The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

Definitions.

* 1.1A parte is a lesse magnitude in respect of a greater magni∣tude, when the lesse measureth the greater.

As in the other bookes before, so in this, the author first setteth orderly the definitions and declarations of such termes and wordes which are necessarily re∣quired to the entreatie of the subiect and matter therof, which is proportion and comparison of proportions or proportionalitie. And first he sheweth what a parte is. Here is to be considered that all the definitions of this fifth booke be general to Geometry and Arithmetique, and are true in both artes, euen as proportion and proportionalitie are common to them both, and chiefly appertayne to number, neither can they aptly be applied to matter of Geometry, but in respect of number and by number. Yet in this booke, and in these definitions here set, Euclide semeth to speake of them onely Geometrically, as they are applied to quantitie continu∣all, as to lines, superficieces, and bodies: for that he yet continueth in Geometry. I wil notwithstanding for facilitie and farther helpe of the reader, declare thē both by example in number, and also in lynes.

For the clearer vnderstandyng of a parte, it is to be noted,* 1.2 that a part is taken in the Mathematicall Sciences two maner of wayes.* 1.3 One way a part is a lesse quan∣titie

in respect of a greater, whether it measure the greater o•• no. The second way,* 1.4 a part is onely that lesse quantitie in respect of the greater, which measureth the greater. A lesse quantitie is sayd to measure or number a greater quantitie,* 1.5 when it, beyng oftentymes taken, maketh precisely the greater quantitie without more or lesse, or beyng as oftentymes taken from the greater as it may, there remayneth nothyng. As suppose the line AB to contayne 3. and the lyne CD to contayne 9. thē doth the line AB measure the line

[illustration]

CD: for that if it be take certayne times, namely, 3. tymes, it maketh precisely the lyne CD, that is 9. without more or lesse. Agayne if the sayd lesse lyne AB be taken from the greater CD, as often as it may be, namely, 3. tymes, there shall remayne nothing of the greater. So the nū∣ber 3. is sayde to measure 12. for that beyng taken certayne tymes, namely, foure tymes, it maketh iust 12. the greater quantitie: and also beyng taken from 12. as of∣ten as it may, namely, 4. tymes, there shall remayne nothyng. And in this meaning and signification doth Euclide vndoubtedly here in this define a part:* 1.6 saying, that it is a lesse magnitude in comparison of a greater, when the lesse measureth the greater. As the lyne AB before set, contayning 3. is a lesse quantitie in compa∣rison of the lyne CD which containeth 9. and also measureth it. For it beyng cer∣tayne tymes taken, namely, 3. tymes, precisely maketh it, or taken from it as often as it may, there remayneth nothyng. Wherfore by this definition the lyne AB is a part of the lyne CD. Likewise in numbers, the number 5. is a part of the number 15. for it is a lesse number or quantitie compared to the greater, and also it measu∣reth the greater: for beyng taken certayne tymes, namely, 3. tymes, it maketh 15. And this kynde of part is called commonly pars metiens or mensurans,* 1.7 that is, a mea¦suryng part: some call it pars multiplicatina:* 1.8 and of the barbarous it is called pars aliquota,* 1.9 that is an aliquote part. And this kynde of parte is commonly vsed in A∣rithmetique.* 1.10

The other kinde of a part,* 1.11 is any lesse quantitie in comparison of a greater, whe∣ther it be in number or magnitude, and whether it measure or no. As suppose the line AB to be 17. and let it be deuided into two partes in the poynt C, namely, in∣to the line AC, & the

[illustration]

line CB, and let the lyne AC the greater part containe 12. and let the line BC the lesse part contayne 5. Now eyther of these lines by this definition is a part of the whole lyne AB. For eyther of them is a lesse magnitude or quātity in cōparisō of the whole lyne AB: but neither of thē measureth the whole line AB: for the lesse lyne CB contayning 5. taken as oftē as ye list, will neuer make precisely AB which contayneth 17. If ye take it 3. tymes it maketh only 15. so lacketh it 2. of 17. which is to litle. If ye take it 4. times, so maketh it 20. thē are there thre to much, so it neuer maketh precisely 17. but either to much or to litle. Likewise the other part AC measureth not the whole lyne AB: for takē once, it maketh but 12. which is lesse then 17. and taken twise, it maketh 24. which are more then 17. by ••. So it neuer precisely maketh by takyng therof the whole AB, but either more or lesse. And this kynde of part they commonly call pars constituens, or componens:* 1.12 Because that it with some other part or partes, maketh the whole. As the lyne CB together with the line AC maketh the whole lyne AB. Of the barbarous it is called pars aliquanta.* 1.13 In this signification it is taken in B••rla∣•••• in the beginnyng of his booke, in the definition of a part, when he saith: Euery

lesse number compared to a greater, is sayd to be a part of the greater, whether the lesse mea∣sure the greater, or measure it not.

Multiplex is a greater magnitude in respect of the lesse, when the lesse measureth the greater.* 1.14

As the line CD before set in the first example, is multiplex to the lyne AB. For that CD a lyne contayning 9. is the greater magnitude, and is compared to the lesse, namely, to the lyne AB contayning 3. and also the lesse lyne AB measureth the greater line CD: for taken 3. tymes,

[illustration]

it maketh it, as was aboue sayde. So in numbers 12. is multiplex to 3: for 12 is the greater number, and is compared to the lesse, namely, to 3. which 3. also measu∣reth it:* 1.15 for 3 taken 4 tymes maketh 12. By this worde multiplex which is a terme proper to Arithmetike and number, it is easy to consider that there can be no ex∣act knowledge of proportion and proportionalitie, and so of this fifth booke wyth all the other bookes followyng, without the ayde and knowledge of numbers.

Proportion is a certaine respecte of two magnitudes of one kinde,* 1.16 according to quantitie.

Euclide as in the first definition, so in this & the other following, and likewise in all his Propositions of this booke, mentioneth onely magnitudes, and geueth his examples and demonstrations of lines: for that hetherto in the 4. bookes before he hath entreated of lines & figures, and so cōtinueth in his sixth booke following after this, comparing figure to figure, and sides of figures to sides of figures, with∣out mention of number at all. Notwithstanding as it is sayd they are generall to all kinde of quantitie, both discrete and continuall, namely, number and magnitude: and neede for the young reader and studient in these artes to be de∣clared in both. For, the opening of them in numbers (in which they are first and naturally founde) geueth a great and marueilous light to their declaration in magnitudes.* 1.17 Proportion (sayth he) is a certaine behauiour, that is, a certaine respect or comparison of two quantities of one kinde: as of one line to an other, and one figure generally to an other, and one number to an other, as touching quantitie, that is to say, that the quantitie compared,* 1.18 is to that wherunto it is compared, eyther equall, or greater, or lesse then it. For after these three maners may thinges be compared the one to the other. But quantities of diuers kindes can not be compared together. A superficies can not be compared to a line: nor number to a body: nor a body to a line or number: for that they are not of one kinde. For example of this definition,* 1.19 take two quantities, namely, two lines AB and CD, and compare the one to the other, namely, AB to

[illustration]

CD according to some certaine respect of greatnes, or lessenes, or equalitie, namely, in this example, let AB be greater then CD, & containe it twise. Now thys comparison, relation, or respect of AB to CD, and generally of any one quantitie to any other, is called proportion. Likewise is

it, if AB be equall to CD, as in the second example: or if AB be lesse then CD, as in the third example. The like is it also in numbers

[illustration]

comparing 5. to 5. equall to equall,* 1.20 or 6. to 3. the grea∣ter to the lesse, or 4. to 8. the lesse to the greater. So to the accomplishing of any Proportion there are re∣quired two quantities,* 1.21 and also a comparing or respect of the one to the other. The quantities compared to∣gether are commonly called the Termes of the Pro∣portion. And in this booke of Euclide, and also in o∣ther writers of Geometrie, the first Terme, namely, that which is compared is called the antecedent,* 1.22 whether it be equall, greater, or lesse then the other: And the second Terme, namely, that wherunto the comparison is made, is called the consequent.* 1.23 As in the former example, the line AB compa∣red to the line CD is antecedent: and the line CD is consequent: And contra∣riwise if the line CD be compared to the line AB, then is the line CD antece∣dent, and the line AB consequent. Albeit in Arithmeticke Boetius and others call the terme compared Dux,* 1.24 and the ••rme to whom the cōparison is made they call Comes.* 1.25 This booke hath bene accompted of all men one of the hardest and most intricate of all Euclides bookes.* 1.26 And proportion is a generall knowledge to all learninges, chiefly to the Mathematicalls. Wherefore it shall be very necessary some litle briefe instruction and induction to be here added in the beginning here∣of: of the knowledge and nature of proportions and what they are, and of how many kindes: which thinges are here of Euclide supposed to be before knowen, and therefore maketh no mention so distinctly of them.

