And though I, here, speake onely of two thyngs Miscible: and most commonly, mo then three, foure, fiue or six, (&c.) are to be Mixed: (and in one Compound to be reduced • & the Forme resultyng of the same, to serue the turne) yet these Rules are sufficient: duely repeated and iterated. Note. In procedyng first, with any two: and then, with the Fonne 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 varie • ies, 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 w • thall: And other (perchaunce) with a proude snuffe will disdaine this litl •• 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 Match • and maryed with the most simple, easie, and short way of the noble Rule of Algiebar. Who can remaine, therfore vnpersuaded, to loue, alow, and honor the excellent Science of Arithmetike? For, here, you may perceiue that the litle finger of Arithmetike, is of more might and contriuing then a hunderd thousand mens wittes, of the middle sorte, are hable to perfourme, or truely to conclude, with out helpe thereof.

Now will we f • rder, 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 〈 in non-Latin alphabet 〉 . 〈 in non-Latin alphabet 〉 . That is, the Skill of O • dring Souldiers in Battell ray after the best maner to all purposes. This Art so much dependeth vppon Numbers vse, and the Mathematicals, that AElianus (the best writer therof,) in his worke, tö 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 AEneas: Gyneas of Thessaly: Pyrrhus Epirota: and Alexander his sonne: Clearchus: Pausanias: Euangelus: Polybi • s, 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 • apitaine will affirme the Science of Arithmetike, to be one of his chief Counsaylors, directers and aiders. Which thing (by good meanes) was euident to the Noble, the Couragious,☞ the loyall, and Curteous Iohn, late Earle of Warwicke. Who was a yong Gentleman, throughly knowne to very few. Albeit his lusty valiantnes, force, and Skill in Chiualrous feates and exerci • es: 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, forecas • inges 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 Fa •• Immortall.

To our purpose. This Iohn, by one of his actes (besides many other: both in England and Fraunce, by me, in him noted.) did dislose 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 de • ised) for ordring of all Companies, summes and Number • of mē, (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 s • em few, being many. To marche in battaile or Iornay: with many such feate • , to Foughten f •• ld, Skarmoush, or Ambushe appartaining: And of all these, liuely desi • nementes (most curiously) to be in velame parchement described: with Notes & peculier markes, as the Arte requireth: and all these Rules • and descriptions Arithmeticall,This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age • hauing no issue by his wife: Daughter to the Duke of Somerset. inclosed in a riche Case of Gold, he vsed to weare about his necke • as his Iuell most precious, and Counsaylour most trusty. Thus, Arithmetik • , of him was shryned in gold: Of Numbers frute, he had good hope. Now, Numbers therfore innumerable, in Numbers prayse, his sh •• ne shall finde.

What nede I, (for farder • rofe 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 Pap •••• anus institute a law of partition, and allowance, betwene man and wife a ••• a ••• orce: But how Acc • rsius, Bald • s, Bartolus, Iason, Alex • nder, and finally Alciat •• (bei • g o ••• rwise, notably well learned) do iumble, gesse, and erre, from the • quity • • rt • nd I •• ent 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: • nd with Accursius Glosse, much cumbred: burst out, and sayd: Nulla est i • 〈◊〉 libr • , 〈◊〉 glossa difficili • r: Cui • s c • mputationem nec Scholas • ici • ec Doct • res intellig •• t. &c. That is • In the whole booke, there is no • losse harder then thi •• Whose accoump • or reckenyng, neither the Scholers, nor the Doctours vnderstand. &c. What can they say of I • lianus law, Si ita S • ript • m. &c. Of the Testators will iustly performing, betwene the wife, Sonne and daughter? How can they perceiue the ••••• tie of Aphricanus, Arithmeticall Reckening, where he treateth of Lex Falcid • a • How •• n they deliuer him, from his Reprouers: and their maintainers: as I •••• es, Accursius Hypolitus and Alciatus? How • ustly and artificially, was African • s 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 cōtributed by the legatari • s 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 Iudgemēt, & Iust determinatiō of the Auncient Romaine Lawmakers. But much more expert ought he to be, who should be hable, to decide with aequitie, 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, to be that vertue, Iustice. by which, to euery one, is rendred, that to him appertaineth. God challengeth this at our handes, to be honored as God: to beloued, as a father: to be feared as a Lord & master. Our neighbours proportiō, 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 sam • ) 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, br • efly, of other Sciences discoursing, findeth them, not hable to bring it to passe: But of the Science of Numbers, he sayth. Illa, qua numerum mortalium generi d •• n, id profecto efficiet • Deum antem aliquem, magis quam fortunam, • d sa •• tem nostram, hoc m • nus nobis arbitror contulisse. &c. Nam ipsum • onorum omnium Authorem, cur non maximi boni, Prudentiae 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 th • greatest good thyng, namely, Wisedome? There, at length, he proueth Wisedome to be atteyned, by good Skill of N • mbers. 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 of 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.

