number is made of vnities and therfore cannot a point be a common part of all lines and measure them as vnitie is a common part of all numbers and measureth them Vnitie taken certayne tymes maketh any number For there are not in any number infinite vnities but a point taken certayne tymes yea as often as ye list neuer maketh any line for that in euery line there are infinite pointes Wherfore lines figures and bodies in Geometry are oftentymes incommensurable and irrationall Now which are rationall and which irrationall which commensurable and which incommensurable how many and how sundry sortes and kindes there are of them what are their natures passions and properties doth Euclide most manifestly shew in this booke and demonstrate them most exactly This tenth booke hath euer hitherto of all men and is yet thought accompted to be the hardest booke to vnderstand of all the bookes of Euclide Which coÌ„mon receiued opinion hath caused many to shrinke and hath as it were deterred them from the handeling and treatie thereof There haue bene in deede in times past and are presently in these our dayes many which haue delt and haue taken great and good diligence in commenting amending and restoryng of the sixe first bookes of Euclide and there haue stayed themselues and gone no farther beyng deterred and made afrayde as it seemeth by the opinion of the hardnes of this booke to passe forth to the bookes following Truth it is that this booke hath in it somewhat an other strauÌ„ger maner of matter entreated of theÌ„ the other bokes before had and the demonstrations also thereof the order seeme likewise at the first somewhat straunge and vnaccustomed which thinges may seeme also to cause the obscuritie therof and to feare away many from the reading and diligent study of the same so much that many of the well learned haue much complayned of the darkenes and difficultie thereof and haue thought it a very hard thing and in maner impossible to attayne to the right and full vnderstanding of this booke without the ayde and helpe of some other knowledge and learnyng and chiefly without the knowledge of that more secret and subtill part of Arithmetike commonly called Algebra which vndoubtedly first well had and knowne would geue great light therunto yet certainly may this booke very well be entred into and fully vnderstand without any straunge helpe or succour onely by diligent obseruation of the order and course of Euclides writinges So that he which diligently hath perused and fully vnderstandeth the 9. bookes goyng before and marketh also earnestly the principles and definitions of this â—enth booke he shal well perceiue that Euclide is of himselfe a sufficient teacher and instructer and needeth not the helpe of any other and shall soone see that this tenth booke is not of such hardnes and obscuritie as it hath bene hetherto thought Yea I doubt not but that by the trauell and industry taken in this translation and by addicions and emendations gotten of others there shall appeare in it no hardnes at all but shall be as easie as the rest of his bookes are Definitions 1 Magnitudes commensurable are suchwhich one and the selfe same measure doth measure First he sheweth what magnitudes are commensurable one to an other To the better and more cleare vnderstanding of this definition note that that measure whereby any magnitude is measured is lesse then the magnitude which it measureth or at least equall vnto it For the greater can by no meanes measure the lesse Farther it behoueth that that measure if it be equall to that which is measured taken once make the magnitude which is measured if it be lesse then oftentimes taken and repeted it must precisely render and make the magnitude which it measureth Which thing in numbers is easely sene for that as was before said all numbers are commensurable one to an other And although Euclide in this definition comprehendeth purposedly onely magnitudes which are continuall quantities as are lines superficieces and bodies yet vndoubtedly the explication of this and such like places is aptly to be sought of numbers as well rationall as irrationall For that all quantities commensurable haue that proportion the one to the other which number hath to numbers In numbers therfore 9 and 12 are commensurable because there is one common measure which measureth them both namely the number 3. First it measureth 12 for it is lesse then 12. and being taken certaine times namely 4 times it maketh exactly 12 3 times 4 is 12 it also measureth 9 for it is lesse then 9. and also taken certaine times namely 3 times it maketh precisely 9 3 times 3 is 9. Likewise is it in magnitudes if one magnitude measure two other magnitudes those two magnitudes so measured are said to be commensurable As for example if the line C being doubled do make the line B and the same lyne C tripled do make the line A then are the two lines A and B lines or magnitudes commensurable For that one measure namely the line C measureth theÌ„ both First the line C is lesse theÌ„ the line A and alsolesse theÌ„ the line B also the line C taken or repeted certaine times namely 3 times maketh precisely the line A and the same line C taken also certain times namely two times maketh precisely the line B. So that the line C is a common measure to them both and doth measure them both And therfore are the two lines A and B lines commensurable And so imagine ye of magnitudes of other kyndes as of superficiall figures and also of bodies 2 Incommensurable magnitudes are such which no one common measure doth measure This diffinition neadeth no explanation at all it is easely vnderstanded by the diffinition going before of lines commensurable For contraries are made manifest by comparing of the one to the other as if the line C or any other line oftentimes iterated doo not render precisely the line A nor the line B theÌ„ are the lines A and B incommensurable Also if the line C or any other line certayne times repeted doo exactly render the line A and doo not measure the line B or if it measure the line B and measureth not also the line A the lines A and B are yet lines incoÌ„mensurable so of other magnitudes as of superficieces and bodyes 3 Right lines commensurable in power are such whose squares one and the selfe same superficies area or plat doth measure To the declaration of this diffinition we must first call to minde what is vnderstanded ment by the power of a line which as we haue before in the former bookes noted is nothing ells but the square thereof or any other plaine figure equall to the square therof And so great power habilitie â—s a line said to haue as is the quantitie of the square which it is able to describe or a figure superficial equal to the square
therof 〈◊〉 This iâ— also to be noted that of lines some are commensurable in length the one to the other and some are commensurable the one to the other in power Of lines commensurable in length the one to the other was geuen an example in the declaration of the first diffinitioÌ„ namely the lines A and B which were commensurable in length one and the selfe measure namely the line C measured the length of either of them Of the other kinde is geuen this diffinition here set for the opening of which take this example Let there be a certaine line namely the line BC and let the square of that line be the square BCDE Suppose also an other line namely the line FH let the square thereof be the square FHIK and let a certayne superficies namely the superficies A measure the square BCDE taken 16. times which is the number of the litle areas squares plats or superficieces coÌ„tained and described within the sayd squares ech of which is equall to the superficie A. Agayne let the same superficies A measure the square FHIK 9. times taken according to the number of the fieldâ—s or superficieces contayned and described in the same Ye see theÌ„ that one and the selfe same superficies namely the superficies A is a common measure to both these squares and by certayne repeticions thereof measureth them both Wherefore the two lines BC and FH which are the sides or lines producing these squares and whose powers these squares are are by this diffinition lines commensurable in power 4 Lines incommensurable are such whose squares no one plat or superficies doth measure This diffinition is easy to be vnderstanded by that which was sayd in the diffinition last set before this and neadeth no farther declaration And thereof take this example If neither the superficies A nor any other superficies doo measure the two squares B CDE and FHIK or if it measure the one â—â—rely BCDE and not the other FHIK or if it measure the square FHIK and not the square BCDE the two lines BC and FH are in power incommensurable and therfore also incommeÌ„surable in length For whatsoeuer lines are incommeÌ„surable in power the same are also incommensurable in length as shall afterward in the 9. proposition of this booke be proued And therfore such lines are here defined to be absolutely incommensurable These thinges thus standing it may easely appeare that if a line be assigned and layd before vs there may be innumerable other lines commensurable vnto it and other incommensurable vnto it of commensurable lines some are commensurable in length and power and some in power onely 5 And that right line so set forth is called a rationall line Thus may ye see how to the supposed line first set may be compared infinite lines some commensurable both in length power and some commensurable in power onely and incommensurable in length and some incommensurable both in power in length And this first line so set whereunto and to whose squares the other lines and their squares are compared is called a rationall line commonly of the most part of writers But some there are which mislike that it should be called a rationall line that not without iust cause In the Greeke copy it is called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 rete which signifieth a thing that may be spokeÌ„ expressed by word a thing certayne graunted and appoynted Wherefore Flussates a man which bestowed great trauell and diligence in restoring of these elementes of Euclide leauing this word rationall calleth this line supposed and first set a line certaine because the partes thereof into which it is deuided are certaine and known and may be expressed by voyce and also be coumpted by number other lines being to this line incommensurable whose parts are not distinctly known but are vncertayne nor can be expressed by name nor assignd by number which are of other men called irrationall he calleth vncertaine and surd lines Petrus Montaureus although he doth not very wel like of the name yet he altereth it not but vseth it in al his booke Likewise wil we doo here for that the word hath bene and is so vniuersally receiued And therefore will we vse the same name and call it a rationall line For it is not so great a matter what names we geue to thinges so that we fully vnderstand the thinges which the names signifie This rationall line thus here defined is the ground and foundation of all the propositions almost of this whole tenth booke And chiefly from the tenth proposition forwardes So that vnlesse ye first place this rationall line and haue a speciall and continuall regard vnto it before ye begin any demonstration ye shall not easely vnderstand it For it is as it were the touch and triall of all other lines by which it is known whether any of them be rationall or not And this may be called the first rationall line the line rationall of purpose or a rationall line set in the first place and so made distinct and seuered from other rationall lines of which shall be spoken afterwarde And this must ye well commit to memory 6 Lines which are commensurable to this line whether in length and power or in power onely are also called rationall This definition needeth no declaration at all but is easily perceiued if the first definition be remembred which â—heweth what magnitudes are commensurable and the third which â—heweth what lines are commensurable in power Here notâ— how aptly naturally Euclide in this place vseth these wordes commensurable either in length and power or in power onely Because that all lines which are commensurable in length are also commensurable in powerâ— when he speaketh of lines commensurable in leÌ„gth he euer addeth and in power but when he speaketh of lines commensurable in power he addeth this worde Onely and addeth not this worde in length as he in the other added this worde in power For not all lines which are commensurable in power are straight way commensurable also in length Of this definition take this example Let the first line rationall of purpose which is supposed and laide forth whose partes are certaine known and may be expressed named and nuÌ„bred be AB the quadrate wherof let be ABCD then suppose againe an other lyne namely the line EF which let be commensurable both in length and in power to the first rationall line that is as before was taught let one line measure the length of eche line and also lâ—t one superâ—icies measure the two squares of the said two lines as here in the example is supposed and also appeareth to the eie then is the line E F also a rationall line Moreouer if the lyne EF be commensurable in power onely to the rationall line AB first set and supposed so that no one line do measure the two lines AB and EF As in example yâ— see to be for
that the line EF is made equall to the line AD which is the diameter of the square ABCD of which square the line AB is a side it is certayne that the â—ide of a square is incoÌ„meÌ„surable in leÌ„gth to the diameter of the same square if there be yet founde any one superficies which measureth the two squares ABCD and EFGH as here doth the triangle ABD or the triangle ACD noted in the square ABCD or any of the foure triangles noted in the square EFGH as appeareth somwhat more manifestly in the second example in the declaration of the last definition going before the line EF is also a rational line Note that these lines which here are called rationall lines are not rational lines of purpose or by supposition as was the first rationall line but are rationall onely by reason of relation and comparison which they haue vnto it because they are commensurable vnto it either in length and power or in power onely Farther here is to be noted that these wordes length and power and power onely are ioyned onely with these wordeâ— commensurable or incommensurable and are neuer ioyned with these woordes rationall or irrationall So that no lines can be called rational in length or in power nor like wise can they be called irrationall in length or in power Wherin vndoubtedly Campanus was deceiued who vsing those wordes speaches indifferently caused brought in great obscuritie to the propositions and demonstrations of this boke which he shall easily see which marketh with diligence the demonstrations of Campanus in this booke 7 Lines which are incommensurable to the rationall line are called irrationall By lines incommensurable to the rationall line supposed in this place he vnderstandeth such as be incommensurable vnto it both in length and in power For there are no lines incommensurable in power onely for it cannot be that any lines should so be incommenâ—urable in power onely that they be not also incommensurable in length What so euer lines be incommeâ—surable in power the same be also incommensurable in length Neither can Euclide here in this place meane lines incommensurable in length onely for in the diffinition before he called them rationall lines nâ—ither may they be placed amongst irrationall lines Wherfore it remayneth that in this diffintion he speaketh onely of those lines which are incommensurable to the rationall line first geuen and supposed both in length and in power Which by all meanes are incommensurable to the rationall line therfore most aptly are they called irrationall lines This diffinition is easy to be vnderstanded by that which hath bene sayd before Yet for the more plainenes see this example Let the â—â—rst rationall line supposed be the line AB whose square or quadrate let be ABCD. And let there be geuen an other line EF which lâ—t be to the rationall line incommensurable in length and power so that let no one line measure the length of the two lines AB and EF and let the square of the line EF be EFGH Now if also there be no one superficies which measureth the two squares ABCD and EFGH as is supposed to be in this example theÌ„ is the line EF an irrationall line which word irrational As before did this word rational misliketh many learned in this knowledge of Geometry Flussates as he left the word rationall and in steade thereof vsed this word certaine so here he leaueth the word irrationall and vseth in place thereof this word vncertaine and euer nameth these lines vncertaine lines Petrus Montaureus also misliking the word irrationall would rather haue them to be called surd lines yet because this word irrationall hath euer by custome and long vse so generally bene receiuedâ— he vseth continually the same In Greeke such lines are called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 alogoi which signifieth nameles vnspeakeable vncertayne in determinate and with out proportion not that these irrationall lines haue no proportion at all either to the first rationall line or betwene them selues but are so named for that theyr proportions to the rationall line cannot be expressed in number That is vndoubtedly very vntrue which many write that their proportions are vnknowne both to vs and to nature Is it not thinke you a thing very absurd to say that there is any thing in nature and produced by nature to be hidde from nature and not to be knowne of nature it can not be sayd that their proportions are vtterly hidde and vnknowne to vs much lesse vnto nature although we cannot geue them their names and distinctly expresse them by numbers otherwise should Euclide haue taken all this trauell and wonderfull diligence bestowed in this bookeâ— in vaine and to no vseâ— in which he doth nothing ellâ— but teach the proprieties and passions of these irrationall linesâ— and sheweth the proportions which they haue the one to the other Here is also to be noted which thing also Tartalea hath before diligently notedâ— that Campanus and many other writers of Geometryâ— ouer much â—â—â—ed and were deceiued in that they wrote and taught that all these lines whose squares were not sâ—gnified and mought be expressed by a square number although they mighâ— by any other number as by 11. 12. 14. and such others not square numbers are irrationall lines Which is manifestly repugnant to the groundes and principles of Euclide who wil that all lines which are commensurable to the rationall line whether it be in length and power or in power onely should be rationall Vndoubtedly this hath bene one of the chiefest and greatest causes of the wonderfull confusion and darkenes of this booke which so hath tossed and tormoyled the wittes of all both writers and readers masters and scholers and so ouerwhelmed them that they could not with out infinite trauell and sweate attayne to the truth and perfect vnderstanding thereof 8 The square which is described of the rationall right line supposed is rationall Vntill this diffinition hath Euclide set forth the nature and proprietie of the first kinde of magnitude namely of lines how they are rationall or irrationall now he bâ—ginneth to â—hew how the second kinde of magnitudes namely superficies are one to the other rationall or irrationall This diffinition is very playne Suppose the line AB to be the rationall line hauing his parts and diuisions certaynely knowne the square of which line let be the square ABCD. Now because it is the square of the rationall line AB it is also called rationall and as the line AB is the first rationall line vnto which other lines compared are coumpted rationall or irrationall so is the quadrat or square thereof the â—irst rationall superficies vnto which all other squares or figures compared are coumpted and named rationall or irrationall 9 Such which are commensurable vnto it are rationall In this diâ—â—inition where it is sayd such as are commensurable to the square of the rationall line are not vnderstand onely other squares or
quadrates but all other kindes of rectiline figures playne plats superficieses What so euer so that if any such figure be coÌ„mensurable vnto that rationall squareâ— it is also rationall As suppose that the square of the rationall line which is also rationall be ABCD suppose 〈◊〉 so some other square as the square EFGH to be commensurable to the same theÌ„ is the square EFGH also rational So also if the rectiline figure KLMN which is a figure on the one side longer be commensurable vnto the sayd square as is supposed in this exampleâ— it is also a rational superficies and so of all other superficieses 10 Such which are incommensurable vnto it are irrationall Where it is sayd in this diffinition such which are incommensurable it is generally to be taken as was this word coÌ„mensurable in the diffinitioÌ„ before For al superficieses whether they be squares or figures on the one side longer or otherwise what maner of right lined figure so euer it be if they be incommensurable vnto the rationall square supposed theÌ„ are they irrationall As let thâ— square ABCD be the square of the supposed rationall line which square therefore is also rationall suppose also also an other square namely the square E suppose also any other figure as for example sake a figure of one side longer which let be F Now if the square E and the figure F be both incommensurable to the rationall square ABCD then is 〈◊〉 of these figures E F irrationall And so of other 11 And these lines whose poweres they are are irrationall If they be squares then are their sides irrationall If they be not squares but some other rectiline figures then shall the lines whose squares are equall to these rectiline figures be irrationall Suppose that the rationall square be ABCD. Suppose also an other square namely the square E which let be incoÌ„meÌ„surable to the rationall square therefore is it irrationall and let the side or line which produceth this square be the line FG then shall the line FG by this diffinition be an irrationall line because it is the side of an irrationall square Let also the figure H being a figure on the one side longer which may be any other rectiline figure rectangled or not rectangled triangle pentagone trapezite or what so euer ells be incommensurable to the rationall square ABCD then because the figure H is not a square it hath no side or roote to produce it yet may there be a square made equall vnto it for that all such figures may be reduced into triangles and so into squares by the 14. of the second Suppose that the square Q be equall to the irrationall figure H. The side of which figure Q let be the line KL then shall the line KL be also an irrational line because the power or square thereof is equal to the irrationall figure H and thus conceiue of others the like These irrationall lines and figures are the chiefest matter and subiect which is entreated of in all this tenth booke the knowledge of which is deepe and secret and pertaineth to the highest and most worthy part of Geometrie wherein standeth the pith and mary of the hole science the knowlede hereof bringeth light to all the bookes following with out which they are hard and cannot be at all vnderstoode And for the more plainenes ye shall note that of irrationall lines there be diâ—ers sortes and kindes But they whose names are set in a table here following and are in number 13. are the chiefe and in this teÌ„th boke sufficiently for Euclides principall purpose discoursed on A mediall line A binomiall line A first bimediall line A second bimediall line A greater line A line containing in power a rationall superficies and a mediall superficies A line containing in power two mediall superficieces A residuall line A first mediall residuall line A second mediall residuall line A lesse line A line making with a rationall superficies the whole superficies mediall A line making with a mediall superficies the whole superficies mediall Of all which kindes the diffinitions together with there declarations shal be set here after in their due places ¶ The 1. Theoreme The 1. Proposition Two vnequall magnitudes being geuen if from the greater be taken away more then the halfe and from the residue be againe taken away more then the halfe and so be done still continually there shall at length be left a certaine magnitude lesser then the lesse of the magnitudes first geuen SVppose that there be two vnequall magnitudes AB and C of which let AB be the greater Then I say that if from AB be taken away more then the halfe and from the residue be taken againe more then the halfe and so still continually there shall at the length be left a certaine magnitude lesser then the lesse magnitude geueÌ„ namely then C. For forasmuch as C is the lesse magnitude therefore C may be so multiplyed that at the length it will be greater then the magnitude AB by the 5. definition of the fift booke Let it be so multiplyed and let the multiplex of C greater then AB be DE. And deuide DE into the partes equall vnto C which let be DF FG and GE. And from the magnitudes AB take away more then the halfe which let be BH and againe from AH take away more then the halfe which let be HK And so do continually vntill the diuisions which are in the magnitude AB be equall in multitude vnto the diuisions which are in the magnitude DE. So that let the diuisions AK KH and HB be equall in multitude vnto the diuisions DF FG and GE. And forasmuch as the magnitude DE is greater then the magnitude AB and from DE is taken away lesse then the halfe that is EG which detraction or taking away is vnderstand to be done by the former diuision of the magnitude DE into the partes equall vnto C for as a magnitude is by multiplication increased so is it by diuision diminished and from AB is taken away more then the halfe that is BH therefore the residue GD is greater then the residue HA which thing is most true and most easie to conceaue if we remeÌ„ber this principle that the residue of a greater magnitude after the taking away of the halfe or lesse then the halfe is euer greater then the residue of a lesse magnitude after the taking away of more then the halfe And forasmuch as the magnitude GD is greater then the magnitude HA and from GD is taken away the halfe that is GF and from AH is taken away more then the halfe that is HK therefore the residue DF is greater then the residue AK by the foresayd principle But the magnitude DF is equall vnto the magnitude C by supposition Wherefore also the magnitude C is greater then the magnitude AK Wherefore the magnitude AK is lesse then the magnitude C. Wherefore of the magnitude