Ye must vnderstand that there are of proportions two generall kindes,* 1.27 the one is called rationall, certaine, and knowen, and the other irrationall, vncertaine and vnknowen. Such magnitudes or quantities, which may be expressed by numbre, are called rationall magnitudes or quantities••* 1.28 As suppose a line, namely, the line AB to containe 5. inches, & compare it to the line CD,

[illustration]

contayning 3. inches: these quantities ye see may be ex∣pressed by numbers, namely, by these numbers 5. and 3: and therefore are rationall, and haue the same proporti∣on, that number hath to number, namely, that the number 5. hath to the number 3: and therefore the proportion of the one to the other, is a rationall, certaine, and knowen proportion. And generally when soeuer one number is compared to an other, or two lines or other magnitudes, both which may be expressed by num∣ber, the proportion betwene them is euer rationall, and onely the proportion of such quantities is rationall. So that in Arethmeticke all proportions are rationall,* 1.29 for that therein euer one number is compared to an other.

There are certaine lines magnitudes or quantities which cā not be named and expressed by number, and therefore commonly are called Surd lines or magni∣tudes.* 1.30 As suppose the square ABCD to containe 16, then the side or roote ther∣of, namely, the line AB containeth 4, and the diameter of the

[illustration]

same square, namely, the line BC shall be •• 32, which is a surd number, and can nor be expressed by any determinate and certaine number, but onely by this maner of circumlocu∣tion Roote square of 32. Now if ye compare the line AB to the line BC, or contrariwise the line BC to the line AB, for that one of them is a surde quantitie, neither can ech of them be expressed by number (and therefore can not haue that

proportion that number hath to number) the proportion betwene them is irratio∣nall, confused, vnknowen, vncertaine, and surd. And this kinde of proportion is found onely in magnitudes, as in lines and figures (and not in numbers) of which he of purpose entreateth in his tenth booke. Wherfore I wil here omit to speake of it, and remit it to his due place. And somewhat will I now say for the elucidation of the first kinde.

* 1.31Proportion rationall is deuided into two kindes, into proportion of equalitie, and into proportion of inequalitie. Proportion of equalitie is, when one quanti∣tie is referred to an other equall vnto it selfe: as if ye compare 5 to 5, or 7 to 7, & so of other. And this proportion hath great vse in the rule of Cosse.* 1.32 For in it all the rules of equations tende to none other ende but to finde out and bring forth a nū∣ber equall to the number supposed, which is to put the proportion of equalitie.

* 1.33Proportion of inequalitie is, when one vnequall quantity is compared to an o∣ther, as the greater to the lesse, as 8. to 4: or 9. to 3: or the lesse to the greater as 4. to 8: or 3. to 9.

* 1.34Proportion of the greater to the lesse hath fiue kindes, namely, Multiplex, Super∣particular, Superpartiens, Multiplex superperticular, and Multiplex superpartiens.

Multiplex, is when the antecedent containeth in it selfe the consequent cer∣tayne times without more or lesse:* 1.35 as twice, thrice, foure tymes, and so farther. And this proportion hath vnder it infinite kindes. For if the antecedent contayne the consequent iustly twise, it is called dupla proportion,* 1.36 as 4 to 2. If thrice tripla,* 1.37 as 9. to 3. If 4. tymes quadrupla as 12. to 3. If 5. tymes quintupla as 15. to 3. And so infinitely after the same maner.

* 1.38Superperticular is, whē the antecedēt containeth the consequent only once, & moreouer some one part therof as an halfe, a third, or fourth, &c. This kinde also hath vnder it infinite kindes. For if the antecedent containe the consequent once and an halfe, therof it is called Sesquialtera,* 1.39 as 6. to 4: if once and a third part Sesqui∣tertia,* 1.40 as 4. to 3: if once and a fourth part Sesquiquarta,* 1.41 as 5. to 4. And so in like ma∣ner infinitely.

* 1.42Superpartiens is, whē the antecedent cōtaineth the consequent onely once, & moreouer more partes then one of the same, as two thirdes, three fourthes, foure fifthes and so forth. This also hath infinite kindes vnder it. For if the antecedent containe aboue the consequent two partes, it is called Superbipartiens,* 1.43 as 7. to 5. If 3. partes Supertripartiens as 7. to 4.* 1.44 If 4. partes Superquadripartiens,* 1.45 as 9. to 5. If 5. partes Superquintipartiens as 11. to 6.* 1.46 And so forth infinitely.

* 1.47Multiplex Superperticular is when the antecedent containeth the consequent more then once, and moreouer onely one parte of the same. This kinde likewise hath infinite kindes vnder it. For if the antecedent containe the consequent twise and halfe therof, it is called dupla Sesquialtera,* 1.48 as 5. to 2. If twise and a third Dupla Sesquitertia as 7. to 3.* 1.49 If thrice and an halfe Tripla sesquialtera as 7. to 2.* 1.50 If foure times and an halfe Quadr••pla Sesquialtera, as 9. to 2. And so goyng on infinitely.

* 1.51Multiplex Superpartient, is when the antecedent contayneth the consequent more then once, and also more partes then one of the consequent. And this kinde also hath infinite kindes vnder it. For if the antecedent containe the consequent twise, and two partes ouer, it is called dupla Superbipartiens as 8. to 3.* 1.52 If twice and three partes, dupla Supertripartiens as 11. to 4.* 1.53 If thrice and two partes, it is named Tripla Superbipartiens as 11. to 3.* 1.54 If three tymes and foure partes Treble Super∣quadripartiens as 31. to 9.* 1.55 And so forth infinitely.

Here is to be noted that the denomination of the proportion b••twene any two numbers, is had by deuiding of the greater by the lesse. For the quotient o••

number produced of that diuision is euen the denomination of the proportion.* 1.56 Which in the first kinde of proportion, namely, multiplex, is euer a whole number, and in all other kindes of proportion it is a broken number.

As if ye will know the denomination of the proportion betwene 9 and 3. De∣uide 9. by 3. so shall ye haue in the quotient 3. which is a whole number, and is the denomination of the proportion: and sheweth that the proportion betwene 9. & 3. is Tripla. So the proportion betwene 12. and 3. is quadrupla, for that 12. beyng deuided by 3. the quotient is 4. and so of others in the kinde of multiplex. And al∣though in this kinde the quotient be euer a whole number, yet properly it is refer∣red to vnitie, and so is represented in maner of a broken number as 5/•• and 4/•• for v∣nitie is the denomination to a whole number.

Likewise the denomination of the proportion betwene 4 and 3 is 1. 1/•• for that 4 deuided by 3. hath in the quotient 1 1/•• one and a third part, of which third part, it is called sesquitercia: so the proportion betwene 7 and 6. is 1 ⅙ one and a sixt, of which fixt part it is called sesquisexta, and so of other of that kinde. Also betwene, 7 and 5 the denomination of the proportion is 1 ••/•• one and two fifthes, which de∣nomination cōsisteth of two parts, namely, of the munerator and denominator of the quotient of 2. and 5: of which two fifthes it is called superbipartiens quintas: for 2 the numerator sheweth the denomination of the number of the partes, and 5. the denominator, sheweth the denominatiō, what parts they are, & so of others. Also the denomination betwene 5 and 2. is 2 ½ two and a halfe, which consisteth of a whole number and a broken, of 2. the whole number it is dupla, and of the halfe, it is called sesquialtera, so is the proportion dupla sesquialtera.

Agayne the denomination of the proportion betwene 11. and 3. is 3 ••/•• three and two thirdes, consisting also of a whole number and a broken, of 3. the whole numbre it is called tripla, and of ••/•• the broken number, it is called Superbipartiens tertias, so the proportion is tripla superbipartiens tertias. Thus much hetherto tou∣ching proportion of the greater quantitie to the lesse.

