, commendable in duty, and for commonwealths they are the life, strength, support and flourishing. Aristotle in his Ethics (to fetch the seed of justice and the light of direction, to use and execute the same) had to turn to the perfection and power of numbers: for arithmetic and geometric proportions. Plato in his book called Epinomis (which book is the treasury of all his doctrine), where his purpose is to seek a science that, when a man had it perfectly, would make him seem and so actually be wise. He, briefly discussing other sciences, finds them not able to bring it about; but of the science of numbers he says: That science, surely, which has taught mankind number, will be able to bring it about. And I think that a certain god, rather than fortune, has given us this gift for our happiness. For why should we not judge him who is the author of all good things to be also the cause of the greatest good thing, namely wisdom? There, at length, he proves wisdom to be attained by good skill in numbers. With that great testimony, and the many proofs and reasons expressed before, you may be sufficiently and fully persuaded of the perfect science of arithmetic, and come to this account: that of all sciences, next to theology, it is most divine, most pure, most ample and general, most profound, most subtle, most useful and most necessary. Its next sister is the absolute science of magnitudes; of which (by the direction and help of him whose magnitude is infinite and of us incomprehensible) I now intend so to write that, both with the multitude and with the magnitude of marvelous and fruitful truths, you (my friends and countrymen) may be stirred up and awakened to see what certain arts and sciences (for our unspeakable benefit) our heavenly Father has prepared and revealed for us by various philosophers and mathematicians.\n\n### Arithmetic, 3\n\nBoth number and magnitude have a certain original seed (as it were) of incredible property, which man is never able to declare fully. For number, a unit, and for magnitude, a point, seem to be very much alike as original causes; but the difference is nevertheless great. We defined a unit to be a mathematical thing indivisible; a point, likewise, we said to be a mathematical thing indivisible. Further, a point may have a certain fixed situation: that is, we may assign and set a point to be here, there, over there, etc. Here, notice, our unit is free, and can endure no bondage, nor be tied to any place or seat, divisible or indivisible. Again, because a point may have a situation limited to it, a certain motion (to a place and from a place) belongs to a point. But a unit cannot be imagined to have any motion. A point, by its motion, produces mathematically a line (as we said before), which is the first kind of magnitudes and the simplest; a unit cannot produce any number. A line, though it is produced by a point moved, does not consist of points; number, though it is not produced by a unit, does consist of units as a material cause. But formally,\n\nNumber.\n\nNumber is the union and unity of units. This uniting and joining is the workmanship of our mind, which, out of distinct and separate units, makes a number by the uniformity resulting from a certain multitude of units. And so every number can be given its least part, namely a unit; but of a magnitude (not even of a line) can the least part be given, because its division can be conceived without end. All magnitude is either a line, a plane or a solid. A line, plane or solid can be perceived by no sense, nor represented exactly by hand in any way, nor produced by nature; but as number came to our perception by degrees, so we are helped by visible forms to imagine what our mathematical line is, and what our point is. Our magnitudes are so precise that one line is no broader than another, for they have no breadth; nor do our planes have any thickness; nor do our bodies have any weight, however large in dimension. We can have bodies smaller than anything art or nature can produce, and greater also than all the world can contain. Our least magnitudes can be divided into as many parts as the greatest. For a line an inch long (with us) can be divided into as many parts as the diameter of the whole world from east to west, or extended any way. What privileges, above all manual art and nature's power, have our two mathematical sciences: to exhibit and deal with things of such power, liberty, simplicity, purity and perfection; and in them to proceed so certainly, so orderly, so precisely? Excellent is that mechanical workman judged who can come nearest to representing works mathematically demonstrated.\n\nAnd our two sciences, remaining pure and absolute in their proper terms and in their own matter, to have and allow only such demonstrations as are plain, certain, universal and of eternal truth?\n\n### Geometry\n\nThis science of magnitude, its properties, conditions, and appurtenances, has commonly been called geometry, now and from the beginning, by all philosophers. But truly, that name is too base and too small for a science of such dignity and breadth. And perhaps that name has been allowed to remain, by the common and secret agreement of all wise people, so that it carries a perpetual memory of the first and greatest benefit that this science showed to common people: which was when the boundaries and markers of land were lost and confused (as in Egypt every year, with the overflowing of the Nile, the greatest and longest river in the world), or when bequeathed land had to be assigned, or sold land had to be laid out, or (when disorder prevailed) commons had to be divided into separate holdings. For, on these and similar occasions, some by ignorance, some by negligence, some by fraud, and some by violence, wrongly limited, measured, encroached on, or claimed (under the pretense of a fair content and measure) those lands and grounds: great loss, unrest, murder, and war very often followed. Until, by God's mercy and human effort, the perfect science of lines, planes, and solids (like a divine judge) gave to each person what was their own. The people, pleased by this art and greatly helped by the fair measuring of their lands, and other philosophers writing rules for land measuring, between them both confirmed the name Geometria: that is, according to the very etymology of the word, land measuring. In this, the people knew no further use of magnitudes than in planes; and the philosophers had no fit hearers or students to disclose anything further to them than flat, plain geometry. And though these philosophers knew of further use and best understood the etymology of the word, yet they applied the name Geometria generally to all sorts of magnitudes: unless, at times, Plato and Pythagoras, when they wanted to state their own teaching precisely. Then, was\n\n(* Plato, Republic 7.)\n\n* Geometria, with them, the study concerned with the plane. But you can see clearly enough from Euclid's Elements that our science is wider than measuring planes, and that nothing less is taught in it (on purpose) than how to measure land. So another name must be had for our mathematical science of magnitudes: one that cares nothing for clod or turf, hill or dale, earth or heaven, but is absolute Megethologia: not creeping along the ground and dazzling the eye with pole, perch, rod, or line, but \"lifting the heart above the heavens by invisible lines, and\n\nimmortal beams meet with the reflections of the incomprehensible light, and so bring joy and unspeakable perfection.\" Of this true use of our Megethica, or Megethologia, divine Plato seemed to have good taste and judgment: and (by the name of geometry) he noted it in that way, and warned his students of it, as may clearly be seen in his seventh dialogue, of the Commonwealth. Where it is thus, and well translated: Truly (says Plato) whoever has tasted even a very little of geometry will not deny us this: that this science is of a quite contrary condition to what those who are occupied in it say about it. And it follows, of our geometry: It must be confessed (says Plato) that [geometry] is learned for the knowledge of that which always is, and not of that which, in time, both comes into being and comes to an end. etc. Geometry is the knowledge of what is everlasting. It will therefore lift up our mind to the Truth (O Gentle Sir), and by that means prepare the thought for the philosophical love of wisdom: so that we may turn or convert toward heavenly things [both mind and thought], which now, contrary to what is fitting, we cast down onto base and inferior things. etc. Commandment must therefore chiefly be given, that those who inhabit this most honorable City by no means despise geometry. For even those things [done by it] which seem, in a manner, to lie beside the purpose of geometry are of no small importance. etc. And besides the manifold uses of geometry in matters relating to war, he adds more: a second, unintended fruit and benefit arising from geometry, saying: But also we know that, for the easier learning of all arts, it much matters whether one has any knowledge of geometry or not. etc. Let us therefore make an ordinance or decree, that this science be learned by young men in the second place. This was divine Plato's judgment, both of the intended, chief, and perfect use of geometry, and of its second, dependent, derivative benefits. And for us, Christians, there are a thousand thousand more occasions to need the help of *\n\n(I. D. * In this, I would gladly shake off the earthly name of geometry.)\n\nMegethologicall Contemplations: by which to train our imaginations and minds, little by little, to forsake and abandon the gross and corruptible objects of our outward senses; and to grasp, by sure demonstrative teaching, mathematical things. And by them, to be readily helped and guided to conceive, reason about, and conclude on things intellectual, spiritual, eternal, and such as concern our everlasting bliss: which otherwise (without a special privilege of illumination or revelation from heaven) no mortal person's wit can naturally reach or grasp. And truly, by my small talent from above, I am able to prove and testify that the literal text and order of our divine Law, Oracles, and Mysteries require more skill in numbers and magnitudes than expositors have commonly stated: they have rather only, at most, warned of this and shown their own lack in it. (To name any is needless, and to note the places is no part of this: but if I am properly asked, my answer is ready.) And without the literal, grammatical, mathematical, or natural truths of such places, perceived by good and certain art, no spiritual sense (proper to those places, by absolute theology) will depend on them.\n\n\"No one, therefore, can doubt that, toward the attaining of incomparable knowledge and heavenly wisdom, mathematical speculations, both of numbers and magnitudes, are means, aids, and guides: ready, certain, and necessary.\" From now on, in this preface of mine, I will shape my talk for Plato's runaway students: or rather, for those who can well, and also will, use their outward senses, to the glory of God, the benefit of their country, and their own private contentment or honest advancement on this earthly stage. To them I will orderly recite, describe, and declare a great number of arts, derived from our two mathematical fountains into the fields of Nature. By which such seeds and roots as lie deep hidden in the ground of Nature are refreshed, quickened, and provoked to grow, shoot up, flower, and bear fruit, infinite and incredible. And these arts will be such as depend more on the properties of magnitudes than on number. And with good reason we may call them arts, and mathematically derivative arts: for (at this time) I define\n\n(An Art.)\n\nAn art, to be a methodical and complete body of teaching, having plenty of sufficient and particular matter to deal with, by the allowance of the metaphysical philosopher: knowledge of which is necessary to human life. And that I count,\n\n(A Mathematically Derivative Art.)\n\nAn art mathematically derivative, which by a mathematical demonstrative method, in numbers or magnitudes, orders and confirms its teaching, as much and as perfectly as the subject matter allows. And because I intend to use the name and property of a\n\n(A Mechanician.)\n\nMechanician, otherwise than it has been used so far, I think it good (for the sake of distinction) to give you also a brief description of what I mean by it. A mechanician, or a mechanical workman, is a person whose skill is to work and finish any sensible work perfectly, without knowledge of mathematical demonstration, by the principal or derivative mathematician, demonstrated or demonstrable. I know very well that whoever invents or makes these demonstrations is generally called a speculative mechanician, which differs nothing from a mechanical mathematician. So, in view of different actions, one person may hold the names of several arts: at one time a logician, at another (when the same matter is handled otherwise) a rhetorician. Of these trifles I make little account (for now, in view of my preface): I leave them for the fine handling of subtle, exacting disputers. In other places they may command me to give a good reason; but here I will not be unreasonable.\n\n(1.)\n\nFirst, then, from the purity, absoluteness, and immateriality of principal geometry is derived that kind of geometry which is commonly counted as geometry: and it is the art of measuring perceptible magnitudes, their exact quantities and contents.\n\n### Common Geometry\n\nThis teaches measuring up close: the practitioner is at the thing being measured, and so, by correctly applying a compass, rule, square, yard, ell, perch, pole, line, gaging rod, (or such like instrument) to the length, plane, or solid being measured,\n\n(1.)\n\n* to learn either the length, perimeter, or linear distance; and this is called mecometry. Or\n\n(2.)\n\n* to learn the content of any plane surface: whether it is ground surveyed, board measured, or glass measured, or such like thing; which measuring is named embadometry.\n\n(3.)\n\n* Or else to understand the solidity and content of any bodily thing: as of timber and stone, or the content of pits, ponds, wells, vessels, small and great, of all shapes. Where the measuring of wine, oil, beer, or ale vessels, etc. commonly has a particular name, and is called gaging. And the general name of these solid measures is stereometry.\n\n(2.)\n\nOr else, this common geometry also aims to teach the practitioner how to measure things from a good distance, keeping apart from the thing measured, and by that to understand either\n\n(1.)\n\n* how far away a thing seen (on land or water) is from the measurer; and this may be called apomecometry:\n\n(2.)\n\nOr, how high or deep, above or below the level of the measurer's standing, anything seen on land or water is; called hypsometry.\n\n(3.)\n\n* Or, it tells the measurer how broad anything is that lies in the measurer's view, as long as it is situated on land or water; and may be called platometry. Though I make it a condition here that the thing measured be situated on land or water:\n\nyet know for certain that the various heights of clouds, blazing stars, and the moon can (by these means) have their distances from the earth determined; and that the solidity of the blazing stars and the moon can be measured as well as their distances. But because these things are not commonly taught, nor so easily carried out by an ordinary practitioner, I prefer to count such measures as belonging to some of our other arts, which deal more purposely with things up high, rather than to this common land-measuring geometry: as in perspective and astronomy, etc.\n\nFrom these feats (applied further) has sprung the feat of geodesy, or land measuring: to measure and survey land, woods, and waters more skillfully from afar. More skillfully, I say. But God knows (so far, in these realms of England and Ireland, whether through ignorance or fraud, I cannot tell in every particular)\n\nhow much wrong and injury has been done (in my time) by untrue measuring and surveying of land or woods, in any way. And this I am sure of: the value of the difference between the truth and such surveys could have kept (forever) an excellent mathematics reader in each of our two universities; allowing each (yearly) a hundred marks of lawful money of this realm, which indeed would seem to be needed here (though provided for in other ways), just as the famous University of Paris has two mathematics readers, each with two hundred French crowns a year, from the French king's magnificent generosity alone. Now, returning again to our purpose: moreover, from the earlier geometric knowledge have grown the skills of geography, chorography, hydrography, and Stratarithmetrie.\n\n### Geography\n\n\"Geography teaches the ways in which, in various forms (as spherical, plain, or other), the situation of cities, towns, villages, forts, castles, mountains, woods, harbors, rivers, creeks, and such other things, on the outer face of the earthly globe (either in the whole, or contained in some principal member and portion of it) may be described and drawn, in proportions analogous to nature and truth, and may be represented most fitly to our view.\" Of this art, how great pleasure and how many benefits come to us daily and hourly is perceived by most people. While some use them to beautify their halls, parlors, chambers, galleries, studies, or libraries; others for things past, such as battles fought, earthquakes, heavenly fires, and such occurrences as are mentioned in histories, by that to view, as it were, the place, the neighboring region, the distance from us, and such other circumstances. Some others, to view at once the large dominion of the Turk: the wide Empire of the Muscovite; and the little piece of ground where Christendom (by profession) is certainly known. Little, I say, in respect of the rest. etc. Some, either to direct their own journeys into far lands, or to understand other people's travels. To conclude, some for one purpose and some for another like, love, get, and use maps, charts, and geographic globes. Of whose use, to speak fully, would require a book of its own.