the line AB is rationall by the definition Wherfore by the definition also of rationall figures the parallelogramme CD shall be rationall Now resteth an other caâ—e of the thirde kinde of rationall lines commensurable in length the one to the other which are to the rationall line AB first set commensurable in power onely and yet are therfore rationall lines And let the lines CE and ED be coÌ„mensurable in length the one to the other Now then let the selfe same construction remaine that was in the former so that let the lines CE and ED be rationall commensurable in power onely vnto the line AB But let them be commensurable in length the one to the other Then I say that in this case also the parallelogramme CD is rationall First it may be proued as before that the parallelogramme CD is commensurable to the square DF. Wherfore by the 12. of this booke the parallelogramme CD shall be commensurable to the square of the line ABâ— But the square of the line AB is rationall Wherefore by the definition the parallelograÌ„me CD shall be also rationall This case is well to be noted For it serueth to the demonstration and vnderstanding of the 25. Proposition of this booke ¶ The 17. Theoreme The 20. Proposition If vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth thereof shall be a rationall line and commensurable in length vnto that line wherupon the rationall parallelogramme is applied SVppose that this rationall rectangle parallelogramme AC be applied vpon the line AB which let be rationall according to any one of the foresaid wayes whether it be the first rationall line set or any other line commensurable to the rationall line first set and that in length and in power or in power onely for one of these three wayes as was declared in the Assumpt put before the 19. Proposition of this booke is a line called rationall and making in breadth the line BC. Then I say that the line BC is rationall and commensurable in length vnto the line BA Desrcribe by the 46. of the first vpon the line BA a square AD. Wherfore by the 9. definitioÌ„ of the tenth the square AD is rationall But the parallelogramme AC also is rationall by supposition Wherefore by the conuersion of the definition of rationall figures or by the 12. of this booke the square DA is commensurable vnto the parallelogramme AC But as the square DA is to the parallelogramme AC so is the line DB to the line BC by the first of the sixt Wherfore by the 10. of the tenth the line DB is commensurable vnto the line BC. But the line DB is equall vnto the line BA Wherefore the line AB is coÌ„mensurable vnto the line BC. But the line AB is rationall Wherefore the line BC also is rationall and commensurable in length vnto the line BA If therefore vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth therof shall be a rationall line commensurable in length vnto that line whereupon the rationall parallelogramme is applied which was required to be demonstrated ¶ An Assumpt A line contayning in power an irrationall superficies is irrationall Suppose that the line AB coÌ„taine in power an irrationall superficies that is let the square described vpon the line AB be equall vnto an irrationall superficies Then I say that the line AB is irrationall For if the line AB be rationall theÌ„ shall the square of the line AB be also rationall For so was it put in the definitions But by supposition it is not Wherefore the line AB is irrationall A line therefore contayning in power an irrationall superficies is irrationall ¶ The 18. Theoreme The 21. Proposition A rectangle figure comprehended vnder two rationall right lines commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall is called a mediall line SVppose that this rectangle figure AC be comprehended vnder these rationall right lines AB and BC commensurable in power onely Then I say that the superficies AC is irrationall and the line which contayneth it in power is irrationall and is called a mediall line Describe by the 46. of the first vpon the line AB a square AD. Wherefore the square AD is rationall And forasmuch as the line AB is vnto the line BC incommensurable in length for they are supposed to be commensurable in power onely and the line AB is equall vnto the line BD therefore also the lineâ— BD is vnto the line BC incommensurable in length And 〈◊〉 â—hâ— lin◠〈…〉 is to the line â— C so 〈◊〉 the square AD to the parallelogramme AC by the first of the fiuâ— Wherefore by the 10. of the tenth the square DA is vnto the parallelogramme AC incommensurable But the square DA is rationall Wherefore the parallelogramme AC is irrationall Wherefore also the line that contayneth the superficies AC in power that is whose square is equall vnto the parallelogramme AC is by the Assumpt going before irrationall And it is called a mediall line for that the square which is made of it is equall to that which is contayned vnder the lines AB and BC and therefore it is by the second part of the 17. of the sixt a meane proportionall line betwene the lines AB and BC. A rectangle figure therefore comprehended vnder rationall right lines which are commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall and is called a mediall line At this Proposition doth Euclide first entreate of the generation and production of irrationall lines And here he searcheth out the first kinde of them which he calleth a mediall line And the definition therof is fully gathered and taken out of this 21. Proposition which is this A mediall line is an irrationall line whose square is equall to a rectangled figure contayned of two rationall lines commensurable in power onely It is called a mediall line as Theon rightly sayth for two causes first for that the power or square which it producethâ— is equall to a mediall superficies or parallelogramme For as that line which produceth a rationall square is called a rationall line and that line which produceth an irrationall square or a square equall to an irrationall figure generally is called an irrationall line so iâ— thaâ— line which produceth a mediall square or a square equall to a mediall superficies called by speciall name a mediall line Secondly it is called a mediall line because it is a meane proportionall betwene the two lines coÌ„mensurable in power onely which comprehend the mediall superficies ¶ A Corollary added by Flussates A rectangle parallelogramme contayned vnder a rationall line and an â—rrationall line is irrationall For if the line AB be rationall and
if the line CB be irrationall they shall be incommensurable But as the line BD which is equall to the line BA is to the line BC so is the square AD to the parallelograÌ„me AC Wherefore the parallelogramme AC shall be incommensurable to the square AD which is rationall for that the line AB wherupon it is described is supposed to be rationall Wherefore the parallelogramme AC which is contayned vnder the rationall line AB and the irrationall line BC is irrationall ¶ An Assumpt If there be two right lines as the first is to the second so is the square which is described vpon the first to the parallelograme which is contained vnder the two right lines Suppose that there be two right lines AB and BC. Then I say that as the line AB is to the line BC so is the square of the line AB â—â— that which is contained vnder the lines AB and BC. Describe by the 46. of the first vpon the line AB a square AD. And make perfect the parallelograme AC Now for that as the line AB is to the line BC for the line AB is equall to the line BD so is the square AD to the parallelograme CA by the first of the sixâ— and AD is the square which is made of the line AB and AC is that which is contained vnder the lines BD and BC that is vnder the lines AB BC therfore as the line AB is to the line BC so is the square described vppon the the line AB to the rectangle figure contained vnder the lines AB BC. And conuersedly as the parallelograme which is contained vnder the lines AB and BC is to the square of the line AB so is the line CB to the line BA ¶ The 19. Theoreme The 22. Proposition If vpon a rationall line be applied the square of a mediall line the other side that maketh the breadth thereof shal be rationall and incommensurable in length to the line wherupon the parallelograme is applied SVppose that A be a mediall line and let BC be a line rationall and vpon the line BC describe a rectangle parallelograme equall vnto the square of the line A and let the same be BD making in breadth the line CD Then I say that the line CD is rationall and incoÌ„mensurable in length vnto the line CB. For forasmuch as A is a mediall line it containeth in power by the 21. of the tenth a rectangle parallelograme comprehended vnder rationall right lines commensurable in power onely Suppose that is containe in power the parallelograme GF and by supposition it also containeth in power the parallelograme BD. Wherefore the parallelograme BD is equall vnto the parallelograme GF and it is also equiangle vnto it for that they are ech rectaÌ„gle But in parallelogrames equall and equiangle the sides which containe the equall angles are reciprocall by the 14. of the sixt Wherfore what proportioÌ„ the line BC hath to the line EG the same hath the line EF to the line CD Therefore by the 22. of the sixt as the square of the line BC is to the square of the line EG so is the square of the line EF to the square of the line CD But the square of the line BC is commensurable vnto the square of the line EG by supposition For either of them is rationall Wherefore by the the 10. of the tenth the square of the line EF is commensurable vnto the square of the linâ— CD But the square of the line EF is rationall Wherefore the square of the line CD is likewise rationall Wherefore the line CD is rational And forasmuch as the line EF is incoâ—mensurable in length vnto the line EG for they are supposed to be commensurable in power onely But as the line EF is to the line EG so by the assumpt going before is the square of the line EF to the parallelograme which is contained vnder the lines EF and EG Wherefore by the 10. of the tenth the square of the line EF is incommensurable vnto the parallelograme which is contained vnder the lines FE and EG But vnto the square of the line EF the square of the line CD is commensurable for it is proued that â—ither of them is a rationall linâ— And that which is contained vnder the lines DC and CB is commensurable vnto that which is contained vnder the lines FE and EG For they are both equall to the square of the line A. Wherefore by the 13. of the tenth the square of the line CD is incommensurable to that which is contained vnder the lines DC and CB. But as the square of the line CD is to that which is contained vnder the lines DC and CB so by the assumpt going before is the line DC to the line CB. Wherefore the line DC is incommensurable in length vnto the line CB. Wherefore the line CD is rationall and incommensurable in length vnto the line CB. If therefore vpon a rationall line be applied the square of a mediall line the other side that maketh the breadth thereof shal be rationall and incommensurable in length to the line whereupon the parallelogramme is applied which was required to be proued A square is sayd to be applied vpon a line when it or a parallelograme equall vnto it is applied vpon the sayd line If vpon a rationall line geuen we will apply a rectangle parallelograme equall to the square of a mediall line geuen and so of any line geuen we must by the 11. of the sixt finde out the third line proportionall with the rationall line and the mediall line geuen so yet that the rationall line be the first and the mediall line geuen which containeth in power the square to be applied be the second For then the supeâ—ficies contained vnder the first and the third shal be equall to the square of the midle line by the 17. of the sixt ¶ The 20. Theoreme The 23. Proposition A right line commensurable to a mediall line is also a mediall line SVppose that A be a mediall line And vnto the line A let the line B be commensurable either in length in power or in power only Then I say that B also is a mediall line Let there be put a rationall line CD And vpon the line CD apply a rectangle parallelograme CE equall vnto the square of the line A and making in breadth the line ED. Wherefore by the proposition going before the line ED is rationall and incommensurable in length vnto the line CD And againe vpon the line CD apply a recâ—angle parallelograme CF equall vnto the square of the line B and making in breadth the line DF. And forasmuch as the line A is commensurable vnto the line B therefore the square of the line A is commensurable to the square of the line B. But the parallelograme EC is equall to the square of the linâ— A and the parallelograme CF is equall to
the square of the line B wherefore the parallelograme EC is coÌ„mensurable vnto the parallelograme CF. But as the parallelograme EC is to the parallelograme CF so is the line ED to the line DF by the first of the sixt Wherefore by the 10. of the tenth the line ED is commensurable in length vnto the line DF. But the line ED is rationall and incoÌ„mensurable in length vnto the line DC wherefore the line DF is rationall and incommensurable in length vnto the line DC by the 13. of the tenth Wherefore the lines CD and DF are rationall commensurable in power onely But a rectangle figure comprehended vnder rationall right lines commensurable in power onely is by the â—1 of the tenth irrationall and the line that containeth it in power is irrationall and is called a mediall line Wherefore the line that containeth in power that which is comprehended vnder the lines CD and DF is a mediall line But the line B containeth in power the parallelograme which is comprehended vnder the lines CD and DF Wherefore the line B is a mediall line A right line therfore commensurable to a mediall line is also a mediall line which was required to be proued ¶ Corollary Hereby it is manifest that a superficies commensurable vnto a mediall superficies is also a mediall superficies For the lines which containâ— in power those superficieces are commensurable in power of which the one is a mediall line by the definitioÌ„ of a mediall line in the 21. of this tenth wherefore the other also is a mediall line by this 23. propositioÌ„ And as it was sayd of rationall lines so also is it to be sayd oâ— mediall lines namely that a liâ—e commensurable to a mediall line is also a mediall line a line I say which is commensurable vnto a mediall line whether it be commensurable in length and also in power or ells in power onely For vniuersally it is true that lines commensurable in length are also commensurable in power Now if vnto a mediall line there be a line commensurable in power if it be commensurable in length theÌ„ are those lines called mediall lines commensurable in length in power But if they be commensurable in power onely thâ—y are called mediall lines commensurable in power onely There are also other right lines incommensurable in length to the mediall line and commensurable in power onely to the same and these lines are also called mediall for that they are commensuâ—able in power to the mediall line And in aâ— muâ—h as they are mediall lines they are commensurable in power the one to the other But being compared the one to the other they may be commensurable either in length and theâ—efoâ—e in power or ells in power onely And then if they be commensurable in length they are called also mediall lines commensuâ—able in length and so consequently they are vnderstanded to be commensurable in power But iâ— they be commensurable in power onely yet notwithstanding they also are called mediall lines commensurable in power onely Flussates after this proposition teacheth how to come to the vnderstanding of mediall superficieces and lines by surd numbers after this maner Namely to expresse the mediall superficieces by the rootes of numbers which are not square numbers and the lines coÌ„taining in power such medial superficieces by the rootes of rootes of numbers not square Mediall lines also commensurable are expressed by the rootes of rootes of like sâ—perficial numbers but yet not square but such as haue that proportion that the squares of square numbers haue For the rootes of those numbers and the rootes of rootes are in proportion as numbers are namely if the squares be proportionall the sides also shal be proportionall by the 22. of the sixt But mediall lines incommensurable in power are the rootes of rootes of numbers which haue not that proportion that square numbers haue For their rootes are the powers of mediall lines which are incommensurable by the 9. of the tenth But mediall lines commensurable in power onely are the rootes of rootes of numbers which haue that proportion that simple square numbers haue and not which the squares of squares haue For the rootes which are the powers of the mediall lines are commeÌ„surable but the rootes of rootes which expresse the sayd mediall lines are incommensurable Wherefore there may be found out infinite mediall lines incommensurable in powâ—r by comparing infinite vnlike playne numbers the one to the other For vnlike playne numbers which haue not the proportion of square numbers doo make the rootes which expresse the superficieces of mediall lines incoÌ„mensurable by the 9. of the tenth And therefore the mediall lines containing in power those superficieces are incoÌ„mensurable in length For lines incommensurable in power are alwayes incommensurable in length by the corrollary of the 9. of the tenth ¶ The 21. Theoreme The 24. Proposition A rectangle parallelogramme comprehended vnder mediall lines coÌ„mensurable in length is a mediall rectangle parallelogramme SVppose that the rectangle parallelogramme AG be comprehended vnder these mediall right lines AB and BC which let be commensurable in length Then I say that AC is a mediall rectangle parallelogramme Describe by the 46. of the first vpon the line AB a square AD. Wherefore the square AD is a mediall superficies And â—orasmuch as the line AB is commensurablâ— in length vnto the line BC and the line AB is equall vnto the line BD therefore the line BD is commensurable in length vnto the line BC. But 〈◊〉 the line DB is to the line BC so is the square DA to the parallelogramme AC by the first of the sixt Wherfore by the 10. of the tenth the square DA is commensurable vnto the parallelogramme AC But the square DA is mediall for that it is described vpon a mediall line Wherefore AC also is a mediall parallelograÌ„me by the former Corollary A rectangleâ— c which was required to be proued ¶ The 22â— Theoreme The 25. Proposition A rectangle parallelogramme comprehended vnder mediall right lines commensurable in power onely is either rationall or mediall And now if the line HK be commensurable in length vnto the line HM that is vnto the line FG which is equall to the line HM then by the 19. of the tenth the parallelogramme NH is rationall But if it be incommensurable in length vnto the line FG then the lines HK and HM are rationall commensurable in power onely And so shall the parallelograÌ„me HN be mediall Wherefore the parallelogramme HN is either rationall or mediall But the parallelogramme HN is equall to the parallelogramme AG. Wherefore the parallelogramme AC is either rationall or mediall A rectangle parallelogramme therefore comprehended vnder mediall right lines commensurable in power onely is either rationall or mediall which was required to be demonstrated How to finde mediall lines commensurable in power onely contayning a rationall parallelogramme and also other mediall lines commensurable in power contayning a mediall