Proportion of the lesse quantitie to the greater hath as many kindes, as that of the greater to the lesse, which kindes are in the same order:* 1.57 and haue also the selfe same names, but that to the names afore put ye must adde here this word sub. As comparing the greater to the lesse, it was called multiplex, superparticular, superpar∣tient, multiplex superparticular, and multiplex superpartient, now comparing the lesse quantitie to the greater, it is called submultiplex,* 1.58 subsuperparticular,* 1.59 subsuperpartient,* 1.60 submultiplex superparticular, and submultiplex superpartient. And so in like maner to all the inferior kindes of all sortes of proportion ye shall adde that worde sub. The examples of the former serue also here, onely transposing the termes of the pro∣portion making the antecedent consequent, and the consequent the antecedent. As 4. to 2. is dupla proportion: so 2. to 4. is subdupla. As 9. to 3. is tripla: so is 3. to ••. subtripla. And as 9. to 6. is sesquialtera, so 6. to 9. is subsesquialtera. As 7. to 5. is superbipartiens quintas, so is 5. to 7. subsuperbipartiens quintas. As 5. to 2. is dupla sesquialtera, so is 2. to 5. subdupla sesquialtera. And also as 8. to 3. is dupla superbi∣partiens tertias, so is 3. to 8. subdupla superbipartiens tertias. And so may ye pro∣cede infinitely in all others. Thus much thought I good in this place for the ease of the beginner to be added touching proportion.

* 1.61Proportionalitie, is a similitude of proportions.

As in proportion are compared together two quantities, and proportion is no∣thing els but the respect and comparison of the one to the other, and these quanti∣ties are the termes of the proportion: so in proportionallitie are compared toge∣ther two proportions. And proportionallitie is nothing els, but the respect & com∣parison of the one of them to the other. And these two proportions are the termes of this proportionallitie. He calleth it the similitude, that is, the likenes or idempti∣tie of proporti∣ons:* 1.62

[illustration]

As if ye wil cōpare the pro∣portion of the line A contay∣nyng 2. to the line B contayning 1, to the proportion of the line C contayning 6. to the line D contayning 3, either proportion is dupla. This likenes, idemptitie, or equallitie of proportion is called proportionallitie.* 1.63 So in number 9. to 3. and 21. to 7. either pro∣portion is tripla. Where note that proportions compared together, are sayd to be like the one to the other:* 1.64 but magnitudes compared together, are said to be equall the one to the other.

* 1.65Those magnitudes are sayd to haue proportion the one to the other, which being multiplied may exceede the one the other.

Before he shewed and defined, what proportion was, now by this definition he declareth betwene what magnitudes proportion falleth, saying: That those quanti∣ties are said to haue proportion the one to the other, which being multiplyed, may excede the one the other.* 1.66 As for that the

[illustration]

line A being multiplied by what soeuer multiplication or nūber, as taken twise, thrise, or foure, fiue, or more times, or once and halfe, or once and a third, & so of any other part, or partes, may excede and become greater then the line B or contrariwise, then these two lines are said to haue proportion the one to the other. And so ye may see that betwene any two quātities of one kinde, there is a propor∣tion. For the one remayning vnmultiplied, & the other being certaine times mul∣tiplied, shall be greater then it. As 3. to 24. hath a proportion, for leauing 24. vnmul∣tiplied, and multiplying 3. by 9, ye shall produce 27: which is greater then 24, and excedeth it. Here is to be noted, that Euclide in defining what quantities haue pro∣portion,* 1.67 was compelled to vse multiplication, or els should not his definition be generall to either kinde of proportion: namely, to rationall and irrationall: to such proportion I say which may be expressed by number, and to such as cannot be ex∣pressed by any determinate number, but remaineth surd and innominable. In rati∣onall quantities which haue one common measure, the excesse of the one aboue the other is knowen, and by it is knowen the proportion, which may be expressed by some determinate number. But in irrationall quantities which haue no cōmon measure, it is not so. For in them the excesse of the one to the other is euer vn∣knowen,

& therefore is surd, and innominable. As betwene the side of a square and the diameter therof•• there is vndoubtedly a proportion, for that the side certaine times multiplied may excede the diameter. Likewise betwene the diameter of a circle and the circumference therof there is certainlie, by this definition, a propor∣tion, for that the diameter certaine times multiplied may excede the circumference of the circle: although neither of these proportions can be named & expressed by number. For this cause therefore vsed Euclide this maner of defining by multipli∣cation.

Magnitudes are sayd to be in one or the selfe same propor∣tion,* 1.68 the first to the second, and the third to the fourth, when the equimultiplices of the first and of the third beyng compa∣red with the equimultiplices of the second and of the fourth, according to any multiplication: either together exceede the one the other, or together are equall the one to the other, or together are lesse the one then other.

In the definition last going before, he shewed what magnitudes haue propor∣tion the one to the other, & now this diffinition sheweth what magnitudes are in one and the selfe same proportion,* 1.69 and how to know whether they be in one and the self same proportion, or not. It is plaine that euery proportiō hath two termes, so that when ye compare proportion to proportion, ye must of necessitie, haue 4. termes, that is, an antecedent and a consequent, to either of the proportions. As sup∣pose A, B, C, D, to be foure magnitudes, A the first, B the second, C the third, and

[illustration]

D the fourth now if ye take the equimultiplices of A and C the first & the third, that is, if ye multiply A and C by one and the selfe same number, as let the multi∣plex of A be E, and let the equimultiplex of C be F. Likewise also if ye take the equimultiplices of B and D, the second and the fourth, that is if ye multiply them by any one number, whether it be by that number wherby ye multiplied A & C, or by any other number greater or lesse, as let the multiplex of B be G, and the e∣quimultiplex of D be H: how it the equimultiplices of A and C be both greater hen the eqnimultiplices of B and D, that is if the multiplex of A be greater then the multiplex of B, and the multiplex of C be greater then the multiplex of D, or if they be both lesse then they: or both equall to them,* 1.70 then are the magnitudes A, B and ••, D in one and the selfe same proportion.

Likewi•••• in numbers 8. to 6. hath a proportion, also 4. to 3. hath a proportion:

now to see whether they be in one and the selfe same proportion or not, set them in order as in the example here written, 8 the first, 6 the second, 4 the third, and 3. the fourth. Now take the equemultiplices of 8 and 4. the first, and the third, that is, multiply them by one and the selfe

[illustration]

same number, suppose it be by 3. so the tri∣ple of 8 is 24. & the triple of 4. is 12: like∣wise take the equimultiplices of 6 and 3. the second and the fourth, multiply••ng them likewise by one and the selfe same number, suppose it be also by 3 as before ye did, the triple of 6 is 18. and the triple of 3 is 9. Now ye see that the triple of 8 the first, namely, 24. excedeth the triple of 6. the second, namely, 18: likewise the triple of 4 the third number, namely, 12. excedeth the triple of 3. the fourth, namely, 9. Wherefore by the first part of this definition, the numbers 8 to 6. and 4 to 3. are in one and the selfe same proportiō, because that the equemultiplices of 8 and 4. the first & the third, do both exceede the equimultiplices of 6 and 3. the second and the fourth.

* 1.71Againe, take the same numbers and try the same after this maner. Take the equimultiplices of 8. and 4. the first and the third, multiplieng eche by 3. as before ye did, so shall ye haue 24 for the tri∣ple

[illustration]

of 8. and 12 for the triple of 4. as ye had before. Then take the equimulti∣plices of 6 and 3. the second and the fourth, multipliyng them by some one number, but not by 3 as before ye did: but by 4. so for the quadruple of 6 the second number, shall ye haue 24. and for the quadruple of 3 the fourth number, ye shall haue 12. And now ye see that the equimultiplices of 8 and 4. the first and the third, namely, 24 and 12. are both equall to the multiplices of 6 and 3. the second and the fourth, namely, to 24 and 12. Wherfore the numbers geuen, are by the second part of this definition in one and the selfe same proportion, because the equimultiplices of 8 and 4 the first and the third, are both equall to the equimultiplices of 6 and 3. the seconde and the fourth.

