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

¶The 7. Theoreme. The 9. Proposition. [ 1] Squares described of right lines commensurable in length, haue that pro∣portion the one to the other, that a square number hath to a square number. [ 2] And squares which haue that proportion the one to the other that a square number hath to a square nūber, shall also haue their sides cōmensurable in [ 3] length. But squares described of right lines incommensurable in length, haue not that proportion the one to the other, that a square number hath to [ 4] a square number. And squares which haue not that proportion the one to the other that a square nūber hath to a square number, haue not their sides commensurable in length.

SVppose that A and B be lines

[illustration]

cōmensurable in length.* 1.1 Then I say that the square of the line A hath vnto the square of the line B, that proportion that a square num∣ber hath to a square number. For foras¦much as the lines A and B are commen¦surable in length: therefore the line A hath vnto the line B that proportion that number hath to number (by the 5. of this tenth). Let the line A haue vnto the line B that proportion, that the number C hath to the number D, Now for that as the line A is to the line B, so is the number C to the number D: but the square of the line A is vnto the square of the line B in double proportiō of that which the line A is vnto the line B (for like rectiline figures (by the first corollary of the 20. of the sixt) are in double proportion of that which the sides of like proportion are) and likewise the square number produced of C is to the square number produced of D, in double proportion of that which the number C is to the number D (for by the 11. of the eight betwene two square numbers there is one meane proportionall number & a square number is to a square number in double proportion of that which side is vnto side). Wherefore as the square of the line A is to the square of the line B, so is the square number produced of the number C, to the square number produced of the number D.

An other demonstration to proue the same.

Forasmuch as the lines A and B are commensurable, therefore (by the 5 of this tenth) A hath vnto B the same proportion that number hath to number.* 1.2 Let them haue that proportiō that the number C hath to the number D. And let the number C multiplying himselfe pro∣duce the number E, and multiplying the number D, let it produce the number F: and let the number D multiplying himselfe produce the number G. And forasmuch as the number C multiplying himselfe produced the number E, and multiplying the number D it produced the number F: therefore (by the 17. of the seuenth) as the number C is to the number D, that is, the line A to the

[illustration]

line B so is the num∣ber E to the nūber F. But as the line A is to the line B, so is the square of the line A to the parallelograme contained vnder the lines A and B (by the first of the sixt) Wher¦fore as the square of the line A is to that which is contayned vnder the lines A

and B, so is the number E to the number F. Agayne for as muche as the number C multi∣plying the number D produced the number F, & the number D multiplying himselfe pro∣duced the number G: therefore (by the 17. of the seuenth) as the number C is to the num∣ber D, that is, as the line A is to the line B, so is the number F to the number G. But as the line A is to the line B, so is parallellograme contained vnder the lines A and B to the square of the line B (by the first of the sixt). Wherefore as that which is contained vn∣der the lines A and B is to the square of the line B, so is the number F to the number G. But as the square of the line A is to that which is contayned vnder the lines A and B, so is the number E to the number F. Wherefore of equalitie (by the 22. of the fifte) as the square of the line A is to the square of the line B, so is the number E to the number G. But either of these numbers E and G is a square number. For E is produced of the number C multiplyed into him selfe, and G is produced of the number D multiplyed into him selfe. Wherefore the square of the line A hath vnto the square of the line B that proportion that a square number hath to a square number: which was required to be demonstrated.

An other demonstration of the same first part after Montaureus.

Suppose that there be two lines commensurable in length A and B. Then I say that the squares described of those lin••s shalbe in proportion the one to the other as a square number is to a square nū∣ber.* 1.3 For forasmuch as the lines A and B are commēsurable in length, they shalbe in proportion the one to the other as number is to number (by the 5. of this booke). Let A be to B in duple proportion, which is in such proportion as number is to number, namely, as 4. is to 2, and 6. to 3, and so of many other. And (by the s••cond o•• the eight) take three least numbers in continuall proportion, and in duple propor¦tion, & let the same be the numbers 4.2.1: whe••fore by