to the number CE so is the square of the line AB to the square of the line AF therefore by conuersion by the corollary of the 19. of the fifte as the number CD is to the number DE so is the square of the line AB to the square to the line FB But the number C D hath not to the number DE that proportion that a squarâ— nâ—mbeâ— hâ—th to a square number Wherefore neither also the square of the line AB hath to the square of the line BF that proportion that a square number hath to a square number Wherefore the line AB is by the 9 of the tenth incommensurable in length to the line BF And the line AB is in power more then the line AF by the square of the right line BF which is incommensurable in length vnto the line AB Wherfore the lines AB and AF are rationall commensurable in power onely And the line AB is in power more then the line AF by the square of the line FB which is commensurable in length vnto the line AB â— which was required to be done ¶ An Assumpt If there be two right lines hauing betwene them selues any proportion as the one right line is to the other so is the parallelograme contained vnder both the right lines to the square of the lesse of those two lines Suppose that these two right AB and BC be in some certaine proportion Then I say that as the line AB is to the line BC so is the parallelograme contained vnder AB and BC to the square of BC. Describe the square of the line BC and let the same be CD and make perfect the parallelograme AD now it is manifest that as the line AB is to the line BC so is the parallelograme AD to the parallelograme or square BE by the first of the sixt But the parallelograme AD is that which is bontained vnder the lines AB and BC for the line BC is equall to the line BD and the parallelograme BE is the square of the line BC. Wherefore as the line AB is to the line BC so is the parallelograme coutained vnder the lines AB and BC to the square of the line BC which was required to be proued ¶ The 8. Probleme The 31. Proposition To finde out two mediall lines commensurable in power onely comprehending a rationall superficies so that the greater shall be in power more then the lesse by the square of a line commensurable in length vnto the greater LEt there be taken by the 29. of the tenth two rationall lines commensurable in power onely A and B so that let the line A being the greater be in power more then the line B being the lesse by the square of a line commensurable in length vnto the line A â— And let the square of the line C be equall to the parallelograme contained vnder the lines A and B which is done by finding out the meane proportionall line namely the line C betwene the lines A and B by the 13. of the sixt Now the parallelograme contained vnder the lines A and B is mediall by the 21. of this booke Wherefore by the corollary of the 23. of the tenth the square also of the line C is mediall Wheâ—fore the line C also is mediall Vnto the square of the line B let the parallelograme contained vnder the lines C and D be equall by finding out a third line proportionall namely the line D to the two lines C and B by the 11. of the sixt But the square of the line B is rationall Wherfore the parallelograme contained vnder the line C and D is rationall And for that 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 assumpt going before But vnto the parallelograme contained vnder the lines A and B is equall the square of the line C and vnto the square of the line B is equal the parallelograme contained vnder the lines C and D as it hath now bene proued therefore as the line A is to the line B so is the square of the line C to the parallelograme contained vnder the lines C D. But as the square of the line C is to that which is contayned vnder the lines C and D so is the line C to the line D. Wherefore as the line A is to the line B so is the line C to the line D. But by supposition the line A is commensurable vnto the line B in power onely Wherefore by the 11. of the tenth the line C also is vnto the line D commensurable in power onely But the line C is mediall Wherefore by the 23â— of the tenth the line D also is mediall And for that as the line A is to the line B so is the line C to the line D but the line A is in power more then the line B by the square of a line commensurable in length vnto the line A by supposition Wherefore the line C also is in power more then the line D by the square of a line commensâ—rable in length vnto the line C. Wherefore there are found out two mediall lines C and D commensurable in power onely comprehending a rationall superfiâ—ies and the line C is in power more then the line D by the square of a line commensurable in length vnto the line C. And in like sort may be found out two mediall lines commensurable in power onely contayning a rationall superficies so that the greater shal be in power more theÌ„ the lesse by the square of a line incoÌ„mensurable in leÌ„gth to the greater namely when the line A is in power more theÌ„ the line B by the square of a line incoÌ„mensuraâ—le in length vnto the line A which to do is taught by the 30. of this booke The selfe same construction remaining that part of this proposition froÌ„ these wordes And for that as the line A is to the line B to these wordes But by supposition the line A is commensurable vnto the line B may more easely be demonstrated after this maner The lines C B D are in continuall proportion by the second part of the 17. of the sixt But the lines A C D are also in continuall proportion by the same Wherefore by the 11. of the fifth as the line A is to the line C so is the line B to the line D. Wherfore alternately as the line A is to the line B so is the line C to the line D. c. which was required to be doone ¶ An assumpt If there be three right lines hauing betwene them selues any proportion as the first is to the third so is the parallelograme contained vnder the first and the second to the parallelograme contained vnder the second and the third Suppose that these three lines AB B C and CD be in some certayne proportion Then I say that as the line AB is to
the line CD so is the parallelograme contayned vnder the lines AB and BC to the parallelograme contayned vnder the lines BC and CD From the point A raise vp vnto the line AB a perpendicular line AE and let AE be equall to the line BC and by the poynt E draw vnto the line AD a parallel line EK and by euery one of the poyntes B C and D draw vnto the line AE parallel lines BF CH and DK And for that as the line AB is to the line BC so is the parallelograme AF to the parallelograme BH by the first of the sixt and as the line BC is to the linâ— CD so is the parallelograme BH to the parallelograme CK Wherefore of equalitie as the line AB is to the line CD so is the parallelograme AF to the parallelograme CK But the parallelograme AF is that which is contayned vnder the lines AB and BC for the line AE is put equall to the line BC. And the parallelograme CK is that which is contained vnder the lines BC and CD for the line BC is equal to the line CH for that the line CH is equall to the line AE by the 34. of the first If therefore there be three right lines hauing betwene them selues any proportion as the first is to the third so is the parallelograme contained vnder the first and the second to the parallelogramme coÌ„tained vnder the second and the third which was required to be demonstrated ¶ The 9. Probleme The 32. Proposition To finde out two mediall lines commensurable in power onely comprehending a mediall superâ—icies so that the greater shall be in power more then the lesse by the square of a line commensurable in length vnto the greater LEt there be taken three rationall lines coÌ„mensurable in power onely A B C so that by the 29. of the tenth let the line A be in power more then the line C by the square of a line commensurable in length vnto the line A. And vnto the parallelogramme contayned vnder the lines A B let the square of the line D be equall But that which is contayned vnder the lines A and B is mediall Wherefore by the Corollary of the 23. of the tenth the square of the line D also is mediall Wherefore the line D also is mediall And vnto that which is contayned vnder the lines B and C let be equall that which is contayned vnder the lines D and E which is done by â—inding out a fourth line proportionall vnto the lines D B C which let be the line E And for that by the Assumpt going before as that which is contayned vnder the lines A and B is to that which is contayned vnder the lines B and C so is the line A to the line C. But vnto that which is contayned vnder the lines A B is equall the square of the line D and vnto that which is contayned vnder the lines B C is equall that which is contayned vnder the lines D and E. Wherefore as the line A is to the line C so iâ— the square of the line D to that which is contayned vnder the lines D and E. But as the square of the line D is to that which is contayned vnder the lines D and E so is the line D to the line E by the Assumpt put before the 22. of the tenth Wherefore as the line A is to the line C so is the line D to the line E. But the line A is vnto the line C commensurable in power onely Wherefore the line D is vnto the line E commensurable in power onely But D is a mediall line Wherefore by the 23. of the tenth E also is a mediall line And for that as the line A is to the line C so is the line D to the line E and the line A is in power more then the line C by the square of a line commensurable in length vnto the line A. Wherefore by the 14. of the tenth D is in power more then E by the square of a line commensurable in length vnto the line D. I say moreouer that that which is contayned vnder the lines D and E is mediall For forasmuch as that which is contayned vnder the lines B C is equall to that which is contayned vnder the lines D and E but that which is contayned vnder the lines B and C is mediall Wherefore that which is contayned vnder the lines D and E is also mediall Wherefore there are found out two mediall lines D and E coÌ„mensurable in power onely comprehending a mediall superficies so that the greater is in power more then the lesse by the square of a line commensurable in length to the greater which was required to be done And thus iâ— iâ— euâ—deâ—t how in like sorte may be founde out two mediall lines commâ—â—surable in power onely contayning a mediall superficâ—es so that the greater shall be in power more theâ— the lesse by the square of a line incommensurable in length vnto the greater When the line A is in power more then the line C by the square of a line incoÌ„mensurable in length vnto the line A as the thirteth teacheth vs. 1. ¶ An Assumpt Suppose that there be a rectangle triangle ABC hauing the angle BAC a right angle And â—y the 12. of the first from the poynt A to the right line BC a perpendicular line being drawen AD then I say first that the parallelogramme contayned vnder the lines Câ— and BD is equall to the square of the line BA Secondly I say that the parallelogramme contayâ—ed vnder the lines BC and CD is equall to the square of the line CA. Thirdly I say that the parallelogramme contayned vnder the lines BD and DC is equall to the square of the line AD. And fourthly I say that the parallelogramme contayned vnder the lines BC AD is equall to the parallelogramme coÌ„tayned vnder the lines BA AC As touching the second that the parellelogramme contained vnder the lines Bâ— and CD is equall to the square of the line AC is by the selfe same reason proued For the triangle ABC is like to the triangle ADC Wherefore as the line BC is to the line AC so is the line AC to the line DC Wherefore the parallelogramme contained vnder the lines BC and CD is equall to the square of the line AC As touching the third that the parallelogramme contained vnder the lines BD and DC is equall to the square of the line DA is thus proued For forasmuch as if in a rectangle triangle be drawn froÌ„ the right angle to the base a perpendicular line the perpendicular so drawen is the meane proportional betwene the segmeÌ„ts of the base by the corollary of the 8. of the sixt therâ—ore as the line BD is to the line DA so is thâ— line AD to the line DC Wherefore by the 1â— of the sixt the parallelogramme
contayned vnder the lines BD and DC is equall to the square of the line DA. As touching the fourth that the parallelogramme contained vnder the lines BC and AD is equall to the parallelogramme contained vnder the lines BA and AC is thus proued For forasmuch as as we haue already declared the triangle ABC is like and therefore equiangle to the triangle ABD therefore as the line BC is to the line AC so is the line BA to the line AD by the 4. of the sixt But if there be foure right lines proportionall that which is contained vnder the first and the last is equall to that which is contained vnder the two meanes by the 16. of the sixt Wherefore that which is contained vnder the lines BC and AD is equall to that which is contayned vnder the lines BA and AC I say moreouer that if there be made a parallelogramme complete contained vnder the lines BC and AD which let be EC and if likewise be made complete the parallelogramme contained vnder the lines BA and AC which let be AF it may by an other way be proued that the parallelogramme EC is equall to the parallelogramme AF. For forasmuch as either of them is double to the triangle ACB by the 41. of the first and thinges which are double to one and the selfe same thing are equall the one to the other Wherefore that which is contained vnder the lines BC and AD is equall to that which is contained vnder the lines BA and AC 2. ¶ An Assumpt If a right line be deuided into two vnequall partes as the greater part is to the lesse so is the parallelogramme contayned vnder the whole line and the greater part to the parallelogramme contayned vnder the whole line and the lesse part This Assumpt differeth litle from the first Proposition of the sixt booke 3. ¶ An Assumpt If there be two vnequall right lines and if the lesse be deuided into two equall partes the parallelogramme contained vnder the two vnequall lines is double to the parallelogramme contained vnder the greater line halfe of the lesse line Suppose that there be two vnequall right lines AB and BC of which leâ— AB be the greater and deuide the line BC into two equall partes in the point D. Thâ—n I say that the parallelogramme contained vnder the lines AB BC is double to the parallelogramme contained vnder the lines AB and BD. From the point B raise vp vpon the right line BC a perpendicular line BE and let BE be equall to the line BA And drawing from the point C and D the lines CG and DF parallels and equall to BE and then drawing the right line GFE the figure is complete Nâ—â— for that aâ—â—he line DB is to the line DC so is the parallelogramme BF to the parallelogramme DG by the 1. of the sixt therâ—ore by composition of proportion as the whole line BC is to the line DC so is the parallelogramme BG to the parallelogramme DG by the 18. of the fift But the line BC is double to the line DC Wherefore the parallelogramme BG is double to the parallelogramme DG But the parallâ—logramme BG is contained vnder the lines AB and BC for the line AB is equall to the line BE and the parallelogramme DG is contayned vnder the lines AB and BD for the line BD is equall to the line DC and the line AB to the line DF which was required to be demonstrated ¶ The 10. Probleme The 33. Proposition To â—inde out two right lines incommensurable in power whose squares added together make a rationall superficies and the parallelogramme contained vnder them make a mediall superficies TAke by the 30. of the tenth two rationall right lines commensurable in power onely namely AB and BC so that let the line AB being the greater be in power more then the line BC being the lesse by the square of a line incommensurable in length vnto the line AB And by the 10. of the first deuide the line BC into two equall partes in the point D. And vpon the line AB apply a parallelogramme equall to the square either of the line BD or of the line DC and wanting in figure by a square by the 28. of the sixth and let that parallelogramme be that which is contained vnder the lines AE and EB And vpon the line AB describe a semicircle AFB And by the 11. of the first from the point E raise vp vnto the line AB a perpendiculer line EF cutting the circumference in the point F. And draw lines from A to F and from F to B. And forasmuch as there are two vnequall right lines AB and BC and the line AB is in power more then the line BC by the square of a line incommensurable in lâ—ngth vnto AB and vpon the line AB is applied a parallelograme equall to the fourth part oâ— the square of the line BC that is to the square of the halfe of the line BC and wanting in â—igure by a square and the said parallelogramme is that which is contained vnder the lines AE and EB wherfore by the 2. part of the 18. of the tenth the line AE is incommeÌ„surable in length vnto the line EB But as the line AE is to the line EB so is the parallelogramme contained vnder the lines BA and AE to the parallelogramme contayned vnder the lines AB and BE by the second assumpt before put And that which is contained vnder the line BA and AE is equall to the square of the line AF by the second part of the first assumpt before put And that which is contained vnder the lines AB and BE is by the first part of the same assumpt equall to the square of the line BF Wherfore the square of the line AF is incommâ—nsurable to the square of the line BF Wherfore the lines AF and BF are incommensurable in power And forasmuch as AB is a rationall line by supposition therfore by the 7 definition of the tenth the square of the line AB is rationall Wherefore also the squares of the lines AF and FB added together make a rationall superficies For by the 47. of the first they are equal to the square of the line AB Again forasmuch as by the third part of the first assumpt going before that which is contained vnder the lines AE and EB is equall to the square of the line EF. But by supposition that which is contained vnder the lines AE and EB is equall to the square of the line BD. Wherfore the line FE is equall to the line BD. Wherfore the linâ— BG is double to the line â— E. Wherfore by the third assumpt going before that which is contained vnder the lines AB and BC is double to that which is contained vnder the lines AB and EF. But that which is contained vnder the lines AB and BC is by supposition mediall
Wherfore by the corollary of the 23. of the tenth that which is contained vnder the lines AB and EF is also mediall but that whiche is contayned vnder the lines AB and EF is by the last parte of the first assumpt goyng before equall to that which is contained vnder the lines AF and FB Wherefore that which is contained vnder the lines AF FB is a mediall superficies And it is proued that that which is composed of the squares of the lines AF and FB added together is rationall Wherfore there are found out two right lines AF and FB incommensurable in power whose squares added together make a rationall superficies and the parallelogramme contained vnder them is a mediall superficiâ—sâ— which was required to be done ¶ The 11. Probleme The 34. Proposition To finde out two right lines incâ—â—mensurable in power whose squares added together make a mediall superficies and the parallelogramme contayned vnder them make a rationall superficies TAke by the 31. of the tenth two mediall lines AB and BC commensurable in power onely comprehending a rationall superficies so that let the line AB be in power more then the line BC by the square of a line incommensurable in length vnto the line AB And describe vpon the line AB a semicircle ADB And by the 10. of the first deuide the line BC vnto two equall partes in the point E. And by the 28. of the sixt vpon the line AB apply a parallelogramme equall to the square of the line BE and wantyng in figure by a square and let that parallelogramme be that which is contayned vnder the lines AF and FB Wherfore the line AF is incommensurable in length vnto the line FB by the 2. part of the 18. of the tenth And from the point F vnto the right line AB raise vp by the 11. of the first a perpendiculer line FD and draw lines from A to D and from D to B. And forasmuch as the line AF is incommensurable vnto the line FB but by the second assumpt going before the 33. of the tenth as the line AF is to the line FB so is the parallelogramme contayned vnder the lines BA and AF to the parallelogramme contained vnder the lines BA and BF wherfore by the tenth of the tenth that which is contained vnder the lines BA and AF is incommensurable to that which is contayned vnder the lines AB and BF but that which is contained vnder the lines BA and AF is equall to the square of the line AD and that which is contained vnder the lines AB and BF is also equall to the square of the line DB by the second part of the first assumpt going before the 33. of the teÌ„th wherfore the square of the line AD is incoÌ„mensurable to the square of the line DB. Wherefore the lines AD and DB are incommensurable in power And forasmuch as the square of the line AB is mediall therefore also the superficies made of the squares of the lines AD and DB added together is mediall For the squares of the lines AD and DB are by the 47. of the first equall to the square of the line AB And forasmuch as the line BC is double to the line FD as it was proued in the proposition going before therefore the parallelogramme contained vnder the lines AB and and BC is double to the parallelogramme contained vnder the lines AB and FD by the third assumpt going before the 33. proposition wherefore it is also commensurable vnto it by the sixt of the tenth But that which is contained vnder the lines AB and BC is supposed to be rationall Wherfore that which is contained vnder the lines AB and FD is also rationall But that which is contained vnder the lines AB and FD is equall to that which is contained vnder the lines AD and DB by the last part of the first assumpt going before the 33. of the tenth Wherfore that which is contayned vnder the lines AD and DB is also rationall Wherefore there are â—ound out two right lines AD and DB incommensurable in power whose squares added together make a mediall superficies and the parallelogramme coÌ„tayned vnder them make a rationall superficies which was required to be done ¶ The 12. Probleme The 35. Proposition To finde out two right lines incommensurable in power whose squares added together make a mediall superficies and the parallelogramme contained vnder them make also a mediall superficies which parallelogramme moreouer shall be incommensurable to the superficies made of the squares of those lines added together TAke by the 32. of the tenth two mediall lines AB and BC commensurable in power onely comprehending a mediall superficies so that let the line AB be in power more then the line BC by the square of a line incommensurable in length vnto the line AB And vpon the line AB describe a semicircle ADB and let the rest of the construction be as it was in the two former propositions And forasmuch as by the 2 part of the 18. of the tenth the line AF is incommensurable in length vnto the line FB therfore the line AD is incommensurable in power vnto the line DB by that which was demonstrated in the propositioÌ„ going before And forasmuch as the square of the line AB is mediall therefore that also which is composed of the squares of the lines AD and DB which squares are equall to the square of the line AB by the 47. of the first is mediall And forasmuch as that which is contained vnder the lines AF and FB is equall to either of the squares of the lines EB and FD for by supposition the parallelogramme contained vnder the lines AF and FB is equall to the square of the line EB and the same parallelogramme is equall to the square of the line DF by the third part of the first assumpt going before the 33. of the teÌ„th Wherfore the line BE is equall to the line DF. Wherfore the line BC is double to the line FD. Wherefore that which is contained vnder the lines AB and BC is double to that which is contained vnder the lines AB and FD. Wherfore they are commensurable by the sixt of this boke but that which is contained vnder the lines AB and BC is mediall by supposition Wherfore also that which is contained vnder the lines AB and FD is mediall by the corollary of the 23 of the tenth but that which is contained vnder the lines AB and FD is by the fourth part of the first assumpt going before the 33. of the tenth equall to that which is contained vnder the lines AD and DB wherfore that which is contained vnder the lines AD and DB is also mediall And forasmuch as the line AB is incommensurable in length vnto the line BC. But the line BC is commensurable in length vnto the line BE. Wherfore by the 13â— of