* 1.72Agayne to shew the same, and for the fulnes of the diffinition, take the same numbers 8, 6, 4, 3. and take the equimultiplices of 8 and 4. the first and the thirde, multiplieng eche by 2. so haue ye 16 for the duple of 8, the first number, and 8 for the duple of 4 the third number: then take also the equimultiplices of 6 and 3, the second and the fourth, multipliyng eche by 3. so haue ye 18 for the triple of 6 the second, and 9 for the triple of 3. the

[illustration]

fourth number. And now ye see that the equimultiplices of 8 and 4. the first and the third, namely, 16. and 8 are both lesse then the equimultipli∣ces of 6 and 3. the second & the fourth namely 18 and 9. For 16 are lesse then 18, and 8 are lesse then 9. Wherefore by the third part of this diffinition, the numbers proposed are in one and the selfe same proportion, for that the equimultiplices of 8 and 4 the first and the third are both lesse then the equimultiplices of 4 and 3 the second and the fourth.

Farther in this diffinition, this particle (according to any multiplication) is most diligently to be considered,* 1.73 which signifie••h by any multiplication indiffe∣rently whatsoeuer. For whensoeuer the quantities be in one and the selfe same proportion, then by any multiplication whatsoeuer, the equimultiplices of the f••rst and the third, shall exceede the equimultiplices of the second and the fourth, or shall be equall vnto them, or lesse then them. Yet it may so happen by some one multiplication; that the equimultiplices of the first and the third, do exceede the e∣quimultiplices of the second and the fourth, and yet the quantities geuen shal not be in one and the selfe same proportion. As in this example here set, where the e∣quimultiplices of 6 and 5, the first and the thirde, namely, 18. and 15. doo both exceede the equimultiplices of 4 and

[illustration]

3. the second and the fourth, namely, 8 and 6.* 1.74 yet are not the numbers ge∣uen in one and the selfe same propor∣tion. For 6 hath not that proportion to 4. which 5. hath to 3. In this exam∣ple 6 and 5 the first and the third were multiplied by 3. which made their equimul∣tiplices 18 and 15. which exceede the equimultiplices of 4 and 3, the second and the fourth beyng multiplied by 2. namely, 8 and 6: but if ye shall multiply 6 and 5 the first and the thirde by 2. ye shall produce 12 and 10 for their equimultipli∣ces, and then if ye multiply 4 and

[illustration]

3. the second and the fourth by 3. so shall ye produce for their equimulti∣plices 12 and 9. Now ye see that by this multiplication the equimultipli∣ces of the first and the thirde doo not both exceede the equimultiplices of the second and fourth: for 12 the multiplex of 6 doth not exceede 12 the multiplex of 4. and therfore the numbers or quanti∣ties ar•• not in one and the selfe same proportion, for that it holdeth not in all mul∣tiplications whatsoeuer.

And because this diffinition requireth all maner of multiplicatiōs to bring forth the excesses, equalities, and wantes of the antecedents aboue, to, or vnder the con∣sequents; to auoide the tediousnes and infinite labour therof, I haue set forth a rule much to be made of and estemed, wherby ye may in any rationall proportion pro∣duce equimultiplices of the first and the third equall to the equimultiplices of the second and the fourth. The rule is this,* 1.75 take two numbers whatsoeuer in that pro¦portion in which your quantities are, & by the number which is antecedent mul∣tiply the consequents of your proportions, namely, the second and the fourth: and by the number which is the consequent multiply the antecedentes of your pro∣portions, namely, the first and the third•• then necessarily shalbe produced the equi∣multiplices of the first and the third equall to the equimultiplices of the second & the fourth. As by example, take 6 to 2. and 3 to 1, which are in one & the selfe same proportiō, & taking these two nūbers 9 & 3. which are in the same proportiō, now by 9 the antecedent multiply the cons••quēts 2 & 1. and so shal ye haue 18 & 9 for the equimultiplices of the second & the fourth,* 1.76 then by 3 the consequent multiply the antedēts 6 & 3, so shal ye haue 18 & •• for the equimultiplices of the first & the third, which are equal to the former equimultiplices of the secōd & fourth. Wher∣of it foloweth that if ye multiply 18 & 9 the equimultiplices of the first and the third by any nūber greater thē 3. wherby they were now multiplied, they shal both euer exceede the equimultiplices of the second & the fourth: & if ye multiply thē

by any number lesse then 3. they shall euer both want of them. So that whatsoe∣uer multiplication it be, they shall euer both exceede, be equal, or want aboue, to, or from 18. ••nd 9. the equimultiplices of the second and fourth.

* 1.77Magnitudes which are in one and the selfe same proportion, are called Proportionall.

As if the lyne A, haue the same proportion to the line B, that the lyne C hath to the lyne D, then are the

[illustration]

said foure magnitudes A, B, C, D, called proportionall. Also in numbers for that 9. to 3. hath that same proportiō that 12 hath to 4:* 1.78 therefore these foure nūbers 9.3.12.4.* 1.79 are said to be proportionall. Here is to be noted that this likenes or idemptitie of proportiō which is called, as before was said proportionalitie, is of two sortes: the one is continuall, the other is discontinuall.* 1.80 Continuall proportionalitie is, when the magnitudes set in lyke proportion, are so ioyned together, that the second which is consequent to the first, is antecedent to the third, and the fourth which is consequent to the third, is antecedent to the fift, and so continually forth. So eue∣ry quantitie or terme in this proportionalitie, is both antecedent and consequent (consequent in respect of tha which went before,* 1.81 & antecedent in respect of that which followeth) except the first, which is onely antecedent to that which follow∣eth, and the last which is onely consequent to that which went before. Take an ex∣ample in these numbers, 16.8.4.2.1.* 1.82 In what proportion 16. is to 8, in the same is 8. to 4, in the same also is 4. to 2, and likewise 2. to 1. For they all are in duple pro∣portion: 16. the first is antecedent to 8, and 8. is consequent vnto it: and the selfe same 8. is antecedent to 4: which 4 beyng consequent to 8. is antecedent to 2, which 2 likewise is consequent to 4. and antecedent to 1: which because he is the last, is onely consequent, and antecedent to none, as 16. because it was the first, was antecedent onely, and consequent to none. Also in this proportionalitie all the magnitudes must of necessitie be of one kynde,* 1.83 by reason of the continuation of the proportions in this proportionalitie, because there is no proportion betwene quantities of diuers kyndes. Discontinuall proportionalitie is,* 1.84 when the magni∣tudes which are set in lyke proportion, are not continually set, as before they were, hauyng one terme referred both to that which went before, and to that which fo∣loweth, but haue their termes distinct and seuered asonder: as the first is antece∣dent to the second, so is the third antecedent to the fourth. Example in numbers, as 8 is to 4.* 1.85 so is 6. to 3. for either proportion is duple. Where ye see, how ech pro∣portion hath hys owne antecedent and consequent distinct from the antecedent and consequent of the other, and no one number is antecedent and consequent in diuers respectes. And by reason of the discontinuaunce of the proportions in this proportionalitie,* 1.86 the quantities compared, may be of diuers kyndes, because the consequent in the first proportion is not the antecedent in the second proportion. So that ye may compare superficies to superficies, or body to body in the selfe same proportion that ye do lyne to lyne.

When the equemultiplices being taken,* 1.87 the multiplex of the first excedeth the multiplex of the second, & the multiplex of the third, excedeth not the multiplex of the fourth: then hath the first to the second a greater proportion, then hath the third to the fourth.

