Part 2

In lyke sorte, the necessary, wonderfull and Secret doctrine of Proportion, and proportionalytie hath purchased vnto it selfe a peculier maner of handlyng and workyng: and so may seme an other forme of _Arithmetike_.

[3.]

Moreouer, 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._

[4.]

Practise hath led _Numbers_ farder, and hath framed them, to take vpon them, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_ and _Irrationalitie_. (For in 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 them: as 20 + ²√24. ³√16 + 33 - ²√10. ⁴√44 + 12¼ + ³√9. And so, infinitely, 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,

[* Anno. 1550.]

of 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:

[5.]

and 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_.

And first: who, nerer at hand, can be a better witnesse of the frute receiued by _Arithmetike_, then all kynde of Marchants? Though not all, alike, either nede it, or vse it. How could they forbeare the vse and helpe of the Rule, called the Golden Rule? Simple and Compounde: both 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.

The Mintmaster, and Goldsmith, in their Mixture of Metals, either of diuerse kindes, or diuerse values: how are they, or may they, exactly be directed, and meruailously pleasured, if _Arithmetike_ be their guide? And the honorable 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

[R. B.]

(aboue 200. yeares ago) inuented. And forasmuch as I am vncertaine, who hath the same: or when that litle Latin treatise, (as the Author writ it,) shall come to be Printed: (Both to declare the desire I haue to pleasure my Countrey, wherin I may: and also, for very good profe of Numbers vse, in this most subtile and frutefull, Philosophicall Conclusion,) I entend in the meane while, most briefly, and with my farder helpe, to communicate the pith therof vnto you.

First describe a circle: whose diameter let be an inch. Diuide the Circumference into foure equall partes. 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.

[* Take some part of Lullus counsayle in his booke de Q. Essentia.]

Now, 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_

[* Note.]

(for the * _Temperament_ is counted none. And for it, you must put a Ciphre, if at any time, it be in mixture).

HOTE +C | | + | | + | | +E | MOIST A TEMPERATE B DRYE +------+------+------+------+------+------+------+------+ |D | + | | + | | + | | + COLD

Counting 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:

[Note.]

I 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.

[The Second Rule.]

“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. ___________________________ | | | | {P}. _2._ | _Hote. 4._ | | | | | {P}. _1._ | _Hote. 3._ | |____________|______________|

Now 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.{P}. is double to 1.{P}. 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}-3=8. And now adding to eche part 3: you haue (as afore) 3{x}=11. _________________________ | | | _ | {P}. _2._ | _Hote. 4._ | ⅓ _ _The forme_ | | | _ _3⅔ resulting._ | {P}. _1._ | _Hote. 3._ | _ ⅔ |___________|_____________|

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 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.

Now 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

[Τακτικὴ.]

Τακτικὴ. “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

[☞]

_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 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_.

To 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:

[This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age: hauing no issue by his wife: Daughter to the Duke of Somerset.]

And 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.