\n\n### Chorography\n\nChorography seems to be a lesser thing, and a twig, of geography: and yet nevertheless it is manifold in practice and very wide in use. \"This teaches, by analogy, to describe a small portion or circuit of ground, with its contents: not regarding what commensuration it has to the whole, or any parcel contained outside it. But in the territory or parcel of ground which it takes in hand to describe, it leaves out (or undescribed) no notable or odd thing above the ground that is visible. And sometimes of things under ground it gives some particular mark or warning: as of metal mines, coal pits, stone quarries, etc.\" Thus, a duchy, a shire, a lordship, or less, may be described distinctly. But it is marvelous pleasant and profitable to display to our eye, and measure in proportion, the plan of a city, town, fort, or palace, in true symmetry: without going close to any of them, and from out of gunshot, etc. By that the architect can furnish himself with as many patterns as he likes: to his great instruction, even in those things that are proportioned outwardly: either simply in themselves, or in relation to hills, rivers, harbors, and woods adjoining. Some also call this particular description of places topography.\n\n### Hydrography\n\n\"Hydrography delivers to our knowledge, on globe or in plain, the perfect analogous description of the ocean sea coasts, through the whole world: or in the chief and principal parts of it:\" with the islands and chief particular places of danger contained within the bounds, and sea coasts described: as of quicksands, banks, pits, rocks, races, countertides, whirlpools, etc. This deals chiefly with the element of water: as geography principally took the description of the element of earth (with its appurtenances) to task. And besides this, hydrography requires a particular register of certain landmarks (where marks can be had) from the sea, well able to be sighted: at what point of the sea compass they appear, and what apparent shape, situation, and bigness they have, in respect of any dangerous place in the sea, or near to it, assigned. And on all coasts, what moon makes full sea, and what way the tides and ebbs come and go, the hydrographer ought to record. The soundings likewise, and the channels' ways: their number and depths ordinarily at ebb and flood, the hydrographer ought to have certainly known by observation and careful measuring. And many other points belong to perfect hydrography, and to making up sailing directions, of which I need not speak here: such as describing, in any place, on globe or plain, the 32 points of the compass truly (of which scarcely four in England have right knowledge, because its lines are neither straight lines nor circles). Making a due projection of a sphere on a plain. The variation of the compass from true north: and such like matters (all of great importance) I will not speak of here: because I may seem already to have enlarged the bounds and duty of a hydrographer much more than anyone (to this day) has noted or prescribed. Yet I am well able to prove that all these things pertain, and are also proper, to the hydrographer. The chief use and end of this art is the art of navigation: but it has other diverse uses: even to be enjoyed by those who never lose sight of land.\n\n### Stratarithmetrie\n\nStratarithmetrie is the skill, used in war, by which a man can set any set number or total of men into a figure like any given geometric figure, provided that figure can hold them. Two things limit it: the usual spaces allowed between soldiers, and the fact that men cannot be divided into fractions. Even so, he can arrange the given number of men into the largest figure they can fill, and report the surplus, if there is any. He can also report the next set number that, with the surplus, will make up a figure exactly like the figure assigned. The same skill works the other way. Given any army or company of men, whose figure and sides in orderly standing or array are known, he can state the exact number of men contained in that figure, or that could be contained in it in order.\n\nHe can also know this figure and its sides, whether he is near at hand or far off. That is as far as the description and the nature of Stratarithmetrie goes, enough for this time and place.\n\nThe difference between Stratarithmetrie and tactics.\n\n\"It differs from the art of tactics (drawing up battle lines), because there you need the wisdom and foresight to know what purpose you are ordering the men for, and the skilled ability to design and use, for any occasion or purpose, the fittest and most necessary order, array, and figure for your company and total of men.\" By figure I mean a square, triangle, circle, oval, rectangle (the Greeks call it Eteromekes), rhombus, rhomboid, crescent, ring, serpentine, and such other geometric figures, which have been and are to be used in war for convenience, necessity, and advantage, etc. He must have no small skill who would give a true report, or one near the truth, of the numbers and totals of foot soldiers or horsemen in the enemy's formation. To estimate from far off, down to close limits of more and less, is not common, even among those who would gladly do it.\n\nI. D. — my friend, you will find it hard to carry out my description of this work. But by chorography you can help yourself somewhat, where the figures known by their sides and angles are not regular, and where resolving them into triangles can serve, etc. And yet you will find it strange to deal so generally with arithmetic figures, and for a battle array, since their contents differ so much from those of like geometric figures.\n\nCaptains can make good use of policy, at fitting times and in suitable places: using figures that make the greatest show of the number they have, and, taking advantage of the three usual kinds of space between foot soldiers or horsemen, taking the largest; or, when they want to seem few though they are many, doing the opposite in both figure and space. The herald, pursuivant, sergeant royal, captain, or anyone careful to come near the truth here, besides the judgment of his expert eye, his skill in tactical ordering, and the help of his geometric instrument — a ring or an astronomical staff, built conveniently for carrying and use — can wonderfully help himself with perspective glasses, in which, I trust, our descendants will prove more skillful and expert, and for greater purposes, than in these days can almost be credited as possible.\n\nThus I have quickly passed over the artificial feats that depend chiefly on common geometry, and that are usually and generally counted under the name of geometry.\n\n### The Mixed Sciences and Perspective\n\nBut there are very many other methodical arts. Departing from the purity, simplicity, and immateriality of our principal science of magnitudes, they still make great use of the help, direction, and method of that principal science, and they have their own names, distinct both from the science of geometry, from which they are derived, and from one another: such as perspective, astronomy, music, Cosmography, astrology, statics, anthropography, Trochilike, Helicosophy, Pneumatithmie, Menadrie, Hypogeiodie, Hydragogie, horometry, painting, architecture, navigation, Thaumaturgike, and Archemastrie. I think it necessary to describe these in order, and also to touch on some of their useful purposes, and so to make this Preface a little sweet, pleasant nosegay for you: to comfort your spirits when you are almost out of courage and in despair, through a brutish way of thinking, imagining that geometry had only served to build a house, or a fine bridge, or the roof of Westminster Hall, or some clever little device or engine belonging to a carpenter or a joiner, etc. The truth is far different from what the world has commonly judged, by word and deed, to this day, and good proof of that will be made.\n\nAmong these arts, perspective rightly ought to come first, before any perfect knowledge of astronomical appearances can be attained. And because of the special place of light, being the first of God's creations, and because the eye is the light of our body, our most powerful sense, and our most skillful and geometric organ, we will begin with perspective. Perspective is a mathematical art that demonstrates the manner and properties of all radiations: direct, broken, and reflected. This description is brief, but it reaches as far as the world is wide. It concerns all creatures, and all actions and passions performed by the emission of rays. By rays, or natural lines, I mean here not only rays of light or color — though these give the eye the appearance, witness, and proof on which to ground the art — but also the certain and determined active radial emissions of other forms, both substantial and accidental. By this art, leaving aside the highest points, we can use our eyes and the light with greater pleasure and better judgment: both of things seen in light and of other things, which work and produce their effects by the same order of light's radiations. We should be ashamed not to know the cause of the many ways our eye is deceived and misled: how the eye takes a round globe or sphere, seen from far off, to be a flat plain circle, and likewise judges a plain square to be round; supposes parallel walls to draw together from far away; and takes the roof and floor, the one to bend downward and the other to rise upward, when they are a little distance from you. Again, of things moving at the same speed, it thinks the nearer moves faster and the farther much slower. Or, of two things, one of which moves incomparably faster than the other, it judges the slower to move very fast and the other to stand still: what an error of our eye! Of the rainbow — its colors, the order of the colors, its size, its place and height, etc. — is it not pleasant and necessary to know the demonstrative causes? Of two or three suns appearing, of blazing stars, and such like things, brought to pass by natural causes, and yet at the same time signifying matters beyond them: is it not useful for people to know their true and natural cause and occasion? In fact, is it not greatly beneath the dignity of human nature to be so tricked and misled by things at hand, before one's eyes? Such as a peacock's tail and a dove's neck; or a whole oar held in water, seeming broken. Things far off seeming near, and near seeming far off. Small things seeming great, and great things small. One man seeming like an army. Or a person terribly afraid of his own shadow. So much so that, if you are alone near a certain mirror and try to thrust at it with a dagger or sword, you will suddenly be moved to give back, in a manner, by reason of an Image,\n\nA marvelous mirror.\n\nappearing in the air between you and the mirror, with a like hand, sword or dagger, and with like quickness, thrusting at your very eye, just as you do at the mirror. This is strange to hear of, but more marvelous to see than these words of mine can express. And yet by optical demonstration the order and cause of it are just as certain as the effect that follows. I dare take it upon myself even further, to satisfy the noble courage that longs ardently for the wisdom of natural causes: to let him understand that in London he can have proof of what I have said here with his own eyes. A gentleman, who, for his good service to his country, is famous and honorable:\n\nand for skill in the mathematical sciences and languages, is the extraordinary man of this land, etc. He himself is able, and I am sure will very willingly, let the mirror and the proof be seen, and so I ask it of him here: for the increase of wisdom among the honorable, for stopping the mouths of the malicious, and for repressing the arrogance of the ignorant. You can easily guess what I mean. This art of perspective is of such excellence, and can lead to the certifying and carrying out of such things as no one would easily believe without actual proof. I say nothing of natural philosophy, which without perspective cannot be fully understood nor perfectly attained. Nor of astronomy, which without perspective cannot be well grounded; nor of astrology, naturally verified and confirmed. That part of it which deals with glasses — where glass is a general name in this art for anything from which a ray rebounds — is called catoptrics, and has so many uses, both marvelous and profitable, that it would take me too long to note the principal conclusions already known. And also, perhaps, some things might lack due credit with you, and I would lose my labor, and you would slip into a hasty judgment,\n\nBefore you have learned enough of the power of nature and art.\n\n### Astronomy\n\nNow, to proceed: Astronomy is a mathematical art that demonstrates the distances, magnitudes, and all natural motions, appearances, and conditions proper to the Planets and fixed Stars: for any time past, present, and to come, with respect to a given Horizon, or without respect to any Horizon. By this art we learn the distance of the Starry Sky, and of each Planet from the Center of the Earth, and the size of any Fixed star seen, or Planet, in respect to the Earth's size. For example, we are sure, by this art, that the Solidity, Mass, and Body of the Sun contains the quantity of the whole Earth and Sea a hundred and thirty-two times, less by ⅛, one eighth part of the earth. But the body of the whole earthly globe and Sea is bigger than the body of the Moon forty-three times, less by ⅛ of the Moon. Therefore the Sun is bigger than the Moon 7000 times, less by 59 39/64, that is, precisely 6940 25/64 bigger than the Moon. And yet an unskilled person would judge them the same size. Therefore one of them must be much farther from us than the other. The Sun, when he is farthest from the earth (which now, in our age, is when he is in the 8th degree of Cancer), is 1179 semidiameters of the Earth distant. And the Moon, when she is farthest from the earth, is 68 semidiameters of the earth and ⅓. The nearest the Moon comes to the earth is 52¼ semidiameters. The distance of the Starry Sky from us, in semidiameters of the earth, is 20081½ — twenty thousand eighty-one and a half. Subtract from this the Moon's nearest distance from the Earth, and there remain 20029¼ semidiameters of the earth — twenty thousand twenty-nine and a quarter.\n\nSo thick is the heavenly Palace that the Planets have all their exercise in, and most marvelously perform the Command and Charge given to them by the omnipotent Majesty of the King of kings. This is what in Genesis is called Ha Rakia. Consider it well. The semidiameter of the earth contains 3436 4/11 of our common miles, three thousand four hundred thirty-six and four elevenths of one mile, and so the whole earth and sea round about is 21600 of our miles, twenty-one thousand six hundred, allowing sixty miles for every degree of the greatest circle. Now if you weigh well with yourself just this little portion of the fruit of astronomy, concerning the size and distances of Sun, Moon, and Starry Sky, and the huge mass of Ha Rakia, will you not find your conscience moved, with the kingly Prophet, to sing the confession of God's Glory, and say, \"The heavens declare the glory of God, and the firmament [Ha Rakia] shows the works of his hands.\" And so on, for those first five stanzas of that kingly Psalm. Well, well, it is time for some people to lay hold of wisdom, and to judge things truly, and not to explain the Holy word all by allegories, so as to neglect the wisdom, power, and goodness of God that is to be seen and learned in and by his Creatures and Creation. Through parables and analogies of their natures and properties, the course of Holy Scripture also declares very many Mysteries to us. The whole frame of God's creatures, which is the whole world, is to us a bright mirror: from which, by reflection, beams and radiations bounce back to our knowledge and perception, representing the image of his infinite goodness, almighty power, and wisdom. And we are by that taught and persuaded to glorify our Creator as God, and to be thankful for it. If the Heathens could find these uses in these most pure, beautiful, and mighty bodily creatures, shall we — after the true Sun of righteousness has risen above the Horizon of our temporal Hemisphere, and has so abundantly streamed the direct beams of his goodness, mercy, and grace into our hearts, a heat that all creatures feel, spiritual and bodily, visible and invisible — shall we, I say, look upon the Heaven, Stars, and Planets as an ox and a donkey do, caring and inquiring no further about what they are, why they were created, and how they carry out what they were created for? Since all creatures were created for our sake, and both we and they were created chiefly to glorify the Almighty Creator, and to do that by every means possible to us. Do not be ignorant (says Plato in Epinomis) that astronomy is a thing of excellent wisdom. Astronomy was commended to us from the beginning, and in a manner commanded by God himself, since he made the Sun, Moon, and Stars to be for us Signs, and knowledge of Seasons, and for Distinctions of Days and years. Many words are not needed. But I wish everyone would weigh this word: Signs. And besides that, compare it also with the tenth chapter of Jeremiah. And though some think that they have found a rod there, yet Modest Reason will be an impartial judge of who ought to be beaten with it, as far as our purpose is concerned. Leaving that, please understand this: without great diligence of observation, examination, and calculation, their periods and courses — by which the Distinction of Seasons, years, and New Moons could be precisely known — could not be exactly certified. To do that is the art we have here defined to be astronomy. By it we may have the distinct Course of Times, days, years, and Ages: both for considering Sacred Prophecies foretold and accomplished in due time, and for holding high Mystical Solemnities, and for all other human affairs, conditions, and covenants at a set time between one person and another: with many other great uses: in all of which, truly, there would be great uncertainty, confusion, untruth, and brutish barbarism without the wonderful diligence and skill of this art: continually learning and fixing Times and periods of Time by the record of the heavenly book in which all times are written, and to be read with an astronomical staff instead of a pointer stick.