the tenth the line AB is incommensurable in length vnto the line BE. Wherefore the square of the line AB is incommensurable to that which is contained vnder the lines AB and BE by the first of the sixt and 10. of this booke But vnto the square of the line AB are equall the squares of the lines AD and DB added together by the 47. of the first and vnto that which is contayned vnder the lines AB and BE is equall that which is contained vnder the lines AB and FD that is which is contained vnder AD and DB. For the parallelogramme contained vnder the lines AB and FD is equall to the parallelogramme contained vnder the lines AD and DB by the last part of the first assumpt going before the 33. of this tenth booke Wherfore that which is composed of the squares of the lines AD and DB is incommensurable to that which is contained vnder the lines AB and DB. Wherefore there are found out two right lines AD and DB incommensurable in power whose squares added together make a mediall superficies and the parallelogramme contayned vnder them make also a mediall superficies which parallelogramme moreouer is incommensurable to the superficies composed of the squares of those lines added together which was required to be done The beginning of the Senaries by Composition ¶ The 2â— Theoreme The 36. Proposition If two rationall lines commensurable in power onely be added together the whole line is irrationall and is called a binomium or a binomiall line 〈…〉 B and BC is incommensurable to the square of the line BC. But vnto the parallelograme contained vnder the lines AB and BC is commensurable the parallelograme contained vnder AB and BC twise by the 6. of the tenth wherefore that which is contained vnder AB and BC twise is incommensurable to the square of the line BC by the 13 of the tenth But vnto the square of the line BC is commensurable that which is composed of the squares of the lines AB and BC by the 15. of the tenth for by supposition the lines AB and BC are commensurable in power onely Wherefore by the 13. of the tenth that which is composed of the squares of the lines AB and BC added together is incommensurable to that which is contained vnder the lines AB and BC twise Wherefore by the 16. of the tenth that which is contained vnder AB and BC twise together with the squares of the lines AB and BC which by the 4. of the second is equall to the square of the whole line AC is incommensurable to that which is composed of the squares of AB and BC added together But that which is composed of the squares of AB and BC added together is rationall for it is commensurable to either of the squares of the lines AB and BG of which either of them is rationall by supposition wherfore the square of the line AC is by the 10. definition of the tenth irrationall Wherefore the line AC also is irrationall and is called a binomiall line This proposition sheweth the generation and production of the second kinde of irrationall lines which is called a binomium or a binomial line The definition whereof is fully gathered out of this proposition and that thus A binomium or a binomiall line is an irrationall line composed of two rationall lines commensurable the one to the other in power onely And it is called a binomium that is hauing two names because it is made of two such lines as of his partes which are onely commensurable in power and not in length and therefore ech part or line or at the least the one of them as touching length is vncertaine and vnknowne Wherefore being ioyned together their quantitie cannot be expressed by any one number or name but ech part remayneth to be seuerally named in such sort as it may And of these binomiall lines there are sixe seuerall kindes the first binomiall the second the third the fourth the fifth and the sixt of what nature and condition ech of these is shal be knowne by their definitious which are afterward set in their due place ¶ The 25. Theoreme The 37. Proposition If two mediall lines commensurable in power onely containing a rationall superficies be added together the whole line is irrationall and is called a first bimediall line LEt these two mediall lines AB and BC being commensurable in power onely and contayning a rationall superficies the 27. of the tenth teacheth to finde out two such lines be composed Then I say that the whole line AC is irrationall For as 〈◊〉 sayd in the proposition next going before that which is composed of the squares of the 〈◊〉 AB and BC is incommeÌ„surable to that which is contained vnder the lines AB and BC twisâ— wherefore by the 16. of the tenth that which is composed of the squares of the lines AB and BC together with that which is contained vnder the lines AB and BC twise that is the square of the line AC is incommensurable to that which is contayned vnder the lines AB and BC twise But that which is contayned vnder the lines AB and BC twise iâ— commensurable to that which is contayned vnder the lines AB and BC once by the 6. of the tenth wherefore the square of the whole line AC is by the 13â— of the tenth incâ—mmensurable â—o that which is contained vnder the lines AB and BC once But by supposition the lines AB and BC comprehend a rationall superâ—icies Wherefore the square of the whole line AC is irrationall wherefore also the line AC is irrationall And it is called a first bimediall line The third irrational line which is called a first bimediall line is shâ—wed by this proposition and the definition thereof is by it made manifest which is this A first bimediall line is an irrationall line which is composed of two mediall lines commensurable in power onely contayning a rationall parallelograme It is called a first bimediall line by cause the two mediall lines or partes whereof it is composed contayne a rationall superficies which is preferred before an irrationall ¶ The 26. Theoreme The 38. Proposition If two mediall lines commensurable in power onely contayning a mediall superficies be added together the whole line is irrationall and is called a second bimediall line LEt these two medial lines AB and BC being commensurable in power onely and contayning a mediall superficies the 28. of the tenth teacheth to findâ— out two such lines be added together Then I say that the whole line AC is irrational Take a rationall line DE. And by the 44. of the first vpon the line DE apply the parallelograme DF equal to the square of the line AC whos 's other side let be the line DG And forasmuch as the square of the line AC is by the 4. of the second equall to that which is composed of
the squares of the lines AB and BC together with that which is contained vnder the lines AB and BC twise but the square of the line AC is equall to the parallelograme DF. Wherefore the parallelograme DF is equall to that which is composed of the squares of the lines AB and BC together with that which is contayned vnder the lines AB and BC twise Now then agayne by the 44 of the first vpon the line DE apply the parallelograme EH equall to the squares of the lines AB and BC. Wherefore the parallelograme remayning namely HF is equall to that which is contained vnder the lines AB and BC twise And forasmuch as either of these lines AB and BC is mediall therefore the squares of the lines AB and BC are also mediall And that which is contained vnder the lines AB and BC twise is by the corollary of the â—4 of the tenth mediall For by the 6. of this booke it is commeÌ„surable â—â— that 〈◊〉 is contained vnder the lines AB and BC once which is by supposition medial 〈…〉 squares of the lines AB and BC is equall the parallelograme EH and vnto that 〈◊〉 contayned vnder the lines AB and BC twise is equall the parallelograme HE 〈…〉 either of these parallelogrames HE and AF is mediall and they are applyed vpon the rationall line ED. Wherefore by the 22. of the tenth either of these lines DH and HG is a rationall line and incâ—mmensurable in length vnto the line DE. And forasmuch as by supposition the line AB is incommensurable in length vnto the line BC. But as the line AB is to the line BC â— so is the square of the line AB to the parallelograme which is contayned vnder the lines AB and BC by the first of the sixt Wherefore by the 10 of this bookâ— the square of the line AB is incommensurable to the parallelograme contayned vnder the lines AB and BC. But to the square of the line AB is commensâ—able that which is composed of the squares of the lines AB and BC by the 15. of the tenth For the squares of the lines AB and BC are commensurable when as the lines AB and BC are put to be commensurable in power onely And to that which is contayned vnder the lines AB and BC is commensurable that which is contained vnder the lines AB and BC twise by the 6 of the tenth wherefore that which is composed of the squares of the lines AB and BC is incommensurable to that which is contayned vnder the lines AB and BC twise But to the squares of the lines AB and BC is equall the parallelograme EH And to that which is contayned vnder the lines AB and BC twise is equall the parallelograme FH Wherfore the parallelograme FH is incommensurable to the parallelograme HE. Wherfore the line DH is incommensurable in length to the line HG by the 1 of the sixt and 10 of this booke And it is proued that they are rationall lines Wherefore the lines DH HG are rationall commensurable in power onely Wherefore by the 36. of the tenth the whole line DG is irrationallâ— And the line DE is rationall But a rectangle superâ—icies comprehended vnder a rationall line and an irrationall line is by the corollary added after the 21 of the tenth irrationall Wherefore the superâ—icies DF is irrationall And the line also which containeth it in power is irrational But the line AC containeth in power the superficies DF. Wherefore the line AC is irrationall And it is called a second bimediall line This Proposition sheweth the generation of the fourth irrationall line called a second bimediall line The definition wherof is euident by this Proposition which is thus A second bimediall line is an irrationall line which is made of two mediall lines commensurable in power onely ioyned together which comprehend a mediall superficies And it is called a second bimediall because the two mediall lines of which it is composed coÌ„taine a mediall superficies and not a rationall Now a mediall is by nature in knowledge after a rationall ¶ The 27. Theoreme The 39. Proposition If two right lines incoÌ„mensurable in power be added together hauing that which is composed of the squares of them rationall and the parallelograÌ„me contayned vnder them mediall the whole right line is irrationall and is called a greater line LEt tâ—ese two right lines AB and BC being incommensurable in power onely and making that which is required in the Proposition The 33. of the tenth teacheth to finde out two such lines be added together Then I say that the whole line AC is irrationall For forasmuch as by supposition the parallelogramme contained vnder the lines AB and BC is mediall therefore the parallelogramme contained twise vnder the lines AB and BC is mediall For that which is contained vnder AB and BC twise is commensurable to that which is coÌ„tained vnder AB and BC once by the 6. of the tenth Wherefore by the Corollary of the 23. of the tenth that which is contained vnder AB BC twise is mediall But by supposition that which iâ— composed of the squares of the lines AB and BC is rationall Wherefore that which is contained vnder the lines AB and BC twise is incommensurable to that which is composed of the squares of the lines AB and BC. Wherfore by the 16. of the tenth that which is composed of the squares of the lines AB and BC together with that which is contayned vnder the lines AB BC twise which is by the 4. of the second equall to the square of the line AC is incommensurable to that which is composed of the squares of the lines AB and BC. But that which is composed of the squares of the lines AB and BC is rationall Wherefore the square of the whole line AC is irrationall Wherefore the line AC also is irrationall And is called a greater line And it is called a greater line for that that which is composed of the squares of the lines AB BC which are rationall is greater then that which is contayned vnder the lines AB and BC twise which are mediall Now it is meete that the name should be geuen according to the propertie of the rationall An Assumpt This Proposition teacheth the production of the fift irrationall line which is called a greater line which is by the sense of this Proposition thus defined A greater line is an irrationall line which is composed of two right lines which are incommensurable in power the squares of which added together make a rationall superficies and the parallelogramme which they containe is mediall It is therefore called a greater line as Theon sayth because the squares of the two lines of which it is composed added together being rationall are greater then the mediall superficies contained vnder them twise And it is conuenient that the denomination be taken of the proprietie of the
rationall part rather then of the mediall part ¶ The 28. Theoreme The 40. Proposition If two right lines incoÌ„mensurable in power be added together hauing that which is made of the squares of them added together mediall and the parallelogramme contayned vnder them rationall the whole right line is irrationall and is called a line contayning in power a rationall and a mediall superficies In this Proposition is taught the generation of the sixt irrationall line which is called a line whose power is rationall and mediall The definition of which is gathered of thys Proposition after this maner A line whose power is rationall and mediall is an irrationall line which is made of two right lines incommensurable in power added together whose squares added together make a mediall superficies but that supersicies which they containe is rationall The reason of the name is before set forth in the Proposition ¶ The 29. Theoreme The 41. Proposition If two right lines incommensurable in power be added together hauyng that which is composed of the squares of them added together mediall and the parallelogramme contayned vnder them mediall and also incommensurable to that which is composed of the squares of them added togetherâ— the whole right line is irrationall and is called a line contayning in power two medials In this proposition is taught the nature of the 7. kinde of irrationall lines which is called a line whose power is two medials The definition whereof is taken of this proposition after this maner A line whose power is two medials is an irrationall line which is composed of two right lines incommensurable in power the squares of which added together make a mediall superficies and that which is contained vnder them is also mediall and moreouer it is incommensurable to that which is composed of the two squares added together The reason why this line is called a line whose power is two medials was before in the ende of the demonstration declared And that the said irrationall lines are deuided one way onely that is in one point onely into the right lines of which they are composed and which make euery one of the kindes of those irrationall lines shall straight way be demonstrated but first will we demonstrate two assumptes here following ¶ An Assumpt Take a right line and let the same be AB and deuide it into two vnequall partes in the point C and againe deuide the same line AB into two other vnequal partes in an other point namely in D and let the line AC by supposition be greater then the line DB. Then I say that the squares of the lines AC and BC added together are greater then the squares of the lines AD and DB added together Deuide the line AB by the 10. of the first into two equall partes in the point E. And forasmuch as the line AC is greater then the line DB take away the line DC which is common to them both wherfore the residue AD is greater then the residue CB but the line AE is equall to the line EB Wherfore the line DE is lesse then the line EC Wherfore the pointes C and D are not equally distant from the point E which is the point of the section into two equall partes And forasmuch as by the 5. of the second that which is contayned vnder the lines AC and CB together with the square of the line EC is equall to the square of the line EB And by the same reason that which is contayned vnder the lynes AD and DB together with the square of the line DE is also equall to the self same square of the line EB wherfore that which is contained vnder the lines AC and CB together with the square of the line EC is equall to that which is contained vnder the lines AD and DB together with the square of the line DE of which the square of the line DE is lesse then the square of the line EC for it was proued that the line DE is lesse then the line EC Wherfore the parallelogramme remayning contayned vnder the lines AC and CB is lesse theÌ„ the parallelogramme remayning contayned vnder the lines AD and DB. Wherfore also that which is contayned vnder the lines AC and CB twise is lesse then that which is contayned vnder the lines AD and DB twise But by the fourth of the second the square of the whole line AB is equall to that which is composed of the squares of the lines AC and CB together with that which is contained vnder the lynes AC and CB twise and by the same reason the square of the whole line AB is equall to that which is composed of the squares of the lines AD and DB together with that which is contayned vnder the lynes AD and DB twise wherfore that which is composed of the squares of the lynes AC and CB together with that which is contayned vnder the lynes AC and CB twise is equall to that which is composed of the squares of the lynes AD and DB together with that which is contayned vnder the lynes AD and DB twise But it is already proued that that which is contayned vnder the lynes AC and CB twise is lesse then that which is contayned vnder the lines AD DB twise Wherfore the residue namely that which is composed of the squares of the lines AC and CB is greater then the residue namely then that which is composed of the squares of the lines AD and DB which was required to be demonstrated ¶ An Assumpt A rationall superficies exceedeth a rationall superficies by a rationall superficies Let AD be a rationall superficies and let it exceede AF being also a rationall superficies by the superficies ED. Then I say that the superficies ED is also rationall For the parallelogramme AD is commensurable to the parallelogramme AF for that either of them is rationall Wherefore by the second part of the 15. of the tenth the parallelogramme AF is commensurable to the parallelogramme ED. But the the parallelogramme AF is rationall Wherfore also the parallelogramme ED is rationall ¶ The 30. Theoreme The 42. Proposition A binomiall line is in one point onely deuided into his names SVppose that AB be a binomiall line and in the point G let it be deuided into his names that is into the lines wherof the whole line AB is composed Wherefore these lines AC and CB are rationall commensurable in power onely Now I say that the line AB cannot in any other point besides C be deuided into two rationall lines commensurable in power onely For if it be possible let it be deuided in the point D so that let the lines AD and DE bâ— rationall commensurable in power onely First this manifest that neithâ—â— of these poinâ—es C and D deuideth the right line AB into two equall partes Otherwise the lines AC and CB should be rationall commensurable in
length and so likewise should the lines AD and DB be For euery line measureth it selfe and any other line equall to it selfe Moreouer the line DB is either one and the same with the line ACâ— that is is equall to the line AC oâ— els it is greater-then the line AC either els it is lesse then it If DB be equall to the line AC then putting the line DB vpon the line AC eche endes of the one shall agree with eche endes of the other Wherfore putting the point B vpon the point A the point D also shall fall vpon the point C and the line AD which is the rest of the line AC shall also be equall to the line CB which is the rest of the line DB. Wherfore the line AB is deuided into his names in the point C. And so also shal the line AB being deuided in the point D be deuided in the self â—ame point that the self same line AB was before deuided in the point C which is coÌ„trary to the suppositioÌ„ For by suppositioÌ„ it was deuided in sundry pointes namely in C D. But if the line DB be greaterâ— the the line AC let the line AB be deâ—ided into two equal partes in the point E. Wherfore the points C D shal not equally be distant froÌ„ the point E Now by the first assupt going before this propositioÌ„ that which is coÌ„posed of the squares of the lines AD DB is greater theÌ„ that which is composed of the squares of the lines AC CBâ— But that which is composed of the squares of the lines AD DB together with that which is coÌ„tained vnder the lines AD DB twise is equall to that which is composed of the squares of the lines AC CB together with that which is contained vnder the lines AC and CB twise for either of them is equall to the square of the whole line AB by the 4. of the second wherefore how much that which is coÌ„posed of the squares of the lines AD and DB added together is greater then that which is composed of the squares of the lines AC and CB added together so much is that which is contained vnder the lines AC and CB twise greater then that which is contained vnder the lines AD and DB twise But that which is composed of the squares of the lines AD and DB excedeth that which is composed of the squares of the lines AC and CB by a rationall superficies by the 2. assumpt going before this propositionâ— For that which is composed of the squares of the lines AD and DB is rationall and so also is that which is composed of the squares of the lines AC and CB for the lines AD and DB are put to be rationall commensurable in power onely and so likewise are the lines AC and CB. Wherfore also that which is contained vnder the lines AC and CB twise exceedeth that which is contained vnder the lines AD DB twise by a rational superficies wheÌ„ yet notwithstaÌ„ding they are both medial superficieces by the 21. of the tenth which by the 26. of the same is impossible And if the line DB be lesse then the line AC we may by the like demonstration proue the selfe same impossibilitie Wherfore a binomiall line is in one point onely deuided into his names Which was required to be demonstrated 〈…〉 ollary added by Flussates Two ration 〈…〉 surable in power onely being added together cannot be equall to two other rationall line 〈…〉 in power onely added together For either of them should make a binomia 〈…〉 so should a binomiall line be deuided into his names in moe poyntes then onâ—â—â—ch by this proposition is proued to be impossible The like shall follow in the fiue 〈◊〉 irrationall lines as touching their two names ¶ The 31. Probleme The 43. Proposition A first bimediall line is in one poynt onely deuided into his names SVppose that AB be a first bimediall line and let it be deuided into his partes in the point C so that let the lines AC and CB be mediall coÌ„mensurable in power onely and containing a rationall superficies Then I say that the line AB can not be deuided into his names in any other poynt then in C. For if it be possible let it be deuided into his names in the poynt D so that let AD DB be mediall lines commensurable in power onely comprehending a rationall superficies Now forasmuch as how much that which is contayned vnder the lines AD and DB twise diâ—ferreth from that which is contayned vnder the lines AC and CB twise so much differreth that which is composed of the squares of the lines AD and DB from that which is composed of the squares of the lines AC and CB but that which is contayned vnder the lines AD and DB twise differreth from that which is contayned vnder the lines AC and CB twise by a rationall superficies by the second assumpt going before the 41. of the tenth For either of those superficieces is rationall Wherefore that which is composed of the squares of the lines AC and CB differeth from that which is composed of the squares of the lines AD and DB by a rationall superficies when yet they are both mediall superficieces which is impossible Wherefore a first bimediall line is in one poynt onely deuided into his names which was required to be proued ¶ The 32. Theoreme The 44. Proposition A second bimediall line is in one poynt onely deuided into his names SVppose that the line AB being a second bimediall line be deuided into hys names in the poynt C so that let the lines AC and CB be mediall lines commensurable in power onely comprehending a mediall superficies It is manifest that the poynt C deuideth not the whole line AB into two equall partes For the lines AC and CB are not commensurable in length the one to the other Now I say that the line AB cannot be deuided into his names in any other poynt but onely in C. For if it be possible let it be deuided into his names in the poynt D so that let not the line AC be one and the same that is let it not be equall with the line DB. But let it be greater then it Now it is manifest by the first assumpt going before the 42. proposition of this booke that the squares of the lines AC and CB are greater then the squares of the lines AD and DB. And also that the lines AD and DB are mediall lines commensurable in power onely comprehending a mediall supersicies Take a rationall line EF. And by the 44. of the first vpon the line EF apply a rectangle parallelograme EK equall to the square of the line AB From which parallelograme take away the parallelograme EG equall to the squares of the lines AC and CB Wherefore the residue namely the parallelograme HK
