CHAP. I. The definition and subiect of Arithmeticke.

ARithmetick is the Art of num­bring well.

This definition is short & plaine, being drawen from the end of Arithmetike, af­ter which sort other Arts al­so are defined.

To number vvell, is to expresse and pra­ctise the vvhole force, properties, and vse of num­bers.

It appeareth then that the Subiect of A­rithmetike (that is to say, the thing wherea­bout that Art is wholly occupied and where­unto all the rules thereof are to be referred) is number.

Number is a Multitude consisting of vni­ties.

It is the definition of the seuenth booke of Euclide. Aristotle in his Categories seemeth [Page 2] to define it thus. A number is a discreet quan­titie not hauing position but order. The which definition agreeth with that of Euclid, sauing that it distinguisheth an vnitie from a point by a certaine contrary qualitie which is in them; attributing to a point position or place, to an vnitie only order & consequence. Whereby it appeareth that a number is the ioyning togither of things following one an other in order not touching one the other much lesse cleauing togither, and that in such sort that you can not determine any bond or common knot in them, whereby they should be knit togither and continued.

The beginning of number in an Vnitie.

An vnitie is that, whereby euery thing that is, is said to be one.

An vnitie properly is no number, because it is no multitude, for multitudes onely are numbred: neither is it a part of a number be­cause that euery part of a number ought also to be a number. It answeareth in proportion to a moment, and to a point: whereof the one is the beginning of time, the other of magni­tude, and yet no part of them: as Aristotle proueth in the fifth and in the beginning of the sixth booke of Phisicks: yet in Arithme­ticall numeration it is wont to be taken for a number, and that the least that may be.

The diuision of number shall be drawen out of numeration.

The Scholemen deuide numbers into Di­gites, [Page 3] articles & mixt numbers: to this end as I think that they might make a distinction in the naturall consequence of numbers: wherin first of all the nine simple figures are to be cō ­sidered, the which the number 10. succeedeth as it were a knot knitting them to the mixt numbers following. But for so much as this is no lawfull diuision of numbers, we will draw an other out of the last part of Numeration, which we call Diuision, the vse whereof is exceeding great. In the meane time if the for­mer two numbers metaphorically so termed (which haue giuen occasion to the rest vsed of the Latines who say Arithmeticum paucio­res habere digitos, plures autem articulos homi­ne) if that diuision I say do displease you, you may peraduenture more fitly deuide num­bers into simple, decades and compounde numbers.

CHAP. II. Of Notation, and of the first part thereof.

• Simple, consisting of
  • Notation which hath two parts
    • One more generall, wherein the figures are consi­dered,
      • By them­seluyes, and so there are
        • Nine signifying figure.
        • One without any signi­fication.
      • Or
      • One with another, where ther are to be cōsidered,
        • Degrees, which are
          • Simple, containing single numbers.
          • Multiplied cōtaining
            • Tens.
            • hun­dreds.
        • and
        • Periods, whereof some are
          • Single
          • Com­pound
            both being
            • perfect
            • or
            • imper­fect.
    • The other more spe­cial, wherin the nū bers to be cōsidered are either of
      • One terme, as whole nūbers
      • More termes, and they be either
        • Of one kind, as partes, o­therwise called fractions.
        • Of diuerse kindes, as mixt numbers.
  • Numerration. A.
• Compared. E.

ARithmeticke is double: Simple and com­pared, Simple is that vvhich teacheth the practise of simple numbers vvithout compa­rison.

This partition springeth of the diuersitie of the subiect. For numbers are considered ei­ther by themselues alone absolutly, or else one with an other in respect of their proportio­nalitie, or proportion which they haue one to an other. Hereon commeth this foresayd di­uision containing all things belonging to this Art.

Simple Aruthmeticke consisteth of two parts, Notation and Numeration.

Notation is the vvay havv to set downe numbers in vvriting, and to expresse them being vvritten.

The Greekes called it Semeiosis, and it is more fitly called Notation then NUmeration, because it teacheth vs how to note, that is, to write euery number, and to expresse it when it is written.

There be two parts of it. The first expresseth generally the value of the figures vvherewith the numbers are vvritten.

The figures vvherewith numbers are written are ten.

As 1. 2. 3. 4. 5. 6. 7. 8. 9. 0. Neither the Greekes nor the Latins vsed these figures. For the Ro­m [...]nes vsed onely thesE seuen letters, M. D. C. L. X. V. I but the Grecians vsed all the letters of the Alphabet, being marked with certaine [Page 6] prickes. Many therefore thinke that these fi­gures were inuented by the Arabians, the which people in old time practised this Art and the whole science of the Mathematickes very diligently, where of beside the word Al­gorithme, whereby they expressed the Art of numbring, many other words left vnto vs in this kinde of knowledge may be a sufficient testimonie. For the Latins in the former rude and barbarous age forsaking the wel-springs of the Grecians, followed after the small ri­uers of the Arabians. Valla and Cardane thinke them to be inuented by the Indians. Some there be that attribute thē to the Phoe­nicians. They haue bin retained by the com­mon consent almost of all people, because they be most conuenient to number withall, and most fit to receaue all kinde of practise vsed in this Art.

The figures are considered eyther by them­selues alone, or one with another. By themselues alone, when they are without composition, or as it were without composition, for then the first nine haue a certaine value.

The tenth which hath the forme of a circle signifieth nothing standing by it selfe alone, but be­ing placed with the rest, it maketh to the increase of their value.

As, 1. is asmuch as one, 2. two, 3. three, 4. four, 5. fiue, 6. six, 7. seuē, 8. eight, 9. nine, 0. of it own nature is without value, yet hauing his place wherein none of the other signifying figures [Page 7] is placed, it encreaseth the value of thē which follow: for it is neuer set in the last place. It is called commonly a Cyphar.

Figures are considered one with an other be­ing ioyned together, and then the processe and or­der of their places encreaseth their value.

For so much as euery number may be en­creased infinitely, therefore by reason of their diuers encrease amoūting to tens, hundreds, thousands ten thousands, &c. it is vnpossible to write euery one of thē with a diuers figure. For then should the varietie of the figures be also infinite. But what is wanting in the fi­gures, that their places through often repea­ting of them, maketh vp. In continuance of which places the figures take vnto them a di­uers signification, & that so much the greater by how much the more in ioyning the num­bers togither they come nearer to the left hād.

The places of numbers are distinguished by degrees and periods. A degree is a place where­by the value of each number is gathered.

A degree is of two sorts: the first is of sim­ple numbers, the other of multiplyed numbers, to wit, the second degree is of tens, the thirde of hundreds, and so foorth after the same manner in the rest.

Numbers according to the custome of all people proceede from the left hand toward the right, so that the least num­ber occupieth the last place next to the right hand. Wherefore the last figure in value is [Page 8] the first, & euery figure in the first place kee­peth his first and proper signification. In the other places euery one is armed with a dou­ble power, to wit, with his owne and with an other borrowed for a time. For this consi­deration is generally to be obserued in the places: that, that which followeth is twise as much as that which went before.

As for example: the figure 4. by it selfe a­lone is foure: and so much also it signifieth in the first degree. But being remoued into the second, it signifieth fortie, as you see here, 40. Againe, being set in the third degree it is ten times so much as it was in the second: that is, foure hundred, as here, 400. And so forth af­ter the same manner both in the other figures and degrees.

A periode is that vvhich comprehendeth the degrees, and is eyther perfect, consisting of three degrees, or imperfect contayning lesse then three.

To conclude, that Period vvhich consisteth but of one trinitie or ternarie of degrees is single, but that vvhich consisteth of more then one, is compound.

As 1 2 3 is a perfect single period: 1 2 3 4 5 6 is a perfect compounde periode. Also 1 2 is an vnperfect single periode: 1 2 3 4, 1 2 3 4 5 is an vnperfect cōpound period. 1 2 3 4 5 6 7 8. is an vnperfect triple periode. In numbers the degrees are to the periodes, as the commata & cola are the periodes in an oration.

CHAP. III. Of the other part of Notation.

THus much concerning Notation in gene­rall: now follovveth the other part which expresseth the numbers vvritten more parti­cularly.

And here vve haue to deale vvith numbers, either of one terme, or more. Whole numbers are vvritten vvith one terme.

As 20, 24, 360, 1234, 1578, they be also called numbers of one bande or rancke.

Hovv vvhole numbers of a single periode should be set downe, may be gathered by tha vvhich hath beene saide before. But that the compound numbers may be expressed, they must vvith prickes or small lines be deuided into single periods, and be so vttered that at the first pricke vve name a thousand, at the second a thousand thousand, at the third a thousand thousand thou­sand, and so forth going forward after the same order, till vve haue numbred the vvhole.

Numbers be they neuer so great may easily be expressed, if distinguishing them with pe­riodes you vtter them by partes, in such sort that both the value of the figures and the power of the degrees, may gouerne them when they be vttered. So that at the end of the first periode you shall name a thousand, at the end of the second a thousand thousand (which some call a million) and to be short, at the end of the third periode, you shall name [Page 10] a thousand thousand thousand: obseruing the same encrease in all the periodes following, be they neuer so many. Hereupon it com­meth to passe that the least signifying figure of the periode following, is greater then any of the periode going before. Moreouer the ready vttering of numbers maketh a man to write them speedily, & contrariwise the spee­dy writing of them maketh a man to vtter them readily. For example, hauing distin­guished this number 34˙567˙890 with prickes you shall expresse it thus beginning at the left hand. At the first periode you shall say thirtie foure thousand thousand or else thirty foure millions. At the second, fiue hū ­dred sixtie seuen thousand. At the third eight hundred and nintie.

The number following you shall expresse thus 2˙016˙542˙009˙873. Two thou­sand thousand thousand thousand: then six­teen thousand thousand thousand, fiue hun­dred fourtie two thousand thousand, 9. thou­sand eight hundred seuentie three.

The Romaines were wont to number by hundred thousands. The which fashion if you list to follow you shal set down a pricke next to the second degree of the second periode & expresse the other numbers following, as you did the first. As in this number 77˙89˙320, you shall say thus seuenty seuē hundred foure score and nine thousand three hundred and twentie. After this manner Plinie in his se­cond [Page 11] booke 108. Chap. affirmeth according to the saying of Artemidorus that the longi­tude of the earth inhabited West ward from the pillers of Hercules in India, is eighty fiue hundred seuentie and eight thousand miles. The which number must be set downe thus.

85|78|000.

Thus much for the notation of numbers of one terme: now follow the numbers of manie termes. And they be eyther of one kinde as partes (which we commonly call fractions,) or of diuerse kinds as mixt numbers.

Partes are set downe with two termes deui­ded one from an other with a line: whereof the vppermost numbreth the parts, the neathermost nameth them.

Although an vnitie by it self of it own na­ture can not be deuided, yet if we regarde the subiect thereof, or the magnitude whereto it is applied, it may be deuided. And yet the vni­ties of the number proceeding of this diuisi­on do not arise of the very vnitie it selfe, but of the deuiding of the magnitude, that is: they come not of the vnitie as it is by it selfe alone, but as it is ioyned to some other thing. As a shilling can not be deuided as it is one, but as it is a peece of money, which customablie is deuided into testons, groates, three pences, two pences, pence and half pence, &c. Wher­fore although partes be called by the name of numbers, and are written with the figures be­longing [Page 12] to whole numbers: yet in very deede they are no numbers, if Euclides definition of a number be true. But in the notation of these partes there are two things to be consi­dered, the quantitie of the parts (for a whole number may haue many parts, both great and finall) and the number of them. Hereupon it commeth that when they be written they require two termes or numbers: whereof the vppermost declareth the number of the parts the which are to be found out, whereupon it is called the Number, or Numerator. But the neathermost naming the parts of the whole, expresseth their quantitie: wherefore it is called the name, or the Denominator. As 1/2 signifieth one part of the whole deuided into two halfes, that is; one halfe. 1/3 one third part 3/4 three fourth partes, or three quarters. 5/6 fiue sixth partes, that is, fiue partes of the whole deuided into sixe parts.

Hereby it appeareth that the greater the Denominator is, the lesse is the quantitie of the partes.

Partes are eyther principall, or of the second sort. The one haue their originall of the first di­uision of the vvhole, the other are parcels of the principall parts.

They may be called Simple & Deriuatiue parts: whereof the one ariseth of the first di­uision of whole numbers, the other of brea­king the parts into smaller parcels. Their no­tation and value differeth much. The princi­pall [Page 13] parts are written after the maner before named. But the parcels, or the parts of parts called commonly fractions of fractions, are set downe by the first toward the left hand, with­out any line betweene them for difference sake, as 5/12 1/3 fiue twelue parts of one third part, 3/4 7/8 three fourth parts, of seuen eight parts: or else otherwise by the preposition 5/6 out of 3/10.

Moreouer, partes are eyther proper or im­proper. They are lesse then the vvhole: these are eyther equall vnto, or greater then the vvhole.

True and proper partes in deede are lesse then the whole, and therefore that which they signifie can not be expressed by an vnitie, whereupon they haue the numerator alwaies lesse then the Denominator. But those parts whose Numerator is equall or bigger then the Denominator, are improper partes, for that they may be expressed by an whole or mixt number: as 3/3 are as much as one whole: 4/2 are two whole, 6/5 are one whole, and 1/5 more then the whole.

Wherefore although whole numbers be sometime set downe as partes, yet when the Numeration is once finished, they are neuer written after this manner, but must be redu­ced to whole numbers: for you cannot fitly say, 2/2 two halfes, or 4/3 foure third partes, or 8/4 eight quarters: but one whole, one whole with a third part, &c.

There remaine as yet mixt numbers, which are whole numbers with partes. But their no­tation may be easily gathered out of the former kindes.

In Geometrie they are not accounted for numbers: but when the measure of a thing cannot be expressed but by a mixt number, it is called irrationall, as though it were not indeede to be expressed by a number. Here­vpon commeth the name of Surde numbers.

CHAP. IIII. Of Numeration, the other part of simple Arithmeticke, and of the first kind thereof.

• Simple, or of nūbers of one kind which is either of
  • Whole nūbers, and is either
    • Prime, as
      • Addition.
      • Subductiō.
    • Second as
      • Multipli­cation.
      • Diuision.
  • Parts.
• Mixt, or of numbers of diuerse kindes.

HItherto we haue entreated of Notation, nowe followeth Numeration: which of [Page 15] two numbers giuen findeth out the third.

As the framing of an argument is in Lo­gicke, euen so is numeration in this art. For as the Logicians in reasoning doe inferre the conclusion by the premisses, euen so the A­rithmeticians by numbering doe inferre a di­uerse number from the numbers giuen.

The numbers giuen are either of one and the selfe same kind, as whole numbers, or partes, or of diuerse kinds as mixt numbers.

Numeration of whole numbers is when the numbers giuen are whole numbers.

And this Numeration is double. Prime: which numbreth one number with an other one­ly once, and that eyther by adding or subtracting them: whereupon it is called Addition or Sub­traction.

As if these numbers 360. and 15. were giuen either to be added or subtracted, you must either adde 360. once to 15. or else sub­tract them once and not many times as in the second numeration. Wherefore this may well be called single numeration, and the other manifold numeration.

Addition is a Prime Numeration, which ioy­ning one number to an other, findeth out the totall of the numbers giuen.

Touching Addition of simple figures the rule of reason teacheth vs how to doe it, but the addition of compound figures may be learned by this rule.

If, proceeding orderly from the first of the [Page 16] numbers giuen to those which folow, you set the particular summes (being lesse then ten) of the figures of euery degree added together alone by themselues, each one vnder his owne degree, putting the ouerplus (that is, for eue­rie ten, an vnitie) to the degree following: you shall finde out the totall of the numbers gi­uen.

Addition of single nūbers may be wrought by the direction of nature only, wherefore it may be practised in our mindes, and brought hither for the speedie dispatching of our worke. As 5 and 6 are 11, 7 and 8 are 15, 9 and 9 are 18. But the Addition of com­pound numbers goeth on by peecemeale frō part to part, so that it requireth the helpe of art. The first precept to be obserued in this place, is to number all the figures of euerie place alone by themselues, as though they were but single figures: and this must be done in all the partes of Numeration following. The seconde is to number the figures of one and the same degree onely together, that you confound not the one with the other. Last of all, you must set downe the totall made of the parts, in such sort, that if they increase to ten or more, onely the figure next to the right hand must be written vnder that degree, the other which remaineth must be added to the degree following. As for example, adde 983. to 402. First, setting downe the numbers in such sort one vnder an other, that the figures [Page 17] of euery degree may answere may answere one to an other, I must beginne at the first and say, 2 and 3 are 5, the which I set downe vnder that degree: then 0 and 8 are 8, the which I write vnder the second place. Then 4 and 9 are 13, which number for that it is in the last place, I set it downe whole. So that the totall of the num­bers giuen is, 1385. The example standeth thus.

Item adde 89647 vnto 78450. The num­bers being duely set downe one vnder an o­ther, according to the order of their degrees, and a line drawne vnderneath them, I begin to number at the right hand saying, 0 & 7 are 7. Then 5 and 4 are 9. Item 4 and 6 are 10. But for so much as the totall is written with two figures, I set downe the cypher on­ly vnder that degree, and keepe the vnitie in my minde. Then 8 and 9 are 17, and the v­nitie which I kept maketh 18, where againe I set downe 8 because it is next to the right hande and keepe the vnitie next to the left hand in my minde. Lastly 7 and 8 are 15, and one which I kept are 16, the which I set down wholly. So the totall of the numbers giuen is, 168097. The example is to be set downe in this order.

In this part of numeration and in the rest, there are said to be but only two numbers gi­uen, although sometime there be more to be added together, for that the two first numbers onely are added together, and then if there be more, the totall is added to that which fol­loweth, as in the example following.

Here, I say, 2 and 6 are 8, 8 and 1 are 9, ta­king 8, which is the totall made of 2 and 6, for one simple number, and adding it to 1 stan­ding aboue it, &c.

Subduction is a prime Numeration, vvhich taking one number from another, findeth out the Remainder of the numbers giuen.

The vse of it is to find out the difference be­twene two numbers: whereupon it is requi­site that the numbers giuen, should be vne­quall, and that from which the subtraction is made, must be the greater.

Subduction of simple numbers may be drawen of the table of our mind. But cōpound nūbers are to be subducted by the the Theoreme following.

If proceeding orderly from the first of the numbers giuen, you set the remainder of eue­ry figure of one & the same degree, subtracted one from the other (either by it selfe alone, or by borrowing ten of the degree folowing) se­uerally ech one vnder his own degree: the re­mainder of the numbers giuen may be found.

In euery part of numeration there must a certaine practise go before it: the which must not then be to seeke when we should worke, but it must be drawen out of the threasure of our minds, and ministred vnto vs. So that in subductiō there is required of vs a ready fore­sight, that we may know what nūber will re­maine, when any one of the figures is taken frō another by it selfe alone, or els added vnto ten. As 5 from 8, there remaine 3, 4 from 9 there remaine 5. Or by adding ten to the sin­gle figure: as 9 from 17, there remaine 8, 6 from 15, there remaine 9, &c.

In compound and great numbers, it is most conuenient to place the greater vppermost, and the lesse beneath, and then to number by the parts going on towarde the left hande. For that way is more easie then the other which proceedeth from the left hande to the right, as in the examples following.

In the first example euery thing is plaine.

In the second I beginne to number, 2 from 4 remaine 2, 3 from 3 remaine 0. Then for so much as 8 can not be taken from 7, the figure of the same degree, I borrow an v­nitie of the degree following, and ioyne it to 7 thus 17, and drawe 8 out of 17 there re­maines 9 to be noted vnder neath. This may be done, if (as some are wont) I take the di­stance of the neather number from 10. & adde it to the vppermost number, or set downe the vppermost number only, if the neathermost be 10. As for example, because 8 differeth 2 from 10, I take 2 and adde them to 7 which make 9 to be written vnder the third degree. But for so much as I borrowed 1 of 8 in the vppermost number following, there remaine but 7, I draw therefore 9 from 17 (taking an vnitie from 5 following) there remaine 8. In the last degree, because there is no figure be­neath which aunswereth to that aboue, I set not downe the whole 5 but 4, because I bor­rowed one of it before.

In the third example, that the signifying figures may be subducted out of the cyphers, we must borrow of the numbers following an vnitie, as we did before. First take 4 from 5 there remaines 1, then because you cannot take 8 from 6, you must subtract it from 16, there remaines 8. For although the figures following, from whence I borrow the vnitie, be a cyphar, and therefore without significa­tion, [Page 21] yet for so much as it hath a signifying figure before it, I may borrow an vnitie of it. Then I go on and take 9 from 9, there remai­neth nothing. Moreouer I take 1, not from 10, because there was an vnitie borrowed of it before, but from 9, there remaine 8. And for so much as there is no figure to be drawen out of that which followeth, I set downe 9. Likewise I write downe the last figure lesse by one then it is, because I presuppose that it wanteth that vnitie which I tooke from the first cyphar.

CHAP. V. Of the second kind of Numeration.

HEtherto of prime Numeration. Now fol­loweth second Numeration, vvhich num­breth one number vvith an other, so often as one of the two numbers giuen requireth, and that ey­ther by multiplying or diuiding: vvhereupon it is called Multiplication or Diuision.

Second Numeration is either often Addi­tion, as Multiplication: or often Subtraction, as Diuision. For one of the numbers giuen is so often eyther encreased or diminished, as the other, eyther the multiplyer or diuisor re­quireth, according to the number of the vni­ties contayned in them.

Multiplication is a seconde Numeration, vvhich ioyning together the multiplicande, so often as there bee vnities in the Multiplier, [Page 22] bringeth foorth the facit.