BOth, 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 〈1 page〉 pretence of iust content, and me • sure • 〈◊〉 lan • es and groundes • • re •• 〈◊〉 , ••• quietnes, murder, and warre did (full oft) ensue: • ill •• by Gods 〈◊〉 and 〈◊〉 industrie, The perfect Science of Lines, Plaines, and Solides (lik • • diuine 〈◊〉 ,) 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, con •• rmed the name of G •• m • tria, that is, (according to the very etimologie of the word) Land measuring. Wherin, the p • ople knew no farder, of Magnitudes vse, but in Plai ••• : and the Philosophers, of thē, had no feethearers, or Scholers • • a • der to disclose vnto, then of • lat, plaine Geometrie. And though, these Philosophers, knew of farder vse, and best vnderstode the etymologye of the worde, yet this name G ••• etria, 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 Plato. 7. de Rep. 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, th • rfore, must nedes be had, for our Mathematicall Science of Magnitudes: which regardeth neither clod, no • turff • neither hill, nor dale • neither ••• th nor heauen: but is absolute Megeth • logia: not creping on ground, and dissellng the eye, with pole perche, rod or lyne:☞ but liftyng the hart aboue the heauens, by 〈◊〉 lines, and immortall beames • meteth with the reflexions, of the light incomprehensible • and so procureth Ioye, and perfection vnspeakable. Of which true vse of our Meg • thica, 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 • uidently be sene. Where (in Latin) thus it is: right well translated: 〈…〉 quid Geometri • 〈…〉 loquuntur, qui in ipsa vers •• t •• . In English, 〈◊〉 o • Verely (sayth Plat • ) whosoeuer haue, (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, Quod q •• ritur cognoscendi illius gratia, quod se • per est, 〈…〉 & Interit. Geometria, eius quod est semper, Cognit •• est. 〈…〉 Veritatem, • nimum: at que ita, ad Philosophandum 〈…〉 qua, nunc, contra quam decet, ad inf • ri • r • 〈…〉 &c. 〈…〉 praecipiendum est, vt qui praclarissimam hanc 〈…〉 spernant. Nam & quoe pr • ter ipsius propositu • , 〈…〉 sunt. &c. It must nedes be confessed (saith Plat • ) That [ 〈◊〉 ] i • learned, for the knowyng of that, which is 〈◊〉 : 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 lif • 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 con • ert, toward heauenly thinges [ 〈…〉 ] which now, otherwise then b • comme ••• vs, we cast 〈…〉 or inferior things &c. Chiefly, therfore, Comma •• dement must be giuen, that such as do • nhabit this most honorable Citie, by no meanes, despise Geometrie. For euen those thinges [done by •• ] which, in manner, seame to be, beside the purpose of Geometrie: are of 〈1 page〉 〈1 page〉 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 foūd (foreuer) 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 famou • ••• ersit • e 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 S ••• rithmetrie.

Geographie teacheth wayes, by which, in sūdry formes, (as Spharike, Plaine or other), the Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, & such other things, vpō the outface of the earthly Globe (either in the whole, or in some principall mēber and portion therof cō tayned) may be described and designed, in cōmensurations 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, • arthquakes, 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 m • n • trauaile • . To conclude, some, for one purpose: and some, for another, liketh, loueth, ge ••• th, and vse •• , Mappes, Chartes, & Geographicall Globes. Of whose vse, to speake su •• iciently, would require a booke peculi • r.

Chorographie 〈…〉

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 • les 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.