\n\n### Music\n\nMusic, as a kind of motion, has its original cause. So, after the swiftest and slowest motions in the firmament, which nature performs and the astronomer considers, I will now speak of another kind of motion: sound-producing, audible, and countable by human beings. By music here I mean the science that the Greeks call Harmonice. I will not take up the dispute between the ancient harmonists and canonists. Music is a mathematical science. By sense and reason it teaches us to judge and order the differences between high and low sounds perfectly. Astronomy and music are sisters, Plato says. The eyes were made for astronomy, and the ears for harmonious motion. But astronomy has a more divine contemplation and usefulness than any mortal eye can perceive. So music is to be considered:\n\n(1.)\n\nthat the mind may be preferred to the ear. And from audible sound we should go on to examine which numbers are in harmony and which are not. And why the one are and the other are not. I could at length,\n\n(2.)\n\nin the heavenly motions and distances, describe the marvelous harmony of Pythagoras' harp\n\n(3.)\n\nwith eight strings. Something could also be said of Mercury's two harps,\n\n(4.)\n\neach with four elemental strings. And very strange things could be said about the harmony,\n\n(5.)\n\nthat belongs to our spiritual part. As can be seen in Ptolemy's third book, in the fourth and sixth chapters.\n\n(6.)\n\nAnd what causes the fitting bond and friendly fellowship between the intellectual and mental part of us and our gross and corruptible body, but a certain middle and harmonious spiritual nature, in which both share and which (in a way) results from both? In\n\n(7.)\n\nthe tune of the human voice, and also\n\n(8.)\n\nthe sound of instruments: of what could be said about harmony, no ordinary musician would readily believe.\n\n(I. D. Read Aristotle's 8th book of Politics: the 5, 6, and 7 chapters. There you will have further occasion to think about music than is commonly thought.)\n\nBut of the various mixture (as I may call it), the coming together, the different comparison, and the application of these harmonies: as of three, four, five, or more, the effects have been marvelous; and the like could still be found and produced, if we take some proportional account of our time and our situation, compared with the state of the things then, in which and by which those wonderful effects were produced. Democritus and Theophrastus said that by music the griefs and diseases of mind and body could be cured or brought on. And we find in the record that Terpander, Arion, Ismenias, Orpheus, Amphion, David, Pythagoras, Empedocles, Asclepiades, and Timotheus, by harmonic consonance, have done and brought about things more than marvelous to hear of. I will say no more about them here. I am sure that ordinary music, as commonly used, is found by musicians and listeners to be so useful and pleasant that, if I were to say and argue only this much—that it ought to be used differently from the way it is—I would find more critics than I could find people who understood what I meant. So in things that are evident and better known than I can express, and that are accepted and approved (as I wish some other things had the same luck), I will not extend my text any further, but will go on with my purpose.\n\n### Cosmography\n\nHere I will give you a brief account of cosmography. Cosmography is the whole and complete description of both the heavenly and the elemental part of the world, and how they correspond to each other, with the comparison between them that is needed. This art requires astronomy, geography, hydrography, and music. So it is no small art, nor as simple as it is casually taken to be in common practice. It matches heaven and earth in one frame, and fits their corresponding parts together properly, so that the heavenly globe can be drawn, in practice, on the geographic and hydrographic globe. There we can consider the equinoctial circle, the ecliptic line, the colures, the poles, the stars in their true longitudes, latitudes, declinations, and altitudes, as well as the climes and parallels. And with an attached horizon and the rotation of the earthly globe (as the heaven is carried around by the primum mobile, the outermost moving sphere, in 24 equal hours), we can learn the risings and settings of the stars (as prescribed by Virgil in his Georgics; by Hesiod; by Hippocrates in his medical sphere, to Perdicca, king of the Macedonians; by Diocles, to King Antigonus; and by other famous philosophers): a thing necessary for working the earth properly, for navigation, and for changes in the human body, whether healthy, sick, wounded, or bruised. By the rotation, too, or the movement of the cosmographic globe, the rising and setting of the sun, the lengths of days and nights, and the hours and times (both night and day) are known, along with very many other pleasant and necessary uses. Of these, some are known; better ones remain for those who know how to use them:\n\nNote this.\n\nwho can make a wonderful bonfire from a spark of true fire, by applying the right material properly.\n\n### Astrology\n\nHere I make astrology an art separate from astronomy: not by any new device, but by good reason and authority. For astrology is a mathematical art that reasonably demonstrates the workings and effects of the natural beams of light and the secret influence of the stars and planets, in every element and in every elemental body, at all times, in any given horizon. This art is furnished with many other great arts and experiences: with perfect perspective, astronomy, cosmography, natural philosophy of the 4 elements, the art of graduation, and some good understanding of music; and moreover with another great art that follows later, though I set this one before it here, for reasons that move me. You can see that there is enough material to make this rare and secret art from, and that it is hard enough to shape into a syllogistic conclusion. Yet the many and unceasing labors of the most ancient and wise philosophers, to attain this art, and to confirm it by examples of its effects, have left us enough proof and witness: and we can also see every day that the human body, and all other elemental bodies, are altered, disposed, ordered, pleased, and displeased by the influential working of the sun, the moon, and the other stars and planets. So says Aristotle, in the first of his Meteorological books, in the second chapter: this elemental world is of necessity almost next adjoining the heavenly motions, so that from there all its force may be governed. For that is to be taken as the first cause of all things, from which the beginning of motion is. And again, in the tenth chapter: We must therefore also take up the principles of these things, and the causes of all things alike. The principle, then, as the mover, the chief and first of all, is that circle in which the sun's motion is plain, etc. And so on. His Meteorological books are full of arguments and effective demonstrations of the force, working, and power of the heavenly bodies in and upon the four elements, and upon other bodies made of them, whether perfectly or imperfectly composed. And in his second book, On Generation and Corruption, in the tenth chapter: the uppermost motion is not the cause of generation and corruption, but the motion of the zodiac is, because that motion is both continuous and caused by two movements. And in his second book, and the second chapter of his Physics: For man, says he, and the sun, are the cause of man's generation. Very many authorities can be brought for that conclusion, of 1000, 2000, even 3000 years of antiquity, from great philosophers, expert, wise, and godly men: a conclusion that we can discern and perceive daily and hourly by sense and reason. All beasts feel it and simply show it in their actions and feelings, outward and inward. All plants, herbs, trees, flowers, and fruits. And finally, the elements, and all things made of the elements, give testimony, as Aristotle said, that their whole dispositions, forces, and natural motions depend on the activity of the heavenly motions and influences. So, besides the specific order and form due to every seed, and besides the nature proper to the individual matrix of the thing produced, what the heavenly impression will be is what the perfect and careful astrologer has to conclude: not only (by apotelesms) τὸ ὁτὶ, but by natural and mathematical demonstration τὸ διότι. What sciences are needed for this, without exception, I have partly warned you of here; and in my Propædeumes, besides other matter disclosed there, I have set out the whole method mathematically: something that in our age has not been so carefully handled by anyone I ever saw or heard of. I was,\n\n(Years 1548 and 1549, in Louvain.)\n\n(for 21 years ago) by certain earnest disputations with the learned Gerardus Mercator, Antonius Gogaua, and others, so provoked to this; and by my constant and unbreakable zeal for the truth, in observations of heavenly influences down to the minute of time, then so diligent; and chiefly by the supernatural influence from the Star of Jacob, so directed: that any modest and sober student who carefully and diligently seeks the truth will both find and confess the truth in my words here, and will also become a reasonable reformer of three sorts of people who go greatly wrong about these influential workings.\n\nThe one sort is light believers, the other light despisers, and\n\nthe third light practitioners. The first and most common sort think that the heavens and stars are answerable to any of their doubts or desires:\n\n(1.)\n\nThat is not so, and in fact they reach much too far. The second sort think that no influential force from the heavenly bodies has any sway in generation\n\n(2.)\n\nand corruption in this elemental world. And to the sun, the moon, and the stars—so many, so pure, so bright, so wonderfully big, so far away, so varied in their motions, so constant in their periods, and so on—they assign a slight, simple office or two, and so allow them, according to their capacities, as much influential force and power as the sign of the Sun, Moon, and Seven Stars hung up as signs in London to tell the houses apart, and other such crude helps in our worldly affairs. And they do not understand, or will not understand, the other workings and forces of the heavenly sun, moon, and stars: not as much as the sailor or the farmer; no, not as much as the elephant, or the Cynocephalus, or the porcupine. Nor will they allow these perfect and uncorrupted mighty bodies as much forceful radiation and strength as they see in a small piece of magnet stone, which shows its working at a great distance. And perhaps they think that the sea and rivers, like the Thames, are some living thing, and so ebb and flow, and run in and out, of themselves, at their own fantasies. God help, God help. Surely these men fall short, and are either too dull, or willfully blind, or perhaps too malicious. The third sort is the common and ordinary astrologer or practitioner, who, though not duly, skillfully, and perfectly\n\n(3.)\n\nfurnished, yet goes wrong, either for vain glory or gain, or like a simple fool and blind Bayard, in both matter and manner: to the discredit of the wary and modest astrologer, and to the robbing of those most noble bodily creatures of their natural force: they being most mighty, most beneficial to all elemental generation, corruption, and their appurtenances, and most harmonious in their monarchy. For these things, known and used with due restraint, we might highly and continually glorify God, with the princely prophet, saying: \"The heavens declare the glory of God: who made the heavens in his wisdom: who made the sun to have dominion over the day: the moon and stars to have dominion over the night: so that day to day utters speech, and night to night declares knowledge. Praise him, all you stars, and light. Amen.\"\n\n### Statics, 1\n\nIn order, now follows, of Statics: some words on what we mean by that name, and what benefit depends on such an art. Statics is a mathematical art. It shows the causes of heaviness and lightness in all things, and the motions and properties that belong to heaviness and lightness. And because the balance, or scales, is the chief instrument of the senses, and experience of these demonstrations can be had through it, we call this art statics: that is, experiments with the balance. If only people knew what profit, in every way, this art could bring to a capable examiner and a diligent practitioner.\n\n\"You alone know all things exactly (O God), who have made weight and balance your judgment, who have created all things in number, weight, and measure, and have weighed the mountains and hills in a balance, who have weighed both heaven and earth in your hand. We are therefore warned by your sacred word to consider your creatures, and by that consideration to gain a glimpse, as it were, or a shadow of understanding, that your wisdom, power, and goodness are infinite and unspeakable, declared in your creatures. And being further warned by your merciful goodness that three principal means were used in the creation of all your creatures, namely number, weight, and measure; and since the two arts of number and measure, ancient, famous, and most necessary for human uses, are already sufficiently known and available; we ask you, through your accustomed goodness, that this third key may come to the full and sufficient knowledge of such servants of yours as would gladly find in your workmanship your true designs (made purposely for that use), by which we should glorify your name and show your wondrous wisdom and goodness to those who are weak in faith. Amen.\"\n\nDo not be surprised at this outburst, dear friend, you gentle and zealous student. Another day you may see what moved me to it. For now I will give you some basis, and also some show, of the benefits that come from this art. And because this art is rare, my words and practices might be too obscure, unless some light were held before the matter. That will best be done by giving you a few main conclusions out of Archimedes' demonstrations, as follows.\n\n1.\n\nThe surface of any liquid, standing by itself and at rest, is spherical. Its center is the same as the center of the Earth.\n\n2.\n\nIf solid bodies of the same size, or volume, as any liquid, and with the same weight, are lowered into that liquid, they will settle downward so that no part of them lies above the surface of the liquid, and yet they will not sink all the way down, or drown.\n\n3.\n\nIf any solid body lighter than a liquid is lowered into that liquid, it will settle down into it so far that the amount of liquid equal in volume to the part of the body settled into it weighs as much as the whole solid body.\n\n4.\n\nAny solid body lighter than a liquid, pushed down into that liquid, will move upward with a force equal to how much heavier the liquid of the same volume as the whole body is than that body itself.\n\n5.\n\nAny solid body heavier than a liquid, lowered into that liquid, will sink all the way down. In that liquid it will be lighter by exactly the weight of the liquid that has the same size, or volume, as the solid body.\n\n6.\n\n(John Dee. A sphere can be cut to any assigned proportion by this proposition, mechanically, by mixing a liquid to a certain weight compared with the weight of the sphere floating in it.)\n\nIf any solid body lighter than a liquid is lowered into that liquid, the weight of that body is to the weight of the liquid (which is equal in volume to the whole body) in the same proportion as the part of the body that has settled down is to the whole body.\n\nFrom these truths, great errors can be set right in people's opinions about the natural motion of light and heavy things. Almost everyone in natural philosophy allows these errors, trusting too much to authority and to false assumptions. For example, that of two bodies, the heavier moves downward faster than the lighter.\n\n(A common error, noted.)\n\nThis error was not first pointed out by me, but by a man named John Baptist de Benedictis. His chief proposition is this, and it seems like a paradox:\n\nIf there are two bodies of one shape and one kind, equal in volume or unequal,\n\n(A paradox.)\n\nthey move through equal spaces in equal times, whether both are moving in air, or both in water, or in any one medium.\n\nHereupon, in the art of gunnery,\n\ncertain good discussions, otherwise, may receive great improvement and advancement.\n\n(The wonderful uses of these propositions.)