This is the 15. definition of the 7. Booke of Euclide. It maketh no matter which num­ber you make the multiplicande, or which number you make the multiplier, as it appea­reth by the 16. prop. of the seuenth Booke of Euclide, which sayth: If two numbers mul­tiplied together, the one into the other, pro­duce any numbers, the numbers produced are equall the one to the other: that is, they make one and the same number. Notwithstanding the Scholemen set the greatest number vp­permost for the multiplicande, and the least neathermost for the multiplier, for that it see­meth most conuenient for young beginners.

Multiplication of single numbers is to be con­ceiued in our mindes by this Theoreme.

If eyther of the two numbers giuen be di­uided into certaine partes, the product com­ming of the whole numbers multiplied toge­ther, is equall to the product made by one of the whole numbers and the parts of the other number so diuided. The 1. proposition of the second booke, or else, if both the numbers gi­uen be diuided into certaine parts, that which is made of the whole number, is equall to the product made of their parts.

That we may multiply easily and readily, we must haue in our mindes, a table whereby we may know what the pruduct of euery sim­ple figure is, being multiplied one by an o­ther: the which thing may be easily done by [Page 23] the rule of whole numbers and their partes. Euery child by the direction of nature can tell how many twise foure, or foure times fiue, or thrise sixe doe make. But if you happen to sticke in greater numbers, a litle exercise will make this table very readie. As if you would knowe how much seuen times eight is: di­uide eyther of the two numbers giuen, into as many partes as you list, as 7. into 2. 3. 2. Then multiply 8. by euery one of these parts, and adde the particular productes together, and you shall haue 56. How many are eight times 9. diuide 9. into 3. 3. 3. and multiply 8. by those partes, adding the products toge­ther, so you shall make 72. The same may be done if you diuide both the numbers. The examples must be set downe thus.

[Page 24]

The Scholemen frame this Table by this Theoreme. Two numbers being giuē which ioyntly together are more then tenne, if you multiply the difference of each of them from tenne one by the other, & then subduct cross­wayes one of the differences out of one of the numbers giuen, the product and the remain­der parted into diuerse degrees, shall be the product of the numbers giuen. As you see in these examples.

Multiplication of compound figures is drawen out of the Theoreme following.

If proceeding from the first of the nūbers giuen toward the left hand, you multiply the figures of the multiplier, into euery one of [Page 25] the multiplicand, and ioyne together the par­ticular products being lesse then ten, setting them orderly vnder their multiplier, and put the ouerplus to the degree following: the to­tall made of the partes, is the product of the numbers giuen.

For example sake, multiply, 365 by 3. Here the numeration must proceede from part to part. Hauing therefore set downe the numbers, as you did in the former kindes of numeration, multiply the neathermost figure seuerally into euery one of the vppermost thus: three times 5 are 15, I set downe 5, and keepe 1: thrise 6 are 18, and one which I kept, are 19, I set downe 9 and keepe one to be ad­ded to the degree following: thrise 3 are 9, and one are 10, which I set downe wholy. The example standeth thus.

Now take an example of multiplication to be wrought by the parts of both the num­bers giuen. Multiply 1568 by 54: Item mul­tiply 3508476 by 2509. In both these num­bers you shall haue so many rowes of num­bers, as there be figures in the multiplier. Wherefore we must take diligent heede, that we confound not the particular products, and beware that in distinguishing the products, we set the first figure of euery one of them [Page 26] vnder his particular multiplier, as you see here. [...]

An abridgement of multiplication.

Neither are the shortest wayes to be neg­lected: this therefore is an especiall abridge­ment of multiplication.

If one, or both the numbers giuen, haue cyphers in the beginning, thē if multiplying only the signifying figures together, you put the cyphers to the totall, you shall haue the product of the numbers giuen.

The correllary of this abridgement may be this.

If the last figure of the multiplier be an v­nitie, and the other cyphers, then setting the cyphers before the multiplicand, you shall haue the product of the numbers giuen. The exsamples.

Diuision is a numeration, vvhich drawing one number from another, as often as may be, findeth out the quotient of the greater.

In diuision there are three numbers to be considered, the Diuidende, the Diuisor, & the Quotient, the which must be so placed, that the Diuidend may stande aboue, the Diuisor beneath, and the Quotient at the side, or be­twene them both. The vse of diuision is to declare how many times the lesse is contained in the greater. The Artisicers terme diuision by a Geometricall phrase, calling it Compari­son, which is the applying of a measure giuen to any right line, as here,

A—|—|—|—|—B

C—D the line CD, is compared with the line AB to see how many times it may be cōtained in it. Or it is the ioyning of one side to an other side, giuen to make a rightangled parallelogram. As therfore in multiplication, the multiplying of two sides togither maketh a right angled figure, if the lines meete per­pendicularly, euen so in diuision, the diuiding of the area or platforme of the right angled fi­gure giuen by the length as it were by the di­uisor findeth out the breadth which is repre­sented by the quotient, as you see here.

Multiply the side A C, by the side C D, the product will be the area or platforme of the right angled figure A B C D. Likewise diuide the platforme of the right angled fi­gure E F G H, by the side G H which is the length, you shall find out the other side E G which is the breadth which answereth to the quotient in diuision. Hereupon it is called the breadth of the comparison or of the com­pared figure.

A quotient is a part of the diuidende, hauing the same denomination vvith the diuisor.

This definition is taken out of the 39 prop. of the seuenth booke of Euclide which sayth: If a number measure any number, the number measured shall haue a part of the same Denomi­nation with the number which measureth it. And contrariwise, as it is in the 40. prop. If it haue a part, the number vvhereof the part taketh his denomination, shall measure it. For Euclide ta­keth measuring here for diuiding: as for ex­ample, because 48 may be diuided by 6 into 8 the quotient, I say that 8 being the quotient, is the 6 part of 48, and taketh his denomi­nation of the Diuisor. And contrariwise, be­cause 8 is the 6 part of 48, therefore 6 diuide 48. Likewise if 6 diuide 54, the quotient will be 9, and 9 therefore is the 6 part of 54, and because 9 is the sixth part of 54, therefore 6 diuide 54, &c.

A part of a number is a lesse number in re­spect of a greater, when the lesse measureth the [Page 29] greater.

This is the first definition of the 5 and the third def. of the seuenth booke. Euclide will not haue euery number that is lesse then a greater number, to be a part of the greater, but onely that which being taken oftentimes by it selfe alone, measureth the greater, that is, eyther maketh it, or taketh it cleane away: As two is the third part of 6, fiue is the halfe part often. For that 2 being three times taken, eyther maketh or taketh away 6, and 5 twice takē make 10 or destroyeth ten. This kind of part is called commonly, pars metiens, or men­surans. A measuring part and of the barba­rous it is called pars aliquota, an aliquot part, that is, a quotient: but the number which by it selfe alone being oftentimes taken, maketh not a number without the helpe of some other part or number, is the parts of the whole, v­sing the worde parts in the plurall number for distinction sake. As 2 are the parts of fiue, because two make not fiue alone, without the helpe of three, or some other numbers. This kind of part they commonly call, pars consti­tuens, or componens: of the barbarous it is cal­led, pars aliquanta.

Diuision of a simple quotient may be taken out of the table of Multiplication. For by the same numbers that a number is made, by the same it may be diuided.

The first principle to be obserued in diui­sion is to consider wittily, and to know rea­dily [Page 30] by what number, and what number eue­ry one of the single figures do diuide another, as sixe times 7 are 42, therefore 7 diuide 42 by 6, and 6 diuide it by 7. Item seuen times 9 are 63, therefore 9 diuide 63, and the quotient is 7, or 7 diuide it, and the quotient is 9.

The diuision of a number that hath many fi­gures, is wrought by the Theoreme follow­ing.

If, beginning at the last figure of the num­bers giuen, you multiply the particular quo­tient of the diuisor contayned in the diuidend (setting it downe aside by it selfe) into the Diuisor, and then subtract the product from the diuidend, doing this as often as may be, by setting forwarde the Diuisor towarde the right hand, till you come to the first figure of the diuidend, the quotient of the parts will be the quotient of the numbers giuen.

Although diuision commonly be wrought by finding out the quotient by multiplica­tion & subduction: yet the principall worke wherein the whole force or vertue of diuision consisteth, is onely the finding out of the quo­tient, which being once found, the diuision in minde is supposed to be done. But when we haue found out the quotient, we are to thinke and consider with our selues into how many partes the diuisor cutteth the number which is set ouer it, and how often it may be drawen out of it. That this therefore may be set before our eyes, and that in remouing [Page 31] forward the Diuisor, the remainder (wherein afterwarde the quotient must be sought for) may more manifestly appeare, the truth of that which we conceiued before, is proued by multiplication and subtraction.

To expresse this by examples, diuide 4936 by 4, we must seeke how often the Diuisor 4 being set vnder the last figure of the diui­dend, is contayned in the same. I say then that 4 is contayned in 4 once: wherefore I set downe 1 in the quotient, and take 4 out of 4, there remaines nothing. Then set I the Di­uisor one degree forwarde towarde the right hand, where 4 are contayned in 9 twice: wher­fore I note 2 in the quotient, and multiply the Diuisor by it, the product is 8, which being subducted from 9, there remaine 1. The figures out of which the subtraction is made, must straight way be blotted out. Againe set forwarde the diuisor: there 4 may be taken from 13 three times, wherefore I set 3 in the quotient, and multiply 4 by it, the product is 12, which being drawen out of 13, there remaines 1. To conclude, I find 4 in 16 foure times, therefore I set downe 4 for the quoti­ent, and multiply 4 by it, which being ta­ken out of 16, there is nothing left. The ex­ample.

Againe deuide 1008 by 36. In this exam­ple there be many things to be taken heede of. First, because the last figure of the diui­dend is lesse then the last of the Diuisor, ther­fore I set not 2 vnder 1, but one place farther towarde the right hande vnder the o. And first I consider with my sefe, how often three may be had in ten, I find it to be thrise, and there remaineth one. But for so much as I cannot subduct the Product made by multi­plying the quotient into the whole diuisor, out of the number standing aboue it, to wit, 108 out of 100, therefore I take a quotient lesse then that by one, that is 2, whereby the diuisor being multiplied, there arise 72, which being taken out of 100, there remaine 28 to be written ouer the head. Then the diuisor being set forward, I see three to be contained in 28 nine times, but because I must haue re­gard also of the figure following, I set but 8 in the quotient, which being multiplied into the diuisor, make 288 to be drawen out of the number set ouer it. The example is thus.

It appeareth therefore by this, that the que­stion must be made not of the whole Diuisor, vnlesse it be a single number, but only of the last figure of the Diuisor, and euery particu­lar must be but a single figure, as if you shold seeke how often 2 were in 21, you can not take it tenne times, but 9 times, but 9 times at the most. Moreouer the nature of the thing requireth that we find out alwayes such a quotient, as being multiplied by the Diuisor, maketh no greater a number then the diuidend is. See the examples following.

An abridgement of diuision.

If the Diuisor end in cyphers, the worke may be wrought by the signifying figures a­lone setting the cyphers in the meane time, vnder the vtmost figures of the diuidend, next to the right hand. But if the last figure of the Diuisor be an vnitie, and the rest cy­phers, then setting the Diuisor as before, and [Page 34] taking the figures that haue no cyuphers vn­derneath them for the quotient, the diui­sion is dispatched. the figures that remaine after the diuision is done, must be set downe as the partes are with the diuisor vnderneath them for their denominator.

As for example, diuide 165968 by 360. Item 6734 by 100: the example shall stand thus.

CHAP. VI. Of the double diuision of numbers.

• First they are either
  • Odde.
  • Or
  • Euen: which may be diuided either,
    • One way only, and are either,
      • Euenly e­uen.
      • Or
      • Oddely euen.
    • Many wayes, and are both euenly, and oddely euen.
• Secondly, as they be cōsidered, either
  • By thēselues alone, and then they are
    • Prime.
    • or
    • Compound.
  • One with an other, and then also they are
    • Prime.
    • or
    • Compound.

NVmbers are diuided two manner of wayes, first into euen and odde numbers. An euen number is that which may be diuided by 2. An odde number is that vvhich can not be diuided by 2.

This is the 6 & 7 definition of the seuenth booke of Euclide, whence this difference of numbers drawen out of their diuision, is ta­ken, as that also which followeth, whose vse in Arithmeticke is very great. Euen numbers are as these following, 2. 4. 6. 8. 10. 12. 14. and so forth after the same order, alwaies omit­ting one. Odde numbers are, as 3. 5. 7. 9. 11. 13. 15. &c.

Of euen numbers some may be diuided but one way onely, other some may be diuided many waies. Those which may be diuided but one way onely, are euenly euen, or oddely euen.

A number euenly euen, is that vvhich an euen number diuideth by an euen number.

This is the eight definition of the seuenth, such are 4. 8. 16. 32. 64. &c. that is to say, all the numbers from 2 vpward doubled by 2. As appeareth by the 32 prop. of the 9 booke.

A number odly euen, is that vvhich an odde number measureth by an euen number.

As it is in the 9 definition of the seuenth booke. Euclide calleth it an euenly odde nū ­ber, as 6. 10. 14. 18. 22. For 3 an odde number diuideth 6 by 2 an euen nūber: 5 an odde nū ­der diuideth 10. by 2 an euen nūber. Such are all the numbers whose moytie or halfe is an [Page 37] odde number, as appeareth by the 33 prop. of the seuenth booke.

The numbers vvhich may be diuided manie vvayes, are euenly euen and odde: which may be diuided both by an euen number, and by an odde, into an euen number.

This third kind of numbers, is let passe of Euclide among the definitions of the seuenth booke, but yet not neglected in the proposi­tions of the ninth booke. The examples ther­of are these, 12. 20. 24. 28. 56. 144. As 2 an euen number diuideth 12 by 6 an euen num­ber, and 3 an odde number diuideth it by 4 an euen number. Item 2 an euen number di­uideth 20 into 10 an euen number: and 5 an odde number diuideth it into 4 an euē num­ber. Such are all those numbers which are neither doubled by two from 2 vpward, nor haue their halfe an odde number as it is in the 34. prop. of the 9 booke.

Againe, numbers be distinguished otherwaies being considered both by themselues, and one with another. Being considered by themselues, they are eyther prime, vvhich may be diuided by an vnitie onely: or compound, vvhich may be diui­ded by some other number.

This is the 12 and 13 definition of the se­uenth booke, where Euclide diuideth num­bers into prime & compound numbers. The matter of this diuision is taken out of the 34 prop of the seuenth booke, which sayth: That euery number is eyther a prime number, or else [Page 38] diuided by a prime number: that is, a com­pound number, as appeareth by the 33 prop. of the same booke. A prime number is that which no other number diuideth besides an vnitie, sauing that it measureth it selfe. It may be called an vncompound number for that it is made of no number: as 2. 3. 5. 7. 11. 13. &c. A compound number is that which some other number maketh, being taken certaine times, as 4. 6. 10. 12. Of this sort are first of all, all euen numbers, then all those odde numbers which in the eleuenth definition of the seuenth booke are called oddely odde: which may be diuided by an odde number into an odde number, as 9. 15. 21. 25. 27. &c.

Numbers compared one with an other, some are Prime in respect one of an other, vvhich can not commonly be diuided by any other number, but by an vnitie.

Other some are compound in respect one of an other, vvhich may be commonly diuided by one or moe numbers.

This difference of numbers compared to­geather, is drawen out of the former, and is set downe in the twelfth and eleuenth de­finition of the seuenth booke. Numbers prime in respect one of another, haue no common diuisor beside an vnitie, which mea­sureth all numbers: As 3 and 8, 6 and 7, 5 and 12. Therefore when the numbers giuen are prime one to an other, there can no lesse [Page 39] numbers be giuen in the same proportions. So that prime numbers are the least, and the least are prime numbers, as appeareth in the twentie and twentie foure proposition of the seuenth booke. And therefore, when the numbers giuen are prime, they are also the termes of the proportion giuen, as we shall see afterward. Compound numbers in re­spect one of an other, may commonly be diuided by one, or many numbers, as 3 and 9, 9 and 15: for 3 is the common Diuisor of them both. Item 12, 18, 24, may be com­monly diuided not onely by 6, but also by 3 and 2.

If you vvould knowe vvhether the numbers giuen, be prime or compounde one to an o­ther, you may doe it by the Theoreme follow­ing.

Two vnequall numbers being giuen, if in subducting the lesse from the greater as often as may be, the remainder diuideth not that which went before it, vntill it come to an vnitie, the numbers giuen are prime one to an other. 1. prop. 7.

As in 4 and 11, 18 and 7, 35 and 12, in subducting the one from the other continu­ally, you shall come to an vnitie: wherefore I say, that they be prime in respect one of an other.

And hereby we may easily conclude, if af­ter the continuall subducting of the one from the other, we come not downe to an vnitie, but meete with some one number diuiding that which went before: the numbers giuen are compound in respect one of an other. As in 14 and 6, 30 and 18, 49 and 14.

An abridgement of the former worke.

For so much as diuision is nothing but an often subduction, if you diuide the greater number giuen by the lesse, you may doe the former worke more readily. Yet in diuiding [Page 41] the numbers after this maner, you must haue no regard to the quotient, but compare the Diuisor with the remainder. As I proue that 234 and 17 are prime numbers one to the other, by diuision thus.

In compound numbers take these for ex­ample 144 and 27.

CHAP. VII. Of the greatest common Diuisor, and the least common diuidend.

NVmbers are diuided after the manner be­fore named. Moreouer out of the diffe­rence of numbers compared together, there ari­seth a double inuention: one is the finding out of the greatest common diuisor, the other of the least common diuidend.

For so much as compound numbers may be diuided often times by many diuisors, the drift of the first inuention is out of many to choose the greatest number, which may be [Page 42] the common diuisor, or as Euclide termeth it the common measure of them all. The num­bers therefore giuen

• Two onely, and no more, as in the first Rule. I.
• Three or more, & in them we meete with the greatest common diuisor, either.
  • At the first. II.
  • Or at the se­cond time. III.

The finding out of the greatest common diui­sor is to be learned by the Theoreme following.

I. If after the continuall subduction of two numbers giuen, the one from the other, as often as may be (which may more briefly be done by diuision) there remaine some one number which will exactly diuide them both: The number remaining shall be their grea­test common diuisor.

As in 76 and 20, their greatest common diuisor is 4, as appeareth by this.

Item of these compound numbers 63 and 4, their greatest common diuisor is 7.

II. When there are three numbers giuen, if the greatest cōmon diuisor of the two for­most, being compared with the third, do di­uide it also, that diuisor shall be the greatest common diuisor of them all: and so forth in the rest, be they neuer so many. 3 prop. 7.

Here 7 being the greatest common diui­sor of the two for most 14 and 21, compared with the third number 35, diuideth it also and in like manner the fourth 63: wherefore it is the greatest common diuisor of them all.

III. But when the greatest common di­uisor of the formost, diuideth not those which follow, then comparing the first diui­sor, found out with the third number, the greatest common diuisor of the numbers compared together shall be the common di­uisor of the numbers giuen: and so foorth in the rest be they neuer so many 3 prop. 7.

As betwene 18. 12. 9. The greatest common diuisor betwene 18 & 12, is 6, the which nūber [Page 44] for so much as it cannot diuide 9, compare 9 and 6 together by themselues, and then be­cause the greatest common diuisor of 6 and 9 is 3, it shall be also the greatest common Di­uisor of all the numbers giuen. The example is thus set downe.

See also this example following of foure compound numbers in respect one of an o­ther, wherein there is a double comparison made, and the greatest common Diuisor di­uideth it selfe also.

The greatest common diuisor is found out as is aforesaid. Now followeth the finding out of the least number that may commonly be diuided by the numbers giuen: that is, such a number, then which there cannot be a lesse, vvhich two or more numbers giuen may exactly diuide.

Euclide calleth it the least that may be di­uided by certain numbers. For as many num­bers giuen may make many compound num­bers, [Page 45] euen so also they may diuide them, out of which numbers to chose the least that may be diuided, is greatly auaileable to the readi­nesse and easinesse of numbring.

• Only two a­lone, & those
  • Prime one to another. I.
  • or
  • Compound. II.
• Many, wher­in we finde our the least number that may be diui­ded.
  • At the first. III.
  • Or
  • At the second time. IIII.

The finding out of the least number that may be commonly diuided by certaine numbers giuen, is learned by the Theoremes following.

I. If the two numbers giuen, be prime one to an other, their product is the least number that may be commonly diuided by them.

As 15 the product of 3 and 5, being prime numbers one to an other, is their least cōmon diuidend. Item 63 is the least number com­monly diuided by 9 and 7

II. But if the two numbers giuen, be compound one to an other, if the quotient of the one, found out by the greatest common diuisor, multiply the other, the product shall [Page 46] be the least number that may be diuided by them two. 36 prop. 7.

As for example, I diuide 9 and 12, being compound one to another by 3, their greatest common diuisor, the quotient will be 3 and 4. Then if I multiply either 9 by 4, or 12 by 3 crossewayes, the product 36 shall be the least number that may be commonly diuided by 9 and 12, the numbers giuen:

III. But when there be more then two numbers giuen, if the thirde diuideth that which the two former diuided, the first num­bers found out shall be the least that may commonly be diuided by them all.

As these three numbers 6. 10. 15. compound one to an other being giuen, because 30 be­ing the least number to be diuided by 6 and 10, may also be diuided by 15, which is the third number, therefore it is the least that may be diuided by them all. Likewise in 4. 7. 28, because 28 being the least number that may cōmonly be diuided by 4 and 7, it may also be diuided by the third number 28, as by it selfe, it shall be the least number that may be diuided by the numbers giuen. As in these examples.