[illustration]

the corrollary of the second of the eight, the numbers 4. and 1. shalbe square numbers. (For as 4. is a square number produced of 2. multiplied into him selfe, so is 1. also a square number, for it is produced of vnitie multi∣plied into him selfe.) I say moreouer that those are the square numbers, whose proportion the squares of the lines A and B haue the one to the other. For as the number 4. is to the number 2. so is the line A to the line B (••or either proportion is double by supposition): but as the line A is to the line B, so is the square of the line A to the parallelograme contained vnder the lines A and B (by the first of the sixt). Wherefore as the number 4. is to the number 2: so shall the square of the line A be to the parallelo∣grame contained vnder the lines A and B. Likewise as the number 2. is to the number 1. so is the line A to the line B (For either proportion is duple by supposition): but as the line A is to the line B, so is the parallelograme contained vnder the lines A and B to the square of the line B (by the selfe same first of the sixt). Wherefore as the number 2. is to 1. so is the parallelograme contained vnder the lines A and B to the square of the line B: wherefore of equallitie (by the 22. of the fifth) as the square of the line A is to the square of the line B, so is the number 4. to 1. which are proued to be square numbers.

* 1.4But now suppose that the square of

[illustration]

the line A, be vnto the square of the line B, as the square nūber produced of the number C is to the square number produced of the number D. Then I say that the lines A & B are cōmensurable in length. For for that as the square of the line A is to the square of the line B, so is the square number produced of the number C to the square number pro∣duced of the number D: but the propor¦tion of the square of the line A, is vnto the square of the line B double to that proportion which the line A hath vnto the line B (by the corollary of the 20. of

the sixt). And the proportion of the square number which is produced of the number C to the square number produced of the number D is (by the 11. of the eight) double to that pro∣portion which the number C hath vnto the number D. Wherefore as the line A is to the line B, so is the number G to the number D. Wherefore the line A hath vnto the line B the same proportion that the number C hath to the number D. Wherefore (by the 6. of this booke) the lines A and B are commensurable in length: which was required to be proued.

An other demonstration to proue the same.

••Vppose againe that the square of the line A haue vnto the square of the line B,* 1.5 the same proportion, that the square number E hath to the square number G. Then I say that the lines A and B are commensurable in length. For suppose that the side of the square number E

[illustration]

be the nūber C, & let the side of the square number G be the nū∣ber D: and let the number C multiply∣ing the number D produce the number F. Wherefore these numbers E, F, G, are in continuall propor∣tion, aud in the same proportion that the number C is to the number D (by the 17. and 18. of the seuenth).* 1.6 And forasmuch as the meane proportionall betwene the squares of the lines A and B is that which is contained vnder the lines A and B. (Which though it might briefely be proued: yet we take it as now). And likewise the meane proportionall betwene the numbers E and G is the number F (by the 20. of the seuenth): therefore as the square of the line A is to that which is contained vnder the lines A and B, so is the number E to the num∣ber F, and as that which is contained vnder the lines A and B is to the square of the line B, so is the number F to the number G. But as the square of the line A is to that which is con∣tained vnder the lines A and B, so is the line A to the line B (by the first of the sixt). Where∣fore the lines A and B are in the same proportion that the number E is to the number F, that is, that the number C is to the number D. Wherefore the lines A and B are commensu∣rable in length (by the 6. of this tenth) which was required to be proued.

But now suppose that the lines A and B be incom∣mensurable

[illustration]

in length. Then I say that the square of the line A hath not vnto the square of the line B that proportion that a square number hath to a square nū∣ber.* 1.7 For if the square of the line A haue vnto the square of the line B, the same proportion that a square number hath to a square number, thē shall the lines A and B be cōmensurable in length (by the second part of this proportion). But by supposition they are not. Wherefore the square of the line A, hath not vnto the square of the lyne B that proportion that a square number hath to a square number: which was requi∣red to be proued.* 1.8

Againe suppose that the square of the line A haue not vnto the square of the line B, the

same proportion that a square number hath to a square

[illustration]

number. Then I say that the lines A and B are incom∣mensurable in length. For if the lines A and B be com∣mensurable in length, then the square of the line A should haue vnto the square of the line B, the same proportion that a square number hath to a square number, by the first part of this proposition, but by supposition it hath not, wherfore the lines A and B are not commensurable in length.* 1.9 Wher∣fore they are incomensurable in length. Wherfore squares made of right lines commensura- in length, haue that proportion the one to the other, that a square number hath to a square number. And squares which haue that proportion the one to the other, that a square number hath to a squa••e number, shall also haue the sides commensurable in length. But squares de∣scribed of right lines incommensurable in length, haue not that proportion the one to the o∣ther that a square number hath to a square number. And squares which haue not that pro∣po••tion the one to the other, that a s••uare number hath to a square number, haue not also their sides comm••nsurable in length: which was all that was required to be proued.