\n\nAlso, in the intended purpose, allow something for nature's imperfection, which does not match the precision of a demonstration. Moreover, by the propositions above, wisely used, air, water, earth, and fire can be measured closely: how light or heavy they are naturally, in their assigned parts, or as a whole. Then, turning your practice to elemental things, you can work out the proportions of the elements in compound things. Then the proportions of the humors in a person: their weights, and the weight of their bones and flesh, and so on. Then, by weight, you can consider a person's strength in any way, whole or in part. Then you can consider, with pleasure, how differently ships draw water, in the sea and in fresh water, and how to weigh up anything sunk in the sea or in fresh water, and so on. And, to lift your head up high, by weight you can measure the diameters of the Sun and Moon as precisely as by any other instrument, and so on. Friend, please weigh these things with the fair balance of reason, and you will find wonders upon wonders. You will value one drop of truth, even in natural philosophy, more than whole libraries of opinions that are undemonstrated, or that do not answer to nature's law and your experience. Leaving these things there, I will give you two or three simple practices that serve a great purpose, and so finish my notes on statics. In mathematical matters, with the help of the mechanic, we will see here the value of weight.\n\n(A practical exercise in statics, to find the proportion between the cube and the sphere.)\n\nMake a cube of one uniform, evenly heavy material. From that same material, make a sphere, or globe, with a diameter exactly equal to the edge of the cube.\n\nYour material may be wood, copper, tin, lead, silver, and so on (being, as I said, of the same nature and condition, and of even weight throughout).\n\nAnd with an assay balance you may have prepared a great number of the smallest weights, weights that balance can tell apart or test. In that way you can go on to make up a complete set, a company and number of weights, up to six, eight, or twelve pounds in weight, every one of them most carefully tested. And know the amount of each one in your smallest weighable weight.\n\n[Those who cannot get weights of precision can make a number of weights fairly close to precision from uniform, well-sifted sand, by always halving the sand; in time they will come to a smallest common weight. On that I leave the further matter to the judgment of those whom need compels.]\n\nThe Venetian way of reckoning weight may seem precise enough: by eight steps in a descending series, halving from a single grain.\n\n(John Dee. For so you have 256 parts of a grain.)\n\nYour cube, sphere, suitable balance, and suitable weights being ready, get to work. First, weigh your cube. Note the number of weights. Then weigh your sphere, and note the number of weights again. If you now find the weight of the cube to be to the weight of the sphere as 21 is to 11, then you see how the mechanic and the experimenter, without geometry and demonstration, are taught in effect the proportion of the cube to the sphere, as I have demonstrated at the end of the twelfth book of Euclid. Try often with the same cube and sphere. Then change the sphere and the cube to another material, or another size, until you have made a complete general test of it. You may well come closer to the true terms of the proportion.\n\nWhen you have found this one certain drop of natural truth, go on to draw conclusions, and properly test the matters that depend on it. For it is well demonstrated that a cylinder whose height and the diameter of its base equal the diameter of the sphere is sesquialter to that sphere, that is, as 3 to 2. To the weight of the sphere, add half as much again, and you have the weight of that cylinder. That cylinder is also contained in our earlier cube, so that the base of that cylinder is a circle drawn inside the square that is the base of our cube. But the cube and the cylinder, being the same height, have their bases in the same proportion as they bear to each other in bulk, or solidity. But before that, we have two numbers expressing their bulk, solidity, and quantity by weight; therefore,\n\n(The proportion of the square to the inscribed circle.)\n\nwe have the proportion of the square to the circle inscribed in that same square. And so we come to a sensible and practical knowledge of Archimedes' great secret, sought and found by him through great labor of mind. Therefore, to any given circle, you can give an equal square:\n\n(Squaring the circle mechanically.)\n\nas I have taught in my annotation on the first proposition of the twelfth book. Likewise, to any given square, you can give an equal circle:\n\n(To any given square, to give an equal circle.)\n\nIf you draw a circle that stands in the same proportion to your inscribed circle as the square does to that same circle, you can do this by my annotations on the second proposition of the twelfth book of Euclid, in my third problem there. Your diligence may bring you to a proportion of the square to the inscribed circle nearer the truth than 14 to 11. And notice that you can begin with the circle and the square, and so work out what proportion the sphere and the cube bear, just as you have now come from the sphere to the circle. For, if you make a square figure from silver, or gold, or brass sheets or plates (drawn through one hole, as is the custom), and weigh it; then describe the inscribed circle on it, and cut off and file away, precisely down to the circle, the surplus of the square; then, weighing your circle, you will see whether the weight of the square is to your circle as 14 to 11. As I have noted at the beginning of Euclid's twelfth book, and so on. After that, go to my last proposition, on the last of the twelfth, and help yourself through to the end. And note this in passing.\n\n(Note: squaring the circle without knowing the proportion between the circumference and the diameter.)\n\nThat we can square the circle without knowing the proportion of the circumference to the diameter, as you have just seen. And I can demonstrate it in other ways too. So many people have burdened themselves needlessly by working first on that point, which did not need to come first, and is also very intricate. And you can easily, in several different ways, come to know the circumference, once the circle's size is known first. I leave that to your consideration, and hurry on to dispatch another masterly problem, and to bring it nearer to your knowledge, and easier for you to handle, than the world has had it before this day, so far as I can tell. And that is, a mechanical doubling of the cube, and so on. It can be done like this:\n\n(To double the cube readily, by mechanical art, depending on mathematical demonstration.)\n\nMake a square upright pyramid, or a cone, of copper plates or tin plates, perfectly shaped hollow on the inside. In this, take great care to approach, as near as possible, the mathematical perfection of those figures. Let their bases be open all around; everywhere else, as closed and exact as possible. From the vertex to the circumference of the base of the cone, and to the sides of the base of the pyramid:\n\n(John Dee. The 4 sides of this pyramid must be 4 matching equal isosceles triangles.)\n\nlet 4 straight lines be drawn on the inside of the cone and the pyramid, so that where they fall on the perimeters of the bases they make equal angles on both sides of themselves with those perimeters. Divide these 4 lines (in the pyramid, and as many in the cone): one into 12 equal parts, another into 24, another into 60, and another into 100, counting up from the vertex. Or use other numbers of divisions, as experience teaches you.\n\n(John Dee. In all work with this pyramid or cone, keep their positions alike in every point and condition while you are doing one job. Otherwise you will go wrong.)\n\n### Statics, 2\n\nThen set your cone or pyramid, with the vertex downward, perpendicular to the base. (It does not matter if it is the other way around; it hinders nothing.) So let them be kept as steady as possible. Now, if there is a cube that you want to double: make a small cube of copper, silver, lead, tin, wood, stone, or bone. Or else make a hollow cube, or a cubic box, of copper, silver, tin, or wood, etc. You may size these in respect of your pyramid or cone so that the pyramid or cone can hold 3 or 4 times their weight in water, at least, whatever material they are made of. Do not let your solid angle at the vertex be too sharp, so that the water can easily reach the very vertex of your hollow cone or pyramid. Put one of your solid cubes in a suitable balance and take its weight exactly in water. Pour that water (without loss) quietly into the hollow pyramid or cone. Mark in your lines what numbers the water cuts. Take the weight of the same cube again, in the same kind of water you had before:\n\nJohn Dee. Consider well when you must put your waters together, and when you must empty your first water out of your pyramid or cone. Otherwise you will err.\n\nPut that also into the pyramid or cone where you put the first. Mark again in what number or place of the lines the water cuts them. There are two ways to reach your aim: that is, either by numbers or by lines. By numbers: divide the side of your fundamental cube into as many equal parts as it can conveniently hold, with ease and precision in the division. For as the number of your first and lesser line (in your hollow pyramid or cone) is to the second and greater one, both counted from the vertex, so the number of the side of your fundamental cube is to the number belonging to the radical side of the cube that is double your fundamental cube. Multiply that number cube-wise and it will soon show whether it is double the cube number of your fundamental cube or not. By lines, thus: as your lesser and first line (in your hollow pyramid or cone) is to the second and greater, so let the radical side of your fundamental cube be to a fourth proportional line, by the 12th proposition of the sixth book of Euclid. That fourth line will be the cube root, or the radical side, of the cube that is double your fundamental cube, which is what we wanted.\n\nGod be thanked for this invention and the fruit that follows.\n\nFor this I may say with joy, \"I have found it, I have found it, I have found it,\" thanking the holy and glorious Trinity, having greater cause for it than\n\nVitruvius, book 9, chapter 3.\n\nArchimedes had for finding the fraud used in the king's crown of gold, as anyone can easily judge by the difference of the results that follow from the one and the other. Where I spoke before of a hollow cubic box: the same use is made of it, and without weighing. Thus: fill it with water precisely full, and pour that water into your pyramid or cone. And here note where the lines are cut in your pyramid or cone. Again, fill your box as you did before. Put that water also to the first. Mark the second cutting of your lines. Now, as you proceeded before, so you must proceed here.\n\nNote.\n\nAnd if the cube you are to double is ever so great, you have in this way the proportion (on a small scale) between your two little cubes. And then the side of that great cube (to be doubled), being the third, will have a fourth found that is proportional to it, by the 12th of the sixth of Euclid.\n\nNote concerning the spherical surface of the water.\n\nNote that all this while I have not forgotten my first proposition in statics, restated here: that the surface of the water is spherical. In this use your judgment: to the first line add a small hairbreadth more, and to the second add half a hairbreadth more to its length. For you will easily see that the difference can be no greater in any pyramid or cone you are to handle. And you can try it this way, for finding how much the water swells above the level.\n\n\"Square the semidiameter from the center of the earth to the surface of your first water. Then square half the subtendent of that watery surface (the subtendent must have its measure divided into equal parts, the same as the semidiameter from the earth to your watery surface). Subtract that square from the first. Of the remainder, take the square root. Subtract that root from your first semidiameter from the earth to your watery surface: what remains is the height of the water in the middle, above the level.\" You will find this to be something too small to notice. And even if it were greatly noticeable,\n\nNote.\n\nYet, by help of my sixth theorem on the last proposition of Euclid's twelfth book, noted, you may reduce everything to a true level. But you must take further care against accidental causes of the water swelling: for example, if you dampen your hollow pyramid or cone beforehand a little with a moist sponge, you will prevent an accidental cause of swelling, etc. Experience will teach you in abundance, with great ease, pleasure, and benefit.\n\nThus you can double the cube mechanically, treble it, and so on, in any proportion.\n\nNote this abridgment of doubling the cube, etc.\n\nNow I will shorten your effort, cost, and care in this. Without any preparation of your fundamental cubes, you may work this conclusion in the same way. For that was rather a kind of experimental demonstration than the shortest way, and it all depended on one mathematical demonstration: \"Take water (as much as will conveniently serve your purpose, as I warned earlier about the size of your fundamental cubes). Weigh it precisely. Put that water into your pyramid or cone. Then take again the same kind of water, the same weight you had before, and put that also into your pyramid or cone. For each time, your marking of the lines, how the water cuts them, will give you the proportion between the radical sides of any two cubes, one of which is double the other, working as I taught you before:\n\nNote.\n\nExcept that, for your fundamental cube's radical side, you may take any straight line you like here.\"\n\nYet a step further with our drop of natural truth:\n\nTo give cubes one to another in any proportion, rational or irrational.\n\nYou may now give cubes one to another in any proportion given, rational or irrational, in this way. Make a hollow parallelipipedon of copper or tin, with one base missing, or open, as in our cubic box. From the bottom of that parallelipipedon raise up many perpendiculars on each of its four sides. Now if any proportion is assigned to you in straight lines, cut one of your perpendiculars (or a line equal to it, or shorter than it) likewise, by the 10th proposition of the sixth book of Euclid. Set those two parts on two separate lines of those perpendiculars (or you may set them both on one line), making their beginnings at the base, so that their lengths extend upward.\n\nNow set your hollow parallelipipedon upright, perpendicularly, and steady. Pour in water neatly to the height of your shorter line. Pour that water into the hollow pyramid or cone. Mark the place it rises to. Set your hollow parallelipipedon again. Pour water into it exactly to the height of the second line.\n\nEmpty out the first.\n\nPour that water properly into the hollow pyramid or cone. Mark now again where the water cuts the same line you marked before. For there, as the first marked line is to the second, so the two radical sides will be to each other, of any two cubes that in their solidity have the same proportion that was first assigned, whether rational or irrational.\n\nThus, in various ways, you may equip yourself with such strange and profitable matter, which has long been wished for. And though it is done naturally and mechanically, it still has a good mathematical demonstration.\n\nThe demonstrations of this doubling of the cube, and of the rest.\n\nWhich is this: you always have two similar pyramids, or two similar cones, in the proportions assigned; and similar pyramids or cones stand to one another in the proportion of their homologous sides (or lines) tripled. Therefore, if to the first and second lines found in your hollow pyramid or cone you join a third and a fourth in continued proportion, that fourth line will be to the first as the greater pyramid or cone is to the lesser, by the 33rd of the eleventh of Euclid. If pyramid to pyramid, or cone to cone, is double,\n\nJohn Dee. By this, help yourself become a precise practitioner. And so consider how the convexity of the water does not hinder you at all, in any way you can notice.\n\nThen line to line will also be double, etc. But as our first line is to the second, so is the radical side of our fundamental cube to the radical side of the cube to be made or to be doubled. And therefore, joining a third and a fourth line in continued proportion to those two as well will give a fourth line in that proportion to the first as our fourth pyramidal or conic line was to its first. But that was double, or treble, etc., as the pyramids or cones were to one another (as we have proved). Therefore this fourth will also be double or treble to the first, as the pyramids or cones were to one another. But the cube we make is generated from the second of the four proportional lines:\n\nBy the 33rd of the eleventh book of Euclid.\n\nTherefore, as the fourth line is to the first, so that cube is to the first cube. And we have proved the fourth line to be to the first as the pyramid or cone is to the pyramid or cone. Therefore the cube is to the cube as pyramid is to pyramid, or cone is to cone.\n\nJohn Dee. And your diligence in practice can perform it in this way (by the weight of water). Therefore you are now able to give a good reason for all you do.