IIII. But if the third number doth not diuide that number which the former num­bers did diuide, then compare the least diui­dend found out, with the third number: the number which may commonly be diuided by the numbers thus compared together, shal be the least that may be diuided by them all: and so forth in the rest. 38 prop. 7.

As in 4, 8, 12, the least diuidend is 24. Item in 6, 16, 28, the least common diuidend is, 336, as you see.

So, the least number that may be diuided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 30, is 60, where­vpon it commeth to passe, that this number hath so greatly pleased the Astronomers in numbring their parts.

CHAP. VIII. Of the Accidentall numeration of partes which commonly they call Reduction.

• Accidentall: and is called Reduction, which is of partes
  • vnto parts either of
    • Least termes.
    • One deno­mination.
  • Vnto whole numbers, which is either of
    • Proper parts.
    • Improper parts.
• Or
• Essentiall, which is
  • Prime, as
    • Addition.
    • Subduction.
  • Second, as
    • Multiplicatiō.
    • Diuision.

HItherto we haue spoken of the numeration of vvhole numbers: now followeth the nu­meration of parts, wherein the numbers giuen are onely parts. And it is also double, Accident all and Essentiall.

Accidentall numeration is that which by re­duction chaungeth the forme and fashion of the parts, chaunging their value nothing at all.

This numeration in one word is common­ly called reduction, where the parts are so or­dered, that they only take vnto thē another shape, such an one as is fittest for the worke ensuing.

Reduction is double: the one is of parts vnto parts, the other of parts vnto whole numbers.

Reduction of parts vnto parts, is that vvhich findeth out other parts proportionable to the parts giuen, by reducing them either to lesse termes or to termes of one denomination.

Reduction of parts to lesse termes, is that which diuideth the termes of the partes giuen (being compound numbers one to another) by their greatest common diuisor: and taketh their quo­tients in stead of their numbers giuen.

Seeing that parts haue their value onely of the proportion that is betweene the vpper number and the neathermost, and euery pro­portion is best knowen when it is set downe in least termes: therefore in the partes also it shall not be amisse to reduce them which are written in greater termes (that is, with num­bers compound one to an other) or lesser [Page 51] termes, the which thing is done by the com­mon diuisor. For the quotients haue the same proportion that the numbers diuided haue, as appeareth by the 35 prop. 7. So that their value is not chaunged.

The proofe also of this may be taken out of the 17 prop. 7. which saieth, If a number multiply two other numbers, the products shall haue the same proportion that the multipliers haue. Contrariwise therefore if a number di­uide other two numbers, the quotients and the numbers diuided shall haue one propor­tion the one to the other. As if 4 multiply 6 and 8, their products 24 and 32 shall haue the same proportion which the numbers gi­uen haue. Againe, if you diuide these pro­ducts 24 and 32 by 4, the same numbers to wit, 6 and 8 will returne againe in the quo­tient. If the parts giuen therefore be 6/21 they must not be set downe in these termes, but must be reduced straight wayes to 2/7. Like­wise 4/16 diuided by 4, come to 1/4. So 108/252 may be reduced by 36 their greatest common di­uisor to 3/7.

By this kinde of Reduction not onely the termes of the same partes, but also the termes of diuerse parts may be reduced crossewayes one by an other: that is, the numerator of the one, and the denominator of the other. As 2/3 and 5/8 their termes being reduced crosse­wayes will be 2/3 5/4.

But this rule concerning the Reduction [Page 52] of diuerse partes, is not so generall as the other of one and the same parts. For it is more speciall and peculiar, as it were to mul­tiplication, as we shall see hereafter, where­fore we must not thinke that all such partes giuen may be reduced after this manner.

Reduction of parts to one denomination is that, vvhich (vvhen the parts giuen are of di­uerse denomitations) taketh first the least num­ber vvhich may be diuided by the denominators, for the common Denominator, and then diuiding that number by the denominators of the parts gi­uen, setteth down the products made by the quoti­ents, & by the numerators of the parts giuen for the numerators of the partswhich vve seeke for.

It is a thing of great vse to reduce the partes giuen to one denomination, and to make them proportionable one to an other in the least termes that may be: that is; of partes of diuerse kindes, to make partes of one and the same kind. As for example if 3/4 and 5/6 were to be brought to one denomina­tion, I must first seeke out the least number that may be diuided by 4 and 6, which is 12, then I diuide 12 by their denominators 4 and 6, and multiply 3 and 5 the numerators of the parts giuen by the quotients 3 and 2, the pro­ducts 9 and 10 shall be the numerators of the parts sought out: and the least number which might be diuided by the denominators: to wit, 12, shall be the common denominator to them both. As in this example.

See also the examples following.

In the first example therefore, the parts of one denomination are 9/12 10/12, and they be like­wise the least proportionable to the parts gi­uen. For as in 3/4 4 is to 12, so is 3 to 9: and in 5/6 as 6 is to 12, so is 5 to 10, and so forth in the other examples.

By this rule of reduction we may knowe whether the parts being of diuers denomina­tions be equal or vnequall. For being brought to one denomination, if they haue the same numerators they be equall: if not, they be vn­equall: they that haue the greatest numerator are the greatest, and they that haue the least numerator are the lesse. As if it should fall into question what proportion there were betwene 5/6 and 10/12. Item betwene 5/7 and 13/21, whe­ther they be equall or vnequall, this rule will take away the doubt: and will declare that the former parts are equal, the latter vnequal, and 5/7 to be the greater, 13/21 the lesser, as you see here.

To this reduction of partes to parts, succee­deth [Page 55] the reduction of parts to vvhole numbers, and it is either of proper or improper parts.

The first reduction of proper parts taketh the numerators of many partes, hauing one denomi­nation for the parts giuen.

As 3/5 and 4/5 are reduced to whole numbers when the numerators of the partes giuen 3 and 4, being as it were proportionable to the parts giuen, are taken for the parts them­selues: so likewise we reduce these partes 1/9 4/9 7/9 to whole numbers, when we take 1, 4, 7, the numerators for the parts themselues.

Reduction of improper parts taketh the quo­tient of the numerator diuided by the denomina­tor, for the parts giuen.

This reduction is vsed when the partes are but one, and their value either equall or greater then the whole, as 3/3 10/5 27/9. If you di­uide the numerators of euery part seuerally by his denominator, the quotients will be 1, 2, 3.

Hitherto also may those parts be referred whose denominators doe not exactly diuide the numerators, as 36/7 are reduced to 5 1/7 and 29/6 to 4 5/6.

CHAP. IX. Of the Essentiall numeration of parts.

VVE haue spoken of the Accident all nu­meration of partes. Now followeth the [Page 56] Essentiall numeration, which numbreth the parts either by encreasing, or diminishing them.

There be two kindes of Essentiall numeration. The first giueth the same denominator to the nu­merator found out, vvhich the partes haue that be giuen: and it is either Addition or Subduction.

Addition addeth together the numerators ha­uing one denomination, and setteth the common denominator of the parts giuen vnder the totall.

Partes of diuerse denominations the one being greater or lesse then the other, can not be added together without confounding of the denominators, wherefore it is needfull to reduce them to one denomination. As if you adde 3/5 to 4/5 the totall will be 7/5, or by reducti­on 1 2/5. Item adde 3/4 to 5/8, here if you should confound 4 with 8, the totall should haue his denominator of neither of them both. Wher­fore the parts must first be brought to one denomination, and then be added together. As in this example.

The Reduction. The Addition. The totall.

Likewise in adding 1/2 vnto 5/6 and 3/11 there arise 106/66, or by reduction, 1 20/33. The exam­ple is thus. [Page 57] Reduction. Addition. The parts found out.

Subduction is a numeration vvhich subdu­cting the numerators of the parts giuen, hauing one denomination, one from the other, taketh the common denominator of the partes giuen, for the denominator of the remainder.

As in subducting 3/8 from 7/8 there remaines 4/8 or 1/2. Item subduct 1/3 from 10/13. Here for so much as the parts be of diuerse denominati­ons, you must reduce them to 13/39 and 30/39, then subducting the numerators one from ano­ther, that is, 13 from 30, there remaines 17, vnder which you must set the common de­nominator, so that the partes found out are 17/39. As in this example.

Thus much of the first kind of the Essentiall numeration of parts: the second kinde giueth to the numerator found out an other denominator then the parts haue vvhich are giuen. And it is either Multiplication or Diuision.

Multiplication is that vvhich multiplying the termes of one kinde together (that is the nu­merators by the numerators, and the denomi­nators by the denominators) taketh the products for the termes of the parts found out.

This multiplication is properly the fin­ding out of other parts then are giuen, which are to one of the partes giuen, as the other is to an vnitie: and it is wrought by multiply­ing both the numerators and the denomina­tors of the parts giuen one by an other, wher­by we may necessarily conclude, that the partes found out are alwayes lesse then the parts multiplyed. As if there were 1/2 to be multiplyed by 1/3, multiplying the numerators and the denominators one by the other, the product is 1/6, which is to 1/3, as 1/2 is to 1. For as 1/2 is the halfe of one, so 1/6 is the halfe of 1/3, wherefore 1/6 is lesse then 1/3 or 1/2. This per­happes may be perceiued more easily in the numbring of some certaine thing. As for example, in a degree there be 60 minutes, therefore 1/2 of a degree are 30 minutes, 1/3 20 minutes, 1/6 10 minutes: as therefore 60 are to 30, so are 20 to 10. Againe in a shilling there are twelue pence, therefore 1/2 of a shil­ling are sixe pence, 1/3 foure pence, 1/6 2 pence: [Page 59] as therefore a shilling is to sixe pence, so are foure pence to two pence. Item if 5/7 be mul­tiplied by 3/4 the product will be 15/28.

The abridgement of multiplication.

If the opposite termes of the partes gi­uen be numbers compound one to an other, then in their stead take their quotients found out by their greatest common diuisor: and then worke the multiplication as before.

As, multiply 3/4 by 5/18. Here for so much as 3 and 18 being opposite termes, are com­pound one to another, you shall set downe in their stead their quotients 1 and 6 found out by their common Diuisor, so that the parts thus reduced shall be 1/4 and 5/6. Now multiplying 1 by 5, and 4 by 6, the pro­duct will be 5/24. Likewise if 3/7 should be mul­tiplyed by 14/15, first you must reduce them to 1/1 and 2/5, which being multiplied one by the other, make 2/5. As in these examples.

So if 3/10 5/6 and 1/4 be multiplied together, they giue 1/16. For first 3/10 and 5/6 being multiplied to­gether, doe make 1/4, which being afterwarde multiplied with 1/4 make 1/16. Thus:

Correlarie gathered of the former abridgement.

Of this reduction of numbers this conse­quence is gathered. If the crosse or opposite termes of many parts be equall one to the o­ther, then neglecting them which are equall, the numerator remayning set ouer the deno­minator remaining, is the product of thē all.

As if 4/5 5/6 6/7 7/8, were to be multiplied to­gether, I cast away the middle numbers which are equall, and see 4 the numerator o­uer 8 the denominator, so that the product is 4/8 or by reduction 1/2. The reason hereof is this, because the crosse termes being as it were compound one to the other, are by re­duction to one denominator brought to vni­ties thus, 4/1 1/1 1/1 1/8. But for so much as an vni­tie in multiplication chaungeth not a num­ber, therefore the figure 4 multiplied by the three vnities in the same ranke, maketh but 4, no more doth 8.

Moreouer it appeareth by this proporti­on which is in multiplication, that seconde parts may well be reduced to principall parts by this meanes, as 1/3 out of 3/4 are 3/12 or 1/4 of the whole, which thing may be easily perceiued in the numbring of any thing, as 3/4 of a de­gree are 45 minutes, whose third part are 15 minutes, which are equall to 1/4 of a degree. So 2/3 of [...]/4 parts of a crowne are halfe a crowne. For 3/4 are three shillings nine pence, whose 2/3 parts are two shillings sixe pence. So 5/6 1/2 of 3/8 by multiplication are brought to 15/96 or 5/32. And thus much of multiplication of parts.

Diuision is that vvhich in stead of the partes giuen diuideth vvhole numbers proportionable vnto them.

The diuision of parts is properly the find­ing out of a quotient which is in proportion to the diuidend, as an vnitie is to the diuisor. [Page 62] Whereby it is manifest, that the quotient must needes be greater then the parts which are diuided, otherwise then it is in diuision of whole numbers: for there the quotient is alwayes lesse then the diuidend. This in­uention commeth of the diuision of such numbers as are proportionable to the parts giuen, for so much as the partes themselues can not be diuided.

First therefore diuide these partes of one denomination, to wit, 9/10 by 3/10. Here there­fore you shall reduce them (by the former re­duction to whole numbers) vnto 9 and 3, then shall you diuide 9 by 3, the quotient will be 3, which declareth that 3/10 the diuisor is con­tayned in 9/10, the diuidend three times, and as 3/10 are to one, so are 9/10 to three: the which quotient therefore is bigger then the partes diuided.

Diuided 5/6 by 3/6, that is, see how many times 3/6 are contayned in 5/6. First you shall reduce the partes giuen to whole numbers propor­tionable vnto them, that is, to 5 and 3, then shall you diuide 5 by 3, the quotient will be 1 2/3. Contrariwise, 3/6 diuided by 5/6 giue in the quotient 3/5. For both the denominators be­ing neglected, I set the one numerator ouer the other as parts, because the one cannot be diuided by the other.

Furthermore if you diuide parts of diuerse denominations, as 3/4 by 1/3, you shall first make the partes of one denomination, reducing [Page 63] them to 9/12 and 4/12: then by the former redu­ction you shall bring them to whole num­bers, and diuide 9 by 4, the quotient will be 2 1/4. Diuide 8/9 by 5/6, reduce the partes to one denomination, that is, to 16/18 and 13/18, then re­duce these partes hauing one denomination to whole numbers 16 and 15, which being di­uided one by another, make in their quoti­ent 1 1/15.

CHAP. X. Of mixt numeration.

• Accidentall, and is redu­ction of
  • Whole numbers,
  • Or
  • Mixt numbers.
• Essentiall, wherof there are two kinds
  • Prime, as
    • Addition.
    • Subduction.
  • Second, as
    • Multiplicatiō.
    • Diuision.

HEtherto of simple numeration. It remayneth now to speake of mixt numeration vvherein the numbers giuen are whole numbers and parts together. And this numeration also as that of the partes is double. First Accident all, which beside the former reduction consisteth of the reduction both of vvhole and mixt numbers vnto parts.

Reduction of vvhole numbers vnto parts ta­keth the product made of the vvhole numbers giuen, & the denominator of the parts giuen, for the numerator of the parts which we seeke for.

This reduction may be drawen out of the [...]uction of improper parts, as it were a con­sequence following it. For as you reduce 4/4 by diuision vnto 1 whole, so contrariwise you shall bring one whole to 4 quarters, if you multiply one whole by 4 which is the deno­minator of the parts, for they will return to 4/46

Item, as 30/5 by diuision are reduced to 6 an whole number: so if you multiply 6 by 5 the denominator of the partes, and take the pro­duct 30 for the numerator, and 5 for the de­nominator, there will be againe 30/5 which are equall to the whole numbers giuen. Thus whole numbers as often as the vse of nume­ration requireth, are written as parts, by set­ting an vnitie vnderneath them. As for ex­ample these whole numbers, 2, 5, 9, hauing an vnitie set vnderneath them for their deno­minator shall be as it were partes after this manner, 259/111

Reduction of mixt numbers taketh the pro­duct made of the vvhole numbers giuen, and the denominator of the partes giuen being added to the numerator, for the numerator of the partes found out.

As you may reduce 4 3/7 into partes, if you multiply 4 by 7, the denominator of the parts giuen, and adde 3 the numerator, to 28 the product, taking the totall 31 for the nume­rator, and setting the denominator giuen vn­derneath it, thus, 31/7. Likewise reducing 6 3/4, there come forth 27/4, and so foorth in the rest.

Thus much concerning the accidentall nume­ration of mixt numbers: now remaine the speci­all kindes of numeration, vvhereof the first num­breth the termes of mixt numbers seuerally by themselues alone: that is, the vvhole numbers by themselues, and the parts by themselues, and that by adding or subducting them, vvhereupon it is called addition and subduction.

These kinds of numeration differ nothing from the former, but are in proportion all together like vnto them.

The examples of Addition.

I. Adde 3 and 2 4/5 the totall will be 5 4/5.

II. Item adde 3/4 to 6 6/7, first the parts ad­ded together are 45/28, or by reduction 1 17/28, the totall therefore is 7 17/28.

III. Adde 8 7/9 to 4 1/2, first the parts added together are 23/18, or by reduction 1 5/18, then adde, 1 to 4 make 5, which being added to 8, are 13. So that the totall is 13 5/18.

The examples of Subduction.

I. Take from 3 the whole number 5/9, here I reduce an vnitie of the whole number to 9/9, and there remaines 2. Afterward I subduct 5/9 out of 9/9, there remaine 4/9. The remainder therefore of the numbers giuen to be num­bred is, 2 4/9.

II. Item, take 1/4 from 8 1/5. Here because 1/4 cannot be taken out of 1/5, to wit, the greater from the lesse: I take 1 from 8, there remaine 7, then I bring 1 and 1/5 by the reduction of mixt numbers vnto 6/5, from whence (after I haue brought the parts to one denominati­on) I take 1/4, there remaine 19/10. So that the re­mainder of the numbers giuen is, 7 19/20.

III. Take 4 from 10 1/2, the remainder will be 6 1/2.

IIII. Take 5 7/12 from 7 5/8. Here first you shall make the partes of one denomination, that is, 15/24 14/24. Then taking 5 from 7 there re­maine 2, then also subducting the partes ha­uing one denomination one from the other, there remaines 1/24. The remainder therefore of the parts giuen is, 2 1/24.

The latter kinde of mixt numeration num­breth the termes of the mixt numbers giuen ioyntly together: that is, the vvhole numbers vvith the partes, and that by multiplying or di­uiding them: vvhereupon it is called Multi­plication or Diuision.

If 3/4 were to be multiplyed by 5, setting downe the whole numbers after the manner [Page 67] of parts, I multiply 5/1 by 3/4, the product is 15/4 or by reduction 3 3/4.

The value of the parts is sought out after this manner. As if you would knowe what 5/8 of a circle were, I set downe the whole partes of a circle were, I set downe the whole partes of a circle, that is, 360 degrees after the com­mon order of partes, thus, 360/1: then multiply­ing the nūbers giuen together, the product is 1800/8, or by reduction vnto whole nūbers, 225. So many degrees are 5/8 of a circle.

So likewise if you would haue 1/7 of 10 [...]/8, first you shal reduce the last of the two num­bers giuen by the reduction of mixt numbers into 83/8, which being multiplyed by 1/7, make 83/56, or by reduction 1 27/56.

Likewise if both the numbers giuen be mixt, as 4 1/3 and 5 3/11, the product made by multiplication will be 754/33, or by the reducti­on of improper parts, 22 28/33.

The examples of Diuision.

If 4 were to be diuided by 7/9, you must find out whole numbers proportionable to the numbers giuen: for the whole 4 being redu­ced into 9, the parts will be 36/9 and 7/9, wherby I see that the whole numbers 36 and 7, are proportionable vnto the numbers giuen: and they being diuided one by the other, make in the quotient 5 1/7.

Item if 1/2 be diuided by the whole num­ber 3, 3 being reduced vnto 6/2, I finde the whole numbers 1 and 6 to be propor­tionable to the numbers giuen: which being [Page 68] diuided the one by the other, make in the quotient 1/6. Item, diuide 8 2/3 by 4 3/5: here re­duce the termes of both numbers by the re­duction of mixt numbers to 26/3 and 23/5, then make them of one denomination thus, 130/15 69/15, afterwards reduce them to whole numbers proportionable vnto them, that is, to 130 and 69, which being diuided one by the other, the quotient shall be 1 61/69.

CHAP. XI. Of the kindes of proportionalitie or reason.

• Either of the termes, and is called proportionality or reason, whereof
  • The one is in equall termes: and is called the proportionalitie or reason of equalitie.
  • The other is in vne­quall termes: & is called the proportionalitie or reason of inequa­litie, and is to be con­sidered
    • Either according to the difference of the termes.
    • Or according to their quotient: and herein we are to note,
      • The kinds: for of the proportio­nalitie of inequalitie some is,
        • Of the greater inequality, & that is
          • Either Prime, as
            • Multiplex.
            • or
            • Simple, which is
              • Muitiplex Superpar­ticular.
              • Or
              • Multiplex Superpar­tiens.
        • Of the lesser inequali­ty, which is to be diui­ded into as many kinds as the former is diui­ded into.
      • Or The hand­ling of the kinds wher­in we are to cōsidertheir
        • Notation.
        • Nume­ration, which is,
          • Additiō.
          • Or
          • Subdu­ction.
• Or of the propor­tionalities or rea­sons. F.

HIther to of Simple, now followeth Compare Arithmeticke, vvhich consisteth in repressing the nature and the qualitie of termes com­pared together.

In writing of proportions the numbers whereby they are expressed, are by a Geome­tricall phrase called terms: by which word are to be vnderstood such quantities as are cōpa­red together. Hereupō generally such things as are compared together are called Termes.

The cōparing of termes together, is called Pro­portion, which is a certain respect which the terms haue one to another according to their quantitie.