¶ Corrollary.

Hereby it is manifest, that right lines cōmensurable in length, are also euer commensurable in power. But right lines commensurable in power, are not alwayes commensurable in length.* 1.10 And right lines incōmensurable in lēgth are not alwayes incommensurable in power. But right lines incommensura∣ble in power, are euer also incommensurable in length.

For forasmu••h as squares made of right lines commensurable in length, haue that pro∣portion the one to the other,* 1.11 that a square number hath to a square number (by the first part of this proposition), but magnitudes which haue that proportion the one to the other, that number simply hath to number, are (by the sixt of the tenth) commensurable. Wherfore right lines commensurable in length, are commensurable not onely in length, but also in power.

* 1.12Againe forasmuch as there are certaine squares which haue not that proportion the one to the other that a square number hath to a square number, but yet haue that proportion the one to the other which number simply hath to number: their sides in dede are in power com∣mensurable, for that they describe squares which haue that proportion which number sim∣ply hath to number, which squares are therfore commensurable (by the 6. of this booke): but the said sides are incommensurable in length by the latter part of this proposition. Wher¦••fore it is t••ue that lines commensurable in power, are not straight way commensurable in length also.* 1.13

And by the sel••e same reason is proued also that third part of the corollary, that lines incommensurable in length, are not alwayes incommensurable in power. For they may be in∣comm••nsurable in length, but yet commensurable in power. As in those squares which are in proportion the one to the other, as number is to number, but not as a square number is to a square number.

* 1.14But right lines incommensurable in power, are alwayes also incommensurable in length. For i•• they be commensurable in length, they shal also be commensurable in power by the first part of this Corollary. But they are supposed to be incommensurable in length, which is ab∣surde. Wher••ore right lines incommensurable in power, are euer incommensurable in length••

For the better vnderstanding of this proposition and the other following, I haue here added certayne annotacions taken out of Montaureus.* 1.15 And first as touching the signi••ication o•• wordes and termes herein vsed, wh••ch ar•• such, that vnlesse they be well marked and peysed, the matter will be obscure and hard, and in a maner inexplicable.

First, this ye must note, that lines to be commensurable in length, and lines to be in proportion the one to the other, as number is to number is all one. So that whatsoeuer lines are commensurable in length, are also in proportion the one to the other, as num∣ber is to number. And conuersedly what so euer lynes are in proportion the one to the other, as number is to number, are also commēsurable in length, as it is manifest by the 5 and 6 of this booke. Likewise lines to be incommensurable in length, and not to be in proportion the one to the other, as number is to number is all one, as it is manifest by the 7. and 8. of this booke. Wherfore that which is sayd in this Theoreme, ought to be vnderstand of lines commensurable in length, and incommensurable in length.

This moreouer is to be noted, that it is not all one, numbers to be square numbers, and to be in proportiō the one to the other, as a square number is to a square number. For although square numbers be in proportion the one to the other, as a square num∣ber is to a square number, yet are not all those numbers which are in proportion the one to the other, as a square number is to a square number, square numbers. For they may be like superficiall numbers, and yet not square numbers, which yet are in propor∣tiō the one to the other, as a square number is to a square number. Although all square numbers are like superficiall numbers. For betwene two square numbers there ••alleth one meane proportionall number (by the 11. of the eight). But if betwene two num∣bers, there fall one meane proportionall number, those two numbers are like superfici∣all numbers (by the 20. of the eight). So also if two numbers be in proportion the one to the other, as a square number is to a square number, they shall be like superficiall nū∣bers by the first corollary added after the last proposition of the eight booke.