In the sixt definition was declared what magnitudes are said to be in ••••e and the same proportion: now he sheweth in this definition what magnitudes are said to be in a greater proportion. And here is supposed the same order of multiplicati∣on, that there in that definition was vsed: namely, that the first and the third be e∣qually multiplied, that is, by one & the selfe same nūber: and also that the second and the fourth be equally multiplied by

[illustration]

the same or some other number: and then if the multiplex of the first, excede the multiplex of the second:* 1.88 & the mul∣tiplex of the third; excede not the mul∣tiplex of the fourth, the first hath a grea∣ter proportion to the second, then hath the third to the fourth. As suppose that there be foure quantities, A, B, C, D: of which let A be the first, B the second, C the third, & D the fourth. And let A the first cōmine 6. and let B the second con¦taine 2. & C the third 4. & D the fourth 3: Now take the equimultiplices of A and C the first & the third, which let be E and F, so that how multiplex E is to A so multiplex let F be to C: name∣ly for example sake let either of them be triple: so haue you 18. for the multiplex of A, and 12. for the mul∣tiplex of C. Likewise take the equimultiplices of B & D, the second & the fourth, multiplying them also by one and the self same number, as by 4: so haue ye for the multiplex of B the second 8, namely, the line G, and for the multiplex of D the fourth 12, namely, the line H. Now because the line E multiplex to the first, name∣ly, 18, excedeth the line G multiplex to the second, namely, 8: And the line F mul∣tiplex to the third, namely, 12, excedeth not the line H multiplex to the fourth, namely, 12 (for that they are equall) the proportiō of A to B the first to the second, is greater then the proportion of C to D the third to the fourth. So likewise in nū∣bers: take 11. to 2. & 7. to 3. and mul∣tiply

[illustration]

11. & 7.* 1.89 (the first, and the third) by 2, so shall ye haue 22. for the multi∣plex of the first, and 14. for the multi∣plex of the third: and multiply 2. and 3. the second and the fourth by 6: so shall the multiplex of the second be 12. and the multiplex of the fourth be 18:

Now ye see 22. the multiplex of the first, excedeth 12, the multiplex of the second. But 14. the multiplex of the third, excedeth not 18. the multiplex of the fourth: Wherefore the proportion of 11. to 2. the first to the second, is greater then the pro∣portion of 7. to 3, the third to the fourth. And so of all other quantities and num∣bers, which are not in one and the selfe same proportion, ye may know when the first to the second hath a greater proportion then the third to the fourth.

¶An other example.

This example haue I set to declare

[illustration]

that although the proportion of the first to the second be greater then the proportion of the third to the fourth, yet the multiplex of the first excedeth not the multiplex of the secōd. Wher∣fore it is diligently to be noted,* 1.90 that it is sufficient to shew that the proporti∣on of the first to the second is greater thē the proportion of the third to the fourth, if the want or lacke of the multiplex of the first from the multiplex of the second, be lesse then the want or lacke of the multiplex of the third to the multiplex of the fourth. As in this example 16. the multiplex of 8. the first, wanteth of 20. the multiplex of 4. the second, foure: wheras 18. the multiplex of 9, the third, wāteth of 45, the multiplex of 9 the fourth, 27. And so of all others wheras (the proportions being diuers) the equimultiplices of the first and the third are both lesse, then the equimultiplices of the second and the fourth. Likewise if the equimultiplices of the first and the third do both excede the equimultiplices of the second & the first, thē shall the excesse of the multiplex of the first aboue the multiplex of the second, be greater thē the excesse of the multiplex of the third, aboue the multiplex of the fourth. As in these numbers here set, the equimultiplices of 6. and 4. the first and the third, namely, 12. and 8. do both excede the equimultiplices of 2. and 3. the se∣cond and the fourth, namely, 4. and 6. But 12. the multiplex of the first excedeth 4. the multiplex of the second by 4, and 8. the multiplex of the thyrd excedeth 6. the multiplex of the fourth by 2. but 8. is

[illustration]

more then 2. Howbeit this is general∣ly certaine that when soeuer the pro∣portion of the first to the secōd is grea∣ter then the proportion of the third to the fourth, there may be found some multiplication, that whē the equimul∣tiplices of the first and the third shall be compared to the equimultiplices of the second and the fourth, the multiplex of the first shall excede the multiplex of the second, & the multiplex of the third shall not excede the multiplex of the fourth, according to the plaine wordes of the de∣finition.

In like maner when you haue taken the equimultiplices of the first & the third, and also the equimultiplices of the second and the fourth, if the multiplex of the first excede not the multiplex of the second, and the multiplex of the third excede the multiplex of the fourth: then hath the first to the second a lesse proportion, then hath the third to the fourth. As in the example before, if ye chaunge the termes, and make C the first, D the second, A the third, and B the fourth: then shall

F, namely, 12. the multiplex of the first not excede H, namely, 12. the multiplex of the second: but E, namely, 18. the multiplex of the third excedeth G, namely, 8. the multiplex of the fourth. Wherefore the proportion of C to D, the first to the se∣cond, is lesse then the proportion of A to B, the third to the fourth.

Euen so in numbers. As in this ex∣ample,

[illustration]

5. to 4. and 7. to 3. If ye multiply 5. and 7. the first and the third eche by 3, ye shall for the multiplex of 5. the first haue 15. and for the multiplex of 7. the third shall ye haue 21: againe if ye multiply 4. and 3. the second & the fourth by 6, for the multiplex of 4. the second ye shall haue 24, and for the multiplex of 3. the fourth, ye shall haue 18. So ye see that ••5. the multiplex of the first, is lesse then 24, the multiplex of the second: and 21. the multiplex of the third is greater then 18. the multiplex of the fourth. Wherefore the proportion of 5. to 4. the first to the second is lesse then the propor∣tion of 7. to 3. the third to the fourth.

Proportionallitie consisteth at the lest in three termes.* 1.91

Before it was sayd, that proportionalitie is a likenesse or an idemptitie of pro∣portions. Wherfore of necessitie in proportionalitie, there must be two proporti∣ons, and euery proportion hath two termes, namely, his antecedent and conse∣quent. Therfore in euery proportionalitie th••re are foure termes. But for that som∣tyme, one terme supplieth by diuers relations, the roume of two, for in respect to the first it is consequent, and in respect to that which followeth, it is antecedent: therfore three termes at least and not vnder may suffice in proportionalitie, which three are in power foure, and occupy the rome of foure, as is sayd.* 1.92 As suppose that A hath to B that proportion, that B

[illustration]

hath to C: then are these thre quan∣tities A, B, C, set in the lest number of proportionality.* 1.93 Likewise in num∣bers, as 8. 4. 2. and 9. 6. 4.

When there are three magnitudes in proportion,* 1.94 the first shall be vnto the third in double proportion that it is to the second. But when there are foure magnitudes in proportion the first shall be vnto the fourth in treble proportion that it is to the se∣cond. And so alwaies in order one more, as the proportion shall be extended.

This definition is also vnderstand in continuall proportionalitie.* 1.95 As if the thre magnitudes A, B, C, bee proportionall: then shall the proportion of A the

first to C the thirde, bee

[illustration]

double to the proportion which is betwene A & B the first and the seconde, that is the proportiō of A to B taken twise, or added to it self (which is all one) shall make the proportiō of A to C. For the easier vnderstādyng of this & the pra∣ctise therof, it shall be much necessary somwhat to instruct the rude beginner how proportions may be added one to an other. Which is done by this rule.

* 1.96Multiply the antecedent of the one proportion by the antecedēt of the other, and the number produced shall be the antecedent of the proportion which con∣tayneth them both. Likewyse multiply the consequent of the one proportion by the consequent of the other, and the number produced shall be consequent of the proportion which shall contayne them both.

* 1.97An example. If ye will adde the proportion which is betwene 4 and 2. (which is dupla) to the proportion which is betwene 9 and 3. (which is tripla) multiply 9. the antecedent of the first proportion by 4. the antecedent of the

[illustration]

second proportion, and ye shall produce 36. which reserue and kepe for the antecedent of the proportion which ye seeke for. Likewise multiply 3 the consequent of the first proportion, by 2 the conse∣quent of the second, so shall ye haue 6. which 6. shall be consequent to the former antecedēt, namely, to 36. so shal the proportiō which is betwene 36 and 6. namely, sextupla, contayne in it the two proportions geuen, namely, tripla, and dupla. And by this meanes are they added together, & brought into one. And by this may ye adde all other kyndes of proportions whatsoeuer they be. Now for that the diffinition sayth, that if there be three quantities in pro∣portiō, that is, what proportiō the first hath to the secōd, the same hath the second to the third, which for example let be tr••ple, as in these nūbers, 27. 9. 3. adde triple to triple by the rule abouesaid. And forasmuch as it is easier to worke in small nū∣bers then in great, reduce these proportiōs to theyr least denomination:* 1.98 So 27. to 9. reduced to the lest termes in that proportion, is as much as 3. to 1. Likewise 9 to 3 reduced to theyr lest termes are also as much as 3 to 1.* 1.99 now adde together these two triple proportions thus reduced, multipliyng 3 by 3.* 1.100 the one antecedent by the other, so shall ye produce 9 for a new antecedent, then multiply 1 by 1.* 1.101 the one consequent by the other, so shall you produce 1. which let be consequent to 9. your antecedent, so the proportion betwene 9 and 1. which is noncuple contay∣neth both the two triple proportions. And because they were equal the one to the other, it is duple to eche of them. Ye see ••lso that the proportion of 27 to 3. the first to the third, is also noncuple. Wherfore according to the definition, the proporti∣on of the first to the third, is double to the proportion of the first to the second, as 9 to 1. beyng noncuple, is double 3 to 1. which is triple, because it contayneth it twise.