This definition in this place is more general then the 3 def. of the 5 book of Euclide, which faith, that Proportion is a certaine respect of two magnitudes of one kinde onely, according to their quantitie. For in that place it was requisit that there shold be mention made of magnitudes of one kind only, because that magnitudes are not al of one kind. But here those words may be omitted, because al nūbers are of one kind, & euery one may be compared together: and moreouer the analogie or respect which they haue one to another, may be expressed by an other number, which thing cannot be done in all magnitudes, though they be all of one kind itrationall, and surde magnitudes can not be expressed by number.

The comparison of termes, is either equall or vnequall, whereupon it is called Proportion of equalitie or vnequalitie.

Proportion of equalitie is one only, and not di­uided into many kinds.

As the proportion of 5 to 5, of 6 to 6, of 12 to 12. For both in these, and in all other e­quall termes, the respect is all one, neither hath it any more kindes, because that one and the same number cannot be either more or lesse equall to another.

Proportion of inequality is knowne either by the difference of the termes, or by their quotient.

The proportion of difference, is that whereby one terme differeth from another.

The proportion of the quotient is vvhen one terme contayneth another.

The proportion of difference is known by subduction, the proportion of the quotient is knowne by diuision.

As in the naturall order of the numbers the difference betwene 1 and 2, 2 and 3, 3 and 4, 4 and 5, 5 and 6, &c. is all one, that is an vnitie. Againe, the difference betweene 8 and 5 is 3, betweene 6 and 4 it is 2. But the proportion of the quotient is by an especiall phrase simply called proportion. As, the proportion of 2 to 1 is double, of 3 to 2 sesquialter, of 24 to 3 octuple. For the former termes (which cōmonly are called the ante­cedents) being diuided by the termes follow­ing (which are called the cōsequents) the quo­tients are 1, 1 1/2, 8: whereby we may gather that there may be now and then the same diffe­rence betwene the termes though as touching [Page 72] the quotient their proportion differ, as in 4 8 and 12, the difference betwene 4 and 8, and betwene 8 and 12, is all one, yet the propor­tion gathered by their quotients is diuerse, for in the one it is 2, double, in the other it is but sesquialter 1 1/2. Contrariwise also in termes the proportion of the quotient may be all one, though their difference be diuerse: as in 3 and 9, 4 and 12, the quotient is 3 triple in them both: yet the difference is not a like, for in the former it is 6, in the second it is 8.

Prōportion of inequalitie is double, for either the greater terme is antecedent to the lesse, or con­sequent. Hereupon the one is called the proporti­on of the greater inequalitie, the other the pro­portion of the lesser inequalitie.

When two termes are compared together, the one is called the Antecedent, the other the consequent. Therefore, as often as the greater terme is the antecedent and goeth be­fore, it is called the greater inequalitie. As when 6 is compared to 3, and 3 to 2. But if the lesser terme be the Antecedent, and we consider what difference there is betweene that and the greater, it is called the lesser ine­qualitie.

Both these fornamed proportions are either Prime or Seconde.

Prime, is a proportion of one kinde onely, and is either simple or multiplex.

Simple proportion, is when the greater terme [Page 73] contayneth the lesse once and somewhat more, as proportion Superparticular and Superpartiens.

Superparticular, is a simple proportion where­in the great [...] terme contayneth the lesse, and one part more of the lesse.

As when 3 is compared to 2, 6 to 5, 8 to 7. For in these proportions the antecedent contayneth the consequent once, and one part beside of the lesser terme. As it may be perceiued by their quotients when they be diuided thus, 1 1/2, 1 1/5, 1 1/7. This kinde con­tayneth vnder it infinite other kindes, where­of the first is sesquialter, the second sesqui­tertia, the fourth sesquiquarta.

Item there is the same proportion betwene 12 and 8, that is betwene 15 and 10, as appea­reth by their termes reduced to the least de­nomination thus, 1 1/2.

Superpartiens is a simple proportion, wherein the greater terme contayneth the lesse, and [...]er­taine parts of it.

As 5 to 2, 7 to 4, 13 to 8. For the antece­dents being diuided by the consequents, their quotients declare their proportion on this manner, 1 2/3, 1 3/4, 1 5/8. Likewise the propor­tion betwene 15 and 9 is superpartiens, as ap­peareth by the quotient and the parts redu­ced to the least denomination. 1 2/3.

Multiplex is a prime proportion vvhen the greater terme contayneth the lesse certaine times exactly.

Multiplex is defined after the same man­ner [Page 74] in the second defin. of the fifth booke of Euclide, which saith, that Multiplex is a grea­ter quantitie in respect of a lesse, when the lesse measureth the greater.

As the proportions of 4 to 2, 9 to 3, 16 to 4 are multiplices, that is to say, double, triple, quadruple: for their quotients are 2, 3, 4.

Second proportion consisteth of two kindes of proportion, and is either multiplex superparticu­laris, or multiplex superpartiens.

It seemeth that these kinds of proportions are made of the former, by ioyning two of their names together. For the one is the third kinde of proportion ioyned to the first, and the other is the third ioyned to the se­cond.

Multiplex superparticular is is a compound proportion, wherein one terme contayneth ano­ther certaine times, and one part besides.

As 5 to 2, 10 to 3, 13 to 4. For the quo­tients declaring their proportions, are 2 1/29 3 1/3, 3 1/4.

Multiplex superpartiens, is vvhen the one terme containeth the other, & besides more parts then one.

Here we are to vnderstande, that they be the true partes which are not equall to the lesser terme, as in 11 to 4, 12 to 5, 22 to 6. For the quotients are 2 3/4, 2 2/3, 3 2/3.

CHAP. XII. Of Notation and Numeration of proportions.

THus much for the kinds of proportion, now followetg the handling and practise of them: which consisteth in Numeration and Notation.

Notation is the writing of the antecedents a­boue, and the consequents beneath.

The Antecedents 4 8 6 Or 5 1

The Consequents 3 3 2 Or 6 2

The proportions being written, are expressed by the quotient (the parts if there be any, being reduced to the least termes) which declareth as well their proportion as their denomination.

In multiplex proportion it is called dou­ble, triple, quadruple, octuple proportion, where the quotients are 2, 3, 4, 8. In propor­tions of lesser inequalitie this preposition Sub is generally added in them all. As for ex­ample, it is called in Submultiplex, Subdu­ple, Subtriple, Subquadruple, &c. In Super­particular proportion (because the halfe of the greater number is the proportion) it is called after the manner of the Latines Sesqui­altera, Sesquitertia, &c. putting the worde Sequi in the beginning, and the denomina­tor of the partes in the ending as when the quotients are 1 1/2, 1 1/3, 1 1/4. In proportions of the lesser inequalitie we say, Subsesquialte­ra, Subsesquitertia, &c.

Likewise Submultiplex, Superparticular, Submultiplex Superpartiens. For in com­pound proportions the names are also com­pound, as in 3 1/1 triple sesquiquarta, 2 4/4 dupla superquadripartiens, &c.

And thus much for instruction sake. For I know that these wordes are very straunge to vs, and not heard of, nor vsed among the best writers. For they when they would speake of the proportions of things, were wont to expresse them by two numbers. As Archi­medes in expressing the proportion of the circumference of a circle to the Diameter, sayd not that it was triple sesquitertia, but as 22 was to 7.

Moreouer the quotients teach vs the way not onely how to expresse the proportions giuen, but also how to set them downe in the least termes. For if you set an vnitie vnder­neath, or ouer against the quotients being whole numbers, you shall find out the least termes of the multiplex which you seeke for. As in the multiplices, if you set an vni­tie right against 3, 4, 5, the proportion in the least termes will be triple, quadruple, quin­tuple. Againe, if you reduce the quotients being mixt numbers, by the seconde redu­ction of mixt numbers vnto partes, you shall in like manner finde out the least termes in the other kind of proportions: as in Super­particular proportion, if you seeke for the least termes of 1 1/2 sesquialtera, 1 1/3 sesqui­tertia, [Page 77] 1 1/8 sesquioctaua, then the quotients reduced after the foresaide manner, the least termes will be these.

The like is in all the other kinds, as if you would find out the least termes in duple su­perpartiens fiue sixe parts, then shall you of 2 5/6 make the antecedent 17, and the conse­quent 6, and they shall be the least termes in the proportion assigned, and so foorth in the rest.

Thus much for notation. Numeration is the addition or subtraction of proportions.

Addition, is that vvhich taketh the products made by multiplying the like termes together, (that is, by multiplying the antecedents by the antecedents, and the consequents by the conse­quents) for the termes of the compound portion.

The adding of proportions is set downe in the fifth def. of the sixth booke thus. A Proportion is said to be made of proportions, whē the quātities of the proportions multiplyed toge­ther produce another proportion. By the quan­tities of the proportions he meaneth the quo­tients whereof the proportions take their names. As if you adde a sesquialtera to an e­quall proportion, it maketh a sesquialtera. Adde a sesquiquarta to a duple superpartiens two third parts, the product will be a sesqui­tertia: as you see in these examples.

That the products made of multiplying the termes together, haue a compound pro­portion appeareth by the 5 prop. of the eight booke of Euclide, which saith, that Plaine or superficiall numbers, are in that proportion one to the other which is composed of their sides. As 3 and 12 being plaine numbers, haue propor­tion one to another, according to their sides, as you see.

If diuerse proportions be to be added to­gether, first adde two of them, then adde the compound to that which remaineth. As if the proportions giuen were,

First two of them added together, make 18/5 to which the third proportion being added, maketh a Sextuple. 120/22: 6.

Thus euery proportion may be tripled or quadrupled: if the termes of the propor­tion giuen being set downe three or foure [Page 79] times be added together after the foresaide manner. So a double proportion being tri­pled, maketh an octuple. And a sesquialtera being doubled foure times, maketh a quintu­ple sesquidecima, as you see here.

Subduction of proportions, is that which ta­keth the products made of the opposite termes of the proportion giuen, for the termes of the pro­portion remaining.

Which proportion is least, and which may be subduced one from the other in Multipli­ces may be easily knowne by the denomina­tion onely. But in the other (the termes be­ing compared together as partes) it may be gathered by that which hath bene taught be­fore in the partes. Subduction seemeth to be the consequence of that which was set downe in the fifth def of the sixt booke. For if the addition of proportions be wrought by mul­tiplycation, it followeth then that subducti­on must be wrought by diuision. Therefore whether you diuide the quotients one by the other, or multiply the opposite termes of the partes giuen one by the other (for in partes the multiplying of the termes crossewayes is as much as diuision) the termes produced will haue a proportion left like vnto the former.

As take a sesquialtera from a double pro­portion, the remainder will be a sesquitertia. Item, take a sesquiquarta from a double ses­quialtera, there remaineth a double, as in these examples.

There is no multiplication in proportions properly as in numbers, vnlesse you take the continuing of the proportions for their mul­tiplication. Neither is there any diuision re­quired in them beside the finding out of one or more proportionals. Wherefore we will proceede to the rest.

CHAP. XIII. Of proportion and the kinds thereof, but especially of Arithmeticall proportion.

• Either of vnequall reasons, and is called improportion, whereo [...] in the 8 d. 5. with the which Arithmeticke doth not meddle.
• Or of e­quall rea­sons, and is called proportiō, consisting
  • Either in the equa­litie of the differēces: and is cal­led Arith­meticall proportiō, which is
    • Either disiunct,
    • Or continuall, and this againe is,
      • Either simple,
      • Or multi­plex, & is called pro­gression: wherin we are to ob­serue,
        • Either the or­der of the termes,
          • Continu­all, as they follow one another.
          • Or
          • Interrup­ted, when we seeke but for certaine termes.
        • Or the summe of the termes.
  • Or in the equalitie of the quotients: and is called Geometricall proportion. G.

NOw that vve haue spoken of comparing the termes together it remayneth to speake of the comparison of reasons, vvherein first vve haue to deale vvith proportion.

Proportion is the equalitie of reasons.

Euclide in the fourth definition of the fifth booke, defineth proportion to bee a si­militude of reasons. Aristotle in the fifth booke and 3. Chapter of the Ethickes, calleth it an equalitie of reasons. If wee shoulde define it to be the Identitie of reasons, we should not varie much from the meaning of Euclide, who in the sixth definition of the fifth booke speaketh after the same manner, calling them proportionall quantities vvhich haue the same reason one to another. And albeit that these definitions doe properly a­gree to Geometricall proportion, yet if you take the word reason in the largest sense, for any kinde of respect it shall be generall, so that as reason or proportionalitie is a mu­tuall respect of two termes one to another, so proportion shall be a respect of like termes one to another.

Proportion is either Arithmeticall or Geo­metricall.

Arithmeticall proportion is an equalitie of differences.

As in 8, 7, 6, and in 12, 9, 6, 3. Here are two proportions, wherein you may see an equalitie of differences, or an equall ex­cesse [Page 93] of the antecedents. In the first, the difference is 1, in the latter it is 3. So like­wise in 15, 11: 8, 4, as 15 exceede 11 by 4, so 8 exceede 4 by 4.

Arithmeticall proportion is double, for ei­ther it is continued in the termes, or it is seue­red, vvhereupon it is called continuall, or disiunct proportion.

These differences are common both to A­rithmeticall, and to Geometricall propor­tion. That proportion is continuall where­in not onely the extremes are continued to­gether by one meane, but generally euerie proportion, wherein many and sundrie middle termes haue a continuall respect one to another, so that any of them may be ey­ther the antecedent or the consequent. In so much, that not onely the proportion be­twene 3, 2, 1 is continuall: where 2 is the consequent of the former, and the antece­dent of the latter proportion: but this also 5, 7, 9, 11, where the same difference is con­tinued from the first to the last. And any of the middle numbers either 7 or 9, haue the same nature in respect of them that goe be­fore and follow after, that 2 had in the for­mer example.

Proportion is seuered in the termes when the first onely hath the same respect to the second, that the third hath to the fourth, but the second agreeth not pro­portionablie with the thirde: that is, [Page 94] when all the termes in order as they followe are not of like proportion one to the other. As 10, 8, 4, 2, are seuerally proportionall be­cause 2 the first difference is not continued betwene 8 and 4. So that here is a seuering not onely of the termes, but also of the pro­portions. Leauing therefore in this place to speake of disiunct proportion, wee will han­dle that which is continuall.

Continuall proportion is either Simple, where the extreames haue but one middle terme: or manifold, vvherein many middle termes go on in order continually: vvhereupon it is called progression.

Progression is a continuall enlarging of the termes of the proportion giuen: where­in there are two extreames giuen, and manie other numbers in the middle betwene them.

In Arithmeticall progression we are to finde out two things, either the number of the termes as they stand in order, or else their summe.

Sometimes the termes be all sought for as they follow in order, and are found out by the continu­all adding of the difference to the last terme in the progression.

As if you would know the termes of this progression 1, 5, 9, 13, as they follow in or­der, you must adde 4, which is the difference vnto 13, then the fifth terme will be 17, to which if you adde 4 againe, it maketh 21, which is the sixth terme, and so foorth in­finitely.

Sometime breaking off their naturall order, we seeke for some one terme in the progression, which may be found out by the Theoreme following.

The progression increasing, if you take an vnitie from the name of the terme which you seeke for, & adde the product made of the re­mainder, and the common difference of the termes, vnto the first terme, the totall will be that which you desire. The progression de­creasing, if that product be taken frō the first terme the remainder is the terme sought for.

As in this progression 4, 7, 10, where the difference is 3, I desire to know the 12 terme beginning at 4. Here 12 is the name of the terme which I seeke for, from whence I take 1, and multiply 11 which remaine by 3 the difference, the product is 33, whereto I adde 4 the first terme. So the totall is 37 the terme which I sought for. Likewise the 30 terme is 91, and so foorth.

Thus much for the finding out of the termes. The finding out of the summe is the halfe of the product made by multiplying the two extremes added together, and the name of the last terme, as it standeth in order.

As in this example, 1, 5, 9, 13, 17, 21, both the extreames added together make 22, the [Page 96] name of the last terme is 6, for it is the sixt in order. Whereby if you multiply 22, the product will be 132, the halfe whereof (that is) 66, is the summe sought for.

This rule may be proued by the 41 prop. of the first booke of Euclide, which saith, If a parallelogramme and a triangle haue one and the selfe same base, and be in the same altitude, the parallelogramme shall be double to the trian­gle. The numbers then of the progression re­present the triangle: whereof one side is the name of the last terme, or (as they common­ly terme it) the number of the places: the o­ther is the extremes added together. But for so much as the product made of these two sides maketh not a triangle, but a parallelo­gramme, therefore in taking the one halfe I gather by it how bigge the platforme of the triangle is, which is the summe of the pro­gression.

The example of this progression 2, 4, 6, 8, 10, is set downe here as you see:

To set forth the profite and vse of this in resoluing certaine questions, let this be the first example.

For so much as a naturall day hath 24 houres, I demaund how many strokes the clocke vvill strike in a day.

In this example we seeke for the number of the strokes beginning at one, and adding one till you come to the 24 terme. After this manner the clockes in Italy and Norinberge are made. Both the extremes added toge­ther make 25, which being multiplyed by 24, the name of the last place, make 600, whose halfe are 300 strokes, which is the summe sought for.

Another.

There be 30 egges laide in a rancke, euery one three foote from another. I must put them in a basket vvhich standeth 13 foote from the first egge: yet so that at euery time I goe to and fro I must take vp but one. The question is, how farre a man may go before I haue ga­thered them all.

This was wont to be a pastime vsed a­mongst young men, when one gathered the egges and put them in a basket without brea­king them, and another went to a certaine place and came againe, so he which first had done his taske, got the game. Here the que­stion is about the summe of the Arithmeti­call progression.

Both the extremes together are 113, th [...] product made of this summe, and the name of the last terme is 3390, whose halfe is 1693 feete, which make 339 Geometricall paces. This space he goeth and commeth, where­fore the double of it which was first found out, that is 3390 feete, or 678 paces is the sum which we sought for.

CHAP. XIIII. Of Geometricall proportion, and of the golden rule belonging thereunto.

• Disiunct: out of whose proprietie a riseth the golden rule: and that is,
  • Simple: simple proportions is
    • Direct: before which there goeth
      • Either no kind of A­rithmeti­call accoūt Chap. 14.
      • Or some Arithme­ticall ac­count, and that
        • Either simple, as the numerati­on of
          • Whole numers. Chap. 15.
          • Parts or fractions. Chap. 16.
          • Mixt numbers.
        • Chap. 17. Or, com­pared as the pro­portion
          • Arithme­ticall.
          • Geome­tricall.
    • Or Reciprocall. Chap. 18.
  • Or Manifold. H.
• Coniunct or Continuall. L.

THus much of Arithmeticall proportion, Geometricall followeth, vvhich is the equa­litie of reasons.

Here this word Reason is not taken gene­rally, but specially, for the equalitie of the quotients arising of the numbers compared together. As in 2, 6, 4, 12, what reason or pro­portionalitie is betweene 2 and 6, the same is betweene 4 and 12, that is, subtriple. And if foure numbers be directly proportional, they shall be also proportionall backwarde, and crossewayes in this manner.

Geometricall proportion is double: Disiunct, or Continuall.

Disiunct is that, that agreeth neither in termes nor proportionalitie, or reason.

As in 6, 3, 16, 8, for here are two double proportions distinquished both in termes and in reason, because that betweene 3 and 16 there is not the same reason obser­ued, that is betwene 6 and 3 and betweene 16 and 8.

Disiunct proportion consisteth of foure termes at the least. It hath this propertie that the product of the extremes, is equall to the pro­duct made of the middle numbers, and contrari­wise. 16. p. 6. & 19. p. 7.

Among the properties of Geometricall proportion set foorth by Euclide, this is most excellent and necessarie to be knowne, because the force and fruitfulnesse thereof is such, that it may very well be called the head of cyphering, that is, the foundation and off-spring of all rules, as we shall see hereafter. It hath this meaning, that the number made by multiplying of the first in­to the fourth, is equall to the number made by multiplying the seconde into the thirde, as in 6, 4, 18, 12, the product made of 6, multiplyed by 12 is 72, the which num­ber ariseth by multiplying the middle num­bers 4 and 18 one by another. And this thing is proper to continuall, and generally to all Geometricall proportion, as wee shall see hereafter.

This propertie ministreth vnto vs the golden Rule, vvhich three proportionall numbers be­ing giuen, diuiding the product made by multi­plying the third into the second by that vvhich remaineth, findeth out the fourth proportionall number.

Out of the fore named propertie, as out of a most plentifull spring is drawen this Rule, wherein consisteth the chiefest partes of ca­sting accounts. The which some for the ex­cellencie of it doe call the Rule, others the rule of Proportion, other the rule of three, and all men all most in generall, for the ex­cellent vse of it, doe call it the golden Rule. [Page 101] The drift and end of it, is the finding out of a fourth terme, or number, that shall haue the same proportion with the three numbers gi­uen: the which thing it bringeth to passe by multiplying two termes of the propor­tions giuen one by the other, that is, the an­tecedent of the one, and the consequent of the other, and diuiding the product by the other which remaineth: for the quotient will be the fourth number of the proportion giuen.

As in 3, 2, 9, I desire to knowe the fourth that shall be to 9, as 2 to 3, multiply there­fore 2 by 9, and diuide 18 the product by 3 which remaineth, the quotient will be 6, the proportionall number which was sought for.

The cause of this consequence is manifest, because hauing the product of the middle numbers, it is all one as if you should say, that you had the product made of the first and fourth numbers, and to proue that, or to bring to light the fourth number lying hidde in the product, you must diuide the product by the first number, and say, that the quotient is the other number vn­knowen which made the diuidend: because that euerie number is diuided by those num­bers by which it is made, and euery pro­duct diuided by one of the multiplicators bringeth forth the other.