Stratarithmetrie, 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, cā 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. Note. 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. It differreth from the Feate Tacticall, De acic • us instruendis • bycause, The difference betwene Strataruhmetrie and Tacticie. 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. Great pollicy may be vsed of the Capitaines, I.D. F • end •• you will finde it hard, to perform • my descripti • n of • his F • ate. But by Ch • r • graphie • you may helpe your self • some • hat: wher • th • Figures knowne (in Sid •• • nd Angles) are not Regular: And where • Resolution into Triangles can s •• u • . &c. And yet you will finde it strange to deale thus gener • lly with Arithmeticall figures: and, that for Battayle • ay. Their co • tent ••• differ so much from like Geometr • call Figur • s. (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:) contrary wise, 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.

Thus 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. As Perspectiue, Astronomie, Musike, Cosmographie, Astrologie, Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie, Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture, Nauigation, Thaumaturgike and Archemastrie. I thinke it necessary, orderly, 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.

Among 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 em • nations. 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 thing • , in l • ght 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 roūd Globe or Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise iudgeth a plaine Square, to be roūd: 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, A marueilous Glasse. ☞ appearing 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 wish his owne eyes, haue profe of that, which I haue sayd herein. A Gentleman, (which, for his good seruice, S.W.P. done to his Countrey, is famous and honorable: and for skill in the Mathematicall Sciences, and Languages, is the Od man of this kind. &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 no •• 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☞ • Before you haue learned sufficiently the powre of Nature and Arte.

NOw, 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 St • rtes: 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 1/ • 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 1/ • of the Mone. Wherfore the Sonne is bigger then the Mone, 7000 times, lesse, by 59 •• /164 that is, precisely 6940 • 5/ •• 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 1/ • The nerest, that the Mone commeth to the earth, is Semidiameters 52 ¼ The distance of the Starry Skye is, frō vs, in Semidiameters of the earth 20081 1/ • 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 20029 1/ • Twenty thousand nine and twenty and a quarter. Note. So thicke is the heauenly Palace, that the Planetes haue all their exercise in, and most meruailously perfourme the Commaūdement 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; coteineth of our common miles 3436 • / • 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 f • ute Astronomicall, as concerning the bignesse, Dist • nces 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 not so 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 • Omnipotēcy, 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 right wisenesse is risen aboue the Horizon, of our tempor • ll Hemispharie, and hath so abundantly streamed into our hartes, the direct beames of his goodnes, mercy, and grace: Whose heat All Creatures feele: Spirituall and Corpor • ll • 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 Plat • in Epinomis) Astronomiam, Sapientissimū 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. Leaui • g 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 Consideratiō 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 bru • ish 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 Astron • micall • taffe, in stede of a fest • e.

Musike, 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 Ca • onistes. 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 cares were made. But as Astronomie hath a more diuine Contemplation, and cō modity, then mortall eye can perceiue: So, is Musike to be considered, that the 1. Minde may be pref • r • ed, 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, in the heauenly 2. motions and distances, describe a meruallous Harmonie, of Pythagoras. Harpe with • ight stringes. Also, somwhat might be sayd of Mercurius 3. two Harpes, eche of foure Stringes Elementall. And very straunge matter, might be alledged of the Harmo •• e, to our 5. Spirituall part appropriate. As in Pt • lom • us third boke, in the fourth and sixth Chapters may appeare. 6. And 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 (i • a maner) 〈◊〉 ? In the 7. T • ne of Mans voyce, and also 8. the sound of Instrument, what might be sayd, of Harmonie: No common Musicien would lightly beleue. But of the sundry Mixture (as I may terme it) and concurse, 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. 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 Consonācy, 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.

Of 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 Correspō dent: So, as, the Heauenly Globe, may (in practise) be duely described vpon the Geographicall, and Hydrographicall Globe. And there, for vs to consider an AEquinoctiall Circle, an Ecliptike line, Colures, Poles, Sterr • s 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 Pr • mouan • , caried about in 24 • aequall Houres) to learne the Risinges and Settinges of Sterres (of Virgill in his Georgikes: of Hes • od: of Hippocrates in his Medicinall Sphare, to Perdic •• 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 who of a sparke of true fire, can make a wonderfull bonfire, by applying of due matter, duely.☜