And now to know whether two superficiall numbers geuen, be like superficiall num∣bers or no,* 1.16 it is thus found out. First if betwene the two numb••rs geuen, there fall no meane proportionall, then are not these two numbers like superficiall numbers (by the 18. of the eight. But if there do fall betwene them a meane proportionall, then are they like super••iciall numbers (by the 20. of the eight) Moreouer two like superficiall num∣bers multiplied the one into the other, do produce a square number (by the firs•• of the ninth). Wherfore if they do not produce a square number, then are they not like super∣ficiall numbers. And if the one being multiplied into the other, they produce a square number, then are they like superficiall (by the 2. of the ninth). Moreouer if the said two superficial numbers be in superperticular, or superbipartient proportion, then are they not like superficiall numbers. For if they should be like, then should there be a meane proportionall betwene them (by the 20. of the eight). But that is contrary to the Co∣rollary of the 20. of the eight.

And the easilier to conceiue the demonstrations following, take this example of that which we haue sayd.

Suppose that there be a line, namely, C, which imagine to be foure fote long: and let there be an other line D, which let be three foote long. And (by the 13. of the sixt) take the meane propor∣tionall betwene the lines A, D, which let be the line B. Wherefore the square of the line B shall be equall to the rectangle parallelogramme contayned vnder the line C and D (by the 17. of the sixt). Which square shall contayne 12. foote, & so much also shall the parallelogramme described of the lines C & D containe. Take also two other lines E and F, of which let E be 3. foote long, and let F be a foote long. And let the meane proportionall betwene the lines E and F, be the line A. Now then the square of the line A shall containe 3. foote, as also doth the parallelogrāme described of the lines E, F. Thē I say, that the square of the line B, which cōtaineth 12. foote, is to the square of the line A, which contayneth 3. foote, in that proportion that a square number is to a square number. For as the number 12. is to the number 3, so is the square of the line B, which containeth 12. foote, to the square of the line A, which contayneth 3 foote. But the numbers 12. and 3. are like superficiall numbers, for the sides of 12. which are 2. and 6, are proportionall with the sides of 3. which are 1. and 3. Wherefore the square of the line B, which contayneth 12. foote, shall be vnto the square of the line A, which contayneth 3. foote, in that proportion that a like superficiall number is to a like superfici∣••ll number. But like superficiall numbers are in proportion the one to the other, as a square number i••

to a square nūber,

[illustration]

which square num∣bers are 4. and 1. (by the 26. of the eight). Wherefore the square of the line B, which con∣tayneth 12. foote, is to the square of the line A, which cōtai∣neth 3. foote, in that proportion that a square number is to a square number, namely, that the number 4. is to the number 1: which proportion is qua∣druple. For the greater square whi∣che is 12, contay∣neth the lesse square which is 3, foure times. Wherefore the side of the square 12, which is the line B, is double to the side of the square 3, which is the line A. Wher∣fore the line B is to the line A, in that proportion that number is to num∣ber. Whe••fo••e (by the 5. of this booke) the lines B & A are commensurable in length. Which is a supposition necessary to conclude the first part of this Theoreme, namely, that the squares of such lines are in proportion the one to the other, that a square number is to a square number.

So also the nūber which denominateth the greater terme of the proportion of the line B to the line A, which is 2, if it be multiplyed into it selfe, it maketh a square number, namely, 4. Likewise the num∣ber which denominateth the lesse terme, namely, 1. if it be multiplyed into it selfe, it maketh no more but 1. Which vnitie is also in power a square nūber. Wherfore the square of the line B, is to the square of the line A, in that proportion that a square number is to a square number, namely, that 4. is to 1. By this you see (which thing was before noted) that it is not all one, numbers to be square numbers, and to be in proportion the one to the other, as a square number is to a square number. For it is manifest, that the numbers 12. and 3. are not square numbers, when yet the squares expressed by those num∣bers are in that proportion. But the side of the square 12. although it can not of it selfe be expressed by number distinctly, to say that the side thereof is so many foote long, which feete square taken, make the whole square 12: yet b••ing referred or compared to an other thyng, namely, to the side of the square 3, which side also of it selfe can not be expressed by number, it is vnto the sayde side of the square 3, in double proportion. For the one square being quadruple to the other square (as is the square of the line B, which contayneth 12. foote, to the square of the line A, which contay∣neth 3. foote) hath his side double to the side of the other square, by this generall Corollary of the 10. of the sixt, like rectiline figures are in double proportion the one to the other that their sides of like proportion are. Now if a man will say, that the side of the square 12. may be measured, for that hys proportion which it hath to the side of the square 3, is measured by 2 (forasmuch as it is dupla propor∣tion: this is to be considered, that in so saying, you say not, that that magnitude can of it selfe be measu∣red, but the proportion therof. For, that magnitude, namely, the side of the square 12, should by it selfe be measured, when without any respect of the proportion of it to an other thing, we may say that the side of the square, which contayneth 12. foote, is so many foote long, the number of which foote multi∣plyed into it selfe should make that number 12. But this is not possible, for that 12. is not a square num∣ber. Wherefore thus you may say: In asmuch as that square 12. is considered by it self, without hauing any respect of the proportion of it to any other thing, but onely as it is 11. foote, it hath no side which of it selfe can be expressed by number. But if it be compared to any other thing, namely, to the square of 3. foote, then may you say that the side of the square 12. is 2, and the side of the square 3. is 1. But thys is the denomination of that propo••tion which is called duple, which proportion can not be or considered