As in 9, 6, 12, I desire to finde the fourth, multiply therefore 6 by 12, that is the middle [Page 102] number, then I say, the product 72 is the product arising of 9, and the fourth number vnknowen. The product then being had, and one of the multipliers knowen, the other cannot any longer be hid. Wherefore I di­uide 72 by 9 the quotient will be 8, which is the other multiplier, and also the fourth number in proportion: thus, 9, 6, 12, 8.

This rule is vsed in proportion simple or ma­nifolde.

Simple proportion is that which consisteth onely of foure termes or numbers.

The forme vvhereof is double, for either the proportion proceedeth directly forward, or else backewarde, vvherevpon it is called Direct, or Reciprocall.

I call that Direct proportion, vvherein as the first terme is to the second, so the third is to the fourth, and crossewayes.

Which is Direct proportion, may be easily gathered by this, that by howe much the third number is greater or lesse, by so much the fourth in order-shall be greater or lesse, whether the numbers be giuen ordina­rily, or crossewaies, or backwards.

And it is handled two manner of wayes, for sometimes besides the ordering of the terms there is nothing else required: sometimes it requireth some former kind of numeration.

The ordering or disposing of the termes to conclude the proportion attributeth to euery one that is giuen his owne place: that is, to the thing [Page 103] vvhich is called into question, it giueth the third place: the first place to the thing of the same kind: the middle place to that vvhich remaineth.

The lawfull order of the termes is that in distinguishing the reasons, we ioyne the consequent of euery reason to his owne antecedent of the same kind. As if I should say, seeing Iupiter in foure moneths moueth tenne degrees in the Zodiacke: therefore in sixe monethes it moueth 15 degrees. Here the proportion ordinarily shall stand thus, 4, 6, 10, 15. For as 4 monethes is to 6 monethes, so is 10 degrees vnto 15 degrees. For here also the product made of the extreames is equall to the product made of the middle numbers.

But the termes in the Rule of proportions is otherwise ordered, because in things of one kind the question is not so moued, that they should be ioyned together, but crossewayes. For we doe not moue the question customa­bly from the antecedent of the second rea­son to his consequent thus, if 4 moneths are 6 monethes, what are tenne degrees? But the question is moued from the consequent of the first reason to the consequent of the second crossewayes: if 4 monethes giue 10 degrees, then 6 monethes shall giue 15 de­grees. Therefore seeing the proportion is moued crossewayes: and thereby may be handled more conueniently, and may be bet­ter conceiued by this meanes of the learner, [Page 104] therefore in the ordering of the termes, we place them which be of like sort crosse­wayes in sundrie places, not ioyntly toge­ther.

And hereupon the question being moued somewhat confusedly, we must set that terme that hath the question adioyned vnto it in the third place, and the terme of the same kinde, that is the terme which betokeneth the same thing, and is called by the same name in the first place: and the other in the middest: then must we followe the Me­thode and direction of the rule, multiply­ing the thirde by the second, or contrari­wise, the second by the third, and diuiding the product by the first.

The examples.

A traueller is to goe 180 miles, vvhereof he goeth 9 in two dayes, the question is vvhen he shall end his iourney.

Here the question beeing propounded somewhat confusedly, you set in order the termes. And for so much as the question is moued about the 180 miles, in what time they may be gone thorough, you shall set that number in the third place, the 9 miles being a terme of the same sort, shall stande in the first place, the two dayes in the mid­dle. Then multiply 2 by 180, and diuide the product 360 by 9, the quotient wil be 40, [Page 105] the fourth number in proportion answering the question. The example is thus:

If 9 require 2 dayes, ergo 180, 40 daies. If a pole sixe foote long standing vpright vppon the ground casteth a shadowe tenne foote long, I demaund how high that Tower is, vvhich at the same time casteth a shadowe 125 foote long.

Here I see that the numbers giuen are pro­portionall. For as the height of the pole be­ing knowen is vnto the shadowe, so is the hight of the tower vnknowen vnto his sha­dow. The termes therefore being set duely in order, and handled according vnto the rule, the example shall be thus.

If 10 giue 6: ergo 125 shall giue 75.

After the same manner in this rule may the parts be ordered and handled, as may be perceiued by the examples following:

If the sixt part of the Moone encreaseth vvhen she is distant from the Sunne the twelfth part of a circle: how bigge shall she be when she is distant from him halfe a circle.

Here the first terme shall be 1/12 of a circle, the second 1/6 of the Moone, which in that di­stance is lightened with the light of the Sun: the third 1/2, the worke being wrought there ariseth 1, which signifieth that the Moone is at the full when she is halfe a circle off from the sunne. The termes are thus:

1/12 1/6 ergo 1/2 1.

I spend in the fourth part of a moneth 5/6 of a crowne, vvhat doe I spend in 7/10 of a moneth?

Here the termes are giuen in order, wher­fore I multiply the third by the second, the facit will be 7/12, which being diuided by 1/4 af­ter the manner of parts or fractions, the quo­tient will be 2 1/3, so that the example shall be thus:

1/4 5/6: ergo 7/10 2 1/3.

But here if you desire to knowe the value of 1/3 of a crowne or of any other fraction what so euer, you shall search it out in like manner by the rule. For that they teach com­monly, that the number made by multiply­ing of the numerator, and the partes of the whole being knowen, must be diuided by the denominator, whereuppon proceedeth the value of the parts giuen, it is nothing else but to practise the rule of proportion. For it is all one, as if you should set the denomina­tor of the parts in the first place, the nume­rator in the third, and the parts of the whole being knowen in the middlemost place, and so worke accordingly. As, put case three parts of a crowne be 45 pence, you shall find out the value of 1/3 part thus:

If 3 be 45 pence: I shall be 15 pence.

If one or more of the termes be mixt, you must vse the same methode which before was [Page 107] vsed in whole and broken numbers: sauing that both the whole and broken numbers must first be reduced into fractions. As the question being moued, two yardes and 1/6 of cloth are bought for fiue crownes, what shall I pay for 5 3/4 of a yarde: you shall reduce ther­fore the first terme vnto 13/6, the second vnto 5/13 the third vnto 23/4, whereof there amount 13 crownes & 7/26, which is the number required.

Certaine abridgements.

In whole numbers it shall not be amisse to haue regarde vnto the abridgement follow­ing. If of the proportionall numbers giuen, the first and second, or the first and thirde be compound one to another, then they be­ing diuided by the greatest common diuisor: the proportion shall be more briefly conclu­ded by the prime numbers.

The reason of this abridgement is this, be­cause the termes being chaunged after this manner, yet the proportion is not chaunged by the 15 prop of the fifth booke, where it is saide, that Like parts of multiplices, and also their multiplices compared together haue one and the same proportion. As in 12, 32, 15, because 12 and 32 are compound one vnto the other, therfore they being diuided by 4, I take their quotients 3 and 8 in stead of the numbers gluen. Thus, if 3 giue 8, therefore 15 shall giue 40.

Likewise seeing the first and the third that [Page] is to say 12 and 15 are also compound the on [...] vnto the other: taking their quotient 4 and 5, being prime numbers found out by their greatest common Diuisor, for the numbers giuen, the proportion may be easily conclu­ded in this manner.

If 4 be 32: therefore 5 shall be 40.

In the fractions it seemeth a briefe kind of working to dispatch the matter by multipli­cation onely: to wit, if you take the number made by multiplying of the denominator of the first terme, by the numerator of the se­cond terme, and then by the numerator of the third terme, for the numerator of the fourth: and the facit of the numerator of the first terme, by the denominators of the se­cond, and third for the denominator of the 4.

The draught of them is thus:

As if 1/4 giue 2/3, what shall 5/6 giue? Multiply 4 by 2, there ariseth 8, then by 5, the product is 40, which is the numerator of the number which we seeke for. Furthermore multiply 1 by 3, and then by 6, the facit will be 18, the denominator of the fourth terme. The exam­ple is thus:

1/4 2/3 ergo 5/6 40/18, or 2 2/9.

CHAP. XV. Of examples of the rule of proportion, re­quiring some kinde of simple nume­ration before them.

IN the former examples, the rule of proportion was simple.

But oftentimes it requireth some certaine kind of numeration to goe before it: the vvhich nume­ration serueth to the ordering of the termes of the principall proportion, according to the nature and lawes of the question propounded, that ther­by the proportionall number, vvhich vve desire to know, might be vvith more easily found out.

And this numeration is drawen out of both the partes of Arithmeticke: Sometime out of absolute or simple Arithmeticke, and is either simple or mixt.

Great is the varietie of those questions which may be resolued by Geometricall pro­portion. Wherefore here wit is to be requi­red, that by the former conditions and nature of the question offered, wee may be able to perceiue what is meete for the vnknitting and conclusion of the proportion, and howe also (as it were a certaine preparatiue to the future demonstration) the termes of the pro­portion lying hid, are to be vnfolded, order­red, set downe, and to be applyed to the rule it selfe.

First therefore in this example addition of vvhole numbers is required. [Page 110] A Vintener buying at Ansa in Italie 120 hoges­heads of wine for 2400 ss. laid out for the ca­riage of them to Basill 1200 ss. and to the wag­geners for lading and vnlading them 30 ss. for custome in diuerse places he paide 50: his expenses in his iourney came to 64. He desi­reth to gaine 800. I demaund for vvhat he shall sell an hogshed.

In this example Addition ordereth one of the proportionall termes: for adde the price to his expences, and to the gaine, the totall will be 4544. Whereby I perceiue that the number of the hogsheds assigned is vnto this totall summe as one hogshed is to the price which we seeke for. Wherefore the example shall be thus.

If 120 Hog. be worth 4544 sss.: ergo 1 Hog. giues 37 1 [...]/1 [...] sss.

Another.

Two trauellers going from Basill vnto two con­trarie places, doe trauaile a diuerse pase one faster then the other: the one goeth euerie day fiue, the other but three miles: I demaund how many miles they shall be a sunder at nine dayes end.

Here also Addition helpeth to the orde­ring of one of the termes. For adde their daily iourneyes 5 and 3, the totall will be 8, which is the distance for one day, whereupon the rule concludeth,

1 day. giueth 8 miles.: therefore 9 dayes 72. miles.

Examples of subduction going before.

A pipe of vvater voydeth into a cesterne contay­ning 250 firkins, euery houre 24 firkins, now there runne out at another pipe euerie houre 16 firkins, I demaund in vvhat time the Ce­sterne will be filled.

Here you must find out by subduction, one of the termes of the proportion to be ordered according vnto the rule. For reason telleth me that I must seeke out the excesse of the filling aboue the emptying. Subducting ther­fore 16 firkins from 24, there remaines 8 fir­kins, for so many euery houre are left in the cesterne. Hereby the proportion may easily be concluded:

8 Remaine in 1: ergo 250 31 1/4

A certaine man selling an hundred pound of cer­taine kinde of marchandise for 15 crownes, found at the length that in an hundred crowns he had lost so much as he paide for euerie hun­dred vvaight: I demaund vvhat he paid for an hundred.

Here you shall subduct the price of an hundred pound, to wit, 15 crownes, (which number also representeth the losse) out of an hundred crownes, the remainder will be 85, the first terme of the proportion. For euen as 85 crownes which is the stocke diminished, is vnto an 100, which is the whole stocke: [Page 112] euen so 15 the price of an hundred waight which was lost, must be vnto the stocke whereof it was raysed: whereupon the termes shall stand thus: If there remaine

85 Crownes. of 100: ergo 15 de 17 11/17.

Examples vvherein Multiplication goeth before.

A Polonian going vnto the Vniuersitie of Basill to studie, deliuereth vnto a Marchant 126 golden Ducats (euery one being valued at 28 vrsati, to receiue againe of him for them at Basill siluerlings (vvhich commonly are cal­led Thaleri) being vvorth 18 vrsati. The question is how many siluerlings he shall re­ceiue.

Here you shall reduce by multiplication 126 ducats vnto vrsati, accounting to euerie ducate 28, whereof the whole summe amoun­teth vnto 3528 vrsati. Then looke what pro­portion there is betweene 18 vrsati, and one siluerling, the same there is betweene the whole number of the vrsati, and betwene the siluerlings which we seeke for. Wherefore the termes shall be thus:

18 Vrsati are vvorth 1 Thaler.: ergo 3528, 196

This and such like examples may be also resolued by the rule of reciprocall proporti­on, as shall appeare hereafter.

He that hath 34 crownes for his yearely vvages, vvhat hath he for eight dayes?

Here I see there is a question of proporti­on moued, but for so much as the first and third termes be of a diuerse denomination, I reduce them by multiplication to one kind of denomination. And so generally the terms of diuerse kindes are by a former multiplica­tion to be drawen to the least denomination. Therefore I take for a yeare 365 dayes, and that the account might be the more exact, I diuide the 34 crownes into 510 vrsati (recko­ning 15 vrsati to a crowne) or if it please you to bring them into a smaller coyne, you may resolue them into pence, and then the exam­ple shall be thus:

365 510: ergo 8 11 13/73

Sometimes Multiplication goeth before with Addition or Subduction.

The examples of the first.

He that desireth in sixe crownes to gaine 10 vr­sati, how shall he sell a pound of that vvare vvhich cost 20 vrsati?

For so much as the money here set downe is of a diuerse denomination: first of all I re­duce the 6 crownes by multiplication vnto vrsati: valuing euery crowne at 15 vrsati, the facit is 90. Whereunto if I adde the gaine, the whole summe of the vrsati, to wit, 100 shall be the middle terme of the proportion after this manner:

If of 90 there arise 100: then of 20 22 2/9 In the least termes thus: 9 10: ergo 20 22 2/9

A Butcher deliuering vnto his seruant 216 crownes, sent him to market vvilling him to buy as many Oxen, Calues, Weathers, and Hogges of euery one an euen number, as that money vvould afford him to buy: I demaund how many beasts he bought.

First let vs appoint the prices: an Oxe at 11 crownes, a Calfe at two, a Weather at one crowne and 5 vrsati, an Hogge at 3 crownes and tenne vrsati. Here then is there a double multiplication with addition required, to the ordering out of the principall proporti­on. For here you must resolue the 216 crownes by 15 (for at so many vrsati doe Germaines commonly value their crownes) into 3240 vrsari: and likewise reduce the seuerall prices of euerie beast vnto one denomination: to wit, the prices of the Oxen vnto 165, of the Calues vnto 30, of the Weathers vnto 20, of the Hogges vuto 55 vrsati, and then adde to­gether the prices of the same denomination, the totall 270 shall be the first terme of the proportion. For as this number is to one of each kind of beasts, so the whole summo of money is vnto the number of the beastes vn­knowen. The example is thus:

270 giue 1: ergo 3240, 12

The example of the latter.

A theefe hauing by stealth conuayed away cer­taine siluer vessels, in his flight goeth euerie [Page 115] day 128 furlongs. Foure dayes after the owner following the theefe on horsebacke, rideth e­uerie day 174 furlongs. I demaund vvhen he shall ouertake him.

For the resoluing of this question, two termes of the principall proportion must be set downe, the one by multiplication, the o­ther by subduction. For first we must see how farre the theefe was gone before the owner espying the matter entred his iorney. Wher­fore I multiply 128 furlongs by 4 dayes, the facit is 512 furlongs. Secondly, I consider how much the latter outgoeth the former e­uery day. Subducting therefore 128 from 174 the remainder is 46. Then looke what pro­portion the dayly excesse hath vnto a day, the same proportion shall there be betweene the former flight and all those dayes wherein the latter ouertaketh the former. So that the termes of the proportion shall be thus:

46 giue 1: ergo 512 11 4/23.

CHAP. XVI. Of examples of the golden Rule, vvherein there is required some numeration of broken or mixt numbers.

IN the former examples before the proportions could be concluded, there vvas required a cer­taine numeration of vvhole numbers: in those vvhich follow, the numeration is of broken or mixt numbers.

Example. 1.

A certaine man selling his house for 75 crownes, findeth he hath gotten the fourth part of the money vvhich he paide for him, I demaunde vvhat he paide for him?

Here adde vnto the money which he gaue (which is supposed to be 4/4) the games, to wit 1/4, the totall will be 5/4: for he that of his stocke laide out raiseth 1/4, he by 4 gaineth 5. Wherefore the principall proportion shal be thus: 5 75: ergo 4, 60

A Farmer hauing gotten by one yeares haruest certaine sackes of corne, payeth to his Land­lords for his yeares rent, to one 2/5, to another 1/4, and hath remaining 35 sackes: I demaund how many he had at the first.

In this and such like examples of broken numbers hauing diuers denominations, out of the least number which may commonly be diuided by the denominators of the fra­ctions giuen, must be drawen whole nūbers being vnto the number diuided as the parts assigned do require. As here the least number which may be diuided by the denominators set downe which are 5 & 4, is 20: wherof 2/5 are 8, & 1/4 is 5, which being added together, & the cōmon denominator set vnderneath do make 13/20. The farmer then of his whole heap of corn which is supposed to be 20, paid 13, wherefore 13/20 being subducted out of 20/20, there remaine 7/20: wherupon the denominators being omit­ted, the termes shall thus be set together:

7 are 35: ergo 13 were 65

A certaine man selling his horse lost the sixth part and 2/15 of the money which he paid for it. and had remaining 243 crownes: the question is vvhat the house cost him.

This example is like to the former. for you must take the least number which may com­monly be diuided by the denominators of the parts giuen, 6 and 15, to wit, 30: whose parts being of the same denomination, to wit, 1/6 is 5, and 2/15 are 4. The partes added make 9/30, which being subducted from 20/30 leaue 21/30 where vpon the proportionall numbers according to the rule shall be:

21 are 243: ergo 30 347 1/3

So much was the principall stocke wher­fore he lost 104 crownes 1/7.

Sometimes not onely simple, but also mixt nu­meration helpeth oftentimes to the ordering of proportion.

For example.

Of three siluer pots, the first wayeth 12 halfe oun­ces and 2/3, the second 16 1/2, the third 23 1/6. The halfe ounce is worth 11 vrsati 1/2: vvhat then is the price of the pots?

Here of necessity must addition of mixt numbers goe before, although the principall proportion also consisteth of mixt numbers. For the waight of all the pots being added to­gether, you shal haue halfe ounces 52 1/3: wher­upon the proportion shall be thus framed.

If 1 cost 11 1/2, therefore 52 1/3 601 5/6

A Vintener hauing bought 16 hogsheds of wine for 27 3/4 crownes, the price of the vvine falling, could not sell them but for crownes 19 5/6: I demaund vvhat he should haue lost if there had bene 84 hogsheds?

Here subduction of mixt numbers must go before. For the price of the thing sold for lesse then it cost being subducted out of the price which it was bought for, will bring out the second terme of the proportion, so that you may easily answere the question. There­fore 19 5/7 being subducted from 27 3/4 there remaines 7 11/12, the losse in 16 hogsheads of wine. Wherefore the termes of proportion shall thus be set downe:

In 16 hogs. there are lost 7 11/12: ergo 84, 41 9/16

Mixt numeration is more common a great deale then any other, because the measures, prices, &c. of things are seldome contayned in whole numbers: wherefore the studious may here haue plentifull matter to exercise him selfe in.

CHAP. XVII. Of examples of the golden Rule requiring proportion before them.

HItherto numeration vsed in simple and ab­soulte Arithmeticke serued to the finding out of the conclusion in Geometricall proportion. [Page 119] Yet oftentimes some part of compared Arithme­tick is required therunto: as for example, propor­tion either Arithmeticall or Geometricall. First then let these be examples wherein Arithmeti­call proportion must go before.

An Espie enformeth his Captaine of the distance of his enemies campe after this manner. If we take our iourney hence, and goe the first day 30, the second 28, the third 26 furlongs, dimi­nishing after that order euery day two fur­longs, vve shall meete vvith them at fifteene dayes end. Now the enemie knowing of their comming, made toward them, so that at nine dayes end they met together. I demaund how farre the Captaine vvent before he met vvith the enemie.

Here first and formost we must gather the sum of the proportion Arithmeticall (which commonly we call progression) whose termes are 15, and the extremes 30, and 2: the whole summe, then is 240 furlongs, which is the second terme seruing to conclude the Geo­metricall proportion on this manner.

15 240: ergo 9 144

So farre therefore they trauelled, the rest, to wit, 96 furlongs the enemies went.

A labourer vvas hyred for this vvages, to re­ceiue the first day a pennie, the second three, the third fiue, and after the same order his wages should be continued for the dayes fol­lowing: at length being dismissed, he recei­ued [Page 120] 1500 pence, I demaund how many dayes he serued?

Here if you gather together all the termes of the Arithmeticall progression into one summe, you shall make a way to the Geome­tricall progression. As for example, if you adde together his wages for 30 dayes, the first terme of the progression being an vnitie, and the difference 2, the whole summe shall be 900 pence: whereupon I may conclude:

If 900 require 30: ergo 1500, 50.

In the examples following, Geometricall pro­portion must goe before to informe the principall proportion.

When a vveb of cloth contayning 40 yardes is bought for 50 crownes, for how much shall he sell the yard, that in an hundred crownes, de­sireth to gaine 12?

Before you can conclude the question by Geometricall proportion, being as it were the decider of the whole controuersie, you must seeke out the price of one yarde. If 40 yardes are bought for 50 crownes: then one yarde doth cost 11 1/4 of a crowne. This shall be the third terme of the principall propor­tion, in this manner. If of an

100 crownes there arise 112: then of 1 1/4 there arise 1 2/5

A Marchant buying at Venice an hundred yardes of silke for a crowne a yearde, spent [Page 121] in his iourney and for the carriage of them 30 crownes. At Lipsia vvhere the yarde is more by a quarter then the yarde of Ve­nice, he solde euery yard for 3 crownes. The question is vvhat he gayned?