Of Astrologie, here I make an Ar • e, 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 vnderstāding 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 Causa 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. Op • rtet igitur & horum principia sumamus, & causas omnium similiter. Principium igitur vy mouens, praecipuum 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 lati • , Or • us & 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) 〈 in non-Latin alphabet 〉 but by Naturall and Mathematicall demonstration 〈 in non-Latin alphabet 〉 . Whereunto, what Sciences are requis • te (without exception) I partly haue here warned: And in my Brop • de •••• 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, (for Anno. 1548 and 1549. in Louayn. • 1. yeares ago) by certaine earnest disputations, of the Learned G •• ardus M • rc • t •• , and 〈◊〉 Goga • a, (and other,) therto so prouoked: and (by my constant and inuincible zeale to the veritie) in obseruations of Heauenly Influencies (to the Min • te 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 sel • ing for the Truth, will both finde & cōfesse, 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. Wherof, the one, is Light Beleuers, the other, Note. Light Despisers, and the third Light Practisers. The first, & most cōmon Sort, thinke the Heauen and Sterres, to be answerable to any their doutes or desires: which is not so: and, in dede, they, to much, ouer reache. The Second sorte thinke no Influentiall vertue (frō the heauenly bodies) to beare any Sway in Generation and 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 thē (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 wordly 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 Por • entine do • h: 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 s • to ebbe, a • d slow, run in and out, of them selues, at • hei • 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 furnished: 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, Corr • p • ion and the appa ••• nances • and most Harmonious in thei • Monarchie: For which thinges, being • nowen, and modestly vsed: we might highly • nd continually glorifie God, with the princely Prophet saying. The Heauens declare the Glorie of God: who made the Heauēs in his wisedome: who made the Sonne, for to haue dominion of the day: the Mone and Sterres to haue dominion of the nyght: whereby, Day to day •••• reth tal ••• and night, to night declareth knowledge. Prayse him, all ye St • rr • s, and Light. Amen.

IN order, now foloweth, of Statike, somewhat to say, what we meane by • hat name • and what 〈…〉 doth, on 〈◊〉 Art, depend. Statike, is an Arte Mathematicall, which demonstra •• th the causes of heauynes, and lightnes of all thynges: and of motions and properties, to heauynes and lightnes belonging • And for asmuch as by the Bilanx, or Balance (as the chief sensible Instrument,) Experience of these demonstrations may be had: we call this Art, Statike: that is, the Experimentes of the Balance. Oh; that men wist, what proffit, (all maner of wayes) by this Arte might grow, to the hable examiner, and diligent practiser. Thou onely, knowest all thinges precisely (O God) who hast made weight and Balance, thy Iudgement: who hast created all thinges in Number, Waight, and Measure: and hast wayed the mountaines and hils in a Balance: who hast peysed in thy hand, both Heauen and earth. We therfore warned by the Sacred word, to Consider thy Creatures: and by that consideration, to wynne a glyms (as it were,) or shaddow of perceiuerance, that thy wisedome, might, and goodnes is infinite, and vnspeakable, in thy Creatures declared: And being farder aduertised, by thy mercifull goodnes, that, three principall wayes, were, of the, vsed in Creation of all thy Creatures, namely, Number, Waight and Measure, And for as much as, of Number and Measure, the two Artes (auncient, famous, and to humaine vses most necessary,) are, all ready, sufficiently knowen and extant: This third key, we beseche thee (through thy accustomed goodnes,) that it may come to the nedefull and sufficient knowledge, of such thy Seruauntes, as in thy workemanship, would gladly finde, thy true occasions (purposely of the vsed) whereby we should glorifie thy name, and shew forth (to the weaklinges in faith) thy wondrous wisedome and Goodnes. Amen.

Meruaile nothing at this pang (godly frend, you Gentle and zelous Student.) An other day, perchaunce, you will perceiue, what occasion moued me. Here, as now, I will giue you some ground, and withall some shew, of certaine commodities, by this Arte arising. And bycause this Arte is rare, my wordes and practises might be to darke: vnleast you had some light, holden before the matter: and that, best will be, in giuing you, out of Archimedes demonstrations, a few principal Conclusions, as foloweth.

BY these verities, great Errors may be reformed, in Opinion of the Naturall Motion of thinges, Light and Heauy. Which errors, are in Naturall Philosophie (almost) of all mē allowed: to much trusting to Authority: and false Suppositions. As, Of any two bodyes, the heauyer, to moue downward faster then the lighter. This error, is not first by me, Noted: A common error • noted. but by one Iohn Baptist de Benedictis. The chief of his propositions, is this: which seemeth a Paradox.