\n\nBut we suppose pyramid to pyramid, or cone to cone, to be double or treble, etc. Therefore cube is to cube double, or treble, etc. Which was to be demonstrated. And for the parallelipipedon it is evident that the solid parallelipipedons of water stand to one another as their heights do, since they have one base. Therefore the pyramids or cones made from those water parallelipipedons stand to one another as the lines stand to one another between which our proportion was assigned. But the cubes made from lines, in the proportion of the pyramidal or conic homologous lines, stand to one another as the pyramids or cones stand to one another (as we proved before). Therefore the cubes made will stand to one another as the assigned lines stand to one another. Which was to be demonstrated. Note.\n\nNote this corollary.\n\nThis demonstration of mine is more general than only in a square pyramid or cone: consider it well. Thus I have been very long-winded, both mathematically and mechanically; yet, I trust, nothing tedious to those who are well disposed toward these things. And in truth I am forced (to avoid prolixity) to omit several such things, easy to practice: they would be a great treasure to the mathematician, and no small gain to the mechanic.\n\nThe great benefits that follow from these new inventions.\n\nNow, between two given lines, you may find two mean proportionals in continued proportion, by the hollow parallelipipedon and the hollow pyramid or cone. Now, any rectangular parallelipipedon being given, three straight lines may be found, proportional in any proportion assigned, out of which a parallelipipedon equal to the given parallelipipedon will be produced. Of this I noted something on the 36th proposition of the 11th book of Euclid. Now, all those things that Vitruvius in his Architecture specifies as able to be done by doubling the cube, or by finding two mean proportional lines between two given lines, can easily be performed. Now, that problem which I noted to you at the end of my addition on the 34th of the 11th book of Euclid is proved possible. Now, any regular body may be transformed into another, etc. Now, any regular body, any sphere, yes, any mixed solid, and what is more, irregular solids, may be made (in any proportion assigned) like the body first given. Thus, from a little figure (as the Dutch painters call it) in the same symmetry, a giant can be made, and that with any gesture used by the little figure, and the other way around. Now, from any mold or model of a ship you may make one, of the same mold (in any assigned proportion), bigger or smaller.\n\nNow, you may, of any gun, or small piece of ordnance, make another with the same symmetry (in all points) as big and as small as you want. Mark that and think on it. You may apply this infinitely, something so long sought for and now so easily worked out, and at the same time so willingly and freely shared with those who deal faithfully with virtuous studies.\n\nSuch is the fruit of the mathematical sciences and arts.\n\n### Statics, 3\n\nThus the mathematical mind can deal speculatively in its own art, and by good means climb above the clouds and stars. And thirdly, it can, in order, descend, to shape natural things for wonderful uses, and when it wishes return home to its own center, and there prepare more means to ascend or descend by, all to the glory of God and our honest delight on earth.\n\nAlthough the printer has waited for this preface a day or two, I could not take my pen from the paper before I had given you encouraging word and brief instructions on some of the benefits that can be reaped by statics. In the rest I will therefore be as brief as possible, and at the same time describe them somewhat to match. And you will see this in what comes next in order. For although it is so ample and wonderful that one could find fruitful matter in it to speak of for a whole year, and in practice is a treasure without end, yet I will glance over it with very few words.\n\n### Anthropography\n\nThis I call Anthropography. Which is an art restored, and which I put forward for your service. Please think of it as one of the chief points of human knowledge. Although it has only now first been confirmed with this new name, the matter has been considered from the beginning by all perfect philosophers. Anthropography is the description of the number, measure, weight, figure, situation, and color of every diverse thing contained in the perfect body of man: with certain knowledge of the symmetry, figure, weight, characterization, and proper local motion of any assigned part of the said body, and of the numbers belonging to that part. This is the part of the definition that fits this place: enough to show the particularity and excellence of the art, and why it is ascribed here to the mathematical sciences. If the description of the heavenly part of the world has its own art called astronomy: if the description of the earthly globe has its own art called geography: if the matching of both has its own art called cosmography, which is the description of the whole and universal frame of the world: why should not the description of\n\nMan is the lesser world.\n\nhim who is the lesser world, and from the beginning called Microcosmus (that is, the lesser world), and for whose sake and service all other bodily creatures were created, who also shares with spirits and angels, and is made in the image and likeness of God, have its own art, and be called the art of arts, rather than either lack a name or have too base and improper a name? You must borrow or claim for yourself particular parts of this from various professions, and proceed further, as God, nature, reason, and experience shall instruct you. The anatomists will restore some part to you: the physiognomists some: the chiromantists some. The metaposcopists some: the excellent Albert Durer a good part: the art of perspective will help forward the eye somewhat: Pythagoras, Hippocrates, Plato, Galen, Meletius, and many others (in certain things) will contribute. And further, the heavens, the earth, and all other creatures will each show and offer their harmonious service to fill up whatever is lacking here: and concluding with your own experience, you may methodically register the whole for posterity: by which good proof will be had of our harmonious and\n\nMicrocosmus.\n\nMicrocosmic constitution.\n\nThe outward image and view of this, for the art of drawing and painting, for sculpture, and for architecture (for a church, house, fort, or ship), is most necessary and profitable, since it is the chief base and foundation of them.\n\nBook 3, chapter 1.\n\nLook in Vitruvius and see whether I am dealing sincerely for your benefit. Look in Albert Durer, On the Symmetry of the Human Body. Look in the 27th and 28th chapters of the second book, On Occult Philosophy. Consider the ark of Noah. And from that, wade further. Remember the Delphic oracle \"know yourself\", pronounced so long ago, repeated by so many a philosopher, and attempted by the wisest: and then you will see how long ago you have been called to the school where this art could be learned. Well, I am not at all afraid of the disdain of some who think the sciences and arts are only seven. Perhaps such people may, with enough ignorance and shame, fall short of those seven as well: and yet, nevertheless, they cannot set a certain number of arts, nor set in each certain impassable bounds against God, nature, and human industry. New arts rise up daily, and there was no order taken that,\n\nAll arts should be made known to the world in one age, or in one land, or by one man.\n\nLet us embrace the gifts of God, and the ways to wisdom, continually bestowed from above in this time of grace on those who will receive them thankfully: and to the good, all things will work together for good.\n\n### Trochilike\n\nTrochilike is the mathematical art that shows the properties of all circular motions, simple and compound. Because its fruit, as people commonly receive it, is in wheels, it takes the name Trochilike: what you might call wheel art. With this art you can set a wheel to make one full turn in any given time. You can also set two wheels so that their turns in the same time, or in equal times, stand in any proportion you choose. With wheels you can describe a straight line. You can also draw a spiral line on a flat plane, conic section lines, and other irregular lines, as you please. These and similar results are the main conclusions of this art, and they bring forward many pleasant and profitable mechanical works:\n\n(Saw mills.)\n\nSuch as mills that saw great and very long deal boards, with no one standing by. I have seen such in Germany, and in the city of Prague, in the kingdom of Bohemia: coining mills, hand mills for grinding grain, and all manner of mills and wheelwork, driven by wind, smoke, water, weight, springs, people, or animals. Take Agricola's On the Nature of Metals in hand and you will see, in every mine, how great the need for wheelwork is. With wheels, strange and incredible works are done, as will appear in other arts later. A wonderful example of still greater possibility, and of present use, appeared in my time in a certain instrument. Its inventor and craftsman had sold it earlier for twenty talents of gold, and it had then suffered injury and damage by misfortune. Ianellus of Cremona repaired it and presented it to Emperor Charles V. Hieronymus Cardanus can be my witness that in it there was one wheel which turned at such a rate that in 7000 years only it would complete its own period. That is almost incredible. How much further it goes, I keep within my bounds: very many men, still alive, can tell.\n\n### Helicosophy\n\nHelicosophy is a near sister to Trochilike. It is the mathematical art that shows how to design all spiral lines on a flat plane, on a cylinder, a cone, a sphere, a conoid, and a spheroid, with the properties that belong to them. Its use in architecture and in various instruments and engines is essential. In many things the screw does the work that could not otherwise be done. With its help,\n\n(Atheneus, Book 5, chapter 8.)\n\nIt is recorded that, when all the power of the city of Syracuse could not move a certain ship that lay aground, the mighty Archimedes set his screw engine to work and made King Hiero move it himself, at ease, as he wished.\n\n(Proclus, page 18.)\n\nThe king was astonished at this: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει λέγοντι πιϛευτεόν. \"From this day forward,\" said the king, \"Archimedes ought to be believed in whatever he says.\"\n\n### Pneumatithmie\n\nPneumatithmie shows, by closed hollow geometric figures, regular and irregular, the strange properties of water, air, smoke, and fire, in motion or at rest, within their own continuity and as they are joined to the elements next to them. This art is very useful to the natural philosopher, because it proves that a vacuum, or empty space, does not exist in the world. It also shows that all nature dislikes it so much that, contrary to ordinary law, the elements will move or stand still. Water rises rather than leave a space between itself and the air, more than that quantity of air naturally requires or can fill. Again, water hangs and does not fall rather than fall and leave emptiness behind it. Fire and air act the same way: they will descend rather than let their continuity be broken or the element next to them be forced away. And just as they will not stretch into discontinuity, so they will not, and no human force can, be pressed or cramped into a space too small and unfit for their bodily substance. They will use great force and violence to keep their natural right and freedom.\n\n(Going to the bottom of the sea without danger.)\n\nOn this basis, two or three men together can keep air under a great cauldron, press it down in order, and descend to the bottom of the sea without harm, staying there for a time, and so on. Note how the thicker element, water, gives way to the thinner, the air, and in turn takes force from the thinner, and so on. Pumps and all kinds of bellows are founded on this art, along with many other strange devices, such as hydraulica, organs driven by water, and so on. Fine works on this subject, commonly called pneumatics, exist in both Greek and Latin. Old and learned scholars call it the Science of the Full and the Empty.\n\n### Menadrie\n\nMenadrie is a mathematical art. It shows how power and force can be multiplied beyond nature's simple power, so that you can direct, lift, pull toward you, and push or throw any power, weight, or force, simple or multiplied and fixed in amount, that nature would not otherwise let you direct or move.\n\nOther arts help this art a great deal: perspective in some points, statics in others, Trochilike in others, and Helicosophy and Pneumatithmie in others.\n\nBy this art, every crane, gibbet, and engine for lifting something up, or for forcing it in any way, is managed, and the exact cause of its force is known. One example is the force one man has with a Dutch wagon rack, enough to set upright again a heavy loaded wagon that has been overturned.\n\nThe force of the crossbow rack is clearly demonstrated here.\n\nIt also gives the reason why one man can lift with a lever what six men, with only their hands, could not do as easily.\n\nBy this art, in the ordinary cranes in London, where the power is to crane up a weight of 2000 pounds, adding two more wheels in good order, the art concludes that a weight of 200000 pounds can be craned up, etc.\n\nArchimedes knew this art so well that he alone, with his devices and engines, twice or three times routed and defeated the whole army and host of the Romans besieging Syracuse,\n\n(Plutarchus in Marco Marcello.)\n\nMarcus Marcellus the Consul was their general captain.\n\n(Synesius, in his Letters.)\n\nWith his engines he hurled such huge stones out of the city and among them: so many, with such force, and from so far away.\n\n(Polybius.)\n\n(Plinius.)\n\n(Quintilianus.)\n\n(T. Liuius.)\n\nAnd at sea likewise: though their ships could come right up to the walls of Syracuse, he completely routed the Roman fleet. Partly by hurling huge stones:\n\n(Athenæus.)\n\nPartly by casting pikes 18 feet long, made like shafts, which he drove almost a quarter of a mile. Partly by catching hold of their ships, hoisting them up above the water, and suddenly letting them drop back into the sea:\n\n(Galenus.)\n\n(Anthemius.)\n\nPartly by his burning glasses, with which he set their other ships on fire at a distance. And partly by his other schemes, devices, and engines. He conducted himself so bravely that all the force, courage, and strategy of the Romans, for a long time, could accomplish nothing toward taking Syracuse. So the Romans called Archimedes Briareus and Centimanus. Zonaras mentions a certain Proclus, who had understood the art of Menadrie so well, and had invented so well on his own, that with his burning glasses,\n\n(Burning Glasses.)\n\nplaced on the walls of Byzantium, he multiplied the heat of the sun so, and directed its beams against his enemy's fleet with such force and so suddenly, like lightning, that he burned and destroyed both man and ship. Dion also specifies Priscus, a geometrician in Byzantium, who invented and used sundry engines of force multiplied. This was the reason the Emperor Severus spared his life after he had taken Byzantium, because he honored the art, wit, and rare industry of Priscus. But the invention of guns was in no way inferior to the invention of these engines of force.\n\n(Gunnes.)\n\nGuns got the occasion and order of their first invention from an Englishman, though they were first carried out in another land and by other men. And whoever saw the record where the general cause and order of gunnery is first discussed would think that \"small things, slight and common, coming to wise men's consideration and industrious men's handling, may grow to be of force incredible.\"\n\n### Hypogeiodie\n\nHypogeiodie is a mathematical art. It shows how, under the spherical surface of the earth, at any depth, a set course can be laid out and traveled toward any given perpendicular line (whose distance from the perpendicular at the entrance, and its azimuth likewise with respect to that entrance, are known). It shows how any course marked out above the surface of the earth can be kept underground at any set depth, always running directly beneath the course marked on the ground above. And the other way round: given any course under the earth, straight or crooked, how to lay it out on the outer face or surface of the earth, so that perpendiculars drawn from the center of the earth to the spherical surface land exactly on the matching points of those two courses. This art demonstrates all of that, along with every other case and circumstance involved, and everything that belongs to them. The art is very rich in the variety of its conclusions, and it is useful in many ways to the commonwealth. I invented this art at the request of two gentlemen. They had a certain working underground (for gain), and their plots of ground joined above the work; and because of the twists, the different depths, and the heights of the way underground, they were in doubt, and at odds over whose ground the working then lay. Before this, I had published only the name: On the Subterranean Journey. The rest is at God's will. For laborers, miners, and diggers for metals, stone, and coal, and for secret passages underground from place to place (as this land has several), and for other purposes, anyone can easily see both the great fruit of this art and the great help geometry gives to it.