So if there be 4. quantities in continuall proportion, the proportion of the first to the fourth, shall be triple to the proportion which is betwene the first and the second, that is, it shall contayne it three tymes. As for example, Take 4. numbers in continual proportion 8. 4. 2. 1.* 1.102 Ye see that the proportiō of 8 to 1.* 1.103 the first to the fourth,* 1.104 is octupla: the proportion of 8 to 4. the first to the second is dupla, now treble dupla proportion, that is, adde 3. dupla proportions together, by the rule before geuen, as ye see in the example. Multiply all the antecedentes together 2.

the antecedent of the first proportion, by 2. the antecedent of the second, so haue ye 4: which 4. multiply by 2 the antecedent of the third proportiō, so shal ye haue 8 for a new antecedent. In lyke maner multiply all the consequentes together, 1. the consequent of the first proportion by 1. the consequent of the second propor∣tiō, so shal ye haue 1, which 1. multiply agayne by 1. the cōsequent of the third pro∣portion, so shall ye haue agayne 1: which 1. let be consequent to your former ante∣cedent 8: so haue ye 8 to 1. which is octupla, which was also the proportion of the first to the fourth, which octupla is also brought fourth of the addition of thre du∣pla proportions together, and contayneth it three tymes, wherefore octupla is tri∣pla to dupla, and therfore as the diffinition sayth: the proportion of the first to the fourth is tripla to the proportion of the first to the second. And so consequently forth as long as the proportionalitie continueth accordyng to the sentence of the diffinition, the termes of the proportions exceding the number of proportions by one. As if ye haue 5. termes in proportion, the proportiō of the first to the fifth shal be quadrupla to the proportion of the first to the second, and if there be 6. termes, it shall be quintupla and so in order.

Magnitudes of like proportion, are sayd to be antecedents to antecedentes, and consequentes to consequentes.* 1.105

For that before it was sayd, that proportion was a relation or a respect of one quantitie to an other, now sheweth he what magnitudes are sayd to be of like pro∣portion, namely, these whose antecedents haue like respect to their consequentes, and whose consequents receyue

[illustration]

like respectes of their antecedēts.* 1.106 As putting 4. magnitudes A, B, CD. If A antecedent to B, be dou∣ble to B, and C antecedent to D, be double also to D, thē haue the two antecedentes like respectes to their consequents. Likewise if B the consequent be halfe of A, and also D the consequent be halfe of C, then the two consequentes B and D receiue of their antecedentes like respectes and relations. And by this dif∣finition, are these magnitudes A, B, C, D, of like proportion.

Also in numbers, 9. 3. 6. 2: because 9 the antecedent is triple to 3. his conse∣quent, and the antecedent 6. is also triple to 2 his consequent:* 1.107 the

[illustration]

two antecedēts 9 and 6 haue like respectes to their consequentes, and because that 3 the consequent is the subtriple or third part of ••. his antecedent, and likewise 2 the consequent is the subtriple or third part of 6. his antecedent, the two consequentes 3 and 2 receiue also lyke re∣spectes of their antecedentes, and therfore are numbers of like proportion.

Proportion alternate, or proportion by permutation is,* 1.108 when the antecedent is compared to the antecedent, and the conse∣quent to the consequent.

The vnderstanding of this definition & of all the definitions following, depen∣deth of the definition going before, and vse it for a generall supposition, namely, to

haue foure quantities in proportion. Suppose foure magnitudes A, B, C, D, to be in proportion,* 1.109 namely, as A is to B, so let C be to D. Now if ye compare A the ante∣cedent of the first proportion to C the antecedent of the second as to his conse∣quent, & likewise if ye compare

[illustration]

B the cōsequent of the first pro∣portion as an antecedent to D the consequent of the second as to his consequent: then shall ye haue the magnitudes in this sort: as A to C, ante∣cedent to antecedent,* 1.110 so B to D, consequent to consequent, & this is called permu∣tate proportion or alternate. In numbers as 12. to 6, so

[illustration]

8. to 4. either is dupla. Wherefore by permutation of proportion, as 12. to 8. antecedent to antecedent, so is 6. to 4. consequent to consequent, for either is sesquialtera.

* 1.111Conuerse proportion, or propo••tion by conuersion is, when the consequent is taken as the antecedent, and so is compared to the antecedent as to the consequent.

Suppose as before foure magnitudes in proportion, A, B, C, D, as A to B, so C to D:* 1.112 if ye referre B the consequent of the first proportion, as antecedent, to A the antecedent of the first, as to his consequent: and likewise if ye referre D the conse∣quēt of the second proportion as an∣tecedēt

[illustration]

to C the antecedēt of the se∣cond proportiō, as to his cōsequent: thē shall ye haue the magnitudes in thys order. As B to A cōsequent to antecedēt•• so D to C consequent to antecedēt.* 1.113 And thys is called cōuerse proportion. So also in numbers, 9. to 3, as 6. to 2, eyther is tripla, wher∣fore

[illustration]

comparing 3. to 9, the consequent of the first to hys antecedent 9, and also 2. the consequent of the second to hys antecedent 6, by conuerse proporti∣on it commeth to passe as 3. to 9, so 2. to 6: For ei∣ther is subtripla.

* 1.114Proportion composed, or composition of proportion is, when the antecedent and the consequent are both as one compared vnto the consequent.

Suppose that in the former foure

[illustration]

magnituds in proportiō, A, B, C, D, as A is to B, so is C to D:* 1.115 if ye adde A and B the antecedent and the consequent of the first proportion together, and compare them so added as one antecedent to B the consequent of the first proportion as to hys consequent: and likewise if ye adde together

C and D the antecedent and the consequent of the second proportion, and so ad∣ded, compare them as one antecedent to D the consequent of the second propor∣tion, as to his consequent: then shall ye haue the magnitudes in this order. As AB to B, so CD to D, for either of them is tripla. And this is called composed propor∣tion, or composition of proportion. And so also in numbers.* 1.116 As 8. to 4, so 6. to 3: 8. and 4, the antecedent and consequen•• of the first pro∣portion

[illustration]

added together, make 12: which 12. as antecedent cō∣pare to 4. the consequent of the first proportion as to his con∣sequent: so adde together 6. and 3, the antecedent and con∣sequent of the second proportion, they make 9: which 9. as antecedent compare to 3. the consequent of the second proportion, as to his con∣sequent: so shall ye haue by composition of proportion, as 12. to 4, so 9. to 3, for ei∣ther of them is tripla.

Proportion deuided, or diuision of proportiō is,* 1.117 when the ex∣cesse wherein the antecedent excedeth the consequent, is com∣pared to the consequent.

Thys definition is the conuerse of the definition going next before:* 1.118 in it was vsed composition, and in thys is vsed diuision. As before so now suppose foure magnitudes in proportion AB the first, B the second, CD the third, and D the fourth: as AB to B: so CD to D:

[illustration]

AB,* 1.119 the antecedent of the first pro∣portion excedeth B the consequent of the first proportion by the magni∣tude A, wherfore A is the excesse of the antecedent AB aboue the consequent B: so likewise CD the antecedent of the second proportion, excedeth D the conse∣quent of the same proportion, by the quantitie C, wherefore C is the excesse of the antecedent CD aboue the consequent D. Now if ye compare A the excesse of AB the first antecedent, aboue the consequent B, as antecedent to B the con∣sequent, as to his consequent: also if ye compare D the excesse of the second an∣tecedent CD, aboue the consequent D, as antecedent to D the consequent, as to his consequent: then shall your magnitudes be in this order. As A to B, so is C to D: which is called diuision of proportion, or proportion deuided.* 1.120

And so in numbers, as 9. to 6, so 12. to 8, either proportion

[illustration]

is sesquialtera: the excesse of 9. the antecedent of the first proportion aboue 6. the consequent of the same is 3••: the ex∣cesse of 12. the antecedent of the second proportion aboue 8, the consequent of the same, is 4•• then if ye compare 3. the ex∣cesse of 9. the first antecedent aboue the consequent, as antecedent to 6, the conse∣quent, as to hys consequent: and also if ye compare .4 the excesse of 12. the second antecedent aboue the consequent, as antecedent, to 8. the consequent, as to hys consequent, ye shall haue your numbers after this maner by diuision of proporti∣on, as 3. to 6: so 4. to 8: for either proportion is subdupla.