In this example addition of whole num­bers, together with proportion goeth before. For his stocke laide out, and his charges being added, the whole is 130. Besides, to know how many yardes of Lipsia aun­swere the Venice measure, you shall worke by proportion on this manner. If 5/4 of Lipsia measure, make a yarde of Venice, then 80 yardes of Lipsia shall make an hun­dred Venice yardes. So that you may readily inferre the principall proportion thus:

If 1 is solde for 3: ergo 80 240

Whereupon 130 crownes being the stocke and charges subducted out of 240, I see he gayned 110.

Another marchant bought three hogsheads of honie. The first vvayed 349 pound, the se­cond 286 pound, the third 300, out of eue­rie hundred are subducted tenne pound be­cause of the vessell, and a pound of honnie is solde for 5 pence: I demaund vvhat he is to paie.

For so much as the price of one pound of honie is set downe, therefore by proportion [Page 122] seeke out the price of an hundred, 1 pound is solde for fiue pence: ergo 100, for 500 pence, then adde the waight of the vessels, the whole is 110, adde also the waight of the three hogsheds, the totall is 935: whereupon you shall conclude:

110 are sold at 500 pence, ergo 935 at 4250, that is, 170 florentines.

Another buying two yardes of cloth at three crownes, solde sixe yardes afterwardes for 11 crownes. Hauing got by this meanes 46 crownes, there ariseth a question howe many yardes he bought, and how many he sold.

Before you can come vnto the principall proportion, you must order the termes ther­of by a former proportion, and by subducti­on. First therefore you must consider howe much the sixe yardes of cloth which after­wardes were solde at an higher price cost at the first, the which may easily be inferred by that which is presupposed in the question: for if two yardes cost three crownes, then six yardes cost three crownes, then six yardes were sold at nine crownes. This price I subduct from 11, and I see that sixe yardes were sold for two crownes more then they cost. So that it is easie to resolue the doubt thus:

If 2 come of 6, then 46, 138

A Ʋintener bought 1. 24 hogsheds of vvine on this condition, that subducting out of euerie two hogsheds because of the leese fiue gallons, he should pay for euery hogshed 26 crovvines: [Page 123] Tell me vvhat he is to pay.

Proportion going before with multipli­cation and subduction, will set in order the principall proportion. First I seeke howe much is subducted out of al the hogsheds be­cause of the dragges. Out of two hogsheds are drawen 5 gallons, ergo out of 120 hogges­heds are drawen 310. Then I reduce all the foresaid hogsheds vnto gallons, multiplying them by 32, for so many gallons doth euerie hogshed hold, the facit is 3968. Out of these I subduct according to couenant 310 mea­sures: the remainder is 3658 gallons, where­upon the proportion shall be thus:

32 giue 26: ergo 3658 2972 1/8

Of two Carpenters, the one alone vvould frame an house in 30 daies, the other in 40. Now taking another vnto them, they finished the vvorke in 15 dayes. In what time then would the third man haue made it?

Here must you vse proportion with addi­tion, and subduction of fractions. The first would haue done the worke in 30 dayes, ergo in 15 daies he finished 15/30 or 1/2 of the worke, the second would haue wrought it in 40 daies ergo in 15 dayes he wrought 15/40 or 3/8. The partes added together are 7/8 of the worke: which being subducted out of 8/8, that is, out of the whole worke, there remaineth 1/8, the third mans worke. In so much that the pro­portion may thus be concluded:

1/8 is framed in 15: ergo 1/1 120

Put case 14 pound of some vvare were bought for 6 crownes, and euery hundred were sold for 47 crownes: subduct so many pound out of the vvhole as will in the sale amount to 80 crownes gaine. How many pound then vvere there sold?

To vnknit this knot you shall vse propor­tion with subduction of mixt numbers. First I seeke what he paide for an hundred pound. If 14 pound be bought for 6 crownes, then 100 pound giue 42 6/7. Moreouer for so much as an hundred pound were solde for 47 crownes, therefore 42 6/7 being subducted out of 47, the remainder is 4 1/7 the gaine of an hundred: whereupon I say, If 4 1/7 come of 100, then 80 of 1931 1/29.

Sometimes many and sundrie kindes of numeration, as well out of simple as compa­red Arithmeticke, goe before the principall proportion, wherein the diligent young pra­ctitioner may exercise himselfe.

CHAP. XVIII. Of reciprocall proportion.

HItherto of simple direct proportion: now must we come to Reciprocall proportion. Reciprocall proportion is that vvherein the termes being set downe crossewaies, as the third is vnto the second, so reciprocally the first is to the fourth.

A mong the vulgar people we call Reci­procation the going backewarde the same way we begunne: which is euidently seene in reciprocall proportion being compared with direct proportion.

For that way that direct proportion go­eth on forwarde, the same way reciprocall proportion returneth backwarde. For in direct proportion the termes are proportio­nall, not onely as they stand in order, but al­so crossewayes: that is to say, as the first is to the second, so the third is to the fourth, and also as the first is to the third, so the seconde is to the fourth. But in reciprocall pro­portion the termes being taken first order­ly, are altogether without proportion, and crossewayes they differ in all respects in that one is of greater inequalitie, the other al­wayes of lesse: yea rather as the third is to the second, so backwardes, the first is to the fourth, in so much that that which in direct proportion was the first terme, is the thirde in reciprocall proportion: and againe that which is the third in direct proportion, is the first in reciprocall proportion.

For example sake, if twelue hyred ser­uantes doe their worke in 8 dayes, then 24 shall doe it in foure dayes. Here are set downe foure termes of reciprocall pro­portion, which being placed directly shall stande thus: [Page 116] [...] [Page 117] [...] [Page 128] [...] stion I must vse the rule of reciprocall pro­portion on this manner:

If 40 require 7: then 70 how many?

Multiply 40 by 7, the facit shall be 280, which being diuided by 70, giueth 4 or two ounces, which is the waight of the loafe. Three milles in two dayes grinde 20 bushels of corne, in what time then shall fiue milles grind so many bushels?

Here againe by how many more milles there be, in so much lesse time they shal grind the corne. Wherefore the proportion shall be concluded reciprocally in this manner.

3 grinde 2: ergo 5 1 1/5.

A certaine man borrowing 66 crownes of his friend, and repaying them at seuen monethes end, promiseth to pleasure his credit or in as great a matter. Whereupon the creditor af­terward desired to borrow of him 112 crowns, but they doubt how long time he should keepe the money, that the lending of his money might be correspondent.

For so much as the latter man desired to borrow the greater summe, therefore he is bound to restore it sooner then the first shold, whose summe was lesse: whereupon the re­ciprocation shall be concluded thus:

66 giue 7: ergo 112 4 1/8.

So likewise the former example of redu­cing 126 Hungarie ducats (each of them [Page 129] being valued at 28 vrsati) vnto Thaleri be­ing worth 18 vrsati, may be handled and re­solued after this fashion.

28 126: 18 196.

CHAP. XIX. Of manifold proportion, and first of that which is compound by Addition commonly called the rule of Fellowshippe.

• Either cō ­pounded, and is two fold,
  • Prime, wherein there is required
    • Either one kind of numerati­on only, as
      • Addition. Chap. 19.
      • Multipli­cation. Chap. 20.
    • Or many kinds of nu­meration: as that which is compounded by mul­tiplication, and additi­on together. Chap. 21.
  • Or Second, and is called Alligation.
• Or continued. Chap. 27.

HItherto hath bene handled simple proporti­on, as well direct as reciprocall, being set forth by diuerse examples: now followeth mani­fold proportion.

Manifold proportion is that vvhich consisteth of more thenfoure termes.

Simple proportion disiunct, consisteth of foure termes, but manifold proportion con­sisteth of more, as when in one and the same proportion there be manie consequents to one antecedent, or contrariwise, or else when there be two rowes of nūbers proportionall in two numbers, or some such like manner.

Manifolde proportion is double: for either it is compounded in the termes, or els continued.

The first ioyneth together the termes giuen, either by one onely kinde of numeration, to vvit, either by addition or multiplication, or else by more, as by addition and multiplication together.

Compound proportionalitie or propor­tion is said to be that wherein the termes are vnited together. But the things so vnited and ioyned together, are of many brought into one, and heaped together two maner of waies, either by addition or multiplication alone, or else both waies together.

By Addition, as when many as well antecedents as consequents are taken and compared, as one an­tecedent to one consequent.

Although proportions may be compoun­ded many wayes by Addition, yet of them all this is the most vsuall: wherefore this spe­ciall [Page 131] definition seruing most for this purpose is taken in stead of that which is more gene­rall. For example sake, if 3 vnto 2, be as 9 vn­to 6, then by addition 3 and 9 that is 12, shall be in like manner vnto 2 and 6, that is 8, so that the termes of the proportion may stand as you see,

12 vnto 8: are as 3 to 2 and 9 to 6

In this kind of manifold proportion, the rule following is to be vsed,

If in the termes of manifold proportion the consequent of the one proportionality be giuen added together, and the antecedents seuerally by themselues, then as the antece­dents knit together by addition, are vnto the consequents added, so shall the seuerall ante­cedents be vnto their seuerall consequents, and so contrariwise.

This is the 12 p. 7, sauing that it is reuer­sed and applyed some what more fitly to this purpose. For the words of the proposition be these: If there be a multitude of numbers, how many so euer proportionall, as one of the antece­dents is to one of the consequents, so are all the antecedents vnto all the consequents. And the same backeward is true also. By which con­uersion it appeareth how the examples which commonly are referred vnto the rule of fel­lowshippe, hauing many proportions cou­pled by addition, are to be handled and to be [Page 132] resolued by the rule of proportion, in which examples the antecedents of the one propor­tion are set downe seuerally, to wit, the con­tributions of the partners, and the conse­quents are added together, to wit, the com­mon gaine or losse. So that the contributions and common stocke of the partners, or else generally all those numbers into which the other is to bee distributed proportionally must be added together, and the totall must be set in the first place in stead of the ante­cedent of the first proportion, whose conse­quent must be the foresaide number which is taken to be distributed proportionally. In the third place the termes compounded by addition must be placed seuerally, & the rule must be repeated as often as there be seuerall numbers, whereupon I may conclude the proportion. As in the examples following.

Three men being partners laide their money to­gether, the first 60, the second 100, the third 135 crownes. Nowe they gayned after a vvhile 45 crownes. The question is howe much of the gaines each of them by duetie ought to haue.

In this and such like examples equitie and reason requireth, that that which is gotten by the common stocke should also be com­mon, and should be so distributed, that euery mans portion should haue the same propor­tion to the whole gaine, that the mony which [Page 133] he laide downe had vnto the whole stocke: and crossewayes as the whole stocke is vnto the whole gaine, so then euery mans portion laide together, should be vnto his part of the gaines. Wherefore here this question of ma­nifold proportion is manifest. Nowe the termes giuen are these:

The consequents of the second proporti­on are set down but ioyntly together, to wit, 45, the antecedents are seuerall. Wherefore to conclude the seuerall consequents, I set the whole stocke gathered by adding of 60, 100, 135, to wit, 295 in the first place, which is vnto the whole gaine, as the seuerally termes in the third place are vnto the seuerall termes in the fourth place, which we desire to know. The example is thus:

Three partners in a common stocke of 500 crownes, lost 124, to the repayring vvhereof the first vvas bound to be contributarie 20 crownes, the second 43: I demaunde howe much euery one put into the common purse.

Here wee haue of the latter proportion three antecedents, two are assigned in ex­presse words, but the third is set downe som­what obscurely. But for so much as the totall of the three added together is set downe, to wit, 124, the third may easily be knowen by subduction. Adde therefore 20 and 43, the totall is 63, which being subducted out of 124 there remaines 61, which is the third mans portion in the losse. Wherefore as in the former example we gathered euery mans gaine by euery ones contribution, so like­wise in this place by the partes of the losse we may gather each mans stocke. The ex­ample is thus:

Cicero in his Oration for Cecinna sayeth: A woman making her vvill departed this life, she made Cecinna heire of eleuen ounces and an halfe of her goods, and Marcus Fulcinus the free man of her former husbande heire of two fixt parts of an ounce. Ʋnto Eutius she gaue one sixt part. How much then had eue­rie one for his portion.

In this example the whole patrimonie is diuided into eleuen ounces, halfe an ounce, and three sixt partes of an ounce: the which parts are the antecedents of the late propor­tion. But in that they be of diuerse denomi­nations, [Page 135] I resolue them all into the least deno­mination, to wit, into sixts. If then you al­low to an ounce sixe sixths, and to the pound twelue ounces, there shall be in a pound 72 sixths. Whereof 11 ounces and an: halfe are worth 69, and two sixths 2, one 1. Whereup­on supposing the whole patrimonie to be 2500 crownes, the termes shall be thus:

Three couenanted betweene themselues, that of their gaines the first should haue twice as much as the second. Now they had laid downe 24 crownes, and the third 36. But when they had gotten 12 crownes they cannot tell howe to diuide them lawfully according to their co­uenant.

Here that which the first man laide down being couertly expressed, must be gathered out of the couenant. For in that he was to re­ceiue twice so much as the second, it is ma­nifest that that which he laide out was twice as much as that whereof the second was con­tributary, whether it were money, or whe­ther it were his labour, or whether it were valued in both. Wherefore if for the first you take 48 crownes the double of 24, then they being added together, the whole doubt shall be remooued by the rule on this man­ner.

The good man of an house bequeatheth vnto foure men 584 crownes, to be diuided after this rate, that A should receiue 1/2, B 2/3, C 3/4, D 5/6, vvhat vvas euery ones portion?

Here the proportion of the parts vnto the goods which were left is expressed: yet the parts themselues in the goods doe not yet appeare manifest. The least number there­fore that may be diuided by the denomina­tors of the fractions, which is 12, will declare the proportion. For 1/2 of 12, are 6, 2/3 are 8, 3/4 are 9, 5/6 are 10. These being added, the exam­ple shall be thus:

But if he vvilled to make such a diuision a­mong them that were to receiue the legacie that the first should haue 1/2 and 1/5 wanting 20 crownes, the second 1/3 and 30 crownes, the third 3/4 and 10 crownes, the fourth 1/6 wanting 12 crownes: then vvhat should their por­tions be?

In this case that which is more then their portions I take it from the whole substance, that which wanteth I adde vnto it: as 30 and 50 are 80, which being taken from 584, there [Page 137] remaines 504. Hereunto adde 20 and 12, or 32, the totall is 536. Then I seeke out the least number that may be commonly diui­ded by the denominators, to wit, 60, the partes of the same denomination are for the first 30 and 12, for the second 20, for the third 45, for the fourth 10: and then the por­tion of the first being first added together, and afterward all the rest, the example will be thus:

Then that the testator may be satisfied, from the first mans portion I take 20 crowns there remaines 172 48/117, vnto the second adde 30, they make 121 73/117, vnto the third adde 50, his whole portion will be 256 18/117, lastly from the fourth take 12, there remaine vnto him 33 95/117, all which partes are answerable vnto the 584 crownes, which were to be di­uided.

Thre partners laide together three equall por­tions, but they left them there, some for longer time and some for shorter: the first left his money in the stocke two monethes and an halfe, the seconde three monethes and 2/5, the third foure monethes and 2/3. Put case they gained 12 crownes, I demaund what was euery mans portion.

This example being of mixt numeration, requireth no other worke then the former. For the seuerall times being added together, wherein they left their stocke for the com­mon trafficke, the totall will be 317/30: So that the example will be.

CHAP. XX. Of multiplied proportion compound by multi­plication, vvhich commonly is called the double Rule.

THe former examples appertained vnto pro­portion compound by addition: now follow­eth that vvhich is compound by multiplicacion, vvhich in stead of two termes of the proportion, vseth the facit made of them.

This is the right compounding of pro­portion, which gathereth them together by multiplication as was said before, and appea­reth by the fift definition of the sixth booke of Euclide, because it doth not only ioyne to­gether the termes themselues, but also maketh the proportion of the compound termes to comprehend the proportion of the simple. Wherefore hitherto are to be referred, and [Page 139] by this rule of proportion are to be handled these examples, wherein two termes by mul­tiplication must be reduced vnto one: for there proportion as appeareth by the fifth prop. of the eighth booke of Euclide, is ga­thered of the proportion of the termes mul­tiplied one by the other. Of this sort are all those examples which commonly are refer­red to the double rule, wherein two termes are set in the same places, that is, the princi­pall proportion hath a certaine circumstance adioyned vnto it, in so much that here the proportion should be twice concluded, were it not that for those two termes the products made of them are taken as one.

The termes propounded by multiplication, are either of the same place, as in direct proportion, or of diuerse places, as in reciprocall proportion.

The examples of the first are these.

Foure Students spend in 3 monethes 19 crownes: how much therefore shall eight Students spend in 9 monethes.

You see here the principall termes haue annexed vnto them a circumstance of time, & therfore (the maner of ordering the termes in the golden rule being obserued) the first and the third are double, on this fashion:

Wherefore for those two I take the com­pounds by multiplication. As for 4 and 3 [Page 140] I take 12, for 8 and 9, 72, to wit their products: So that the compound multiplication shall be concluded thus:

Here betwene the expences of foure stu­dents in three monethes, there is obserued the same proportion which is betweene the expences of 8 students in 9 moneths, to wit, the proportion compound of the two, to wit, of 4 vnto 8, and 3 vnto 9, or to speake more plainely, the expences are one to an­other, as thrice foure vnto nine times eight. The which may easily be proued, if you put in the meane proportionall number to whom the termes of the antecedents and consequents may belong. So that if you should set it downe that foure student in 9 monethes should make their expence, the example will be thus:

Here then if foure students in three mo­nethes dispende euery one but one, then so many students in nine monethes shall spende thrice as much. Wherefore the proportion [Page 141] of the expences of A vnto B, shall be the same that is betwene three and nine. Like­wise if foure students in nine moneths spend euerie one but one, then 8 students in as ma­nie monethes will spende twise as much. Whereupon there will be the same reason betweene B and C, as is betweene 4 and 8.

Furthermore if you multiply 3 and 9, the termes of the first reason by one and the same number, namely by 4, the productes, namely 12 and 36, shall keepe the same rea­son by the 17 proposition of the 7 of Euclide: so shall 36 and 72 being made of 4 and 8 multiplyed by 9. Whereby it is gathered that as the charges of A is vnto B, so is 12 vnto 36. And in like manner as B is vnto C, so is 36 vnto 72. To conclude, seeing there are here three proportionall numbers, A, B, C, and as many more in number an­swerable vnto them 12, 36, 72, therefore by the 14 of the 7 (taking the extremes the meane proportionals being withdrawen) the charges of A shall be all one to the charges of C, as 12 is vnto 72, whereby the good­nesse of the working doth appeare.

An other example.

Two Printers in foure dayes print 16 formes, ho [...]e many shall seuen Printers print in 14 dayes?

Here the placing of the termes prescribed in the rule of proportion being obserued, they shall be set in this manner:

Then, the first termes and the last being multiplied one by another, the proportion is concluded thus:

8 16: therefore 98 196

Eight yardes of cloth, 4 yardes and 1/4 broad are bought for 11 crownes: therefore how shall 15 yardes of cloth be bought 1 yarde 3/4 broad?

This example of mixt numeration vari­eth nothing from the former, therefore it is thus dispatched:

Three men trading together by ill lucke lost 52 crownes. The first put in 110 crownes for 5 monethes, the second put in 84 crownes, the third 65, I know not for vvhat time. Their traficke being ended, the first found that he had lost 22 crownes, the second found he had lost 18, the third 12. How long therefore was the money of the last two in the common stocke?

I thinke that this example (though it seeme to belong to the Rule of Fellowship) may well be referred to the rule of propor­tion [Page 143] compounded by multiplication, yet it is vnlike to the other, because of the numbers giuen, the middlemost onely is compoun­ded by multiplication, the other are sought out by the rule of three, proceeding from a terme known to those which are vnknowne. Therefore the first mans 22 crownes being multiplied by his 5 moneths, the facit is 550. Whereupon I conclude, if his money, that lost 22 crownes, multiplied by the time doth make 550, what shall the losse of 18 crownes? and what shall the losse of 12 crownes make? The termes shall stand thus:

• 22 550: therefore 18 450
• 22 550: therefore 12 300

The fourth numbers inferred by the rule of three are compounded of ech mans stocke multiplyed by the time for which it was laid out. Therefore ech of them seuerally diui­ded by his owne stocke, the time is seuerally inferred, so that the second mans money was out of his handes fiue monethes and 5/14, the third mans money was out foure and 8/13 of a moneth.

An abridgement of the former worke.

If the termes in the first and third place fall out to be equall, then they being taken a­way, the rest of the termes shall inferre the proportion.

The reason of this abridgement ariseth [Page 144] out of the 17 proposition of the 7, for the numbers giuen keepe the same proportiona­litie which they would haue being multi­plied by one and the same number. As: If the gaine of 25 crownes in foure yeare be eight crownes, vvhat shall 100 crownes yeeld in foure yeare?

Here the termes are giuen as you see.