If there be two bodyes of one forme, and of one kynde, aequall in quantitie or vnaequall, they will moue by aequall space, A paradox. in aequall tyme: So that both theyr mouynges be in ayre, or both in water: or in any one Middle.

Hereupon, in the feate of Gunnyng, certaine good discourses (otherwise) may receiue great amendement, and furderance. N. T. In the entended purpose, also, allowing somwhat to the imperfection of Nature: The wonderfull vse of these Propositions. not aunswerable to the precisenes of demonstration. Moreouer, by the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth, the Fire, may be nerely, knowen, how light or heauy they are (Naturally) in their ••• gned partes: or in the whole. And then, to thinges Elementall, turning your practise: you may deale for the proportion of the Elementes, in the thinges Compounded. Then, to the proportions of the Humours in Man: their waightes: and the waight of his bones, and flesh. &c. Than, by waight, to haue consideration of the Force of man, any maner of way: in whole or in part. Then, may you, of Ships water drawing, diuersly, in the Sea and in fresh water, haue pleasant consideration: and of waying vp of any thing, sonken in Sea or in fresh water &c. And (to lift vp your head a loft:) by waight, you may, as precisely, as by any instrument els, measure the Diameters of Sonne and Mone. &c. Frende, I pray you, way these thinges, with the iust Balance of Reason. And you will finde Meruailes vpon Meruailes: And esteme one Drop of Truth (yea in Naturall Philosophie) more worth, then whole Libraries of Opinions, vndemonstrated: or not aunswering to Natures Law, and your experience. Leauing these thinges, thus: I will giue you two or three, light practises, to great purpose • and so finish my Annotation Staticall. In Mathematicall matters, by the Mechaniciens ayde, we will behold, here, the Commodity of waight. Make a Cube, of any one Vniforme: The practise Staticall, to know the proportion, betwene the Cube, and the Sphare. and through like heauy stuffe: of the same Stuffe, make a Sphaere or Globe, precisely, of a Diameter aequall to the Radicall side of the Cube. Your stuffe, may 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:I. D. by eight descentes progressionall, For, so, haue you. 256. partes of a Graine. halfing, from a grayne. Your Cube, Sphaere, apt Balance, and conuenient waightes, being ready: fall to worke. • . First, way your Cube. Note the Number of the waight. Way, after that, your Sphaere. Note likewise, the Nūber of the waight. If you now find the waight of your Cube, to be to the waight of the Sphaere, 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 Sphaere. Then, chaunge, your Sphaere 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.

When 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 aequall to the Diameter of the Sphaere, is Sesquialter to the same Sphaere (that is, as 3. to 2:) To the number of the waight of the Sphaere, 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, we haue The proportion of the Square to the Circle ins • ribed. 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 aequall: The Squa • ing of the Cir • le. M •• hani • ally. 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 aequall: To any Squir • g • uen • to 〈…〉 . 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 Sphaere, & the Cube, what their proportion is: as now, you came from the Sph • ere to the Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough one hole drawē, 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, Note Squaring of the Circle without knowledge of the proportion betwene Circumference and Diameter. by the way. That 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. To Dubble the Cube redily: by Art Mechanicall: depending vppon Demonstration Mathematicall. Which may, thus, be done: Make 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: Let 4. straight lines be drawen, in the inside of the Cone and Pyramis: I. D. The 4. sides of this Pyrami • must be 4. Isosceles Triangles • lik • and • quall. 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. aequall 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, I. D. as experience shall reach you • Then, In all work • nge • with this Pyramis or Cone. Let their Situation • be in all Pointe • and Conditions, a like, o • all one: while you are about • ne worke. Els you will 〈◊〉 . set your Cone or Pyramis, with the vertex downward, perpendicularly, in respect of the Base. (Though it be otherwayes, it hindreth nothyng.) So let thē 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 Cubi • coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so proportiō in respect of your Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the waight of them, in wa •• , 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: I. D. 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: put that Consider well whan you must put your wate •• togyther: and whan, you must empty you • first water • out of your Pyrami • or Cone. El • you will 〈◊〉 . 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 aequall 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 nūber 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 Fundamētall 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. For this, may I (with ioy) say, EYPHKA, EYPHKA, EYPHKA: thanking the holy and glorious Trinity: hauing greater cause therto, then

Vitruuius. Lib. 9. Cap. 3.

☞ God b • thanked • or this Inuention • & the frui • e • nsuing.