Conuersiō of proportion (which of the elders is commonly cal∣led euerse proportion,* 1.121 or euersiō of proportion) is, whē the an∣tecedent

is compared to the excesse, wherein the antecedent excedeth the consequent.

* 1.122Foure magnitudes supposed as before, AB the first, B the second, CD the third, and D the fourth. As AB to A, so CD to C: AB the antecedent of the first proportion excedeth B the consequent of the same by the magnitude A, where∣fore A is the excesse of the antece∣dent

[illustration]

AB aboue the consequent B: so also the magnitude C is the ex∣cesse of CD the antecedent of the second proportion aboue D the consequent of the same: now if ye referre AB the antecedent of the first proportion, as antece∣dent, to A the excesse therof aboue the consequent B, as to his consequent: if ye compare also CD the antecedent of the second proportion as antecedent to C the excesse therof aboue the consequent D, as to his consequent: then shall your magnitudes come to thys order. As AB to A, so CD to C, and thys is called conuersion of proportion, and of some euersion of proportion. Likewyse in num∣bers, as 9. to 6, so 12. to 8.* 1.123 eyther proportion is sesquialtera:

[illustration]

the excesse of 9. the antecedent of the first proportion aboue 6. the consequent of the same is 3: the excesse of 12. the ante∣cedent of the second proportion aboue 8. the consequent of the same, is 4: now cōpare the antecedent of the first propor∣tion 9. as antecedēt to 3. the excesse therof aboue 6. the consequēt, as to his con∣sequent, likewise compare 12. the antecedent of the second proportion as antece∣dent to 4. the excesse therof aboue 8. the consequent, as to his consequent: so shall your numbers be in thys order by conuersion of proportion: as 9. to 3: so 12. to 4: for either proportion is triple.

* 1.124Proportion of equalitie is, when there are taken a number of magnitudes in one order, and also as many other magnitudes in an other order, comparing two to two beyng in the same pro¦portion, it commeth to passe, that as in the first order of mag∣nitudes, the first is to the last, so in the second order of magni∣tudes is the first to the last. Or otherwise it is a compari∣son of extremes together, the middle magnitudes being taken away.

To the declaration of thys definition are required two orders of magnitudes equall in number, and in lyke proportion:* 1.125 As if there be taken in some deter∣minate number certayne magnitudes, namely, foure, A, B, C, D. And also in the same number be taken other quantities, namely, foure, E, F, G, H: then take the equall proportions by two and two: as A to B, so E to F: as B to C,

so F to G: and as C to D, so G

[illustration]

to H. Now according to the first definition, if A the first magnitude of the first order be to D the last mag∣nitude of the same order, as E the first magnitude of the second order is to H the last magnitude the same, then it is called proportion of equa∣litie, or equall proportion.

By the second definition, which is all one in substance with the first, ye leaue the meane magnitudes in eyther order, namely, B, C, on the one side, and F, G on the other side, and onely compare the extremes of ech side together, which by thys de∣finition shall be in lyke proportion, as A is to D, so is E to H.

Euen so in numbers, take for example these two orders,

[illustration]

27. 9. 12. 24. 25. and 9. 3. 4. 8. 5. there are in eche order as ye see,* 1.126 fiue numbers, then see that all the proportions taken by two & two be like: betwene 27 & 9, numbers of the first or∣der, and betwene 9. and 3, num∣bers of the second order, there is one and the self same propor∣tiō, namely, tripla: also betwene ••, and 12, numbers of the first order, and 3. and 4, numbers of the second order, is like pro∣portion, namely, subsesquitertia proportion: so betwene 12. and 24, numbers of the first order, and 4. and 8, numbers of the second order, is also lyke proportion, namely, subdupla: Last of all, betwene 24. and 15, numbers of the first rowe, and betwene 8. and 5, numbers of the second rowe, the proportion is one, namely, su∣per••ripar••iens quintas. Wherefore by this definition, leauing out all the meane numbers of eche side, ye may compare together onely the e••tremes, and conclude that as 27. of the first row is to 15. the last of the same row, ••o is 9. the first of the second rowe to 5, the last of the same rowe: for the proportion of ech is superqua∣dripartiens quintas.

Here is to be considered,* 1.127 that it is not of necessitie that all the proportione in eche rowe of numbers be set in like order, as in the one so in the other: but it shall be sufficient that the proportions be the same and in equall number in eche rowe. Whether it be in the selfe same order, or in contrary, or inue••••ed order, it maketh no matter. As in these numbers•• 12. 6. 2. in the ••••rst row•• and ••••. 8•• 4.

[illustration]

in the second. As 12. is to 6, the firs•• to the second of the first row, so is 8. to 4. the second to the third of the second row: either i•• duple proportion. And as 6•• to 2•• the second to the third in the first

order: so is 24. to 8. the first to the second in the second order. Where ye see that the proportions are not placed in one and the selfe same order, and course, and yet notwithstanding ye may conclude by equalitie of proportion, leauing the meanes 6. and 8: as 12. to 2. the first to the last of the first or∣der,

[illustration]

so 24. to 4. the first and last of the second order. And so of others whatsoeuer and how soeuer they be placed.

An ordinate proportionality is, when as the antecedent is to the consequent,* 1.128 so is the antecedent to the consequent, and as the consequent is to another, so is the consequent to an other.

For the declaration of this definition are also required two orders of magni∣tudes.* 1.129 Suppose in the first order, that the antecedent A, to his consequent B, haue the same proportion that the antecedent D, hath to his cōsequent E in the second order: and make the consequent B

[illustration]

antecedēt to some other quantitie, as to C. Also make the consequent E antecedēt to an other quātitie, as to F, so that there be the same pro∣portion of B to C, which is of E to to F. And thys disposition of proportions is called ordinate proporti∣onalitie.

[illustration]

Likewise in numbers, 18.9.3 and 6.3.1.* 1.130 As 18. to 9. antece∣dent to consequent, so is 6. to 3. antecedent to consequent: either is dupla proportion: and as 9. the consequent is to an other, namely, to the number 3, so is the consequent 3. to an other, namely, to vnity. And this ordinate proportionalitie may be extēded as farre as ye li••t, as ye may see in the example of numbers in the definition next before.

* 1.131An inordinate proportionality is, when as the antecedent is to the consequent, so is the antecedent to the consequent: and as the consequent is to an other, so is an other to the ante∣cedent.

This definition also as the other before, requireth two orders of magnitudes, Suppose in the first order that the antecedēt A be to the cōsequēt B, as the antece∣dēt C,* 1.132 in the second

[illustration]

order is to the conse∣quent D, & let B the consequēt of the first proportiō be to some other, namely, to the magnitude E, as some other, namely, the magnitude F, is to the antecedent C of the second proportiō:* 1.133 this kinde of proportionalitie is called inordinate or perturbate.

[illustration]

Take also an example in numbers, as 9 to 6. the antecedent to the consequent, so is 3 to 2 the antecedent to the consequent, ei∣ther proportiō is s••squ••ul tera, and as •• the consequent of the first

proportion, is to an other, namely, to the number 3. so is another namely, the num∣ber 6. to 3. the antecedent of the second proportion, for eyther is dupla propor∣tion.

An extended proportionality is, when as the antecedent is to the consequent, so is the antecedent to the consequent,* 1.134 and as the consequent is to an other, so is the consequent to an other. Apertu••bate proportionalitie is, when, thre magnitudes be∣ing compared to three other magnitudes,* 1.135 it cōmeth to passe, that as in the first magnitudes the antecedent is to the conse∣quent, so in the second is the antecedent to the consequent, & as in the first magnitudes the consequent is to an other mag∣nitude, so in the second magnitudes is an other magnitude to the antecedent.