Forsomuch therefore as 25 hath the same reason to 100 which they would haue being multiplyed by 4, as it appeareth by the 17 prop. 7, therefore the equall numbers which are in the first and thirde place, being cast a­way, I conclude the proportion in simple termes thus:

25 8 therefore 100 32

Likewise the question being thus pro­pounded, the gaine of 25 crownes in 4 yeare, is 8 crownes, therefore what shall 100 yeeld in 25 yeares? The termes shall stand thus:

Here againe the alternate termes of the first and third place being omitted, the rest shall conclude the proportion thus:

4 8: therefore 100 200

If the proportion chance to be reciprocall, then are the termes of the first and third place com­pounded by alternate or crosse multiplication, and [Page 145] the question afterward is concluded directly.

Some transpose the principall termes set­ting the first in the third and contrariwise the third in the first place, & then bring the two­fold termes into one by multiplication. The example:

Eight horses in 12 daies eate 9 bushels of oats, in how many dayes shall 18 horse eate 24 bushes?

I easily perceiue that there is here a recipro­cation, for 18 horses will spend in a great deale lesser time that heape of oates, which 8 horse will consume in 12 daies. Therefore by how much the third terme is the greater, by so much the fourth terme is the lesser. Therefore I must take heede that disposing the termes as I did before, I conclude not di­rectly the reciprocall proportion by multi­plying of the first and last termes one by an­other. The termes in the example are thus:

Here therefore I multiply 9 by 18 and make 162 the first terme, likewise I multiply 8 by 24 and make 192 the third terme, whereby the lawfull induction of the proportion is thus dispatched:

162 12: therefore 192 14 2/9

If 100 crownes doe giue 5 crownes interest eue­rie yeare, in vvhat time vvill 56 crownes giue. 12? [Page 146] In this question for so much as a greater interest is sought for by a lesser stocke of mo­ney, the reciprocation is manifest, the which interest craueth so much the more time by how much the more the stocke is lesser. Therefore the termes being set downe as it is conuenient, and multiplied crossewayes, the doubt is answered by the rule of three.

CHAP. XXI. Of manifold proportion compounded by mul­tiplication and addition.

THus much concerning the examples of pro­portion compounded by addition or multi­plication onely: now must vve proceede to that proportion vvhich ioyneth both kindes of nume­ration together.

Therefore proportion compounded by multi­plication and addition, is that vvhich first of all multiplieth the manifold termes giuen one with another, and then addeth their productes toge­ther.

This kinde of proportion by a two folde, yea sometimes by a fourefolde composition, bringeth many termes into one. Vnto this [Page 147] appertaine the examples of the Rule of se­cond fellowship, as they commonly cal them, wherein the time is annexed to the stocke, or some other circumstance to the principall terme. The Theoreme following serueth for the working of those examples.

If among the termes of the manifold pro­portion, the consequents of one reason be gi­uen added together, but some antecedents be giuen seuerally and double: then as the antecedents compounded by multiplication and addition are vnto the consequents added together, so shall the seuerall antecedents compounded by multiplication be vnto the seuerall consequents.

As thus for example.

Two men in common trafficke got 60 crownes, the first brought 50 crownes for two yeare, the second 15 for fiue yeare: vvhat shall be ech man his share.

In this question containing a partnership of equall contributions in regard of the di­uerse times, I see the diuision of the gaine must be made according to the proportion of the time. That this diuision may be made the consequents of the latter reason are gi­uen added together, namely 60, but the ante­cedents are giuen not only seuerally, but also double, as you see here.

Therefore first of all I compound the dou­ble termes by multiplication, so that after a sort I reduce the vnequall times to an equali­tie, the products are 100 and 75, which are all one as if you should say, that the first man should haue as much gaine by 100 crownes in one yeare, as he should haue for 50 crownes in 2 yeare. In like manner the second man should get by 75 crownes in one yeare, as much as he should get by 15 crownes in fiue yeare. Moreouer these two products 100 and 75 set downe seuerally, shall possesse the third place of the proportion: but added together they shall stand in the first place: nowe the rule of three twice repeated, shall deliuer the numbers sought for in this manner:

• 175 60: therefore 100 34 2/7
• 175 60: therefore 75 25 5/7

I doe of set purpose omit the demonstra­tion of this composition of the termes, be­cause it is not much vnlike the former.

A hundred and sixtie footmen and fortie horse­men got a bootie of 138 crownes to be diuided betweene them so, that as often as the footmen receiued one, the horsemen shoulde receiue [Page 149] three. How much were the footemen, howe much vvere the horsemen to haue?

Here likewise I place the consequents of the second reason added together in the se­cond place, namely, 138: in the thirde place I set the products made by multiplying the numbers signifying how often ech one shold take his share by the number of the souldiers: namely, 160 and 120, these added together and making 280, shall haue the first place. The example is thus:

• 280 138: therefore 160 78 6/7
• 280 138: therefore 120 59 1/7

Three Butchers hyred a meddow together, pro­mising to pay yearely rent for it 30 crownes, the first fedde in it 20 Oxen 70 daies, the se­cond fedde 46 Oxen 56 dayes, the third fedde 32 Oxen 60 dayes: How much of the rent shall each partner pay?

Againe therefore I multiply each mans heard by the seueral times, the products 1400, 2576, 1920, shall possesse the third place: and being added together, they shall haue the first place, the yearely rent shall be in the middest thus:

• 5896 30: therefore 1400 7 91/737
• 5896 30: therefore 2576 13 79/737
• 5896 30: therefore 1920 9 567/737

Three marchants were partners for a yeare. The first in the beginning brought 250 crownes, but after 3 moneths he withdrew 100. The [Page 150] second after 2 moneths brought 180 crownes, but after 6 moneths of their partnership were expired, he tooke away 50. The third after three moneths brought in 235 crownes, and after fiue moneths added 45 crownes. Now hauing gotten 68 crownes, what is each mans share?

These kind of examples being in outward shew most intricate, require a litle more la­bor, otherwise they are to be handled by the same art. They wil soone be dispatched, if you indeuor but to seuer euerie mans time accor­dingly as he changed his stocke, and make as many multiplications as there were changes. As in this present example: Because the first man left in the partnership 250 crownes for three moneths, let them be multiplied toge­ther, the product will be 750. Then with­drawing 100 crownes, he left but 150 for the nine moneths remaining. Wherfore these be­ing againe multiplied by their time, make 1350. Which numbers being added, the to­tall summe for the first partner is 2100.

Likewise the second man his 180 crownes being multiplied by the 4 moneths wherein he left them in the common trafficke, doe make 720. Moreouer 50 crownes being sub­ducted from 180 (for so manie he fetcht a­way from the principall after the sixth mo­neth of their partner ship) there remaine 130 crownes, which if you multiply by the sixe moneths remaining, the product will be 780, [Page 151] which added to the former, maketh 1500.

So also multiplie the third man his 235 crownes by 9, they make 2115. Item 45 by 7, they make 315, which added to the for­mer, make 2430. These termes being thus compounded, if you proceed as you did be­fore, the proportion will be inferred, which here you see set downe, with ordering of the whole example.

CHAP. XXII. A treatise of Alligation, vvhereof B. SALIGNA­CVS vvas the Author.

• The proprie­tie which is this: that in alligation,
  • The measures should be like.
  • The price of the mixt measure should be meane in quantitie betweene the prices of the simple measures.
• The kinds which are two
  • Prime: which coū terchaun­geth the differēces betweene the ex­tremes & the mean: & herein we are to consider the
    • Chap. 23. Propriety, which is this, that if the ex­tremes are in number
      • Euen: ech of them seuerally must be cōpared once only with the meane.
      • Odde: then
        • Euery extreame of the greatest number must be compared with the meane once onely.
        • Euery extreame of the least number must be cōpared diuerse times with the meane.
    • KIndes. which are two,
      • The first kind, some part of the nūbers sought for being giuen, inferreth the rest by simple proportion. Chap. 24.
      • The second kind, the to­tall summe made of the numbers sought for being giuen, inferreth all the particulars by proportiō compoūded by addition. Chap. 25.
  • Second: which inferreth the meane by proportion compounded by multi­plication and addition. Chap. 26.

Of the Definition and proprieties of Alligation. Chap. 1.

HItherto we haue spoken of the first kinde of compound proportion: the second followeth, and is called Alligation.

Alligation is an art, vvhich (by the meanes of certaine things giuen) maketh equall the totall made of the prices of the measures mingled toge­ther, vnto the totall made of prices of the mea­sures taken seuerally.

There is 100 pound waight of siluer worth 17 pound to be mingled with other siluer worth 24 pound, so that the totall made of the prices of the waightes mingled together may be equall to the totall made of the pri­ces of the waights taken seuerally. Here all the prices are giuen, the art therefore which by the meanes of these prices giuen maketh equall the totall numbers is Alligation.

Item, Of two kinds of corne mingled to­gether suppose there were 10 bushels worth 16 shillings, and 18 bushels worth 12 shillings: foure bushels of this mingled corne are to be sold, so that the totall summe made of the prices of the 28 bushels mingled together be equall to the totall made of the prices of the 10 and 18 bushels seuerally taken. Here both the number and the price of the simple measures are giuē, but there is but only a cer­tain part giuen of the number of the measures mingled together. The art therefore which by the meanes of the thinges giuen maketh [Page 154] equall the totals propounded in the questiō, is called Alligation.

The propertie of Alligation is this, that the measures in it be alike: but the price of the min­gled measure must be in quantitie meane between the prices of the measures seuerally giuen.

A bushell of corne worth 16 shillings is supposed to bee mingled with a bushell of corne worth 18 shillings. Here by this pro­perty first of all the bushels are like measures. Then the price of a mingled bushell must be in a quantitie meane betweene 16 and 18. that is to say, it must be greater then 16, and lesse then 18. So that the price of a simple measure is called the Extreme, but the price of a mingled measure is the Meane.

CHAP. XXIII. Of Prime Alligation, and the pro­pertie thereof.

ALLigation is two fold. Prime, which doeth counterchange the differences of the termes from the meane.

By the definition of Alligation generally taken it appeareth, that in all alligation there are certaine things giuen. Therfore in Prime alligation generally let all prices be vnder­stood to be giuen. Moreouer to coūterchange the differences of the extremes, is nothing else but to attribute the difference of the lesser extreme to the greater extreme, and contrari­wise [Page 155] to attribute the difference of the greater extreame to the lesser extreame. Nowe this counterchaunge is therfore vsed, that we may thereby make equall the totall made of the prices of the measures seuerally taken vnto the totall of the prices of the measures min­gled together. That this may be the better demonstrated, we must first of all set downe two premisses. The first is this:

If one and the same number doe multiply certaine numbers seuerally, and the totall made of them being added together, the pro­duct made of the totall shall be equall to the totall made of the products of the parts.

As here: let 10 be the totall number made of 4 and 6, let the number multiplying them all, be 2, the products made of 4 and 6, are 8 and 12, and the totall made of them is 20. Therefore the product made of 10 by 2 shall be 20.

The second premisse is this:

If three vnequall numbers being giuen, you multiply any one of them by the other two remayning, and augment the selfe same number by one of them which remaine, and diminish it by the other: and then multiply the number augmented by that which was taken away, and the number diminished by that which was added to: the totall number made of the latter products shall be equall to the totall made of the former products.

As here: Let there be three vnequall num­bers [Page 156] 4, 7, 9, multiply 9 by 4 and 7, and let the products be 36 and 63, then augment and diminish the selfe same number 9 as is afore­said, adde therefore 7 vnto it, and let the to­tall be 16, also from 9 take 4, and let the re­mainder be 5, then multiply 16 by 4, and 5 by 7, and let the products be 64 and 35: here the totall made of 64 and 35, shall be equall to the totall made of 36 and 63.

These things being thus set downe: nowe of two kindes of wine, let one be worth 14 pence, another worth 11 pence, let the diffe­rence be 2 and 1 from the meane, which is 12 pence. The totall made of the differences is 3. The meane price is 12, the extreames are 14 and 11. Therefore multiply the meane which is 12, by the totall made of the diffe­rences, namely by 3, the product is 36, that is to say, three pottles worth 12 pence a peece shall be worth 36 pence, I say therefore that one pottle worth 14 pence, and two pottles worth 11 pence a peece shall be worth 36 pence. That is: I say that the totall made of 1 and 2, by 14 and 11, is 36. For here 3 is the totall made of 1 and 2 as was aforesaid. Ther­fore by the first premisse if you multiply 12 by 2 and 1, the totall made of the productes 24 and 12 shall be equall to the product made of 12 and 3. But 12 multiplied by 3 maketh 36, therefore the totall of the products made of 12 multiplyed by 2 and 1 shall be 36.

Againe these three numbers 11, 12, and 14 [Page 157] are vnequall, the differences of the extremes 14 and 11 from 12, which is the meane, are 2 and 1. The greater extreame 14 is 12 more by 2, the lesser extreme 11 is 12 lesse by 1. Therefore by the second premisse if you mul­tiply 12, augmented by 1, which was taken from it, and 12 diminished by 2, which was added to it: that is to say, if you multiply 14 by 1, and 11 by 2, the totall made of their products shall be equall to the totall made of the products of 12 multiplyed by 1 & 2. But the total of the products made of 12 by 1 and 2 is 36, as it appeared before: therefore the totall of the products made of 1 and 2 by 14 and 11, shall be also 36, which was the thing to be demonstrated. Therfore the reason why with vse this counterchaunging is as you see.

Here this also is to be noted:

If ech number of the extreames be a num­ber of multitude, the differences may be counterchaunged diuersely: and therefore then in one and the same example a diuerse alligation may be made.

But all Alligation maketh equall the to­tall made of the prices of the mixt measures to the totall made of the measures seuerally taken, as appeareth by the definition of ge­nerall alligation: wherefore in the alligation intended when each number of the extremes is a number of multitude, the counterchang­ing of the differences is at our owne choise.

Hereafter followe two proprieties, and as [Page 158] many kindes of prime alligation. The first proprietie is thus:

If the extremes be equall in number, each of them is compared with the meane but once onely.

The second proprietie is thus:

If the extremes be vnequall in number, then each of the extremes whose number is the grea­ter, are compared vvith the meane once onely: but touching the lesser number of extremes, if the extreme be but one onely, then is that one extreme to be cempared vvith the meane so of­ten as there is vnities in the greater number of the extremes.

In this example the greater number of the extremes is 2, and therefore the extreme of the lesser number is twise compared with the meane. Where also you shall note that the totall made of the manifold differences attri­buted [Page 159] to one and the same extreme, is taken for one difference: for, for 7 and 3 wee take 10.

But if the extremes of the lesser number be many, if you compare more then one of them oftentimes vvith the meane, the number arising of the comparison is vvithout arte: but if you compare but one onely vvith the meane, that shall be compared with it so often as the diffe­rence of the vnequall numbers is being added to an vnitie, but the rest shall be compared with the meane but once onely.

In this example the greater number of the extremes is 4, the lesser is 2, the difference of these vnequall numbers is 2, vnto which if you adde 1, the totall will be 3. Now I com­pare onely one extreame of the lesser num­ber, namely 32, oftentimes with the meane 15, and I compare it three times, but I com­pare the extreme which remaineth but once onely with the meane. And then last of all. I take the totall made of the manifold diffe­rences attributed to one extreme, for one difference as I did before.

Thus much concerning the two proprie­ties [Page 160] of prime alligation: now follow the two kindes for the vnderstanding whereof we are to note that the numbers sought for are the numbers of the simple measures.

CHAP. XXIIII. Concerning the first kinde of prime Alligation.

PRime alligation of the first kind, is vvhen some part of the numbers sought for be­ing giuen, vve inferre the rest by simple pro­portion.

Here we take the difference attributed to that extreme, whose measures are by num­ber giuen, for the first terme of proportion, and for the third we take the number of the measures giuen. The which kind of working shall be euidently seene in the examples fol­lowing.

My neighbour mingled 12 bushels of fine vvheat worth 14 pence, with other corne, namely, wheate, barlie, and oates: the bushell of wheat was worth 18 pence, the barlie 11 pence, the oates 9 pence, the bushell of corne mingled to­gether was worth 10 pence. The question is, how much wheat? how much barlie? how much oates was mingled together?

Here a part of the numbers sought for is giuen namely 12 bushels, therefore first of all I counterchaunge the differences of the ex­tremes as you see:

Then for the first terme of the proportion I take the difference attributed to that ex­treme whose measures are by number giuen, namely one bushell worth 14 pence, but for the third terme I take the nūber of the mea­sures giuen, namely 12 bushels worth 14 pence a bushell. And so conclude the questi­on propounded by two simple proportions. The first proportion is thus:

1 Bushell worth 14 pence requireth 13 bu­shels worth 9 pence the bushell: there­fore 12 bushels at 14 pence the bushell require 156 bushels at nine pence the bushell.

The second is thus:

1 bushell worth 14 pence requireth 1 bu­shell of each other kinde: therefore 12 bushels at 14 pence the bushell require 12 bushels of ech other kind.

Wherefore with 12 bushels at 14 pence the bushell were mingled 156 bushels at 9 pence the bushell, and 12 bushell of each of the o­ther kindes.

Suppose there were foure kindes of siluer: let a pound of the one be worth 20 pound, of the other 16 pound, of the third 14 pound, of the fourth 12 pound. These foure kindes of siluer [Page 162] are to be mingled together, so that a pound of the mingled siluer be worth 15 pound. Nowe there were taken 33 pound of the siluer worth 16 pound, the question is how many pound of each other kind is to be taken?

Here a part of the numbers sought for is giuen, namely 33, therefore the differences being counterchaunged as you see:

For the first terme of the proportion, I take the difference added to the extreme whose measures are by number giuen, namely one pound worth 16 pound, but for the third I take the nūber of the measures giuen, name­ly 33 pound at 16 pound the pound. Then I conclude the question propounded by three simple proportions, the first proportion is thus:

1 pound worth 16 pound, requireth one pound worth 14 pound: therefore 33 pound worth 16 pound the pound, re­quire 33 pound worth 14 pound the pound.

The second is thus:

1 pound worth 16 pound, requireth three pound worth 20 pound the pound: therfore 33 pound worth 16 pound the [Page 163] pound require 99 pound worth 20 pound the pound.

The third is thus:

1 pound worth 16 pound, requireth fiue pound worth 12 pound the pound: therfore 33 pound worth 16 pound the pound require 165 pound worth 12 pound the pound.

Therefore if there were 33 pound taken of the siluer worth 16 pound the pound, there might be taken 33 pound of the siluer worth 14 pound the pound, and 99 pound of the siluer worth 20 pound the pound, and 165 pound worth 12 pound the pound, I say they may be taken, but it is not needefull they should be taken. For seeing that here each number of the extremes is a number of mul­titude, therefore the alligation may be mani­folde, according as I noted it before in the de­finition of prime alligation.

The prices of the same kinds of siluer be­ing kept, if there had bene 6 pound taken of the siluer worth 16 pound the pound, and 8 pound of the siluer worth 12 pound the pound, that a pound of the mingled siluer might be worth 15 pound, you shall say thus:

1 pound worth 16 pound requireth 5 pound worth 12 pound: therefore 6 pound worth 16 pound the pound re­quireth 30 worth 12 pound the poūd.

And therfore vnto those 8 pound worth 12 pound the pound, there are to bee added [Page 164] 22 pound of the same price. The rest is easie by that which hath beene said before.

CHAP. XXV. Of the second kind of prime Alligation.

VVE haue spoken of the first kind of prime alligation. Prime alligation of the se­cond kind is that, which by the meanes of the to­tall made of the numbers sought for inferreth all the particulars by the helpe of proportion com­pounded by addition.

This compound proportion is common­ly called the rule of Fellowship. In it for the first terme of proportion we take the totall made of the differences added together: for the second, we take the totall made of the number of the measures, and for the antece­dents of the reasons remaining, the counter­channged differences of the extremes from the meane.

Mine hoast hath two sorts of vvine, one worth sixe pence a quart, another vvorth 12 pence a quart. Of these two he purposeth to drawe sixe quarts at 10 pence: how manie quartes therefore shall hee drawe of each seuerall kinde.

Here the whole number giuen made of the numbers of the simple measures is sixe. Therefore counterchaunging the differences as you see:

For the first terme I take the totall made of the differences which is 6. For the second terme I take 6, and for the antecedents of the reasons remayning I take 2 and 4, and then I make a compounded proportion in this manner:

• 6 6 therefore 2 2
• 6 6 therefore 4 4

Here therefore there must be drawen two quarts at 6 pence the quart, & 4 at 12 pence the quart. Hereby it appeareth that if the to­tall made of the numbers of the simple mea­sures be equall to the totall made of the num­bers of the differences, the numbers of the differences are the numbers of the simple measures.

Hiero King of Syracusa voweth to the goddes for the prosperous successe of his affaires a crowne of gold. Let vs suppose that Hiero for the ma­king of this crowne gaue vnto a vvorkeman 500 pound of gold: and that the workeman made in deede a crowne of iust vvaight, but mingled some siluer in the crowne. The king demaundeth of Archimedes how much golde, how much siluer was in the mingled crowne.

For the resoluing of this question Archi­medes tooke two lumpes of mettall of the same waight with the crown, but of the same [Page 166] kind with the gold and siluer which was in the crowne. I say they must be of the same kind. For let there be two lumps of siluer of equall waight one to an other, but of vne­qall finenesse, the finer lumpe shall fill a lesser place then the other which is not so fine. The same is to be said of the lumpe of gold. So that here will be a most manifest error, if the mettals be they either golde or siluer, be not both of one kind. I say therefore that the lumpe both of golde and siluer was of the same kinde with the golde and siluer in the crowne.