\n\n### Hydragogie\n\nHydragogie shows that water can be led, by nature's law and with artificial help, from any source (a spring, standing water, or running water) to any other assigned place. This art has long been in use, much has been written about it, and very marvelous works have been accomplished in it, as can still be seen in Italy by the ruins of the aqueducts that remain. In other places there are rivers led through the mainland, navigable for many miles. In other places there are the marvelous forcings of water to rise. All of this shows the great skill required of anyone who would be perfect in this art, for every possible case of leading water. I need not speak of allowing fall for every hundred feet, or of vent holes (if the water's journey is long and hard), since many experienced men close at hand (around us) can testify well enough to the practice, even though they do not know the demonstration of why it is necessary. Nor, if they were led up and down and around the mountains from the source of the spring, and then, a place being given, they were asked how low or high that last place is compared with the source from which they had come by such a crooked, up-and-down route, perhaps they would not, or could not, answer that question readily or exactly. So geometry is necessary to Hydragogie. The various ways of forcing water to rise, by tympanum, kettle mills, screw, Ctesibian devices, or the like, can be seen in full in Vitruvius, Agricola, and others. And so it is also perfectly clear how the arts of Pneumatithmie, Helicosophy, statics, Trochilike, and Menadrie supply this art in theory, and serve the good of the commonwealth in practice.\n\n### Horometry\n\nHorometrie is a mathematical art. It shows how, at any appointed time, the exact customary reckoning of time can be known for any given place. These words are smooth, plain, easy English, but their meaning reaches further than you would readily imagine. Some part of this art was called Gnomonice in old times, and more recently Horologiographia, and in English it can be called dialing. The use is ancient, and the invention is older still. The use seems to have been in place at least more than two thousand three hundred years ago:\n\n(4. Reg. 20.)\n\nin King Achaz's dial, then, showing by the sun the divisions of time. This dialing can be done by sun, moon, and stars, and the exact time of day or night known. But drawing these dials demonstratively, of all kinds, requires good skill both in astronomy and in elemental, spherical, phenomenal, and conic geometry. Then, to use the foundations of the art on any regular surface that comes to hand in any place, and on it (in any possible fitting position) to set out every way in which the usual hours can be truly determined by the sun's shadow, will be found to be no easy painter's work. That means painting and setting out the sun's motion to the width of a hair. In this feat (in my youth) I invented a way: in any horizontal, mural, or equinoctial dial, etc., at all hours (when the sun shines), the sign and degree ascending can be known. This is very necessary for the rising of those fixed stars whose influence in the air is plainly very great. I will say no more about its use. But since people's affairs require knowing times and moments when neither sun, moon, nor star can be seen, mechanical skill first invented how the time and hours could be known by water running at a steady rate. The famous Ctesibius was the inventor of this, a man whom Vitruvius rightly praised to the skies. After that, hours were measured by running sand. Then by Trochilike with a weight, and more recently by Trochilike with a spring, without a weight. All of these need oversight and correction by the direction of the sun or the stars (at set times), according to the heavenly equinoctial motion, besides the unevenness of their own working. There remains among the philosophers (not speaking figuratively here),\n\n(A perpetual motion.)\n\na way more excellent, more useful, and more marvelous than all these: to have the motion of the Primouant (or the first equinoctial motion) imitated by nature and art. You will understand more of this later, through further search in weightier studies. And so it is time to finish this discussion of the divisions of time used in our public and private affairs. No man who knows how to spend his time would want to be without its benefit.\n\n### Zography\n\nDrawing is a mathematical art. It teaches and shows how the intersection of all the visual pyramids, made by any assigned plane (with the center, distance, and lights determined), can be represented by lines and the proper colors. This is a remarkable art, and it would take a whole volume to set out its nature and the benefits that follow from it. For perfection, the draftsman needs great skill in geometry, arithmetic, perspective, and anthropography, along with many other particular arts. For the most excellent painter (who is only the draftsman's own craftsman and imitator of what the senses show) has reached such perfection that the senses of men and beasts have judged painted things to be natural rather than made by art: alive and not dead. This mechanical draftsman (commonly called the painter) is marvelous in his skill, and seems to have a certain divine power: he can make absent friends a warm, present comfort; yes, and he can give dead friends a lasting, silent presence, not only with us but with our descendants for many ages. And going on: consider how, in winter, he can show you a lively view of summer's joy and riches, and in summer present the face of winter's dreary state and bareness. Cities, towns, forts, woods, armies, yes whole kingdoms, however far away or great, he can easily bring home with him (into anyone's judgment) as living patterns of the things he has listed. In one little house he can enclose (to the great pleasure of the viewers) a living picture of all visible creatures: those living on the earth or under it; those in the waters lying, creeping, sliding, or swimming; and any bird or fly flying in the air. Even set beside the stars, the sky, and the clouds, yes in the very look of light itself (that divine creature), he can match our eyes' judgment very closely. What a thing this is! Things that do not yet exist he can represent so well that, when they do exist, the picture seems (in a way) to have created them. To what craftsman is painting not a great pleasure and benefit? Which of them all would refuse the guidance and help of painting? The architect, the goldsmith, and the tapestry weaver all think a great deal of painting. Our vivid herbals, our portraits of birds, beasts, and fish, and our detailed anatomical drawings: how are they most perfectly made, or looked at with most pleasure? Is it not by painting alone? And if painting, through the painter's skill, is so useful and marvelous, what should be thought of drawing, the schoolmaster of painting and its chief governor? Though I do not mention sculpture in my table of mathematical arts, everyone can see how painting and sculpture are full sisters, and both very useful in a commonwealth. And excellent craftsmen have written large books in praise of sculpture as well as painting. I take as my witnesses Georgio Vasari, Pittore Aretino, Pomponius Gauricus, and others. To these two arts (and others) a certain odd art called Althalmasat owes much: more than the ordinary sculptor, carver, engraver, cutter, graver, founder, or painter, etc., know how useful their own art is.\n\n### Architecture\n\n(An objection.)\n\nMany people may think that architecture is not worthy, or not fit, to be counted among the mathematical arts. To them I think it good to give some account of my doing so. Not worthy, they will say, because it is only for the building of a house, palace, church, fort, or such coarse works. And you also defined the mathematical arts to be such as dealt with no material or corruptible thing, and also proceeded demonstratively in their faculty, by number or magnitude. First,\n\nThe answer.\n\nyou see that I count architecture here among those mathematical arts which are derived from the principals; and you know that such may deal with natural things and sensible matter. Of which, \"some come nearer to the simple and absolute mathematical speculation than others do.\"\n\n☞\n\nAnd though the architect plans, informs, and directs the craftsman in handiwork, and in the actual building of a house, castle, or palace, and is chief judge of the same, yet with himself (as chief master and architect) remains the demonstrative reason and cause of the craftsman's work: in line, plane, and solid; by geometric, arithmetic, optic, musical, astronomical, cosmographic means (and to be brief) by all the earlier derived mathematical arts, and other natural arts, able to be confirmed and established.\n\nIf this is so, then you may think that architecture has good and proper standing in this honorable company of derived mathematical arts. I will here ask the judgment of two most perfect architects: one is Vitruvius, the Roman, who wrote ten books of it to the Emperor Augustus (in whose days our heavenly Archemaster was born); and the other is Leo Baptista Albertus, a Florentine, who also published ten books of it.\n\nArchitecture, says Vitruvius, is a science adorned with many doctrines and diverse teachings: by whose judgment all works, finished by other craftsmen, are judged.\n\nIt follows: Architecture grows out of framing and reasoning, etc. Reasoning is that which can demonstrate and plainly declare things framed with skill and a sense of proportion.\n\nAgain: In all things, and therefore chiefly in architecture, there are these two: the thing signified, and the thing that signifies. The thing proposed, of which we speak, is the thing signified. But demonstration, set out with the reasons of diverse doctrines, signifies the same thing.\n\nAfter that: An architect, he says, should understand languages, be skilled in painting, well instructed in geometry, not ignorant of perspective, equipped with arithmetic, know many histories, have diligently listened to philosophers, have skill in music, not be ignorant of medicine, know the answers of lawyers, and have good knowledge of astronomy and the courses of the heavens.\n\nHe gives the reasons, in order, why all these arts, doctrines, and instructions are required in an excellent architect. And, for brevity, leaving out the Latin text, he has this:\n\nSecondly, an architect needs the knowledge of painting, so that he can more easily shape out, in painted patterns, the form of whatever work he likes. And geometry gives architecture many helps: first it teaches the use of the rule and the compass, by which (chiefly and easily) descriptions of buildings are drawn up in ground plans, and the setting out of squares, levels, and lines.\n\nLikewise, through perspective, the lights of the sky are well led into buildings, from certain quarters of the world. By arithmetic the costs of buildings are added up, the measures are set out, and the difficult problems of symmetry are worked through by geometric means and methods. etc.\n\nBesides this, philosophy declares the nature of things (which in Greek is called φυσιολογία). It is necessary for an architect to have learned this with diligence, because it has many and various natural questions, especially in aqueducts. For in their courses, the turnings about, over level ground and up the rises, natural spirits or airs are generated in various ways. The hindrances they cause no man can remedy, unless he has learned from philosophy the original causes of things.\n\nLikewise, whoever reads the books of Ctesibius or Archimedes (and of others who have written such rules) cannot think as they do, unless he has received instruction from philosophers in these things.\n\nAnd he must know music, so that he can understand both regular and mathematical music, and tune his ballistas, catapults, and scorpions well. etc.\n\nMoreover, the bronze vessels which in theaters are placed by mathematical order in niches under the steps, and the differences of the sounds (which the Greeks call ηχεῖα) are ordered according to musical symphonies and harmonies, being distributed through the circuits by Diatessaron, Diapente, and Diapason. So that the fitting voice of the players' sound, when it came to these preparations made in order, being increased there, with that increase it might come clearer and more pleasant to the ears of the onlookers. etc.\n\nAnd by astronomy are known the east, west, south, and north; the shape of the sky, the equinox, the solstice, and the course of the stars. Unless one knows these things, one cannot perceive anything at all of the reason of horologies. Since, then, this ample science is adorned, beautified, and stored with so many and various skills and knowledges, I think no one can rightly call themselves an architect all at once. Only those who, from their childhood years, rising by these degrees of knowledge, being fostered in the attainment of many languages and arts, have reached the high tabernacle of architecture. etc.\n\nAnd to those to whom nature has given such quick perception, sharpness of wit, and memory, that they can be thoroughly skilled in geometry, astronomy, music, and the rest of the mathematical arts: such surpass and pass the calling and state of architects:\n\nA mathematician.\n\nand have become mathematicians. etc. And they are rarely found. As in times past there were Aristarchus Samius; Philolaus and Archytas, Tarentines; Apollonius Pergaeus; Eratosthenes Cyreneus; Archimedes and Scopas, Syracusians. These also left to their posterity many engines and gnomonic works, invented and explained by numbers and natural means.\n\nAll this, and the same words (in sense), in one single chapter of this incomparable architect Vitruvius, you will find. And if you were but to take his book in hand and look through it slightly, you would say at once:\n\nVitruvius.\n\nThis is geometry, arithmetic, astronomy, music, anthropography, hydragogy, horometry, etc., and to conclude, the storehouse of all workmanship.\n\nNow let us listen to our other judge, our Florentine, Leo Baptista, and look closely at how he determines of architecture. But before I go any further, he says, I think I ought to set out what man I would have allowed to be an architect. For I will not bring forward a carpenter, as though you could compare him to the chief masters of other arts. For the hand of the carpenter is the architect's instrument.\n\nWho is an architect.\n\nBut I will define the architect as \"that man who has the skill (by a certain and marvelous means and way) both in mind and imagination to determine, and also in work to finish, whatever works, by motion of weight, and coupling and framing together of bodies, may most aptly be useful for the worthiest uses of man.\" And to be able to perform these things, he needs the attainment and knowledge of the best and most worthy things. etc.\n\nThe whole craft of architecture in building consists in lineaments and in framing. And the whole power and skill of lineaments tends to this: that the right and absolute way may be had of fitting and joining lines and angles, by which the face of the building or frame may be comprehended and concluded. And it is the property of lineaments to prescribe to buildings, and every part of them, a fitting place and a certain number: a worthy manner and a seemly order: so that the whole form and figure of the building may rest in the very lineaments. etc. And we may prescribe in mind and imagination the whole forms,\n\nThe immateriality of perfect architecture.\n\nall material stuff being kept out. This point we shall reach by marking and setting out beforehand the angles and lines, by a sure and certain direction and connection. Since, then, these things are thus:\n\nWhat a lineament is.\n\nA lineament shall be the certain and constant prescribing, conceived in mind: made in lines and angles: and finished with a learned mind and wit.\n\nWe thank you, Master Baptist, that you have so aptly brought your art, and the phrase of it, to some mathematical perfection: \"by a certain order, number, form, figure, and mental symmetry;\" all natural and sensible stuff set apart.\n\nNow, then, it is evident, gentle reader, how aptly and worthily I have preferred architecture to be bred and fostered in the dominion of the peerless princess Mathematics, and to be a natural subject of hers. And the name of architecture comes from the principality which this science has above all other arts. And Plato affirms the architect to be master over all who make any work. So he is neither smith nor builder, nor separately any craftsman, but the head, the provost, the director, and judge of all artificial works and all craftsmen. For the true architect is able to teach, demonstrate, distribute, describe, and judge all works wrought. And he alone searches out the causes and reasons of all artificial things. Thus excellent is architecture: though few in our days attain to it, yet the art may not be thought of otherwise than it is in very deed worthy. Nor may we make new and imperfect definitions of ancient arts in our days, for scarcity of craftsmen: no more than we may trim down the definitions of wisdom, or honor, or of friendship or of justice. Nor will I consent to diminish in the least the perfection and dignity (by just cause) allowed to absolute architecture.\n\nUnder the direction of this art are three principal, necessary mechanical arts. Namely, housing, fortification, and shipbuilding. By housing I understand both divine service and man's common use: public and private. Of fortification and shipbuilding strange matter might be told you: but perhaps some will be tired with this long list already recited, and others will pick apart my rough and homely talk with you, done in a hurry, for fear you should miss this true and friendly warning, and foretaste, of mathematical power. Life is short and uncertain: times are dangerous: etc. And still the printer waiting, while my pen holds back: all these things, with further matter of ingratitude, give me occasion to pass on to the other arts remaining, with all speed possible.