which is contained vnder the lines AD and DB twise by a rationall superâ—icies for either of them is rationall Wherfore also the squares of the lines AC and CB added together exceede the squares of the lines AD and DB added together by a rationall superficies when yet ech of them is a mediall superficies which is impossible Wherefore a line containing in power a rationall and a mediall is in one point onely deuided into his names which was required to be demonstrated ¶ The 35. Theoreme The 47. Proposition A line contayning in power two medials is in one point onely deuided into his names SVppose that AB being a line containing in power two medialls be deuided into his names in the point C so that let the lines AC and CB be incommensurable in power hauing that which is composed of the squares of the lines AC CB mediall and that also which is contained vnder the lines AC and CB mediall and moreouer incommensurable â—o that which is composed of the squares of the lines AC and CB. Then I say that the line AB can in no other point be deuided into his names but onely in the point C. For if it be possible let it be deuided into his names in the point D so that let not the line AC be one and the same that is equall with the line DB but by supposition let the line AC be the greater And take a rationall line EF. And by the 43. of the first vpon the line EF apply a rectangle parallelograÌ„me EG equall to that which is coÌ„posed of the squares of the lines AC and CB and likewise vpon the line HC which is equall to the line EF apply the parallelogramme HK equall to that which is contained vnder the lines AC and CB twise Wherefore the whole parallelogramme EK is equall to the square of the line AB Againe vpon the same line EF describe the parallelogramme EL equall to the squares of the lines AD and DB. Wherefore the residue namely that which is contayned vnder the lines AD and DB twise is equall to the parallelogramme remaining namely to MK And forasmuch as that which is coÌ„posed of the squares of the lines AC and CB is by supposition mediall therefore the parallelograÌ„me EG which is equall vnto it is also mediall and it is applied vpon the rationall line EF. Wherefore by the 22. of the tenth the line HE is rationall and incommensurable in length vnto the line EF. And by the same reason also the line HN is rationall and incommensurable in length to the same line EF. And forasmuch as that which is composed of the squares of the lines AC and CB is incommensurable to that which is contained vnder the lines AC and CB twise for it is supposed to be incommensurable to that which is coÌ„tained vnder the lines AC and CB once therefore the parallelogramme EG is incommensurable to the parallelogramme H â— Wherefore the line EH also is incommensurable in length to the line HN and they are rationall lines wherfore the lines EH and HN are rationall commensurable in power onely Wherefore the whole line EN is a binomiall line and is deuided into his names in the point H. And in like sort may we proue that the same binomiall line EN is deuided into his names in the point M and that the line EH is not one and the same that is equall with the line MN as it was proued in the end of the demonstration of the 44. of this booke Wherefore a binomiall line is deuided into his names in two sundry pointes which is impossible by the 42. of the tenth Wherefore a line containing in power two medials is not in sundry pointes deuided into his names Wherefore it is deuided in one point onely which was required to be demonstrated ¶ Second Definitions IT was shewed before that of binomiall lines there were sixe kindes the definitions of all which are here now set and are called second definitioÌ„s All binomiall lines as all other kindes of irrationall lines are coÌ„ceaued coÌ„sidered and perfectly vnderstanded onely in respecte of a rationall line whose partes as before is taught are certayne and knowen and may be distinctly expressed by number vnto which line they are compared Thys rationalâ— line must ye euer haue before your eyes in all these definitions so shall they all be â—asie inough A binomiall line ye know is made of two partes or names wherof the one is greater then the other Wherfore the power or square also of the one is greater then the power or square oâ— the other The three first kindes of binomiall lines namely the first the seconâ— the third are produced when the square of the greater name or part of a binomâ—all eâ—cedeth the square of the lesse name or part by the square of a line which is commâ—nsurable in length to it namely to the greater The three last kindes namely the fourth the â—iâ—t and the sixt are produced when the square of the greater name or part â—â—â—â—edeth the square of the lesse name or part by the square of a line incommensurable in length vnto it that is to the greater part A first binomiall line is whose square of the greater part exceedeth the square of tâ—e lesse part â—y the square of a line commensurable in length to the greater part and the greater part is also commensurable in length to tâ—e rationall line first set As lâ—t the raâ—ionâ—ll line first set be AB whose partes are distinctly knowen suppose also that the line CE be a binomiall line whose names or partes let be CD and DE. And let the square of the line CD the greater part excede the square of the line DE the lesse part by the square of the line FG which line FG let bâ—e commensuâ—able in length to the line CD which is the greater part of the binomiall line And moreouer let the line CD the greater paâ—t be commensurble in length to the rationall line first set namely to AB So by this dâ—â—inition the binomiall line CE is a first binomiall line A second binomiall line is when the square of the greater part exceedeth the square of the lesse part by the square of a line commensurable in length vnto it and the lesse part is commensurable in length to the rationall line first set As supposing euer the rationall line let CE be a binomiall line deuided in the poynt D. The square of whose greater part CD let exceede the square of the lesse part DE by the square of the line FG which line â—G let be coÌ„mensurable in length vnto the line CD tâ—e grâ—ater pâ—â—â— oâ— the binomiall line And let also the line DE the lesse part of the binomiall line be commensuâ—able in lâ—ngth to the rationall line first set AB So by this definition the binomiall line CE is a second binomiall line A third binomiall
to the line BE. But as the line FB is to the line BG so by the 1. of the sixt is the parallelograÌ„me AB which is the square of the line DB to the parallelogramme DG and as the line DB is to the line BE so is the same parallelograÌ„me DG to the parallelogramme BC which is the square of the line BE. Wherefore as the square AB is to the parallelogramme DG so is the same parallelogramme DG to the square BC. Wherefore the parallelogramme DG is the meane proportionall betwene the squares AB and BC. I say moreouer that the parallelogramme DC is the meane proportionall betwene the squares AC and CB. For for that as the line AD is to the line DK so is the line KG to the line GC for they are ech equall to eche Wherefore by composition by the 18. of the fift as the line AK is to the line KD so is the line KC to the line CG But as the line AK is to the line KD so is the square of the line AK which is the square AC to the parallelogramme coÌ„tayned vnder the lines AK and KD which is the parallelogramme CD and as the line KC is to the line CG so also is the parallelogramme DC to the square of the line GC which is the square BC. Wherefore as the square AC is to the parallelogramme DC so is the parallelograÌ„me DC to the square BC. Wherefore the parallelogramme DC is the meane proportionall betwene the squares AC and BC which was required to be demonstrated ¶ An Assumpt Magnitudes that are meane proportionalls betwene the selfe same or equall magnitudes are also equall the one to the other Suppose that there be three magnitudes A B C. And as A is to B so let B be to C. And likewise as the same magnitude A is to D so let D be to the same magnitude C. Then I say that B and D are equall the one the other For the proportion of A vnto C is double to that proportion which A hath to B by the 10. definition of the fift and likewise the selfe same proportion of A to C is by the same definition double to that proportion which A hath to D. But magnitudes whose equemultiplices are either equall or the selfe same are also equall Wherefore as A is to B so is A to D. Wherefore by the 9. of the fift B and D are equall the one to the other So shall if also be if there be other magnitudes equal to A and C namely E and F betwene which let the magnitude D be the meane proportionall ¶ The 36. Theoreme The 54. Proposition If a superficies be contained vnder a rationall line a first binomiall line the line which containeth in power that superficies is an irrationall line a binomiall line SVppose that the superficies ABCD be contained vnder the rational line AB and vnder a first binomial line AD. Then I say that the line which containeth in power the superficies AC is an irrational line and a binomial line For forasmuch as the line AD is a first binomial line it is in one only point deuided into his names by the 42. of this tenth let it be deuided into his names in the point E. And let AE be the greater name Now it is manifest that the lines AE and ED are rationall câ—mmensurable in power onely and that the line AE is in power more then the line ED by the square of a line commensurable in length to the line AE and moreouer that the line AE is commensurable in length to the rationall line geuen AB by the definition of a first Binomiall line set before the 48. proposition of this tenth Deuide by the 10. of the first the line ED into two equall partes in the point F. And forasmuch as the line AE is in power more then the line ED by the square of a line commensurable in length vnto the line AE therefore if vpon the greater line namely vpon the line AE be applied a parallelogramme equall to the fourth part of the square of the lesse line that is to the square of the line EF wanting in forme by a square it shall deuide the greater line namely AE into two partes còmmensurable in length the one to the other by the second part of the 17. of the tenth Apply therfore vpon the line AE a parallelogramme equall to the square of the line EF and wanting in forme by a square by the 28. of the sixt and let the same be that which is contained vnder the lines AG and GE. Wherfore the line AG is commensurable in length to the lyne GE. Draw by the pointes G E and F to either of these lines AB and DG these parallel lines GH EK and FL by the 31. of the first And by the 14. of the second vnto the parallelogramme AH describe an equall square SN And vnto the parallelogramme GK describe by the same an equal square NP. And let these lines MN NX be so put that they both make one right line Wherfore by the 14. of the first the lines RN and NO make also both one right line Make perfect the parallelogramme SP. Wherfore the parallelogramme SP is a square by those thinges which were demonstrated after the determination in the first assumpt going before And forasmuch as that which is contained vnder the lines AG and GE is equall to the square of the line EF by construction therfore as the line AG is to the EF so is the line EF to the line EG by the 14. or 17. of the sixt Wherfore also by the 1. of the sixt as the parallelogramme AH is to the parallelogramme EL so is the parallelogramme EL to the parallelogramme GK Wherfore the parallelogramme EL is the meane proportionall betwene the parallelogrammes AH and GK But the parallelogramme AH is equal to the square SN and the parallelograÌ„me GK is equal to the square NP by coÌ„struction Wherfore the parallelogramme EL is the meane proportionall betwene the squares SN and NP by the 7. of the fifth But by the first assumpt going before the parallelogramme MR is the meane proportionall betwene the squares SN and NP. Wherefore the parallelogramme MR is equall to the parallelogramme EL by the last assumpt going before But the parallelogramme MR is equal to the parallelogramme OX by the 43. of the first and the parallelogramme EL is equall to the parallelograÌ„me FC by construction and by the first of the sixt Wherfore the whole parallelogramme EC is equall to the two parallelogrammes MR OX And the parallelogrammes AH and GK are equall to the squares SN and NP by construction Wherfore the whole parallelogramme AC is equal to the whole square SP that is to the square of the line MX Wherefore the line MX containeth in power the parallelogramme AC I say moreouer that the line MX is a binomiall line For forasmuch as by
the 17. of the tenth the line AG is commensurable in length to the line EG Therefore by the 15. of the tenth the whole line AE is commensurable in length to either of thâ—se lines AG and GE. But by supposition the line AE is commensurable in length to the line AB Wherfore by the 12. of the tenth either of the lines AG GE are commensurable in leÌ„gth to the line AB But the line AB is rationall Wherefore either of these lines AG and GE is rationall Wherfore by the 19. of the tenth either of these parallelogrammes AH and GK is rationall Wherfore by the first of the sixt and 10. of the tenth the parallelogramme AH is commensurable to the parallelogramme GK But the parallelogramme AH is equall to the square SN and the parallelogramme GK is equall to the square NP Wherfore the squares SN and NP which are the squares of the lines MN and NX are rationall and commensurable And forasmuch as by supposition the line AE is incommensurable in length to the line ED. But the line AE is commensurable in length to the line AG. And the line DE is commensurable in length to the line EF for it is double to it by construction Wherfore by the 13. of the tenth the line AG is incommensurable in length to the line EF. Wherfore the parallelogramme AH is incommensurable to the parallelogramme EL. But the parallelogramme AH is equal to the square SN and the parallelogramme EL is equall to the parallelogramme MR. Wherfore the square SN is incommensurable to the parallelogramme MR. But as the square SN is to the parallelogramme MR so is the line ON to the line NR by the 1. of the sixt Wherfore the line ON is incommensurable to the line NR But the line ON is equall to the line MN and the line NR to the line NX Wherfore the line MN is incommensurable to the line NX And it is already proued that the squares of the lines MN and NX are rationall and commensurable Wherefore the lines MN and NX are rationall commensurable in power onely Wherfore the whole line MX is a binomiall line and it containeth in power the parallelogramme AC which was required to be proued ¶ The 37. Theoreme The 55. Proposition If a superficies be comprehended vnder a rationall line and a second binomiall line the line that contayneth in power that superficies is irrationall and is a first bimediall line SVppose that the superficies ABCD be contayned vnder a rationall line AB and vnder a second binomiall line AD. Then I say that the line that containeth in power the superficies AC is a first bimediall line For forasmuch as AD is a second binomiall line it can in one onely point be deuided into his names by the 43. of this tenth let it therefore by supposition be deuided into his names in the poynt E so that let AE be the greater name Wherefore the lines AE and ED are rationall commensurable in power onely and the line AE is in power more then the line ED by the square of a line commensurable in length to AE and the lesse name namely ED is commensurable in length to the line AB by the definition of a second binomiall line set before the 48. propositioÌ„ of this tenth Deuide the line ED by the teÌ„th of the first into two equall partes in the poynt F. And by the 28. of the sixt vpon the line AE apply a parallelogramme equal to the square of the line EF and wanting in figure by a square And let the same parallelogramme be that which is contayned vnder the lines AG and GE. Wherefore by the second part of the 17. of this tenth the line AG is commensurable in length to the line GE. And by the 31. of the first by the poyntes G E F draw vnto the lines AB and CD these parallel lines GH EK FL. And by the 14. of the second vnto the parallelograÌ„me AH describe an equall square SN And to the parallelograÌ„me GK describe an equall square NP and let the lines MN NX be so put that they both make one right line wherefore by the 14. of the first the lines also RN and NO make both one right line Make perfect the parallelogramme SP. Now it is manifest by that which hath bene demoÌ„strated in the propositioÌ„ next going before that the parallelogâ—ame MR is the meane proportionall betwene the squares SN and NP and is equall to the parallelogramme EL and that the line MX contayneth in power the superficies AC Now resteth to proue that the line MX is a first bimediall line Forasmuch as the line AE is incommensurable in length to the line ED and the line ED is commensurable in length to the line AB therefore by the 13. of the tenth the line AE is incommensurable in length to the line AB And forasmuch as the line AG is commensurable in length to the line GE therefore the whole line AE is by the 15. of the tenth commensurable in length to either of these lines AG and GE. But the line AE is rationall wherefore either of these lines AG and GE is rationall And forasmuch as the line AE is incommensurable in length to the line AB but the line AE is commensurable in leÌ„gth to either of these lines AG and GE wherefore either of the lines AG and GE are incommensurable in length to the line AB by the 13. of the tenth Wherefore the lines AB AG and GE are rationall commensurable in power onely Wherefore by the 21. of the tenth either of these parallelograÌ„mes AH and GK is a mediall superâ—icies Wherefore also either of these squares SN and NP is a mediall superâ—icies by the corollary of the 23. of the tenth Wherfore the lines MN NX are mediall lines by the 21. of this teÌ„th And forasmuch as the line AG is coÌ„mensurable in leÌ„gth to the line GE therefore by the 1. of the sixt and 11. of the tenth the parallelograÌ„me AH is coÌ„mensurable to the parallelogramme GK that is the square SN to the square NP that is the square of the line MN to the square of the line NX Wherefore the lines MN and NX are medialls commensurable in power And forasmuch as the line AE is incommensurable in length to the line ED but the line AE is commensurable in length to the line AG and the line ED is commensurable in length to the line Eâ— therefore by the 13. of the tenth the line AG is incommensurable in length to the line Eâ— Whereâ—ore by the â— of the sixt and 11. of the tenth the parallelograme AH is incoÌ„mensurable to the parallelogramme EL that is thâ— square SN to the parallelogramme MR that is the line ON is incommensurable to the line NR that is the line MN to the line NX And it is proued that the lines MN and NX are mediall lines commensurablâ— in power
Wherefore the lines MN and NX are mediall lines commensurable in power onely Now I say moreouer that they comprehend a rational superficies For forasmuch as by supposition the line DE is commeÌ„surable in length to either of these lines AB and EF therefore the line FE is commensurable in length to the line EK which is equall to the line AB by thâ— 12. of the tenth And either of these lines EF and EK is a rationall line Wherefore the parallelograÌ„me EL that is the parallelograÌ„me MR is a rationall superficies by the 19. of the tenth But the parallelogramme MR is that which is contayned vnder the lines MN and NX But if two mediall lines commensurable in power onely and comprehending a rationall superâ—icies be added together the whole line is irrational and is called a first bimediall by the 37. of the tenth Wherefore the line MX is a first bimediall line which was required to be demonstrated ¶ The 38. Theoreme The 56. Proposition If a superficies be contayned vnder a rationall line and a third binomiall line the line that contayneth in power that superficies is irrationall and is a second bimediall line SVppose that the superficies ABCD be comprehended vnder the rationall line AB and a third binomiall line AD and let the line AD be supposed to be deuided into his names in thâ— point E of which let AE be the greater name Then I say that the line that containeth in power the superficies AC is irrationall and is a second bimediall line Let the same construction of the figures be in this that was in the two Propositions nexâ— going before And now forasmuch as the line AD is a third binomiall line therefore these lines AE and ED are rationall commensurable in power onely And the line AE is in power more theÌ„ the line ED by the square of a line coÌ„mensurable in length to the line AE and neither of the lines AE nor ED is commensurable in length to the line AB by the definition of a third binomiall line set before the 48. Proposition As in the former Propositions it was demonstrated so also may it in this Proposition be proued that the line MX containeth in power the superficies AC and that the lines MN and NX are mediall lines commensurable in power onely Wherefore the line MX is a bimediall line Now resteth to proue that it is a second bimediall line Forasmuch as the line DE is by supposition incommensurable in length to the line AB that is to the line EK But the line ED is commensurable in length to the line EF. Wherefore by the 13. of the tenth the line EF is incommensurable in length to the line EK And the lines FE and EK are rationall For by supâ—osition the line ED is rationall vnto which the line FE is commensurable Wherefore the lines FE and EK are rationall lines commensurable in power onely Wherefore by the 21. of the tenth the parallelogramme EL that is the parallelogramme MR which is contayned vnder the lines MN and NX is a mediall superficies Wherefore that which is contayned vnder the lines MN and NX is a mediall superficies Wherefore the line MX is a second bimediall line by the 38. Proposition and definition annexed thereto which was required to be proued ¶ The 39. Theoreme The 57. Proposition If a superficies be contained vnder a rationall line and a fourth binomiall line the line which contayneth in power that superficies is irrationall and is a greater line SVppose that the superficies AC be comprehended vnder a rationall line AB and a fourth binomiall line AD let the binomiall line AD be supposed to be deuided into his names in the point E so that let the line AE be the greater name Then I say that the line which contayneth in power the superficies AC is irrationall and is a greater line For forasmuch as the line AD is a fourth binomiall line therefore the lines AE and ED are rationall commensurable in power onely And the line AE is in power more then the line ED by the square of a line incommensurable in length to AE And the line AE is commensurable in length to the line AB Deuide by the 10. of the first the line DE into two equall partes in the point F. And vpon the line AE apply a parallelogramme equall to the square of EF and wanting in figure by a square and let the same parallelogramme be that which is contayned vnder the lines AG GE. Wherefore by the second part of the 18. of the tenth the line AG is incommensurable in length to the line EG Draw vnto the line AB by the pointes G E F parallell lines GH EK and PL and let the rest of the construction be as it was in the three former Propositions Now it is manifest that the line MX contayneth in power the superficies AC Now resteth to proue that the line MX is an irrationall line and a greater line Forasmuch as the line AG is incommensurable in length to the line EG therefore by the 1. of the sixt and 11. of the tenth the parallelogramme AH is incommensurable to the parallelogramme GK that is the square SN to the square NP. Wherefore the lines MN and NK are incommensurable in power And forasmuch 〈◊〉 the line AE is commensurable in length to the rationall line Aâ— therefore the parallelogramme AK is rationall And it is equall to the squares of the lines MN and NX Wherfore that which is composed of the squares of the lines MN and NX added together is rationall And forasmuch as the line ED is incommensurable in length to the line AB that is to the line EK but the line ED is commensurable in length to the line EF therefore the line EF is incommensurable in length to the line EK Wherefore the lines EK and EF are rationall commensurable in power onely Wherefore by the 21. of the tenth the parallelogramme LE that is the parallelogramme MR is mediall And the parallelograÌ„me MR is that which is contayned vnder the lines MN and NX Wherefore that which is contayned vnder the lines MN and NX is mediall And that which is composed of the squares of the lines MN NX is proued to be rationall the line MN is demonstrated to be incommensurable in power to the line NX But if two lines incommensurable in power be added together hauing that which is made of the squares of them added together rationall that which is vnder them mediall the whole line is irrationall and is called a greater line by the 39. of the tenth Wherefore the line MX is irrationall and is a greater line and it containeth in power the superficies AC which was required to be demonstrated ¶ The 40. Theoreme The 58. Proposition If a superficies be contained vnder a rationall line and a fift binomiall line the line which contayneth in power that superficies is