in fewer termes then two, when as it is a relation of one thing to an other thing: wherefore 2. is not the number of such feete, of which there are 12. in the square. Agayne, if the number 2. shoulde be the side of the square 12, so that that side should be 2, then of the multiplication of 2. into it selfe, shoulde not be made that square 12, but an other square which should be 4. foote: as of the number 2. multiply∣ed into him selfe is produced the square number 4. Neither also if any other number. measure the sid•• of the square 12, and the sayd number be multiplyed into him selfe shall it euer make the number 12. When yet all numbers denominating the side of any square number, if they be multiplyed into them selues, they make the number which denominateth the square, whose sides they denominate. As 2. multiplyed into him selfe maketh 4 : 3. maketh 9 : 4. maketh 16: and so likewise of all others. Where∣fore it is not all one, magnitudes to be in proportion the one to the other, as number is to number, and euery one of them to be measured b•• him selfe without any respect had of the proportion. As here the side of the square 12. can of it selfe by no meanes be measured, but being compared to any other magni∣tude, namely, to the side of the square 3, the proportion thereof is expressed by number. So also the side of the square 3, and of all other square figures, whose areas yet can not be expressed by square numbers. And that which we here say, is manifest euen by the wordes of Euclide in the 5.6. 7. and 8. Theoremes of this booke. Where he sayth not, that magnitudes commensurable and incommensurable are of thē selues or of their owne nature expressed by numbers,* 1.17 but that either they haue or haue not that propor∣tion which number hath to number. Which thing not being well considered, it should seme hath cau∣sed many to erre as hereafter shall be made manifest. And in deede they which haue demonstrated this Theoreme, may seme to some rather to haue demōstrated it particularly & not vniuersally. And doubt∣les I iudge there are some which vnderstand their sayinges otherwise then they ment: when as they thinke, that they suppose certayne lines not onely commensurable in length, as they are supposed to be in the Propositiō, but also such, that ech of

[illustration]

them apa•••• may be expressed by some cer∣tayne number. Wherfore for want of right vnderstāding, this mought they say of their demonstrations: that wheras they thought that they had concluded that generally, which is in this theoreme of Euclide, Squares described of lines cōmensurable in length, are in proportiō the one to the other that a square nū∣ber is to a square number: they conclude par∣ticularly, thys onely: Squares described of lines which may by them selues be expressed by some certaine number, are in proportion. &c. which yet is otherwise, and their demōstra∣tions are right & agreable with the Theoreme. Onely the picture of the figures which the Greeke boke hath, may seeme to bring some doubt. For the squares are so described with certayn litle areas, that the number of them may be denominated by a square number: whereby it mought seeme that the lines A & B which describe the squares, ought to