\n\n### Navigation\n\nThe art of Navigation demonstrates how, by the shortest good way, by the aptest direction, and in the shortest time, a sufficient ship, between any two places (in navigable passage) assigned, may be conducted: and in all storms and natural disturbances happening, how to use the best possible means, by which to recover the place first assigned. What need the master pilot has of the other arts recited here before, it is easy to know: as of hydrography, astronomy, astrology, and horometry. Presupposing continually the common base and foundation of all: namely arithmetic and geometry. So that he be able to understand and judge his own necessary instruments, and necessary equipment: whether they are perfectly made or not: and also can (if need be) make them himself. As quadrants, the astronomer's ring, the astronomer's staff, the universal astrolabe. A hydrographic globe. Hydrographic charts, true (not with parallel meridians). The common sea compass: the compass of variation: the proportional and paradoxical compasses\n\nYear 1559.\n\n(invented by me, for our two Muscovy master pilots, at the request of the Company) clocks with springs: hour, half-hour, and three-hour sandglasses: and various other instruments: and also be able, on a globe or a plane, to describe the paradoxical compass, and duly to use the same, for all manner of purposes to which it was invented. And also be able to calculate the planets' places for all times.\n\nMoreover, with sun, moon, or star (or without them) be able to define the longitude and latitude of the place which he is in: so that the longitude and latitude of the place from which he sailed be given, or known to him. To which belongs an expert means of always being certain of the ship's course. etc. And by foreseeing the rising, setting, non-setting, or midnighting of certain stormy fixed stars, or their conjunctions and angles with the planets, etc., he ought to have expert conjecture of storms, tempests, and waterspouts, and such meteorological effects, dangerous at sea.\n\nFor, as Plato says, To foresee the changes and opportunities of times is fitting no less to the art of war than to husbandry and navigation.\n\nAnd besides such cunning means, more evident tokens in sun and moon ought to be known by him: such as (the philosophical poet) Virgilius teaches in his Georgics. Where he says,\n\nGeorgic 1.\n\nThe sun also, both at his rising and when he sinks into the waves, will give signs; the surest signs follow the sun, etc.—For often we see that on his very face various colors wander. Blue, he announces rain; fiery, the east winds. But if spots begin to mingle with the ruddy fire, then you will see everything alike seething with wind and storm clouds: on such a night let no one urge me to go out over the deep, or to pull up the cable from the land, etc. The sun will give you signs. Who would dare call the sun false?—etc.\n\nAnd so of moon, stars, water, air, fire, wood, stones, birds, and beasts, and of many other things, a certain sympathetic forewarning may be had: sometimes to great pleasure and profit, both on sea and land. It appears sufficiently, for my present purpose, from what has gone before, how mathematical the art of navigation is: and how it needs and also uses other mathematical arts: And now, if I were to go about to speak of the many benefits coming to this land, and others, by ships and navigation, you might think that I was seizing chances to use many words where they are not needed.\n\nYet this one thing I may justly say. In navigation, none ought to take greater care to be skilled than our English pilots. And perhaps some would attempt more, and others would more willingly help, if they knew for certain what privilege God had given this island, by reason of its situation, most convenient for navigation, to places most famous and rich. And though,\n\nYear 1567 S. H. G.\n\n(of late) a young gentleman, a courageous captain, was in great readiness, with good hope and strong grounds of persuasion, to have ventured for a discovery (either westward, by Cape de Paramantia, or eastward, above Nova Zemla and the Cyremisses), and was, at the very near time of attempting, called and employed otherwise (both then and since) in great good service to his country, as the Irish rebels have tasted:\n\nYear 1569\n\nYet I say, though the same gentleman does not deal with it hereafter, someone or other should attend to the matter: and by good advice and careful judgment, little by little, win sufficient knowledge of that trade and voyage: which now I would be sorry (through carelessness, lack of skill, and courage) should remain unknown and unheard of. Since also we are here half challenged by the learned, by a request half published. From it, truly, might grow benefit, to this land chiefly, and to the rest of the Christian commonwealth, far surpassing all riches and worldly treasure.\n\n### Thaumaturgike, 1\n\nThaumaturgike is the mathematical art that gives a set order for making strange works, to be perceived by the senses and to make people greatly wonder. This wonder-work is made in various ways. Some by Pneumatithmie, as the works of Ctesibius and Hero. Some by weight, of which Timæus speaks. Some by tightened strings or springs, with which living motions are imitated. Some by other means, as the images of Mercury, and the bronze head made by Albertus Magnus that seemed to speak. Boethius was excellent at these feats. To whom Cassiodorus, writing, says: Your purpose is to know deep things and to show wonders. By the arrangement of your art, metals bellow; a bronze Diomedes blows a loud trumpet; a bronze serpent hisses; made birds sing sweetly. We tell of small things of you, who can imitate heaven, etc. As for the strange self-moving work, which at Saint Denys, near Paris,\n\nYear 1551\n\nI saw once or twice (Orontius being then with me, in company): it would be too strange to tell. But some have written it. And yet, I hope, it is there to be seen by others. By perspective also strange things are done, as I partly gave you to understand before, in perspective: to see in the air, high up, the living image of another man, either walking back and forth or standing still. Likewise, to come into a house and there see the lively show of gold, silver, or precious stones, and, reaching to take them in your hand, find nothing but air. By this, some men (in all other matters counted wise) have badly overshot themselves, misjudging the means. Therefore said Claudius Cælestinus.\n\n(On those things that happen marvelously in the world. Chapter 8.)\n\nNowadays we see some men, of great learning and reputation, judging certain works to be marvelous, above the power of nature; of which works, one skilled in perspective could easily have given the cause. Of Archimedes' sphere, Cicero bears witness.\n\n(Tusculan Disputations 1.)\n\nThis is very strange to think on. For when Archimedes, says he, fixed into a sphere the movements of the sun, the moon, and the five other planets, he did what the god who (in Plato's Timæus) made the world did: that, with one turning, he should rule motions most unlike in slowness and speed. But we have a greater cause for wonder from Claudianus' report of this, who says this work of Archimedes was made of glass, and discusses it at greater length, which I omit. The wooden dove that the mathematician Archytas made to fly is spoken of by Agellius. Plato reports the strange images of Daedalus. Homer leaves in writing Vulcan's self-movers, by hidden wheels. Aristotle mentions both in his Politics. The workmanship achieved in recent times by good skill in Trochilike was marvelous, etc. For in Nuremberg, an iron fly, let out of the craftsman's hand, would, as it were, fly about among the guests at the table and at last, as though weary, return to its master's hand again. Moreover, an artificial eagle was ordered to fly out of the same town a mighty way, high in the air, toward the emperor coming there, and to follow him until it reached the gate of the town.\n\n☞\n\nThus you see what mathematical art can accomplish when skill, will, industry, and ability are properly put to the proof.\n\nA digression.\n\nAnd for these, and such like marvelous acts and feats, wrought and devised naturally, mathematically, and mechanically:\n\nApologetic.\n\nought any honest student and modest Christian philosopher be counted and called a conjurer? Shall the folly of idiots and the malice of the scornful prevail so much that he who seeks no worldly gain or glory at their hands, but only from God the treasure of heavenly wisdom and knowledge of pure truth, shall he (I say) in the meantime be robbed and stripped of his good name and fame? He who seeks (by Saint Paul's advice) in the properties of creatures and their wonderful virtues just cause to glorify the eternal and almighty Creator: shall that man be condemned in secret as a companion of the hellhounds, and a caller and conjurer of wicked and damned spirits? He who mourns his great lack of time, enough (to his satisfaction) to learn godly wisdom and godly truths in, and sets all his delight only in that: will that man lose and misuse his time in dealing with the chief enemy of Christ our Redeemer, the deadly foe of all humankind, the subtle and shameless perverter of godly truth, the hypocritical crocodile, the envious basilisk, forever desiring, in the blink of an eye, to destroy all humankind, both body and soul, for eternity? Surely, for my part, having something to say here, I have not learned to make so brutish and so wicked a bargain. Should I, for my twenty or twenty-five years of study, for two or three thousand marks spent, for seven or eight thousand miles of going and traveling, only for the sake of good learning, and that in all manner of weather, in all manner of ways and passages, early and late, in danger of violence from people, in danger of destruction by wild beasts, in hunger, in thirst, in perilous heat by day with toil on foot, in dangerous damp cold by night almost taking away life (as God knows), with lodgings often of little comfort and sometimes of less safety? And for much more than all this done and suffered for learning and gaining wisdom, should I (I pray you), for all this, no otherwise nor more warily, or by God's mercy any more luckily, have fished with so large and costly a net, so long in the drawing (and that with the help and advice of Lady Philosophy and Queen Theology), only at length to have caught and drawn up a frog?\n\nA proverb: fair fished, and caught a frog.\n\nOr a devil? For so the common peevish babbler imagines and jangles, and so the malicious scorner secretly wishes, and brazenly faces me down behind my back. Ah, what a miserable thing is this kind of people! How great is the blindness and boldness of the crowd in things above their capacity! What a land, what a people, what manners, what times are these! Have they become devils themselves, and by bearing false witness against their neighbor would they also become murderers? Does God give them so long a respite, to pull themselves back from this horrible slandering of the guiltless, contrary to their own consciences, and still will they not stop? Does the innocent person forbear calling them to answer him legally, according to the rigor of the laws, and will they despise his charitable patience? As they, against him by name, forge, fable, rage, and raise slander by word and print, will they provoke him also, by word and print, to set their names before the world, with their particular devices, fables, beastly imaginations, and un-Christian slanders? Well, well. O you who are such, my unkind countrymen. O unnatural countrymen. O ungrateful countrymen. O brainsick, rash, spiteful, and disdainful countrymen. Why do you oppress me so violently with your slandering of me, contrary to truth and contrary to your own consciences? And I, to this hour, neither by word, deed, or thought, have in any way been harmful, damaging, or wrong to you or yours? Have I, for so long, so dearly, so far, so carefully, so painfully, so dangerously sought and traveled for the learning of wisdom and the gaining of virtue, and in the end (in your judgment) have I become worse than when I began? Worse than a madman? A dangerous member of the commonwealth, and no member of the church of Christ? Do you call this being learned? Do you call this being a philosopher and a lover of wisdom? To leave the straight heavenly way and wallow in the broad way of damnation? To leave the light of heavenly wisdom and lurk in the dungeon of the prince of darkness? To leave the truth of God and his creatures and fawn on the shameless, crafty, stubborn liar who disgraces God's truth to the uttermost of his power? To leave eternal life and bliss and cling to the author of everlasting death, that murderous tyrant most greedily awaiting the prey of a person's soul? Well, I thank God and our Lord Jesus Christ for the comfort I have from the examples of other men before my time, to whom I am not worthy to be compared, either in godliness of life or in perfection of learning; and yet they sustained very like injuries to those I do, or rather greater. Patient Socrates' Apology will testify; Apuleius' Apologies will declare the brutishness of the crowd. Ioannes Picus, Earl of Mirandula, his Apology will show you the raging slander of the malicious and ignorant against him. Ioannes Trithemius' Apology will set out how he had occasion to make a public protestation, both because of the rude and simple and also in regard to such as were counted of the wisest sort of men. \"Many could I recite: but I put off the precise and settled handling of this matter, being reluctant to expose the folly and malice of my native countrymen.\n\n☞\n\nwho can hardly digest or like any extraordinary course of philosophical studies that does not fall within the compass of their capacity, or where they are not let in on the true and secret cause of such wonderful philosophical feats.\" These men are of four sorts, chiefly. The first I may name: vain, pratling busybodies. The second: fond friends. The third: imperfectly zealous. And the fourth: malicious ignorants. To each of these, briefly and in charity, I will say a word or two, and so return to my preface.\n\n(1.)\n\nVain, pratling busybodies: keep your idle gatherings and discussions to something other than talk of matters that are either above your capacity for difficulty or contrary to your conscience in truth.\n\n(2.)\n\nFond friends, stop commending a friend you hardly know out of blind affection, on the grounds that because he knows more than the ordinary student he must therefore be skilled and a doer in such matter and manner as you call conjuring, thinking that by that you advance his fame, and that you make others greatly marvel at your luck in having such a learned friend. Stop attributing impiety where you pretend friendship. For if your tongues were true, then that friend of yours would be untrue both to God and to his sovereign. Such friends and fondlings: I shake you off and renounce you. Shake off your folly.\n\n(3.)\n\nTo you, imperfectly zealous, I say: perhaps you mean well, but you miss the mark, if you will kill a lamb to feed the flock with its blood. Sheep have no natural nourishment from lamb's blood, and no more is Christ's flock properly built up by horrible slanders. Nor is your fair pretense at all well graced by such rough, ragged rhetoric. But those who use me so will find a foul crack in their credit. Speak what you know, and know as you ought: do not know by hearsay when life lies in danger. Search to the quick, and let charity be your guide.\n\n(4.)\n\nMalicious ignorants, what shall I say to you? Keep your tongue from evil. Keep your tongue from slander. Though your tongues are sharpened like serpents, and adder's poison lies in your lips:\n\n(Psalm 140.)\n\nyet take heed, and think in time to yourself: A man of loose tongue shall not be established in the land. Evil shall hunt the violent man until he is cast down. For I am sure that the Lord will do justice for the afflicted, and vengeance for the poor.\n\nThus I ask you, my trusted friends and countrymen (you mathematicians, mechanicians, and philosophers, charitable and discreet), to deal on my behalf with the frivolous and untrue-tongued, my envious adversaries or fond friends. Further, I would wish that at leisure you would consider how Basilius Magnus lays Moses and Daniel before the eyes of those who count all such philosophical studies as mine have been ungodly or unprofitable. Weigh well Saint Stephen's witness of Moses.\n\n(Act. 7. C.)\n\nMoses was instructed in all manner of wisdom of the Egyptians, and he was powerful both in his words and in his works. You see that this philosophical power and wisdom which Moses had was in no way disapproved by the Holy Ghost. Yet Plinius has recorded Moses to be a wicked magician. And that must of necessity be either for this philosophical wisdom learned before his calling to lead the children of Israel, or for those his wonders worked before King Pharaoh after he had the conducting of the Israelites. As for the first, you perceive how Saint Stephen, at his martyrdom (being full of the Holy Ghost), in his recapitulation of the Old Testament made mention of Moses' philosophy with full approval of it, and Basilius Magnus too attests that it was profitable to Moses (and therefore, I say, necessary to the church of God). But as for Moses' wonders done before King Pharaoh: God himself said: See that you do all those wonders before Pharaoh which I have put in your hand. Thus you clearly perceive how rashly Plinius has slandered Moses,\n\n(Book 30, chapter 1.)\n\nof vain and fraudulent magic, saying: There is also another sect of magic, depending on Moses, Jannes, and Iotape, and the Jews, but many thousands of years after Zoroaster. etc.\n\n(1.)\n\nLet all such, therefore, who are far inferior to Plinie in judgment and skill of philosophy, \"take good heed, for fear that they overshoot themselves rashly,\" in\n\n☞\n\njudging of philosophers' strange acts, and the means by which they are done.