irrationall and is a line contayning in power a rationall and a mediall superficies SVppose that the superficies AG be contayned vnder the rationall line Aâ— and vnder a fift binomiall line ADâ—â—nd let the same linâ— AD be supposed to be deuided into his names in the poynt E so that let the line AE be the greater name Then I say that the line which contayneth in power the superficies AC is irrationall and is a line contayning in power a rationall and a mediall superficies Let the selfe same constructions be in this that were in the foure Proposition next going before And it is manifest that the line MX contayneth in power the superficiâ—â— AG. Now testeth to proue that the line MX is a line contayning in power a rationall a mediall superficies Forasmuch as the line AG is incommensurable in length to the line GE therefore by the 1. of the sixt and 10. of the tenth the parallelogramme AH is incommensurable to the parallelogramme HE that is the square of the line MN to the square of the line NX Wherefore the lines MN and NX are incommensurable in power And forasmuch as the line AD iâ— a fifâ— binomiall line and his lesse name or part is the line ED therefore the line ED is commensurable in length to the line AB But the line AE is incommensurable in length to the line ED. Wherefore by the 13. of the tenth the line AB is incommensurable in length to the line AE Wherefore the lines AB and AE are rationall commensurable in power onely Wherefore by the 21. of the tenth the parallelogramme AK is mediall that is that which is composed of the squares of the lines MN NX added together And forasmuch as the line DE is commensurable in length to the line AB that is to the line EK but the line DE is commensurable in length to the line EF wherefore by the 12. of the tenth the line EF is also commensurable in length to the line EK And the line EK is rationall Wherefore by the 19. of the tenth the parallelogramme EL that is the parallelogramme MR which is contayned vnder the lines MN and NX is rationall Wherefore the lines MN and NX are incommensurable in power hauing that which is composed of the squares of them added together Mediall and that which is contayned vnder them Rationall Wherefore by the 40. of the tenth the whole line MX is a line contayning in power a rationall and a mediall superficies and it contayneth in power the superficies AC which was required to be proued ¶ The 41. Theoreme The 59. Proposition If a superficies be contayned vnder a rationall line and a sixt binomiall line the lyne which contayneth in power that superficies is irrational is called a line contayning in power two medials SVppose that the superficies ABCD be contained vnder the rationall line AB and vnder a sixt binomiall line AD and let the line AD be supposed to be deuided â—â—to his names in the point E so that let the line AE be the greater name Then I say that the line that containeth in power the superficies AC is irrationall and is a line contayning in power two medials Let the selfe same constructioÌ„s be in this that were in the former propositions Now it is manifest that the line MX containeth in power the superficies AC and that the line MN is incommensurable in power to the line NX And forasmuch as the line AE is incommensurable in length to the line AB therfore the lines AE and AB are rationall commensurable in power only Wherfore by the 〈◊〉 of the tenth the parallelograÌ„mâ— AK that is that which is composed of the squares of the lines MN and NX added together is mediall Againe forasmuch as the line ED is incommensurable in length to the line AB therefore also the line EF is incoÌ„meÌ„surable in leÌ„gth to the line EK Wherfore the lines EF and EK are rationall commensurable in power onely Wherfore the parallelogramme EL that is the parallelogramme MR which is contained vnder the lines MN and NX is mediall And forasmuch as the line AE is incommensurable in length to the line EF therfore the parallelogramme AK is also incommensurable to the parallelogramme EL by the first of the sixt and 10â— of the tenth But the parallelogramme AK is equal to that which is composed of the squares of the lines MN and NX added together And the parallelogramme EL is equall to that which is coÌ„taiâ—â—d vndâ—r the lines MN and NX Wherfore that which is câ—â—posed of the squares of the lines MN and NY added together is incommensurable to that which is contained vnder the lâ—nes MN and NXâ— and eâ—â—her of them namâ—ly that which is composed of the squares of the lines MN and NX added togâ—â—her and that which is contained vâ—der the lines MN and Nâ— is proued mediall and the lines MN and NX are proued incommensurable in power Wherfore by the 41. of the tenth the whole line MX is a line contayning in power two medials and it containeth in power the superfices AC which was required to be dâ—â—onstrâ—ted An Aâ—â—umpt If a right line be deuided into two vnequall partes the squares which are made of the vnequall partes are greater then the rectangle parallelogramme câ—â—tayned vnder the vnequall partes twise Suppose that AB be a right line and let it be 〈…〉 point C. And let the line AC be the greater part 〈…〉 and â—B are greater theÌ„ that which is contained vnder the lines Aâ— and CB twise Dâ—â—idâ— by the 10. of the first the line AB into two equall partes in the point D. Now forasmuch as the right line AB is deuided into two equall parteâ— in the point D and into two vnequall parteâ— in the point C therfore by the 5. of the second that which is contained vnder the lines 〈…〉 line CD is equall to the square of the line AD. 〈…〉 the lines AC and CB omitting the square of the line CD is lesse then the square of the AD by the 9. common sentence and the seuenth of the fifth Wherefore that which is contained vnder the lines AC and CB twise is lesse then the double of the square of the line AD that is theÌ„ twise the square of the line AD by alternate proportioÌ„ and the 14. of the fift But the squares of the lines AC and CB are double to the squares of the lines AD and DC by the 9. of the secoÌ„d Therfore the squares of AC and CB are more then double to the square of AD alone leauing out the square of DC by the 8. of the fift But the parallelogramme contained vnder the lines AC and â—B twise is proued lesse theÌ„ the double of the square of the line AD. Therfore the same parallelogramme contained vnder the lines AC and CB twise is much lesse then the squares of the lines AC and CB. If a right line
the definition of a first binomiall line seâ— before the 48. proposition of this booke the line DG is a first binomiall line which was required to be proued This proposition and the fiue following are the conuerses of the sixe former propositions ¶ The 43. Theoreme The 61. Proposition The square of a first bimediall line applied to a rationall line maketh the breadth or other side a second binomiall line SVppose that the line AB be a first bimediall line and let it be supposed to be deuided into his partes in the point C of which let AC be the greater part Take also a rationall line DE and by the 44. of the first apply to the line DE the parallelograÌ„me DF equall to the square of the line AB making in breadth the line DG Then I say that the line DG is a second binomiall line Let the same constructions be in this that were in the Proposition going before And forasmuch as the line AB is a first bimediall line and is deuided into his partes in the point C therefore by the 37. of the tenth the lines AC and CB are mediall commensurable in power onely coÌ„prehending a rationall superficies Wherfore also the squares of the lines AC and CB are mediall Wherefore the parallelogramme DL is mediall by the Corollary of the 23. of the tenth and it is applied vppon the rationall line DE. Wherefore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. Againe forasmuch as that which is coÌ„tayned vnder the lines AC and CB twise is rationall therefore also the parallelogramme MF is rationall and it is applied vnto the rationall line ML Wherefore the line MG is rationall and commensurable in length to the line ML that is to the line DE by the 20. of the tenth Wherefore the line DM is incommensurable in length to the line MG and they are both rationall Wherefore the lines DM and MG are rationall commensurable in power onely Wherefore the whole line DG is a binomiall line Now resteth to proue that it is a second binomiall line Forasmuch as the squares of the lines AC and CB are greater then that which is contayned vnder the lines AC and CB twise by the Assumpt before the 60. of this booke therefore the parallelogramme DL is greater then the parallelogrrmme MF Wherefore also by the first of the sixt the line DM is greater then the line MG And forasmuch as the square of the line AC is commensurable to the square of the line CB therefore the parallelogramme DH is commensurable to the parallelogramme KL Wherefore also the line DK is commensurable in length to the line KM And that which is contayned vnder the lines DK and KM is equall to the square of the line MN that is to the fourth part of the square of the line MG Wherefore by the 17. of the tenth the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and the line MG is commensurable in length to the rationall line put namely to DE. Wherefore the line DG is a second binomiall line which was required to be proued ¶ The 44. Theoreme The 62. Proposition The square of a second bimediall line applied vnto a rationall line maketh the breadth or other side therof a third binomiall lyne SVppose that AB be a second bimediall line and let AB be supposed to be deuided into his partes in the point C so that let AC be the greater part And take a rationall line DE. And by the 44. of the first vnto the line DE apply the parallelogramme DF equall to the square of the line AB and making in breadth the line DG Then I say that the line DG is a third binomiall line Let the selfe same constructions be in this that were in the propositions next going before And forasmuch as the line AB is a second bimediall line and is deuided into his partes in the point C therfore by the 38. of the tenth the lines AC and CB are medials commensurable in power only compreheÌ„ding a mediall superficies Wherfore that which is made of the squares of the lines AC and CB added together is mediall and it is equall to the parallelogramme DL by construction Wherefore the parallelogramme DL is mediall and is applied vnto the rationall line DE wherfore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. And by the lyke reason also the line MG is rationall and incommensurable in length to the line ML that is to the line DE. Wherfore either of these lines DM and MG is rational and incommensurable in length to the line DE. And forasmuch as the line AC is incommensurable in length to the line CB but as the line AC is to the line CB so by the assumpt going before the 22. of the tenth is the square of the line AC to that which is contained vnder the lines AC and CB. Wherfore the square of the line AC is incâ—mmmensurable to that which is contayned vnder the lines AC and CB. Wherfore that that which is made of the squares of the lines AC and CB added together is incommensurable to that which is contained vnder the lines AC and CB twise that is the parallelogramme DL to the parallelogramme MF Wherfore by the first of the sixt and 10. of the tenth the line DM is incommensurable in length to the line MG And they are proued both rationall wherfore the whole line DG is a binomiall line by the definition in the 36. of the tenth Now resteth to proue that it is a third binomiall line As in the former propositions so also in this may we conclude that the line DM is greater then the line MG and that the line DK is commensurable in length to the line KM And that that which is contained vnder the lines DK and KM is equall to the square of the line MN Wherfore the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and neither of the lines DM nor MG is commensurable in length to the rational line DE. Wherfore by the definition of a third binomiâ—ll line the line DG is a third binomiall line which was required to be proued ¶ Here follow certaine annotations by M. Dee made vpon three places in the demonstration which were not very euident to yong beginners †The squares of the lines AC and Câ— are medials 〈◊〉 iâ— taught after the 21â— of this tenth and therâ—ore forasmuch as they are by supposition commeÌ„surable th' one to the other by the 15. of the teÌ„th the compound of them both is commensurable to ech part But the partes are medials therfore by the coâ—ollary of the 23. of the tenth the compound shall be
mediall â— For that MX is equall by construction to that which is contayned vnder the lines AC and CB which is proued mediall therfore by the corollary of the 23. of this tenth MX is mediall and therfore by the same corollary his double MF is mediall And it is applied to a rationall line ML beyng equall to Dâ— therfore by the 22. of the tenth the line MG is rationall and incommensurable in length to ML that is to DE. ‡ Because the compound of the two squares of the lines AC and Câ— beyng commensurable one to the other is also to eyther square by the 15. commensurable therfore to the square of AC But the square of AC is proued incommensurable to that which is contained vnder AC CB once Wherfore by the 13. of the tenth the compound of the two squares of the lines AC and CB is incommensurable to that which is coÌ„tained vnder the lines AC and Câ— once But to that which is twise contained vnder the same lines AC and CB the parallelogaÌ„rme once contayned is commensurable for it is as 1. is to 2. therfore that which is made of the squares of the lines AC and CB is incommensurable to the parallelogramme contained vnder AC and CB twise by the sayd 13. of this tenth ¶ A Corollary Hereby it is euident that the squares made of the two partes of a second bimediall line composed is a compound mediall and that the same compound is incommensurable to the parallelogramme contayned vnder the two partes of the second bimediall lyne The proofe hereof is in the first and third annotations here before annexed ¶ The 45. Theoreme The 63. Proposition The square of a greater line applied vnto a rationall line maketh the breadth or other side a fourth binomiall line SVppose that the line AB be a greater line and let it be supposed to be deuided into his partes in the point C so that let AC be the greater part And take a rationall line DE. And by the 44. of the first vnto the line DE apply the parallelogramme DF equall to the square of the line AB and making in breadth the line DG Then I say that the line DG is a fourth binomiall line Let the selfe same construction be in this that was in the former Propositions And forasmuch as the line AB is a greater line is deuided into his partes in the point C therefore the lines AC and CB are incommensurable in power hauing that which is made of the squares of them added together rationall and the parallelogramme which is contayned vnder them mediall Now forasmuch as that which is made of the squares of the lines AC and CB added together is rationall therefore the parallelogramme DL is rationall Wherefore also the line MD is rationall and commensurable in leÌ„gth to the line DE by the 20. of this tenth Againe forasmuch as that which is coÌ„tained vnder the lines AC and CB twise is mediall that is the parallelograÌ„me MF and it is applied vnto the rationall line ML therefore by the â—2 of the tenth the line MG is rationall and incommensurable in length to the line DE Therefore by the 13. of the tenth the line DM is incommensurable in length to the line MG Wherefore the lines DM and MG are rationall commensurable in power onely Wherfore the whole line DG is a binomiall line Now resteth to proue that it is also a fourth binomiall line Euen as in the former Propositions so also in this may we conclude that the line DM is greater then the line MC And that that which is contayned vnder the lines DK and KM is equall to the square of the line MN Now forasmuch as the square of the line AC is incommensurable to the square of the line CB therefore the parallelogramme DH is incommensurable to the parallelogramme KL Wherefore by the 1. of the sixt and 10. of the tenth the line DK is incommensurable in length to the line KM But if there be two vnequall right lines and if vpon the greater be applied a parallelogramme equall to the fourth part of the square made of the lesse and wanting in figure by a square and if also the parallelogramme thus applied deuide the line wherupon it is applied into partes incommensurable in length the greater line sâ—all be in power more then the lesse by the square of a line incoÌ„mensurable in length to the greater by the 18. of the tenth Wherefore the line DM is in power more then the line MG by the square of a line incoÌ„mensurable in length to DM And the lines DM and MG are proued to be rationall coÌ„mensurable in power onely And the line DM is commensurable in length to the rationall line geuen DE. Wherefore the line DG is a fourth binomiall line which was required to be proued ¶ The 46. Theoreme The 64. Proposition The square of a line contayning in power a rationall and a mediall superficies applied to a rationall line maketh the breadth or other side a fift binomiall line SVppose that the line AB be a line contayning in power a rationall and a mediall superficies and let it be supposed to be deuided into his partes in the point C so that let AC be the greater part and take a rationall line DE. And by the 44. of the first vnto the line DE apply the parallelogramme DF equall to the square of the line AB and making in breadth the line DG Then I say that the line DG is a fift binomiall line Let the selfe same coÌ„struction be in this that was in the former And forasmuch as AB is a line contayning in power a rationall and a mediall superficies and is deuided into his partes in the poynt C therefore the lines AC CB are incoÌ„mensurable in power hauing that which is made of the squares of theÌ„ added together mediall and that which is contayned vnder then rationall Now forasmuch as that which is made of the squares of the lines AC and CB added together is mediall therefore also the parallelogramme DL is mediall Wherefore by the 22. of the tenth the line DM is rationall and incommensurable in length to the line DE. Againe forasmuch as that which is contayned vnder the lines AC and CB twise that is the parallelogramme MF is rationall therefore by the 20. the line MG is rationall coÌ„mensurable in length to the line DE. Wherefore by the 13. of the tenth the line DM is incommensurable in length to the line MG Wherefore the lines DM and MG are rationall commensurable in power onely Wherefore the whole line DG is a binomiall line I say moreouer that it is a fift binomiall For as in the former so also in this may it be proued that that which is contayned vnder the lines DK and KM is equall to the square of MN the halfe of the lesse and that the line DK is incommensurable in length
line Deuide the line AB into his partes in the point E. and let the rest of the construction be in this as it was in the former And for that as the line AB is to the line CD so is the line AE to the lyne CF and the line EB to the line FD therfore as the line AE is to the lyne CF so is the line EB to the line FD but the line AB is commensurable to the line CD Wherfore also the lyne AE is commensurable to the lyne CF and likewise the line EB to the line FD. And for thâ—â— as the line AE is to the line CF so is the line EB to the line FD therfore alternately as the lyne AE is to the line EB so is the line CF to the lyne FD. Wherfore by the 22. of the sixt as the square of the lyne AE is to the square of the line EB so is the square of the line CF to the square of the line FD. Wherfore by composition by the 18. of the fift as that which is made of the squares of the lynes Aâ— and Eâ— added together is to the square of the lyne EB so is that which is made of the squareâ— of the lyneâ— Câ— and FD added together to the square of the lyne FD. Wherefore by contrary proportion as the square of the line EB is to that which is made of the squares of the lines AE and Eâ— added together so is the square of the lyne FD to that which is made of the squares of the lynes CF and FD added together Wherfore alternately as the square of the line EB is to the square of the lyne FD so is that which is made of the squares 〈◊〉 the lâ—nes AE and EB added together to that whiche is made of the squares of the lynes CF and FD added together But the square of the lyne EB is coÌ„mensurable to the square of the lyne FD for it hath already bene proued that the lines EB and FD are coÌ„meÌ„surable Wherfore that which is made of the squares of the lines AE EB added together is commeÌ„surable to that which is made of the squares of Câ— FD added together But that which is made of the squares of the lines AE and EB added together is rationall by suppositioÌ„ Wherfore that which is made of the squares of the lynes CF and FD added together is also rationall And as the lyne AE is to the lyne EB so is the line CF to the lyne FD But as the lyne AE is to the lyne EB so is the square of the line A 〈…〉 contayned vnder the lynes AE and EB therfore at the lyne CF is to the lyne FD so is the square of the lyne AE to the parallelogramme contayned vnder the lines AE and EB as the lyne CF is to the lyne FD so is the square of the lyne CF to the parallelograÌ„me contayned vnder the lynes â—F FD. Wherfore as the square of the lyne AE is to the parallelograÌ„me conâ—â—â—â—ed vnder the lines AE and EB so is the square of the lyne CF to the parallelogramme coÌ„tayned vnder the lynes CF and FD. Wherâ—or◠〈◊〉 â—s the square of the line AE is to the square of the lyne CF so is the parallelogramme contained vnder the lynes AE and EB to the parallelogramme 〈◊〉 vndeâ— the lines â—â— and â—â— But the square of the lyne AE is commensurable to the square of the lyne CF for it is already prâ—â—â—d that the lynes AE and CF are commeÌ„surable Wherefore the parallelogramme contayned vnder the lynes AE and EB is commensurable to the parallelogramme contayned vnder the lynes CF and FD. But the parallelogramme contayned vnder the lines AE and EB is mediall by suppoâ—ition Wherfore the parallelogramme contayned vnder the lynes CF and â—D also is mediall And as it hath already bene proued as the line AE is to the lyne EB so is the lyne CF to the lyne FD. But the lyne AE was by supposition incommensurable in power to the line EB Wherfore by the 10. of the tenth the lyne CF is incommensurable in power to the lyne FD. Wherfore the lynes CF and FD are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contayned vnder them mediall Wherfore the whole lyne CD is by the 39. of the tenth a greater lyne Wherfore a lyne commensurable to a greater lyne is also a greater lyne which was required to be demonstrated An other more briefe demonstration of the same after Campane Suppose that A be a greater line vnto which let the line B be commeÌ„surable either in length and power or in power onely And take a rational line CD And vpon it apply the superficies Câ— equall to the square of the line A and also vpoÌ„ the line FE which is equall to the rationall line CD apply the parallelogramme FG equall to the square of the line B. And forasmuch as the squares of the two lines A and â— are commensurable by supposition the superficies Câ— shal be commensurable vnto the superficies FG and therefore by the first of the sixt and tenth of this booke the line DE is commensurable in length to the line GB And forasmuch as by the â—3 of this booke the line DE is a fourth binomiall line therefore by the â—6 of this booke the line GE is also a fourth binomiall line wherefore by the 57. of this booke the line B which contayneth in power the superficies FG is a greater line ¶ The 51. Theoreme The 69. Proposition A line commensurable to a line contayning in power a rationall and a mediall is also a line contayning in power a rationall and a mediall SVppose that AB be a line contayning in power a rationall and a mediall And vnto the line AB let the line CD be commensurable whether in length and power or in power onely TheÌ„ I say that the line CD is a line coÌ„tayning in power a rationall a mediall Duide the line AB into his parts in the poynt E. Wherfore by the 40. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together medial and that which is contayned vnder theÌ„ nationall Let the same construction be in this that was in the former And in like sort we may proue that the lines CF and FD are incommensurable in power and that that which is made of the square of the lines AE and EB is commensurable to that which is made of the squares of the lines CF and FD and that that also which is contayned vnder the lines AE and EB is commeÌ„surable to that which is contayned vnder the lines CF and FD. Wherefore that which is made of the squares of the lines CF and FD is mediall and that which is contayned vnder the lines CF and FD is rationall Wherefore the whole line CD is a line contayning in