[illustration]

be such that they may be expressed by some certaine nūber. As the line A to be 5. foote, and the line B 3. foote. As the two former fi¦gures here set declare. Which thyng yet Eu∣clide supposeth not, but only requireth that they be commensurable in length, as in the former example of the two squares, the whole area of one of which is 12, and the whole area of the other is 3. For although their sides cā not by them selues be expres∣sed by some certayne number, yet are they commensurable in length. Moreouer thys describing of the squares of the lines A and B, deuided by certaine litle areas, may cause this error, that a man should thinke that it is all o••e two numbers to be square numbers, and to be in proportion the one to the other as a square number is to a square number. For the number of the areas in the square of the line A is a square number, namely, 25. produced of the roote 5, which is the length of the line A•• Likewise the number of the areas of the square of the line B, is a square nūber, namely, 9, which is pro∣duced of the roote 3, which is the length of the line B. But we haue before declared that it is not all one, numbers to be called square numbers, a••d to be in proportion the one to the other, as a square nū∣ber is to a square number. Wherefore as touching those areas contayned in the greater square, which is of the line A, and which are in number 25, they do expresse that square number 25, which is produced of the number 5. multiplyed into him selfe, which number 5. is the greater extreme of the proportion betwene 5. and 3, which is the proportion of the lines A and B. And this proportion, namely, of 5. to 3. causeth that the lines A and B are commensurable in length (by the 6. of this booke). The same may be sayd also of the areas of the lesse square. Neither is it of necessitie that you vnderstand those area•• to be squares, as either fe••te square or pases square which make the whole square, although in deede they

may be such, so that the sides of those squares be so many foote long, as 5. foote or 3. foote. Howbeit thys is of necessitie that the numbers which expresse the number of the feete square or pases square, cō∣tayned in the squares, be either both of the

[illustration]

square numbers, as in these square figures of the lines A, B, or that both of them be like supe••ficiall numbers, as in the former squares which were 12. and 3: of which nū∣bers it is manifest by that which hath be∣fore bene said, that they are like superficiall numbers, and therefore haue that propor∣tion the one to the other, that a square nū∣ber hath to a square number. And therfore you may describe the squares of the lines A, B, without any distinction of such litle areas, so that the squares may be voyde and emptie, and contayned onely of foure right lines, as in the figure here put.

¶ An Assumpt.

Forasmuch as in the eight booke in the 26. proposition it was proued, that like playne numbers haue that proportion the one to the other, that a square number hath to a square number: and likewise in the 24. of the same booke it was proued, that if two numbers haue that proportion the one to the other,* 1.18 that a square number hath to a square number, those numbers are like plaine numbers. Hereby it is manifest, that vnlike plaine numbers, that is, whose sides are not proportionall, haue not that proportion the one to the other, that a square number hath to a square number. For if they haue, then should they be like plaine numbers, which is contrary to the supposition. Wherfore vnlike plaine numbers haue not that propor∣tion the one to the other, that a square number hath to a square nūber. And therfore squares which haue that proportion the one to the other, that vnlike plaine numbers haue, shall haue their sides incommensurable in length (by the last part of the former proposition) for that those squares haue not that proportion the one to the other that a square number hath to a square number.

Notes

• * 1.1

  The first part demonstrat••d.
• * 1.2

  An other de∣monstration of the first part.
• * 1.3

  An oth••r de∣mon••••ra••ion o•• the same first part after Montaureus.
• * 1.4

  Demonstra∣tion of the se∣conde part which is the co••••erse of the former.
• * 1.5

  An other de∣monstration of the second part.
• * 1.6

  This Assump•• followeth as a Corollary of the 25: but (so as it might al∣so be here in Methode, pla∣ced) you shall ••inde it after the 53. of this booke, abso∣lutely demon∣strated: for there it ser∣ueth to the 54. his demons••ra∣tion.
• * 1.7

  Demōstratiō of the third part.
• * 1.8

  Demōstratiō of the fourth part which is the cō∣uerse of the ••.
• * 1.9

  Conclusion of the whole pro∣position.
• * 1.10

  A Cor••llary.
• * 1.11

  Pro••e of the first part of the Corollary.
• * 1.12

  Profe of the second part.
• * 1.13

  Profe of the third p••rt.
• * 1.14

  Pro••e o•• the fourth part.
• * 1.15

  Certayne an∣notations ••ut of Montau∣••••us.
• * 1.16

  Rules to know whether two superficiall numbers be like or no.
• * 1.17

  Note.
• * 1.18

  This assumpt is the conuerse of the 26. of the eight.