These two last definitions here put by Zamberte seeme all one with the other two last before set.* 1.136 Wherfore it is not lyke that they were written and set here by Euclide, for that they seeme no•• necessary, but rather superfluous, neither are they found in the Greeke examples commonly set forth in print, nor mentioned of a∣ny that hath written commentaries vpon Euclide, olde or new: Not of Campane, S••••ub••lius, Pellitarius, Orontius, nor Fl••ssates: wherfore it is not of necessitie to adde vnto them any explanation or example either in magnitudes or in numbers. The examples of the two last definitions set before, may likewise serue for them also.

Notes

• * 1.1

  The first defi∣nition.
• * 1.2

  A part taken two maner of wayes.
• * 1.3

  The fi••st way.
• * 1.4

  The second way.
• * 1.5

  How a lesse quantity is sayd to mea∣sure a greater.
• * 1.6

  In what sig∣nificatiō Eu∣clide here ta∣keth a part.
• * 1.7

  Par•• metien•• or mensuran••.
• * 1.8

  Pars multipli∣cati••a.
• * 1.9

  Pars aliquota.
• * 1.10

  This kinde of part cōmonly v∣sed in Arith∣metique.
• * 1.11

  The other kinde of part.
• * 1.12

  Pars constit••∣ens, or compo∣nens.
• * 1.13

  Pars aliquan∣ta.
• * 1.14

  The second definition.
• * 1.15

  Numbers ve∣ry necessary for the vnder∣standing of this booke and the other bookes follo∣wing.
• * 1.16

  The t••ird de∣finition.
• * 1.17

  what propor∣tion is.
• * 1.18

  Thinges com∣pa••ed together this wayes.
• * 1.19

  Example of this definition in magni∣tudes.
• * 1.20

  Example ther¦of in numbers.
• * 1.21

  In proportion•• two quanti∣ties required, which are cal∣led termes.
• * 1.22

  The i. terme called the an∣tecedent.
• * 1.23

  The ij. terme called the con∣sequent.
• * 1.24

  Dux.
• * 1.25

  Comes.
• * 1.26

  This booke counted the hardest of all Euclides bookes.
• * 1.27

  Proportion of two kindes, rational and irrational.
• * 1.28

  Proportion rational.
• * 1.29

  In Arithme∣tique all pro∣portions are rational.
• * 1.30

  Proportion irrational.
• * 1.31

  Rational pro∣portion deuided ••••to two kindes.
• * 1.32

  Proportion of equality.
• * 1.33

  Proportion of inequality.
• * 1.34

  Proportio•• of the greater to the lesse.
• * 1.35

  Multiplex.
• * 1.36

  Duple propor∣tion.
• * 1.37

  Triple, quadru∣ple. Quintuple.
• * 1.38

  Superperticu∣lar.
• * 1.39

  Sesquialtera.
• * 1.40

  Sesquitertia.
• * 1.41

  Sesquiquarta.
• * 1.42

  Superpartiens.
• * 1.43

  Superbiparti∣ens.
• * 1.44

  Supertriparti∣ens.
• * 1.45

  Superquadri∣partiens.
• * 1.46

  Superquinti∣partiens.
• * 1.47

  Multiplex su∣perperticular.
• * 1.48

  Dupla Sesqui∣altera.
• * 1.49

  Dupla sesqui∣tertia.
• * 1.50

  Tripla sesqui∣altera.
• * 1.51

  Multiplex su∣perpartiens.
• * 1.52

  Dupla superbi∣partiens.
• * 1.53

  Dupla supertri∣partiens.
• * 1.54

  Tripla superbi∣partiens.
• * 1.55

  Tripla super∣quad••ipartiens.
• * 1.56

  How to kno•• the denomi∣nation of any proportion.
• * 1.57

  Proportion of the lesse in the greater.
• * 1.58

  Submulti∣plex.
• * 1.59

  Subsuperpar∣ticular.
• * 1.60

  Subsuperper∣tient. &c.
• * 1.61

  The fourth definition.
• * 1.62

  Example of this definition in magni∣tudes.
• * 1.63

  Example thereof in numbers.
• * 1.64

  Note.
• * 1.65

  The fifth defi∣nition.
• * 1.66

  An example of this ••efini∣tion in mag∣nitudes.
• * 1.67

  Why Euclide in defining of Proportion vsed multi∣plication.
• * 1.68

  The sixth de∣finition.
• * 1.69

  An example of this defini∣tiin in magni∣tudes.
• * 1.70

  An example in numbers.
• * 1.71

  An other ex∣ample in numbers.
• * 1.72

  An other example in numbers.
• * 1.73

  Note this par∣ticle according to any multi∣plication.
• * 1.74

  An example where the equimultipli∣ces of the first and third ex∣ceed•• the equi∣multiplices of the second and fourth, and yet the quantities geuen are not in one and the selfe same pro∣portion.
• * 1.75

  A rule to pro∣duce equimul∣tiplices of the first and third equall to the equimultipli∣ces of the se∣cond•• and f••urth.
• * 1.76

  Example thereof.
• * 1.77

  The seuenth definition
• * 1.78

  9 12
• * 1.79

  3 4
• * 1.80

  Proportiona∣lity of two sortes, conti∣••uall and dis∣continuall.
• * 1.81

  An example of continuall proportionali∣tie in num∣bers.
• * 1.82

  16.8.4.2.1.
• * 1.83

  In coutinnall proportionali∣tie the quan∣tities cannot be of one kinde.
• * 1.84

  Discontinuall prop••rtiona∣litie.
• * 1.85

  Example of discontinual proportionali∣ty in numbers.
• * 1.86

  In discōtinual proportionali∣tie the propor∣tions may be of diuers kindes.
• * 1.87

  The eight definition.
• * 1.88

  An example of this defini∣tion in mag∣nitudes.
• * 1.89

  An example in numbers.
• * 1.90

  Note.
• * 1.91

  The ninth de∣finition.
• * 1.92

  An example of this defini∣tion in mag∣nitudes.
• * 1.93

  Example ••n numbers.
• * 1.94

  The tenth definition.
• * 1.95

  Example of this definition in magni∣tudes.
• * 1.96

  A rule to adde proporti∣ons to propor∣tions.
• * 1.97

  Example thereof.
• * 1.98

  27. 9. ••.
• * 1.99

  9
• * 1.100

  3—3
• * 1.101

  1—1
• * 1.102

  8. 4. 2. 1.
• * 1.103

  2 2 2
• * 1.104

  1 1 1
• * 1.105

  The eleuenth definition.
• * 1.106

  Example of this definition in magnitud••.
• * 1.107

  Example in numbers.
• * 1.108

  The twelf••h definition.
• * 1.109

  Example of this de••inition in magnituds.
• * 1.110

  Example in numbers.
• * 1.111

  The thirtenth definition.
• * 1.112

  Example of this definition in magnituds.
• * 1.113

  Example in numbers.
• * 1.114

  The fourtenth definition.
• * 1.115

  Example of this definition in magnituds.
• * 1.116

  Example in numbers.
• * 1.117

  The fi••t••ne definition.
• * 1.118

  This is the cō∣uerse of the former defini∣tion.
• * 1.119

  Example in magnitudes.
• * 1.120

  Example in numbers.
• * 1.121

  The sixtene definition.
• * 1.122

  An example of this defini∣tion in mag∣nitudes.
• * 1.123

  An example in numbers.
• * 1.124

  The seuētenth definition.
• * 1.125

  An example of this defini∣tion in mag∣nitudes.
• * 1.126

  An example in numbers.
• * 1.127

  Note.
• * 1.128

  The eight∣tenth defini∣tion.
• * 1.129

  An example of this defini∣tion in mag∣nitudes.
• * 1.130

  Example in numbers.
• * 1.131

  The nintenth definition.
• * 1.132

  An example of this defini∣tion in magni∣tudes.
• * 1.133

  Example in numbers.
• * 1.134

  The 20. definition.
• * 1.135

  The 2••. defi••i∣tion.
• * 1.136

  These two last definitions not found in the greeke ex∣amplers.