power a rationall and a mediall which was required to be demonstrated An other demonstration of the same after Campane Supose that AB be a line contayning in power a rationall and a mediall whereunto let the line GD be commensurable either in length and power or in power onely Then I say that the line GD is a line contayning in power a rationall and a mediall Take a rational line EZ vpoÌ„ which by the 45. of the first apply a rectangle parallelograÌ„me EZFC equall to the square of the line AB and vpon the line CF which is equall to the line EZ applye the parallelogramme FCHI equall to the square of the line GDâ— and let the breadths of the sayd parallelogrammes be the lines EG and CH. And forasmuch as the line AB is commensurable to the line GD at the least in power onely therefore the parallelogrammes EF and FH which are equall to their squares shal be commensurable Wherefore by the 1. of the sixt the right lines EC and CH are coÌ„meÌ„surable in leÌ„gth And forasmuch as the parallelogramme EF which is equall to the square of the line Aâ— which contayneth in power â— rationall and a mediall is applyed vpon the rationall EZ making in breadth the line EC therefore the line EC is a fifth binomiall line by the 64. of this booke vnto which line EC the line CH is coÌ„meÌ„surable in length wherefore by the 66. of this booke the line CH is also a fifth binomiall line And forasmuch as the superficies CI is contayned vnder the rationall line EZ that is CF and a fifth binomall line CH therefore the line which contayneth in power the superficies CI which by supposition is the line GD is a line contayning in power a rationall and a mediall by the 58. of this booke A line therefore commensurable to a line contayning in power a rationall and a mediall c. ¶ The 52. Theoreme The 70. Proposition A line commensurable to a line contayning in power two medialls is also a line contayning in power two medialls SVppose that AB be a line contayning in power two medialls And vnto the line AB let the line CD be commensurable whether in length power or in power onely Then I say that the line CD is a line contayning in power two medialls Forasmuch as the line AB is a line contayning in power two medialls let it be deuided into his partes in the point E. Wherefore by the 41. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together mediall and that also which is contained vnder them mediall and that which is made of the squares of the lines AE EB is incommensurable to that which is contained vnder the lines AE and EB Let the selfe same construction be in this that was in the former And in like sort may we proue that the lines CF FD are incommensurable in power and that that which is made of the squares of the lines AE and EB added together is commensurable to that which is made of the squares of the lines CF and FD added together and that that also which is contained vnder the lines AE and EB is commensurable to that which is contained vnder the lines CF and FD. Wherefore that which is made of the squares of the lines CF and FD is mediall by the Corollary of the 23. of the tenth and that which is contayned vnder the lines CF and FD is mediall by the same Corollary â— and moreouer that which is made of the squares of the lines CF FD is incommensurable to that which is contained vnder the lines CF and FD. Wherefore the line CD is a line containing in power two medialls which was required to be proued ¶ An Assumpt added by Montaureus That that which is made of the squares of the lines CF and FD added together is incommensurable to that which is contained vnder the lines CF and FD is thus proued For because as that which is made of the squares of the lines AE and EB added together is to the square of the line AE so is that which is made of the squares of the lines CF and FD added together to the square of the line CF as it was proued in the Propositions going before therefore alternately as that which is made of the squares of AE and EB added together is to that which is made of the squares of CF and FD added together so is the square of the line AE to the square of the line CF. But before namely in the 68. Proposition it was proued that as the square of the line AE is to the square of the line CF so is the parallelograÌ„me contained vnder the lines AE and EB to the parallelogramme contained vnder the lines CF and FD. Wherefore as that which is made of the squares of the lines AE and EB is to that which is made of the squares of the lines CF and FD so is the parallelogramme contained vnder the lines AE and EB to the parallelogramme contained vnder the lines CF and FD. Wherefore alternately as that which is made of the squares of the lines AE and EB is to the parallelogramme contained vnder the lines AE and EB so is that which is made of the squares of the lines CF and FD to the parallelogramme contained vnder the lines CF and FD. But by supposition that which is made of the squares of the lines AE and EB is incommâ—nsurable to the parallelogramme contained vnder the lines AE EB Wherefore that which is made of the squares of the lines CF and FD added together is incommensâ—rable to the parallelogramme contained vnder the lines CF and FD which was required to be proued An other demonstration after Campane Suppose that AB be a line contayning in power two medialls wherunto let the line GD be commensurable either in length and in power or in power onely Then I say that the line GD is a line coÌ„tayning in power two medialls Let the same construction be in this that was in the former And forasmuch as the parallelogramme EF is equall to the square of the line AB and is applyed vpon a rationall line EZ it maketh the breadth EC a sixt binomiall line by the 65. of this booke And forasmuch as the parallelogrammes EF CI which are equall vnto the squares of the lines AB and GD which are supposed to be commensurable are commensurable therefore the lines EC and CH are commensurable in length by the first of the sixt But EC is a sixt binomiall line Wherefore CH also is a sixt binomiall line by the 66. of this booke And forasmuch as the superficies CI is contayned vnder the rationall line CF and a sixt binomiall line CH therefore the line which coÌ„tayneth in power the superficies CI namely the line GD is a line contayning in power two medialls by the 59. of
this booke Wherefore a line commensurable to a line contayning in power two medialls c. An Annotation If other to hath bene spoken of sixe Senarys of which the first Senary contayneth the prâ—duction of irrationall lines by composition the second the diuision of them namely that those lines are in one poinâ— onely deuideâ— the third the finding out of binomiall lines of the first I say the second the third the fourth the fift and the sixt after that beginneth the â—ourth Senary containing the difference of irrationall lines betwene them selues For by the nature of euery one of the binomiall lines are demonstrated the differences of irrational lines The fiueth entreâ—teth of the applications of the squares of euery irrational line namely what irrationall lines are the breadthes of euery superficies so applied In the sixt Senary is proued that any line commensurable to any irrationall line is also an irrationall line of the same nature And now shall be spoken of the seuenth Senary wherein againe are plainly set forth the rest of the differences of the said lines betwene them selues And theâ—e is euen in those irrationall lines an arithmeticall proportionalitie And that line which is the arithmeticall meane proportionall betwene the partes of any irrationall line is also an irrationall line of the selfe same kinde First it is certaine that there is an arithmeticall proportion betwene those partes For suppose that the line AB be any of the foresaid irrationall lines as for example let it be a binomiall line let it be deuided into his names in the point C. And let AC be the greater name from which take away the line AD equall to the lesse name namely to CB. And deuide the line CD into two equall partes in the point E. It is manifest that the line AE is equall to the line EB Let the line FG be equall to either of them It is plaine that how much the line AC differeth froÌ„ the line FG so much the same line FG diâ—â—ereth from the line CB for in eche is the difference of the line DE or EC which is the propertie of arithmeticall proportionalitie And it is manifest that the line FG is commensurable in length to the line AB for it is the halfe thereof Wherefore by the 66. of the tenth the line FG is a binomiall line And after the selfe same maner may it be proued touching the rest of the irrationall lines ¶ The 53. Theoreme The 71. Proposition If two superficieces namely a rationall and a mediall superficies be coÌ„posed together the line which contayneth in power the whole superficies is one of these foure irrationall lines either a binomial line or a first bimediall lyne or a greater lyne or a lyne contayning in power a rationall and a mediall superficies But now let the lyne EH be in power more then the line HK by the square of a line incommensurable in length to the line EH now the greater name that is EH is commensurable in length to the rationall line geuen EF. Wherfore the line EK is afourth binomiall line And the line EF is rationall But if a superficies be contained vnder a rationall line and afourth binomiall line the line that containeth in power the same superficies is by the 57. of the tenth irrational and is a greater line Wherfore the line which containeth in power the parallelogramme EI is a greater line Wherefore also the line containing in power the superficies AD is a greater lyne But now suppose that the superficies AB which is rationall be lesse then the superficies CD which is mediall Wherfore also the parallelogramme EG is lesse then the parallelograÌ„me HI Wherfore also the line EH is lesse then the line HK Now the line HK is in power more then the lyne EH either by the square of a line coÌ„mensurable in length to the line HK or by the square of a lyne incommensurable in length vnto the lyne HK First let it be in power more by the square of a line commensurable in length vnto HK now the lesse name that is EH is commensurable in length to the rationall line geuen EF as it was before proued Wherfore the whole line EK is a second binomiall line And the line EF is a rationall line But if a superficies be contained vnder a rationall line and a second binomiall lyne the lyne that contayneth in power the same superficies is by the 55. of the tenth a first bimediall line Wherfore the line which contayneth in power the parallelograme EI is a first bimediall line Wherfore also the line that containeth in power the superficies AD is a first bimediall lyne But now let the line HK be in power more then the line EH by the square of a line incoÌ„mensurable in length to the lyne HK now the lesse name that is EH is coÌ„mensurable in length to the rationall lyne geuen EF. Wherfore the whole line EK is a fift binomiall lyne And the lyne EF is rationall But if a superficies be contayned vnder a rationall lyne and a fift binomiall lyne the line that contayneth in power the same superficies is by the 58. of the tenth a line containing in power a rationall and a mediall Wherefore the lyne that contayneth in power the parallelogramme EI is a line contayning in power a rationall and a mediall Wherfore also the lyne that containeth in power the superficies AD is a lyne contayning in power a rationall and a mediall If therfore a rationall and a mediall superficies be added together the lyne which contayneth in power the whole superficies is one of these foure irrationall lines namely either a binomiall line or a first bimediall line or a greater lyne or a lyne contayning in power a rationall and a mediall which was required to be demonstrated ¶ The 54. Theoreme The 72. Proposition If two mediall superficieces incommensurable the one to the other be composed together the line contayning in power the whole superficies is one of the two irrationall lines remayning namely either a second bimediall line or a line contayning in power two medialls LEt these two mediall superficieces AB and CD being incommensurable the one to the other be added together Then I say that the line which contayneth in power the superficies AD is either a second bimediall line or a line contayning in power two medialls For the superficies AB is either greater or lesse then the superficies CD for they can by no meanes be equall when as they are incommensurable First let the superficies AB be greater then the superficies CD And take a rationall line EF. And by the 44. of the first vnto the line EF apply the parallelogramme EG equall to the superficies AB and making in breadth the line EH and vnto the same line EF that is to the line HG apply the parallelogramme HI equall to the superficies CD making in breadth the line HK And forasmuch as
either of these superficieces AB CD is mediall therefore also either of these parallelogrammes EG and HI is mediall And they are eche applied to the rationall line EF making in breadth the lines EH and HK Wherefore by the 22. of the tenth either of these lines EH and HK is rationall and incommensurable in length to the line EF. And forasmuch as the superficies AB is incommensurable to the superficies CD and the superficies AB is equall to the parallelogramme Eâ— and the superficies CD to the parallelogramme HI therefore the parallelogramme EG is incommensurable to the parallelogramme HI But by the 1. of the sixt as the parallelogramme EG is to the parallelogramme HI so is the line EHâ— to the line HK Wherefore by the 10. of the tenth the line EH iâ—â—spansâ— HKâ— Wherefore the â—â—nes EH â—nd HK are rationall commensurable in power onely Wherfore the whole line EK is a binomiall line And as in the former Proposition so alsâ— in this may it be proued that the line EH is greater then the line HK Wherefore the line EH is in power more then the line HK either by the square of a line commensurable in length to the line EH or by the square of a line incommensurable in length to the line EH First let it be greater by the square of a line commensurable in length vnto the line EH Now neither of these lines EH and HK is commensurable in length to the rationall line geuen EF. Wherefore the whole line EK is a third binomiall line And the line EF is a rationall line But if a superficies be contayned vnder a rationall line a third binomiall line the line that coÌ„taineth in power the same superficies is by the 56. of the tenth a second bimediall line Wherefore the line that containeth in power the superficies EI that is the superficies AD is a second bimediall line But now suppose that the line EH be in power more then the line HK by the square of a line incommensurable in length to the line EH And forasmuch as either of these lines EH and HK is incommensurable in length to the rationall line geuen EF therfore the line EK is a sixt binomiall line But if a superficies be contained vnder a rationall line and a sixt binomiall line the line that containeth in power the same superficies is by the 59. of the tenth a line containing in power two medialls Wherefore the line that containeth in power the superficies AD is a line contayning in power two medialls And after the selfe same maner if the superficies AB be lesse then the superficies CD may we proue that the line that contayneth in power the superficies AD is either a second bimediall line or a line containing in power two medialls If therefore two mediall superficieces incommensurable the one to the other be added together the line contayning in power the whole superficies is one of the two irrationall lines remayning namely either a second bimediall line or a line coÌ„taining in power two medialls which was required to be proued ¶ A Corollary following of the former Propositions A binomiall line and the other irrationall lines following it are neither mediall lines nor one and the same betwene them selues For the square of a mediall line applied to a rationall line maketh the breadth rationall and incommensurale in length to the rationall line wherunto it is applied by the 22. of the tenth The square of a binomiall line applyed to â— rationall line maketh the breadth a first binomiall line by the 60. of the tenth The square of a first bimediall line applied vnto a rationall line maketh the breadth a second binomiall line by the 61. of the tenth The square of a second bimediall line applied vnto a rationall line maketh the breadth a third binomiall line by the 62. of the tenth The square of a greater line applied to a rationall line maketh the breadth a fourth binomiall line by the 63. of the tenth The square of a line containing in power a rationall a mediall superficies maketh the breadth a fift binomiall line by the 64. of the tenth And the square of a line containing in power two medialls applied vnto a rationall line maketh the breadth a sixt binomiall line by the 65. of the tenth Seing therefore that these foresaid breadthes differ both from the first breadth for that it is rationall and differ also the one from the other for that they are binomials of diuers orders it is manifest that those irrationall lines differ also the one from the other Here beginneth the Senaries by substraction ¶ The 55. Theoreme The 73. Proposition If from a rationall line be taken away a rationall line commensurable in power onely to the whole line the residue is an irrationall line and is called a residuall line SVppose that AB be a rationall line and from AB take away a rationall line BC commensurable in power onely to the whole line AB Then I say that the line remayning namely AC is irrationall and is called a residuall line For forasmuch as the line AB is incommensurable in length vnto the line BC and by the assumpt going before the 22. of the tenth as the line AB is to the line BC so iâ— the square of the line AB to that which is contayned vnder the lines AB and BC wherefore by the 10. of the tenth the square of the line AB is incommensurable to that which is contayned vnder the lines AB and BC. But vnto the square of the line AB are commensurable the squares of the lines AB and BC by the 15. of the tenth Wherefore the squares of the lines AB and BC are incommensurable to that which is contayned vnder the lines AB and BC. But vnto that which is contayned vnder the lines AB and BC is commensurable that which is contayned vnder the lines AB and BC twise Wherefore the squares of the lines AB and BC are incoÌ„meÌ„surable to that which is contayned vnder the lines AB and BC twise But the squares of the lines AB and BC are equall to that which is contayned vnder the lines AB and BC twise and to the square of the line AC by the 7. of the second Wherefore that which is contayned vnder the lines AB and BC twise together with the square of the line AC is incommensurable to that which is coÌ„tayned vnder the lines AB and BC twise Wherefore by the 2 part of the 16. of the teÌ„th that which is coÌ„tayned vnder the lines AB and BC twise is incoÌ„meÌ„surable to the square of the line AC Wherefore by the first part of the same that which is contayned vnder the lines AB and BC twise together with the square of the line AC that is the squares of the lines AB and BC are incommensurable to the square of the line AC But the squares of the lines AB and
BC are rationall for the lines AB and BC are put to be rationall wherfore the line AC is irrationall and is called a residuall line which was required to be proued An other demonstration after Campane Campane demonstrateth this Proposition by a figure more briefly after this mâ—ner Let the superficies EG be equall to the squares of the lines AB and BC added together which shall be rationall for that the lines AB and BC are supposed to be rationall coÌ„mensurable in power onely FroÌ„ which superficies take away the superficies DF equall to that which is conâ—â—yaâ—d vnder the lines AB DC twise which shall be mediall by the 21. of this booke Now by the 7. of the second the superficies FG is equall to the square of the line AC And forasmuch as the superficies EG is incommensurable to the superficies DF for that the one is rationall and the other mediall therefore by the 16. of this booke the 〈◊〉 superficies EG is incommensurable to the superficies FG. Wherfore the superficies FG is irrationall And therefore the line AC which contayneth it in power is irrationall which was required to be proued An annotation of P. Montaâ—reâ—s This Theoreme teacheth nothing els but that that portion of the greater name of a binomiall line which remayneth after the taking away of the lesse name from the greater name is irrationall which is called a residuall line that is to say if from the greater name of a binomiall line which greater name is a rationall line coÌ„mensurable in power onely to the lesse name be taken away the lesse name which selfe lesse name is also commensurable in power onely to the greater name which greater name this Theoreme calleth the whole line the rest of the line which remaineth is irrational which he calleth a residuall line Wherfore all the lines which are intreated in this Theoreme and in the fiue other which follow are the portions remayning of the greater partes of the whole lines which were intreated of in the 36.37.38.39.40.41 propositioÌ„s after the taking away the lesse part from the greater In this proposition is set forth the nature of the eight kinde of irrationall lines which is called a residuall line the definition whereof by this proposition is thus A residuall line is an irrationall line which remayneth when from a rationall line geuen is taken away a rationall line commensurable to the whole line in power onely ¶ The 56. Theoreme The 74. Proposition If from a mediall line be taken away a mediall line commensurable in power onely to the whole line and comprehending together with the whole line a rationall superficies the residue is an irrationall line and is called a first mediall residuall line Out of this proposition is taken the definition of the ninth kinde of irrationall lines which is called a first residuall mediall line the difinition whereof is thus A first residuall mediall line is an irrationall line which remayneth when from a mediall line is taken away a mediall line commensurable to the whole in power onely and the part taken away and the whole line contayne a mediall superficies An other demonstration after Campane Let the line DE be rationall vpon which apply the superficies DF equall to that which is contâ—ined vnder the lynes AB and BC twise and let the superficies GE be equal to that which is composed of the squares of the lynes AB and BC wherfore by the 7. of the second the superficies FG is equal to the square of the lyne AC And forasmuch as by supposition the superficies EG is mediall therfore by the 22. of the tenth the lyne DG is rationall coÌ„mensurable in power onely to the rational lyne DE. And forasmuch as by supposition the superficies EH is rational therfore by the 20. of the tenth the line DH is rational commensurable in length vnto the rationall line DE. Wherfore the lynes DG and DH are rationall commensurable in power only by the assumpt put before the 13. of this boke Wherfore by the 73 of this boke the lyne GH is a residuall lyne and is therefore irrationall Wherfore by the corollary of the 21. of this boke the superficies FG is irrational And therfore the line AC which coÌ„tayneth it in power is irrationall and is called a first medial residuall lyne ¶ The 57. Theoreme The 75. Proposition If from a mediall lyne be taken away a mediall lyne commensurable in power only to the whole lyne and comprehending together with the whole lyne a mediall superficies the residue is an irrationall lyne and is called a second mediall residuall lyne SVppose that AB be a mediall line and from AB take away a mediall line CB commensurable in power onely to the whole line AB and comprehending together with the whole line AB a mediall superficies namely the parallelogramme contained vnder the lines AB and BC. Then I say that the residue namely the line AC is irrationall and is called a second mediall residuall line Take a rationall line DI and by the 44. of the first vnto the line DI apply the parallelogramme DE equall to the squares of the lines AB BC and making in bredth the line DG And vnto the same line DI apply the parallelogramme DH equall to that which is coÌ„tained vnder the lines AB BC twise and makyng in breadth the line DF. Now the parallelogramme DH is lesse then the parallelogramme DE for that also the square of the lines AB and BC are greater then that which is contained vnder the lines AB and BC twise by the square of the line AC by the 7. of the second Wherfore the parallelogramme remayning namely FE is equal to the square of the line AC And forasmuch as the squares of the lines AB and BC are mediall therfore also the parallelogramme DE is mediall and is applied to the rationall line DI making in bredth the line DG Wherfore by the 22. of the tenth the line DG is rational and incommensurable in length to the line DI. Againe forasmuch as that which is contained vnder the lines AB and BC is mediall therfore also that which is contained vnder the lines AB and BC twise is mediall but that which is contained vnder the lines AB and BC twise is equall to the parallelogramme DH Wherfore the parallelogramme DH is mediall and is applied to the rationall line DI making in breadth the line DF. Wherfore the line DF is rationall and incommensurable in length to the line DI. And forasmuch as the lines AB and BC are coÌ„mensurable in power onely therfore the line AB is incommeÌ„surable in leÌ„gth to the line BC. Wherfore by the assumpt going before the 22. of the tenth and by the 10. of the tenth the square of the line AB is incommensurable to that which is contained vnder the lines AB and BC. But vnto the square of the line AB are commensurable