\n\n### Thaumaturgike, 2\n\n(2.)\n\nBut much more ought they to beware of forging, devising, and imagining monstrous feats and wonderful works where and when no such were done: not even any spark or likelihood of such things as they, without any shame, report.\n\n(3.)\n\nAnd to conclude, most of all let them be ashamed before humankind, and afraid of the dreadful and just judge: both to devise foolishly or maliciously, and then devilishly to father their new-fangled monsters on me, innocent in hand and heart, for trespassing against the law of God or of humankind in any of my studies or exercises, philosophical or mathematical, as in due time I hope will be more evident.\n\n### Archemastrie\n\nNow I end with Archemastrie. The name is not so new as this art is rare. For another art, a degree below it in skill and power, has already been given this English name. And yet this may serve our purpose well enough at present. This art teaches how to bring to actual sensible experience all worthy conclusions proposed by all the mathematical arts and concluded by true natural philosophy; and it both adds to them a further scope in the terms of the same arts, and also by its own method, in peculiar terms, proceeds with the help of the said arts to the performance of complete experiences which cannot (formally) be challenged from any particular art. If you remember how we considered architecture in respect of all common handiwork, you may get some light from that toward understanding the sovereignty and property of this science. I may call it a science rather than an art, for the excellence and mastery it has over so many and so mighty arts and sciences. And because it proceeds by experiences, and searches out the causes of conclusions by experiences, and also puts the conclusions themselves into experience, some name it the experimental science. Nicolaus Cusanus calls it so in his Experimentes Statikall, and another philosopher,\n\nnative to this land (the flower of whose worthy fame can never die or wither) wrote about it at length at the request of Clement the Sixth. The art carries with it wonderful credit, because it certifies sensibly, fully, and completely, to the utmost power of nature and art. This art certifies by complete and absolute experience; other arts, with their arguments and demonstrations, persuade, and in words prove their conclusions very well.\n\n☞\n\nBut words and arguments are no sensible certifying, nor the full and final fruit of sciences that can be practiced. And though some arts have experiences in them, they are not complete, nor brought to the uttermost they may be stretched to, and applied sensibly. For example: the natural philosopher disputes and makes a goodly show of reason, and the astronomer and the optical mechanician put some things into experience, but neither all that they can, nor yet fully and to the utmost those they do. There the Archemaster steps in and leads the experiences on, by order of his experimental doctrine, to the chief and final power of natural and mathematical arts. Of two or three men in whom this description of Archemastry was verified in practice I have read and heard, and good record exists of such perfection in them. So this art is no fantastic imagination, as some sophister might flourish with his insolubles, dazzle your imagination, and dash your honest desire and courage from believing these things so unheard-of, so marvelous, and of such importance. Well, as you will. I have forewarned you. I have done the part of a friend; I have discharged my duty toward God for the small talent I received from his most merciful hands. To this science the science Alnirangiat does great service. Do not wonder at this name: I do not change the name so used and published in print by others, being a name proper to the science. Under this comes Ars Sintrillia [title unclear in the source], by Artephius, briefly written. But the chief science of the Archemaster (in this world) as yet known is another science, as it were, optical; its name shall be told (God willing) when I have some more just occasion to discourse of it.\n\nHere I must end, thus abruptly (gentle friend, and unfeigned lover of honest and necessary truths). For those who have (for your sake and virtue's sake) requested me (an old, worn-out mathematician) to take pen in hand, through the confidence they placed in my long experience and tried sincerity, for the declaring and reporting somewhat of the fruit and benefit to be attained by the mathematical arts: even they, very much against their own wills, are forced for various reasons to satisfy the workman's request by ending at once. He fears so new an attempt and so costly, and a matter so little (so far) considered or esteemed among the common sort of students.\n\nAnd whereas I was pressed to allege some reason why, in our common speech, this part of the principal science of geometry, called Euclid's Geometric Elements, is published for you to handle, being people without Latin and not university scholars: truly I think it needless.\n\n(1.)\n\nFor the honor and standing of the universities and their graduates is in no way diminished by this, seeing that you receive all this benefit from and through their nurslings, however great it be.\n\n(2.)\n\nNeither are their studies in any way hindered by it, any more than the Italian universities, as the academies of Bologna, Ferrara, Florence, Milan, Padua, Pavia, Perugia, Pisa, Rome, and Siena, or any one of them, find themselves at all disgraced or their studies hindered in the least by Friar Lucas de Burgo or by Nicolaus Tartalea, who in the common Italian language have published not only Euclid's geometry but something of Archimedes, and in arithmetic and practical geometry very large volumes, all in their common speech. Nor have the famous universities in Germany been at all displeased with the geometric institutions of Albertus Durerus in Dutch; nor with the learned translation by Gulielmus Xylander of the first six books of Euclid from Greek into high Dutch; nor with the geometric volume of Gualterus H. Riffius, very diligently translated into the high Dutch tongue and published. Nor do the universities of Spain or Portugal think their reputation decayed, or suppose any of their studies hindered by the excellent mathematical works of P. Nonnius put forth in common speech. Do you not likewise have in French the whole mathematical quadrivium? And yet neither Paris, Orleans, nor any of the other universities of France has at any time been offended with the translators or publishers, nor has any person's study been hindered by it?\n\n(3.)\n\nAnd surely the common and ordinary scholar (and much more the grammarian) before coming to the university can (or may) now be instructed well enough (according to Plato's counsel) in arithmetic and geometry, for the better and easier learning of all manner of philosophy, academic or Peripatetic. And by that means go on more cheerfully, more skillfully, and more quickly in his studies there to be learned, and so profit more in less time than otherwise he should or could do.\n\n(4.)\n\nAlso many good and fertile English wits, of young gentlemen and of others who never intend to meddle with the profound search and study of philosophy (to be learned in the universities), may nevertheless now, with more ease and liberty, have good occasion to put the sharpness of their wits to virtuous use; where otherwise they might (perhaps) spend their time in foolish exercises, or rather lose it, serving neither God nor furthering the common or private good.\n\n(5.)\n\nAnd great comfort with good hope may the universities have from this English geometry and mathematical preface, that they hereafter shall be more regarded, esteemed, and resorted to. For when it shall be known and reported that from the mathematical sciences only such great benefits follow as I have specified, and that indeed some of you students without Latin can be good witnesses of such rare fruit enjoyed by you by that, as either was not heard of before or else not so fully credited: \"Well, may all men conjecture that far greater aid and better furniture to win to the perfection of all philosophy,\n\nUniversities.\n\nmay be had in the universities, being the storehouses and treasury of all sciences,\n\n☞\n\nand all arts necessary for the best and most noble state of commonwealths.\"\n\n(6.)\n\nBesides this, how many a common craftsman is there in these realms of England and Ireland that deals with numbers, rule, and compass, who, with their own skill and experience already had, will be able (by these good helps and information) to find out and devise new works, strange engines, and instruments: for various purposes in the commonwealth, or for private pleasure, and for the better maintaining of their own estate? I will not (therefore) fight against my own shadow. For no man, I am sure, will open his mouth against this enterprise; no one, I say, who either has charity toward his brother (and would be glad to further him in virtuous knowledge), or who has any care and zeal for the bettering of the common state of this realm; neither any who make account of what the wiser sort of men (sage and steady) think of them. To none, therefore, will I make any apology for doing a virtuous act, and for commending or setting forth profitable arts to English people in the English tongue. \"But to God our Creator let us all be thankful: that as he, of his goodness, by his power, and in his wisdom,\n\n☞\n\nhas created all things in number, weight, and measure; so to us, of his great mercy, he has revealed means by which to attain the sufficient and necessary knowledge of the said three principal instruments of his: which means I have abundantly proved to you to be the sciences and mathematical arts.\"\n\nAnd though I have been pressed by the straightness of time, so that in no way could I set the matter down (in my mind) as I determined, hoping for convenient leisure: yet if virtuous zeal and honest intent provoke and bring you to the reading and examining of this compendious treatise, I do not doubt but that, as the truth of it (according to our purpose) will be evident to you, so its pith and force will persuade you, and its wonderful fruit highly please you. And that you may the more easily perceive and better remember the principal points my preface treats of,\n\nThe ground plan of this preface in a table.\n\nI will give you the ground plan of my whole discourse in an annexed table, from the first to the last, somewhat methodically devised.\n\nIf haste has made my poor pen stumble anywhere, you will (I am sure) partly repay me, for my earnest and sincere good will to please you, by considering the rocky, huge mountains and the perilous, untrodden ways it has toiled and labored through (night and day the while) to bring you this good news and comfortable proof of virtue's fruit.\n\nSo I commit you to God's merciful direction for the rest, heartily beseeching him to prosper your studies and honest intents, to his glory and the benefit of our country. Amen.\n\nWritten at my poor house at Mortlake.\n\nIn the year 1570. February 9.\n\nJ. DEE\n\n### The Groundplat of the Mathematical Praeface\n\nHere you have (according to my promise) the ground plan of my mathematical preface, annexed to Euclid, now first published in our English tongue. An. 1570. Febr. 3.\n\nSciences and mathematical arts are either,\n\nPrincipal, which are two only,\n\nArithmetic.\n\nSimple, which deals with numbers only, and demonstrates all their properties and appurtenances, where a unit is indivisible. Mixed, which with the help of principal geometry demonstrates some arithmetic conclusion or purpose.\n\nGeometry.\n\nSimple, which deals with magnitudes only, and demonstrates all their properties, affections, and appurtenances, whose point is indivisible. Mixed, which with the help of principal arithmetic demonstrates some geometric purpose, as Euclid's Elements.\n\nThe use of which is either,\n\nIn supernatural, eternal, and divine things: by application, ascending. In mathematical things: without further application. In natural things: both substantial and accidental, visible and invisible, etc. By application, descending.\n\nThe like uses and applications are (though a degree lower) in the derived mathematical arts.\n\nDerived from the principals; of which some have\n\nThe names of the principals, as:\n\nCommon arithmetic, which considers\n\n—Arithmetic of the most usual whole numbers, and of the fractions belonging to them. —Arithmetic of proportions. —Circular arithmetic. —Arithmetic of radical numbers: simple, compound, mixed, and of their fractions. —Arithmetic of cossic numbers, with their fractions, and the great art of Algiebar.\n\nCommon geometry, which teaches measuring\n\nAt hand\n\nAll lengths—mecometry. All planes, as land, board, glass, etc.—embadometry. All solids, as timber, stone, vessels, etc.—stereometry.\n\nWith distance from the thing measured, as,\n\nHow far from the measurer anything is, seen by him on land or water, called apomecometry. How high or deep, from the level of the measurer's standing, anything is, seen by him on land or water, called hypsometry. How broad a thing is that lies in the measurer's view, as long as it is situated on land or water, called platometry.\n\nFrom which have grown the feats and arts of\n\nGeodesy: to measure and survey lands, woods, waters more skillfully, etc. Geography. Chorography. Hydrography. Stratarithmetrie.\n\nProper names, as:\n\nPerspective,—which demonstrates the ways and properties of all radiations: direct, broken, and reflected.\n\nAstronomy,—which demonstrates the distances, magnitudes, and all natural motions, appearances, and affections proper to the planets and fixed stars, for any time past, present, and to come, with respect to a certain horizon or without respect to any horizon.\n\nMusic,—which demonstrates by reason and teaches by sense how to judge and order perfectly the diversity of sounds, high or low.\n\nCosmography,—which wholly and perfectly describes the heavenly and also the elemental part of the world, and of these parts makes the homologous application and mutual comparison that are necessary.\n\nAstrology,—which reasonably demonstrates the operations and effects of the natural beams of light and the secret influence of the planets and fixed stars in every element and elemental body, at all times, in any horizon assigned.\n\nStatics,—which demonstrates the causes of heaviness and lightness in all things, and the motions and properties belonging to heaviness and lightness.\n\nAnthropography, which describes the number, measure, weight, figure, situation, and color of every diverse thing contained in the perfect body of man, and gives certain knowledge of the figure, symmetry, weight, characterization, and due local motion of any part of the said body assigned, and of the numbers belonging to the said part.\n\nTrochilike,—which demonstrates the properties of all circular motions: simple and compound.\n\nHelicosophy,—which demonstrates the drawing of all spiral lines: in a plane, on a cylinder, cone, sphere, conoid, and spheroid, and their properties.\n\nPneumatithmie,—which demonstrates by close hollow geometric figures (regular and irregular) the strange properties (in motion or rest) of water, air, smoke, and fire in their continuity, and as they are joined to the elements next them.\n\nMenadrie,—which demonstrates how, above nature's virtue and simple power, virtue and force can be multiplied; and so to direct, to lift, to pull toward, and to push or cast from, any multiplied or simple determined virtue, weight, or force that is not naturally so directable or movable.\n\nHypogeiodie,—which demonstrates how, under the spherical surface of the earth, at any depth, to any perpendicular line assigned (whose distance from the perpendicular of the entrance, and the azimuth likewise with respect to the said entrance, being known), a certain way may be laid out and traveled, etc.\n\nHydragogie,—which demonstrates the possible leading of water, by nature's law and by artificial help, from any head (being spring, standing, or running water) to any other place assigned.\n\nHorometry,—which demonstrates how, at all appointed times, the precise usual denomination of time may be known for any place assigned.\n\nPainting,—which demonstrates and teaches how the intersection of all visual pyramids, made by any plane assigned (the center, distance, and lights being determined), may be represented by lines and proper colors.\n\nArchitecture,—which is a science adorned with many doctrines and diverse instructions, by whose judgment all works finished by other workmen are judged.\n\nNavigation,—which demonstrates how, by the shortest good way, by the aptest direction, and in the shortest time, a sufficient ship between any two places (in navigable passage) assigned may be conducted; and in all storms and natural disturbances happening, how to use the best possible means to recover the place first assigned.\n\nThaumaturgike,—which gives a set order for making strange works, to be perceived by the senses, and to make people greatly wonder.\n\nArchemastrie,—which teaches how to bring to actual sensible experience all worthy conclusions proposed by all the mathematical arts and concluded by true natural philosophy; and it both adds to them a further scope in the terms of the same arts, and also by its proper method, in peculiar terms, proceeds with the help of the said arts to the performance of complete experiences, which cannot (formally) be challenged from any particular art.\n\nImprinted by John Day.\n\nAn. 1570. Feb. 25.","download_url":"https://mouseia.org/api/works/dee-praeface/translations/trn_dee_praeface_plain.txt"}],"retex_markdown_url":"https://mouseia.org/api/works/dee-praeface/retex.md"}}