Now let there be 3 such bodies chosen: let the one be a lumpe of gold, the other a lump of siluer, the third the crowne mingled of gold and siluer. And let them be hanged one after another seuerally in a vessell full of wa­ter, and let the waight of the first be 968, of the second 952, of the third 964. Then let the waights be taken for the value of the bodies themselues: the value therefore of the lumpe of gold shall be 968, of the siluer 952, of the crowne 964. Now in the mingled crowne by supposition there are 500 pound waight: therefore here the totall example made of the numbers of the measures shall be 500, so that the differences being counterchaunged as you see:

For the first terme I take the totall made of the differences, namely 16, for the second I take the totall made of the numbers of the measures sought for, namely 500, for the an­tecedents of the reasons remaining. I take 12 and 4, and thereof frame a compound pro­portion in this manner:

• 16 500: therefore 4 125
• 16 500: therefore 12 375

Whereby it is inferred that there was in the mingled crowne 125 pound of siluer, and 375 of gold.

Archimedes in Vitruuius is saide to haue found out the mixture of the golde, groun­ding his reason on the differences of the wa­ter running out of the vessell. But that the art may stand sure, I construe his manner of reason thus: that is to say, that by the knowen differences of the water running out of the vessell Archimedes attained vnto the other differences of the water vnknowen, and then at length answered the question by prime al­ligation of the second kind. Otherwise sup­pose an vnequall running out of the water, that is to say, when the gold was put in, say that there ranne out 20, when the siluer was put in 36, and when the crowne was put in 24. These seuerall quantities of water run­ning out are to be taken for the values of the bodies as they were put in, according to the intent of him which gainesayeth mine asser­tion. [Page 168] Therefore the value of a lumpe of gold of 500 pound, shall be lesser then the value of a masse of siluer of the same waight, which is against reason. Therefore Archimedes by the knowen differences of the water running out, attained to the differences of the water remaining in this manner. Suppose that the vessell into the which the crownes were put were 488 pintes. If therefore the water which ranne out when the lumpe of golde was put in were 20 pintes, the rest was 468, the like we may iudge of the rest.

My hoast mingled foure sortes of vvine to the quantitie of 300 quartes: A quart of the one vvine vvas vvorth 12 pence, of another 10 pence, of the third 9 pence, of the fourth 7 pence: He solde a quart of the mingled vvine for 11 pence, how many quartes of the first? how many of the second? how many of the third? and how many of the fourth kinde did he mingle together?

Here the totall number giuen made of the number of the simple measures is 300. Therefore counterchaunging the differences as you see:

For the first terme I take the totall made [Page 169] of the differences, namely 10, for the second I take 300, for the antecedents of the rea­sons remayning I take 1, 1, 1 and 7, and then I frame a compounded proportion thus:

• 10 300 therefore 7 210
• 10 300 therefore 1 30
• 10 300 therefore 1 30
• 10 300 therefore 1 30

Whereupon I conclude that the quartes herein mingled, were first 210 at 12 pence a quart, then of each of the rest there were 30 quartes.

An Apothecary was to mingle pepper, sugar, cin­namon and ginger to the quantitie of fiftie ounces. An ounce of pepper vvas vvorth 25, sugar 24, cinnamon 22, ginger 18. An ounce of the spice mingled together vvas worth 23: How many ounces of pepper, how many of sugar, how many of each other kinde are to be mingled together?

Here first of all the totall giuen being made of the numbers of the simple measures is 500. Therefore the differences being coun­terchaunged as you see:

For the first terme of the proportion I take the totall made of the differences, namely 9, for the second I take 500, for the antecedents of the reasons remaining I take 5, 1, 1, and 2, and thereof I frame a compounded propor­tion in this manner:

• 9 500 therefore 5 277 7/9
• 9 500 therefore 1 55 5/9
• 9 500 therefore 1 55 5/9
• 9 500 therefore 2 111 1/9

Here therefore may be mingled of pepper 277 ounces and 7/9, of sugar 55 5/9, and so much cinnamon, of ginger 111 and 1/9. I say they may be mingled, but they shall not be ming­led. For in this question each number of the extremes is a number of multitude, and ther­fore a manifold alligatiō may be made there­in, of which manifold alligation I gaue a note before in the first kind of prime alligation.

CHAP. XXVI. Of second Alligation.

HItherto of prime alligation: Second alliga­tion followeth which by proportion cōpoun­ded by multiplication and addition inferreth the meane.

Here both the numbers and prices are gi­uen of each seuerall measure, but there is but some part onely of the mingled measures giuen. Therefore here for the first terme I [Page 171] take the totall made of the numbers of the simple measures, and for the second terme, I take the totall of the products made by mul­tiplying those numbers by their extremes, and last of all for the third terme of the pro­portion, I take the part giuen of the num­ber of the mixt measures.

Of two kinds of corne there were mingled 10 bu­shels at 16 pence a bushell, vvith 18 bushels at 12 pence a bushell: vvhat is the bushell of mingled corne worth?

Here first of all I adde together the num­bers of the simple measures 10 and 18, the to­tall is 28. Then I multiply 10 by 16, and 18 by 12, and make 160 and 216, the totall of them is 376. The totall therefore of the pri­ces of the simple measures is 376 pence. But alligation maketh the totall of the prices of the mixt measures equall to the totall of the simple measures as appeareth by the defini­tion thereof. Therefore these 28 mingled bu­shels shall be worth 376. Therfore I conclude the question thus:

28 mingled bushels are worth 376. there­fore one bushell of mingled corne is worth 13 3/7.

P. Ramus the most famous Philosopher of our time, calleth this alligation, alligation of the meane sought for: and defineth it to be that, which (two extremes being giuen) see­keth out the meane by diuiding the extrems [Page 172] added together by their number. He addeth these wordes as if there be two extremes the diuisor must be 2, if there be three extreames the diuisor must be 3, and so foorth. There­fore by this definition if corne at 16 pence and 12 pence a bushell be mingled together to the quantitie of 28 bushels as is aforesaide, the price of a bushell of mingled corne shall be 14 pence, for here the nūber of the extremes is 2, and the extremes 16 and 12 being added make 28, so that if thou diuide 28 by 2, the quotient is 14.

But let vs see how true this is: Here by the rule of prime alligation the counterchaun­ging of the differences is in this manner:

By the which counterchange it is insinua­ted, that when there are taken two bushels of corne at 16 pence, then must there be two al­so taken of 12 pence the bushell. Therefore when there shall be 10 taken at 16 pence the bushell, then shall there also be 10 taken at 12 pence the bushell. The numbers of the simple measures 10 and 18 being added to­gether make 28. Therefore I reason thus: If when 10 bushels at 16 pence the bushell are mingled with 18 bushels at 12 pence the bu­shell, the meane is then 14: therefore when 28 bushels of these simple seueral cornes are so mingled, that the price of a mingled bushell [Page 173] is 14 pence, then shall 10 bushels at 16 pence the bushell, be mingled with 18 bushels at 12 pence the bushell. (For it is all one way from Newhause to Heidelberge, and from Heidel­berge to Newhause.) But this second asser­tion is false (for here when 10 bushels at 16 pence the bushell are taken, then also there are taken 10 bushels at 12 pence the bushell as appeared before) therefore the first assertion is also false. Whether with 18, or with 10 bu­shels at 12 pence a bushell, you mingle 10 bushels at 16 pence a bushell, the extremes added together in both are 28, and their num­ber is 2. Therefore whether you mingle 10 bushels at 16 pence the bushell, with 18, or 10 bushels at 12 pence a bushell, the price of a bushell of mingled corne shall be all one, and so the price of the cheaper wheate shall be the price of the dearer wheate, which is absurde.

I alwaies reuerenced my master while he was a liue as my duetie required, and much more doe I now imbrace the writings of that most holy Martyr being dead. But it is inci­dent to a man to erre: and therefore he being a Philosopher of most holy memorie, estee­med those men to be madde that thought it an hurtfull thing for a cōmon wealth to haue their faults amended: therefore following his owne decree by my definition, I correct his in this place. For he himselfe either did cor­rect it before his death, or if he had liued any [Page 174] longer, I doubt not but he would willingly haue amended it: if he did correct it, truely it was neuer my chance as yet to see his cor­rection.

Let 6 ounces of cloues at 36, and 8 ounces of cin­namon at 16, be mingled with 4 ounces of pep­per at 15: how shall 4 ounces of the mingled spice be solde?

Here the numbers of the simple measures are 6, 8, and 4, whereof the totall is 18. The products of 36, 16, and 15, made by multiply­ing them by 6, 8, and 4 are 216, 128, and 60. The totall of them is 404. Therefore I will say thus:

18 ounces of mingled spice are worth 404 therefore 4 ounces are worth 89 7/9.

Let the prices of the simple spices be as they were before: and let 1/2 of an ounce of cloues, and 1/8 of an ounce of cinnamon be mingled with 1/4 of an ounce of pepper: vvhat shall an ounce of the mingled spice be worth?

Here first of all the vnlike measures which are giuen 1/2, 1/3, 1/4 must be made like. For the propertie of generall alligation requireth that the measures should be like. Therefore by reduction of fractions to one denomina­tion, for 1/2, 1/3 and 1/4, I finde out 6/12, 4/12, and 3/12. Then that the worke may be more easie for the parts found out, by reduction of fracti­ons to whole numbers, I take 6, 4, and 3. The totall therefore of the numbers of the measures shall be 13. Then multiply 36, 16, & [Page 175] 15, by the numbers of their measures 6, 4, and 3, you shall produce 216, 64, and 45: where­of the totall is 325. Then shall you conclude the question thus:

13 ounces of mingled spice are worth 325, therefore one ounce is worth 25 3/13.

CHAP. XXVII. Of Manifolde proportion continued in the termes.

MAnifolde proportion compounded in the termes hath beene handled hitherto: that remayneth vvhich is continued in the termes.

Manifold proportion continued in the termes is, vvhen vnto the disiunct termes of the reasons giuen, other proportionall termes doe aunswere, vvhereof each middlemost terme ioyneth the an­tecedent reason with the consequent.

As in the examples following

For the reasons giuen in the vppermost rowe of numbers (namely a subsesquialtera, and a subsesquiquarta) are in the nethermost row of numbers so continued and knit one to an other, that of the two middlemost termes there is made one, which according to the reasons giuen in the first ranke of nū ­bers is both the consequent of the former, and the antecedent of the latter. For as 2 is [Page 176] vnto 3, so is 8 vnto 12: and as 4 is vnto 5, so is 12 vnto 15.

In this kind of proportion therefore there is intended an inuention or finding out of the least termes continually proportionall vvith the rea­sons giuen how many so euer.

This inuention is set foorth in the 4 prop. 8, where it is demonstrated by what meanes, how many reasons so euer being giuen in the least termes, you may finde out other least termes continually proportionall, keeping the reasons giuen. Let that probleme there­fore be turned into two theoremes: whereof let the one concerne two reasons giuen, and let the other concerne many reasons giuen in this manner:

CHAP. XXVIII. Of continuall proportion.

• Simple.
• Or
• Manifold: and is called progression, wherein we are to consi­der
  • The or­der of the termes,
    • Entring into the progres­sion.
    • Or
    • Making a new pro­gression.
  • Or
  • The sum of the termes.

HItherto of disiunct proportion: now follow­eth continuall.

Continuall proportion is that vvhose middle termes doe all supply the place of an antecedent, and a consequent.

In continuall proportion the first extreme is onely an antecedent, and the last is onely a consequent: of the other termes, which are in the middest betweene the extremes any of [Page 184] them in respect of that which went before it is a consequent, in respect of that which fol­loweth it is an antecedent, as this is in three termes 9, 6, 4, in foure termes, 8, 12, 18, 27, in fiue termes 32, 16, 8, 4, 2 &c.

Continuall proportion is either simple, or ma­nifold, simple continuall proportion, is that wherin there is but one only meane betwene the extrems.

This is the 9 d. 5. which affirmeth that pro­portion consisteth in three termes at the least, as in 9, 6, 4, where the extremes 9 and 4 are lincked together by one onely terme. This is framed in the least termes, if you multiply the termes of the reason giuen being prime in respect one of another, both by themselues, and one by another, for the products are the termes of the continuall proportion, accor­ding to the reason giuen. As for example, of 2, 1, which are the termes of a double reason, you shall make a simple continuall proporti­on if you multiply 2 by it selfe, and then by 1, and last of all 1 by it selfe, for the products will be 4, 2, 1. Likewise of 3, 2, the termes of a sesquialter reason, you shall by the same meanes make 9, 6, 4.

The propertie of it is this: The product made of the extremes, is equall to the product made of the meane, and contrariwise.

This is the 17 p. 6. concerning magnitudes, and also the 20 p. 7. where this propertie is also attributed vnto nūbers, because in three proportionall numbers, the product made of [Page 185] the extremes, is equall vnto the square made of the meane. As in 9, 6, 4, the product made of 9 and 4, is 36, which is equall to the square of the meane, namely vnto sixe times sixe, or which is all one to the product made of the meane multiplied by it selfe. For it is, as if you shold inferre the proportion disiunctiue­ly in foure termes after this manner:

9 giue 6: therefore 6 giue 4.

Wherefore it is to be gathered, that this propertie is generall to all simple proportion whether it consist of three or foure termes.

Manifold continuall proportion, is that wher­in many meane termes proceede continually after one and the same reason. Hereupon it is called progression.

As 1, 2, 4, 8. Item 16, 24, 36, 54, 81.

In progression we haue aneye to the finding out of the order of the terms making the progression, or else to the summe of them.

The finding out of the termes is two fold: for either the terms giuen enter into the progression, or else new termes are made by their meanes.

All the termes of a proportion giuen, can not be extended and continued. In manifold proportion it may be, but in other reasons it can not so well be done. It shall appeare therefore by the theoreme following, what termes giuen will admit a progression.

If the last terme sauing one doth diuide the product made of the last terme, the quotient shall be the proportionall terme following, [Page 186] Or, If the last terme doth diuide the product made of the last sauing one, the quotient shall be the antecedent proportionall.

This Theoreme is gathered out of the 18 p. 9. It is also deduced as it were a cōsequence out of the proprietie of continuall propor­tion, whereby not onely proportion, but any reason giuen may be continued. As in 3, 9, 27, wherein 9 diuideth 729, the product made of the last number 27, the quotient is 81, which is the fourth proportionall number. Like­wise let 9 which is the second number diuide 9 which is the product made of the first num­ber 3, the quotient 1, is the first proportio­nall number. So that the progression shall be in this manner, 1, 3, 9, 27, 81. This conti­nuation of the termes in manifold propor­tion may be infinitely inferred onely by the multiplication of the last terme by the deno­minator of the reason or proportion:

Yet in those whose reason is manifold, any terme sought for may be found our readily in this manner:

If you diuide the product made of the last number multiplied in it selfe, by any of the numbers which went before it: by how ma­ny degrees the diuisor went before the num­ber multiplied, by so many degrees shall the quotient followe the same number multi­plied.

For example sake, let this double propor­tion be giuen.

And let 16 be the number to be multiplied by it selfe, th product is 256, diuide that by 8 which is the number next before it, the quo­tient 32 is the number which must followe next after 16. But if you diuide the product 256 by 4 which went before 16 in the thirde place, the quotient will be 64, which must follow in the third place after 16. If you make the diuision by 2, which went before 16 in the fourth place, the quotient 128 must follow in the fourth place after 16, and so forth in the rest. For it is as if you should set the rule of proportion thus: If 2 giue 16: then 16 shall giue 128.

Out of the foresaid abridgement, another ariseth, whereby we may finde out any terme of manifold progression, in this manner:

If some certaine termes of continuall pro­gression be giuen, and the numbers follow­ing one another from an vnitie according to their naturall order, doe aunswere the saide termes from the second forward, the quo­tient of the product made by any of the termes whatsoeuer diuided by the first terme, shall be the terme of the progression more by one, then both the numbers answering to the numbers multiplied, doe amount vnto being added together.

As in this triple proportion, let there be giuen some termes, and let the numbers be written vnderneath them in order as you see:

Then if you multiply 27 by 81, and diuide the product 2187 by 3, which is the first nū ­ber in the progression, the quotient 729 shall be the sixth terme of the progrestion which is more by one, then 2 and 3 (that is to say 5) are. Likewise if you multiply 243 by it selfe, and diuide the product by the first, the quo­tient shall be 19683, to be placed in the ninth place, and so forth of the rest.

The termes aswell of manifolde reason as of proportion and progression, may in this manner be continued infinitely. But the na­ture of numbers doth not beare it so well in the other kinds of reason or proportion. For first of all, if of the reason giuen the termes be prime numbers one to another, then can not a third, much lesse a fourth or fifth in whole numbers be adioyned to them in continuall proportion: as it appeareth by the 16 p. of the 9. As in the reason of 3 to 5, which is su­perpartient two fifts, you shall neuer bring forth a continuall proportion, vnlesse you ioyne to it this mixt or surde number 8 1/3, which cannot be expressed by an whole nū ­ber. Likewise how many proportionall num­bers [Page 189] soeuer being giuen, if the extremes be prime one to another, no such number can be giuen at the last, as the second is vnto the first, by the 17 p. of the 9. As in this subsesqui­alter 4, 6, 9, because 4 and 9 are prime one to another, therefore there can be no whole number in such proportionalitie vnto 9, as 4 is vnto 6. Therefore in such kind of pro­portions this Theoreme following taketh place.

If the termes of continuall simple propor­tion, not admitting an vsuall progression be multiplied by the antecedent of the reason whereof they consist, and the last be multi­plied by his consequent: there shall be pro­duced foure least termes continually propor­tionall according to the numbers giuen: and so forth continually by multiplying the pro­ducts by the antecedent giuen, and the last by the consequent giuen, you shall find the least termes how many so euer continually pro­portionall according to the reasons giuen:

As for example, seeing that this sesquiter­tia 16, 12, 9, hath not a fourth number in pro­gression. Therefore first of all, I multiply all the numbers giuen by 4, the antecedent of the reason giuen betweene 4 and 3, whereof they were made: the products shall be 64, 48, 36. Likewise I multiply the last number 9 by 3 the consequent giuen, the product is 27, the which numbers are all continually pro­portionall next vnto the numbers giuen. [Page 190] After this manner many others may be found out.

Behold also this example following of a manifold superparticular reason.

Thus much concerning the ranging of the termes: now followeth the summe of the pro­gression.

The finding out of the summe, is that vvhich (the first terme of the progression increasing be­ing subducted from the sec̄ond and the last) ad­deth vnto the last terme giuen a number, to the which the remainder of the last terme hath such proportion as the remainder of the second terme hath vnto the first.

I said (the progression increasing) that ye may vnderstande that the first terme is here taken for the least, and the last for the grea­test. Now the masterie of this inuention is generall, being not onely of force in mani­fold proportion, as that rule is which the cō ­mon sort of Arithmeticians do prescribe, but all progression of what proportionalitie so [Page 191] euer giuen: It is taken out of the 33 prop. 9. which affirmeth this, of how many propor­tionall numbers soeuer following one ano­ther. If from the second and the last terme there be taken away numbers equall to the first, the remainder of the last terme shall be vnto all the antecedents going before it, as the remainder of the second terme is vnto the first terme. Therefore these three proportio­nall termes being found out, you shall set the excesse or the remainder of the last terme in the third place, and the remainder of the se­cond in the first place, and the first or least terme of the progression in the middlemost place, and then worke by the rule of three, whereby you shall inferre a number contay­ning all them which were antecedents to the last, which being therefore added to the last shall containe the summe of them all.

As in this manifold progression, 2, 4, 8, 16, 32, if 2 be taken from 4, and 32, there re­maine 2 and 30. Now as 2 the remainder of the second terme is vnto the first terme 2, so is 30 vnto all the antecedents. Therefore for the working of the rule, the termes shall stand thus:

2 giue 2: therefore 30 giue 30.

The reason is in ech place alike and equall. Therefore seeing that the totall made of the numbers going before the last is 30, it being added to the last, namely vnto 32, declareth the summe of the progression to be 62.

Likewise in this subtriple progression, 2, 6, 18, 54, let 2 be taken from 6 and 54, the re­mainders shall be 4 and 52. Now as 4 the re­mainder of the second terme is vnto the first terme 2, so is 52 the remainder of the last terme vnto all the antecedents. Wherefore the three termes knowne shall be thus orde­red of the finding out of the fourth terme.

4 2: therefore 52 26

And for so much as this fourth number, is equall vnto all the antecedents, it being added to the last terme 54, the totall 80 shall be the summe of the progression.

Likewise let there be this subsesquialter progression, 16, 24, 36, 54, 81. The remain­ders of the second, and last terme are 8, and 65, whereupon by these three proportionall numbers 8, 16, 65, the fourth number 130 is inferred, which is the totall made of all the antecedents except the last, it being therfore added vnto the last terme 81, declareth the summe of the progression to be 211. Where­by it appeareth that the first, second, and last terme of the progression being knowen the summe cannot lie hid.

An Example.

A certaine man solde his house to be payde for it in wheate after such a manner, that vvhen he that bought it came in at the first doore he should giue him one graine, at the second two, at the third 4, and so foorth proceeding conti­nually [Page 193] by double proportion according to the number of the doores. He that shrunke from his bargaine should pay to the other 12 crowns for a penaltie. Now being 60 doores, the que­stion is how much wheat he was to pay.

A rich man of Basill not ignorant of the huge increase of Geometricall progression, sold his house sometimes vppon this conditi­on being among his cuppes. But the condi­tion of his bargaine being brought vnto an account, the number of the wheat was found to be vnmeasurable, so that all the houses of Basill being turned in garners, were not suf­ficient to receiue it. I thought good to set downe the example. Let the 60 terme of the progression be compendiously found out.

Wherefore the summe of all the graines shall be,

1152921504606846975

The which number is so great, that it soe­meth sufficient to match the sand of the A­driatique sea.