the squares of AB and BC by the 15. of the tenth And vnto that which is contained vnder the lines AB and BC is commeÌ„surable to that which is contained vnder the lines AB and BC twise Wherfore the squares of the lines AB and BC are incommensurable to that which is contained vnder the lines AB and BC twise But vnto the squares of the lines AB and BC is equal the parallelograÌ„me DE and to that which is coÌ„tained vnder the lines AB and BC twise is equall the parallelogramme DH Wherefore the parallelograÌ„me DE is incoÌ„meÌ„surable to the parallelogramme DH But as the parallelogramme DE is to the parallelogramme DH so iâ— the line GD to the line DF. Wherfore the line GD is incommensurable in leÌ„gth to the line DF. And either of theÌ„ is rationall Wherfore the lines GD and DF are rationall commensurable in power onely Wherfore the line FG is a residuall line by the 73. proposition of the tenth And the line DE is a rationall line but a supeâ—ficies comprehended vnder a rationall line and an irrationall line is irrationall by the 21 of the teâ—â—â—â— and the line which containeth in power the same superâ—icies is irrationall by the assumpt going before the same Wherfore the parallelograme FE is irrationall But the line AC containeth in power the parallelogramme FE Wherfore the line AC is an irrationall line and is called a second mediall residuall line And this second mediall residuall line is that part of the greater part of a bimediall line which remayneth after the taking away of the lesse part from the greater which was required to be proued An other demonstrtion more briefe after Campane This proposition setteth forth the nature of the tenth kinde of irrational lines which is called a second residuall mediall line which is thus defined A second residualâ— mediall lyne is an irrationall lyne which remayneth when from a medial line is taken away a mediall lyne commensurable to the whole in power onely and the part taken away the whole lyne contayne a mediall superficies ¶ The 58. Theoreme The 76. Proposition Iâ—â—rom a right line be taken away a right line incommensurable in power to the whole and if that which is made of the squares of the whole line and of the line taken away added together be rationall and the parallelograÌ„me contained vnder the same lines mediall the line remayning is irrationall and is called a lesse line In thys Proposition is contayned the definition of the eleuenth kinde of irrationall lines which is called a lesse line whose definition is thus A lesse line is an irrationall line which remayneth wheÌ„ from a right line is taken away a right line incommensurable in power to the whole and the square of the whole line the square of the part taken away added together make a rationall superficies and the parallelogramme contayned of them is mediall This Proposition may after Campanes way be demonstrated if you remember well the order positions which he in the three former Propositions vsed ¶ The 19. Theoreme The 77. Proposition If from a right line be taken away a right line incommensurable in power to the whole line and if that which is made of the squares of the whole line and of the line taken away added together be mediall and the parallelogramme contained vnder the same lines rationall the line remaining is irrationall and is called a line making with a rationall superficies the whole superficies mediall In this Proposition is declared the nature of the twelueth kind of irrationall lines which is called a line making with a rationall superficies the whole superficies mediall whose definition is thus A line making with a rationall superficies the whole superficies mediall is an irrationall line which remaineth wheÌ„ froÌ„ a right line is taken away a right line incoÌ„mensurable in power to the whole line and the square of the whole line the square of the part taken away added together make a mediall superficies and the parallelogramme contained of them is rationall This Proposition also may after Campanes way be demonstrated obseruing the former caution ¶ The 60. Theoreme The 78. Proposition If from a right line be taken away a right line incommensurable in power to the whole line and if that which is made of the squares of the whole line and of the line taken away added together be medial and the parallelogramme contayned vnder the same lines be also mediall and incommensurable to that which is made of the squares of the sayd lines added together the line remayning is irrationall and is called a line making with a mediall superficies the whole superficies mediall This proposition may thus more briefely be demonstrated forasmuch as that which is composed of the squares of the lines AB and BC is mediall and that also which is contayned vnder them is mediall therefore the parallelogrammâ—s DE and DH which are equall vnto them are mediall but a mediall superficies exceedeth not a mediall superficies by a rationall superficies Wherefore the superficies FE which is the excesse of the mediall superficies DE aboue the mediall superficies DH is irrational And therefore the line AC which contayneth it in power is irrationall c. In this proposition is shewed the conditioÌ„ and nature of the thirtenth and last kinde of irrationall lines which is called a line making with a mediall superficies the whole superficies mediall whose definition is thus A line making with a mediall superficies the whole superficies mediall is an irrationall line which remayneth when from a right liâ—e is taken away a right line incommensurable in power to the whole line and the squares of the whole line and of the line taken away added together make a mediall superficies and the parallelogramme contayned of theÌ„ is also a mediall superficies moreouer the squares of them are incommensurable to the parallelogramme contayned of them An assumpt of Campane If there be fower quaÌ„tities if the difference of the first to the second be as the difference of the third to the fourth then alternately as the difference of the first is to the third so is the difference of the second to the fourth This is to be vnderstand of quaÌ„tities in like sort referred the one to the other that is if the first be greater then the second the third ought to be greater then the fourth and if the first be lesse then the second the third ought to be lesse then the fourth and is also to be vnderstand in arithmeticiall proportionality As for example let the difference of A be vnto B as the difference of C is to D. Then I say that as the difference of A is to C so is the difference of B to D. For by this common seÌ„tence the difference of the extreames is composed of the differences of the extreames to the meanes the difference of A to C is composed of the difference of A
to B and of the difference of B to C. And by the same common sentence the difference of B to D is composed of the difference of B to C and of â—he difference of C to D. And forasmuch as by supposition the difference of A to B is as the difference of C to D and the difference of B to C is common to them both Wherefore it followeth that as the difference of A is to C so is the difference of B to D which was required to be proued ¶ The 61. Theoreme The 79. Proposition Vnto a residual line can be ioyned one onely right lyne rational and commensurable in power onely to the whole lyne LEt AB be a residuall line and vnto it let the line BC be supposed to be ioyned so that let the lines AC and BC be rationall commensurable in power onely Then I say that vnto the line AB cannot be ioyned any other rationall line commensurable in power onely to the whole line For if it be possible let BD be such a line added vnto it Wherfore the lines AD and DB are rationall commensurable in power onely And forasmuch as how much the squares of the lines AD and DB do exceede that which is contained vnder the lines AD and DB twise so much also do the squares of the lines AC and CB exceede that which is contained vnder the lines AC and CB twise for the excesse of eche is one and the same namely the square of the line AB by the 7. of the second Wherâ—ore alternately by the â—ormer assumpt of Campanus how much the squares of the lines AD and DB do exceede the squares of the lines AC CB so much also excedeth that which is contayned vnder the lines AD and DB twise that which is contained vnder the lines AC and CB twise But that which is made of the squares of the lines AD and DB added together exceedeth that which is made of the squares of the lynes AC and CB added together by a rationall superâ—icies for they are either of them rational Wherefore that which is contained vnder the lines AD and DB twise exceedeth that which is contained vnder the lines AC and CB twise by a rationall superficies But that which is contained vnder the lines AD and DB twise is mediall for it is commensurable to that which is contained vnder the lines AD and DB once which superficies is mediall by the 21. of the tenth and by the same reason also that which is contained vnder the lynes AC and CB twise is mediall Wherfore a mediall superficies differeth from a mediall superficies by a rationall superficies which by the 26. of the tenth is impossible Wherfore vnto the lyne AB cannot be ioyned any other rationall line besides BC commensurable in power onely to the whole line Wherfore vnto a residuall line can be ioyned one onely right line rationall and commensurable in power onely to the whole lyne which was required to be demonstrated ¶ The 62. Theoreme The 80. Proposition Vnto a first medial residuall line can be ioyned one onely mediall right lyne commensurable in power onely to the whole lyne and comprehendyng wyth the whole lyne a rationall superficies SVppose that AB be a first mediall residuall line vnto AB ioyne the lyne BC so that let the lynes AC and BC be mediall commensurable in power onely let that which is contained vnder the lines AC and BC be rationall Then I say that vnto the lyne AB cannot be ioyned any other mediall line commensurable in power onely to the whole lyne and comprehending together with the whole lyne a rationall superâ—icies For if it be possible let the line BD be such a line Wherfore the lynes AD and DB are mediall commensurable in power onely and that which is contayned vnder the lynes AD and DB is rational And forasmuch as how much the squares of the lynes AD and DB exceede that which is contayned vnder the lynes AD and DB twise so much also exceede the squares of the lynes AC BC that which is contayned vnder the lynes AC and CB twise for the excesse of eche is one and the same namely the square of the lyne AB Wherfore alternately as it was sayd in the former proposition how much the squares of the lynes AD and DB exceede the squares of the lines AC and CB so much also that which is contained vnder the lines AD and DB twise excedeth that which is contained vnder the lines AC and CB twise But that which is contained vnder the lines AD and DB twise excedeth that which is contained vnder the lines AC and CB twise by a rationall superficies for they are either of them a rationall supersicies Wherfore that which is made of the squares of the lines AD DB excedeth that which is made of the squares of the lines AC CB by a rationall superficies which by the 26. of the tenth is impossible For they are either of them mediall for those foure lines were put to be mediall Wherfore vnto a first mediall residuall line can be ioyned onely one right mediall line commensurable in power onely to the whole line and comprehending with the whole line a rationall superâ—icies which was required to be proued ¶ The 63. Theoreme The 81. Proposition Vnto a second mediall residuall line can be ioyned onely one mediall right line commensurable in power onely to the whole line and comprehending with the whole line a mediall superficies SVppose that AB be a secoÌ„d mediall residual line vnto the line AB ioyne the line BC so that let the lines AC and CB be mediall coÌ„mensurable in power onely and let that which is comprehended vnder the lines AC and CB be mediall Then I say that vnto the line AB can not be ioyned any other mediall right line coÌ„mensurable in power onely to the whole line and comprehending together with the whole line a mediall superficies For if it be possible let the line BD be such a line Wherefore the lines AD DB are mediall commensurable in power onely and that which is contained vnder the lines AD and DB is also mediall Take a rationall line EF. And by the 44. of the first vnto the line EF apply the parallelogramme EG equall to the squares of the lines AC and CB and making in breadth the line EM and from that parallelogramme EG take away the parallelogramme HG equall to that which is contained vnder AC and CB twise and making in breadth the line HM Wherefore the parallelogramme remayning namely EL is by the 7. of the second equall to the square of the line AB Wherefore the line AB containeth in power the parallelograme EL. Againe vnto the line EF apply by the 44. of the first the parallelogramme EI equall to the squares of the lines AD and DB and making in breadth the line EN
But the squares of the lines AD and DB are equall to that which is contained vnder the lines AD and DB twise and to the square of the line AB Wherefore the parallelogramme EI is equall to that which is contained vnder the lines AD DB twise and to the square of the line AB But the parallelogramme EL is equall to the square of the line AB Wherefore the parallelogramme remaining namely HI is equall to that which is contained vnder the lines AD and DB twise And forasmuch as the lines AC and CB are â—â—â—iall thereâ—ore the squares also of the lines AC and CB are mediall and they are equall to the parallelogramme EG wherefore the parallelogramme EG is by that which was spoken in the 75. Proposition mediall and it is applied vnto the rationall line EF making in breadth the line EM Wherefore by the 22. of the tenth the line EM is rationall and inâ—ommensurable in length to the line EF. Againe forasmuch as that which is contayned vnder the lines AC and CB is mediall therefore by the Corollary of the 23. of the tenth that which is contained vnder the lines AC and CB twise is also mediall and it is equall to the parallelogramme HG wherefore also the parallelogramme HG is mediall and is applyed to the rationall line EF making in breadth the line HM Wherefore by the 22 of the tenth the line HM is rationall and incommensurable in length to the line EF. And forasmuch as the lines AC and CB are commensurable in power onely therefore the line AC is incommensurable in length to the line CB. But as the line AC is to the line CB so by the Assumpt going before the 22. of the tenth is the square of the line AC to that which is contayned vnder the lines AC CB. Wherefore by the 10. of the tenth the square of the line AC is incommensurable to that which is contained vnder the lines AC and CB. But vnto the square of the line AC are commensurable the squares of AC CB and vnto that which is contained vnder the lines AC and CB is commensurable that which is contained vnder the lines AC and CB twise Wherefore the squares of the lines AC CB are incommensurable to that which is contained vnder the lines AC and CB twise But vnto the squares of the lines AC and CB is equall the parallelogramme EG and vnto that which is contained vnder the lines AC CB twise is equall the parallelogramme GH Wherefore the parallelogramme EG is incommensurable to the parallelogramme HG But as the parallelogramme EG is to the parallelogrmme HG so is the line EM to the line HM Wherefore the line EM is incommensurable in length to the line HM And they are both rationall lines Wherefore the lines EM and MH are rationall commensurable in power onely Wherefore the line EH is a residuall line and vnto it is ioyned a rationall line HM commensurable in power onely to the whole line EM In like sort also may it be proued that vnto the line EH is ioyned the line HN being also rationall and commensurable in power onely to the whole line EN Wherefore vnto a residuall line is ioyned â—â—re then one onely line commensurable in power onely to the whole line which by the 79. of the tenth is impossible Wherefore vnto a second mediall residuall line can be ioyned onely one mediall right line commensurable in power onely to the whole line and comprehending with the whole line a mediall superficies which was required to be demonstrated ¶ The 64. Theoreme The 82. Proposition Vnto a lesse line can be ioyned onely one right line incommensurable in power to the whole lyne and making together with the whole lyne that which is made of their squares added together rationall and that which is contayned vnder them mediall SVppose that AB be a lesse line and to AB ioyne the line BC so that let BC be such a line as is required in the Theoreme Wherfore the lines AC and CB are incoÌ„mensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder them mediall Then I say that vnto AB cannot be ioyned any other such right line For if it be possible lâ—t the lyne BD be such a line Wherfore the lines AD DB are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder them mediall And for that how much the squares of the lines AD and DB excede the squares of the lines AC and CB so much that which is contained vnder the lines AD and DB twise excedeth that which is contained vnder the lines AC and CB twise by those things which were spoken in the 79. proposition But that which is made of the squares of the lines AD and DB added together excedeth that which is made of the squares of the lines AC and CB added together by a rationall superficies for they are either of them rationall by supposition Wherfore that which is contained vnder the lines AD and DB twise excedeth that which is contained vnder the lines AC and CB twise by a rationall superficies which by the 26. of the tenth is impossible for either of them is mediall by supposition Wherfore vnto a lesse line can be ioyned onely one right line incommensurable in power to the whole line and making together with the whole line that which is made of their squares added together rationall and that which is contained vnder them mediall which was required to be demonstrated ¶ The 65. Theoreme The 83. Proposition Vnto a line making with a rationall superficies the whole superficies mediall can be ioyned onely one right lyne incommensurable in power to the whole lyne and making together with the whole line that which is made of their squares added together mediall and that which is contained vnder them rationall SVppose that AB be a line making with a rationall superficies the whole superficies mediall and vnto it let the line BC be ioyned so that let BC be such a line as is required in the Theoreme Wherfore the lines AC and CB are incommensurable in power haâ—ing that which is made of the squares of the lines AC and CB added together mediall and that which is contained vnder the lynes AC and CB rationall Then I say that vnto the lyne AB cannot be ioyned any other such line For if it be possible let the line BD be such a line Wherfore the lines AD and DB are incommensurable in power hauing that which is made of the squares of the lines AD and DB added together mediall and that which is contained vnder the lines AD and DB rationall Now for that how much the squares of the lines AD and DB exceede the squares of the lines AC and CB so much
that which is contained vnder the lines AD and DB twise exceedeth that which is contayned vnder the lynes AC and CB twise by that which was spoken in the 79. proposition But that which is contained vnder the lines AD and DB twise exceedeth that which is contained vnder the lines AC and CB twise by a rationall superficies for they are either of them rationall by suppositioÌ„ Wherfore that which is made of the squares of the lines AD and DB added together excedeth that which is made of the squares of the lines AC and CB added together by a rationall superficies which by the 26. of the tenth is impossible for they are either of them mediall by supposition Wherfore vnto the line AB cannot be ioyned any other lyne besides BC making that which is required in the proposition Wherfore vnto a line making with a rationall superficies the whole superficies mediall can be ioyned onely one right line incoÌ„mensurable in power to the whole line and making together with the whole line that which is made of their squaâ—es added together mediall and that which is coÌ„tained vnder them rationall which was required to be proued ¶ The 66. Theoreme The 84. Proposition Vnto a line making with a mediall superficies the whole superficies medial can be ioyned onely one right line incommensurable in power to the whole line and making together with the whole line that which is made of their squares added together mediall and that which is contained vnder them mediall and moreouer making that which is made of the squares of them added together incommensurable to that which is contayned vnder them SVppose that AB be a line making with a mediall superficies the whole superficies mediall and vnto it let the line BC be ioyned so that let BC be such a line as is required in the Theoreme Wherefore the lines AC and CB are incommensurable in power hauing that which is made of the squares of the lines AC and CB added together mediall that which is contained vnder the lines AC and CB mediall and moreouer that which is made of the squares of the lines AC and CB is incommensurable so that which is contained vnder the lines AC and CB. Then I say that vnto the line AB can be ioyned no other such line For if it be possible let BD be such a line Wherefore the lines AD and DB are incoÌ„mensurable in power hauing that which is made of the squares of the lines AD and DB added together mediall and that which is contained vnder the lines AD and DB mediall and moreouer that which is made of the squares of the lines AD and and DB added together is incommensurable to that which is contained vnder the lines AD and DB. Take a rationall line EF. And by the 44. of the first vnto the line EF apply the parallelogramme EG equall to the squares of the lines AC and CB and making in breadth the line EMâ— and from the parallelogramme EG take away the parallelogramme HG equall to that which is contained vnder the lines AC CB twise and making in breadth the line HM Wherefore the residue namely the square of the line AB is equall to the parallelogramme EL by the 7. of the second Wherefore the line AB contayneth in power the parallelogramme EL. Againe by the 44. of the first vnto the line EF apply the parallelogramme Eâ— equall to the squares of the lines AD and DB and making in breadth the line EN But the square of the line AB is equall to the parallelograÌ„me EL. Wherefore the residue namely the parallelogramme HI is equall to that which is contained vnder the lines AD and DB twise And forasmuch as that which is made of the squares of the lines AC and CB is mediall and is equall to the parallelogramme EG therefore also the parallelogramme EG is mediall And it is applied vnto the rationall line EF making in breadth the line EM Wherefore by the 22. of the tenth the line EM is rationall and incommensurable in length to the line EF. Againe forasmuch as that which is contained vnder the lines AC and CB twise is mediall and is equall to the parallelogramme HG Wherefore the parallelogramme Hâ—G is mediall which parallelogramme HG is applied to the rationall line EF making