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
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
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 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
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
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
fift binomiall line But if neither of the lines AE nor EB be commensurable in length to the rationall line geuen neither also of the lines CF nor FD shal be commensurable in length to the same and so either of the lines AB and CD shal be a sixt binomiall line A line therefore commensurable in length to a binomiall line is also a binomiall line of the selfe same order which was required to be proued ¶ The 49. Theoreme The 67. Proposition A line commensurable in length to a bimediall line is also a bimediall lyne and of the selfe same order SVppose that the line AB be a bimediall line And vnto the line AB let the lyne CD be commensurable in length Then I say that the line CD is a bimediall line and of the self order that the line AB is Deuide the line AB into his partes in the point E. And forasmuch as the line AB is a bimediall line and is deuided into his partes in the point E therfore by the 37. and 38. of the tenth the lines AE and EB are medials commensurable in power onely And by the 12. of the sixt as the line AB is to the line CD so let the line AE be to the line CF. Wherfore by the 19. of the fift the residue namely the line EB is to the residue namely to the line FD as the line AB is to the line CD But the line AB is commensurable in length to the lyne CD Wherfore the line AE is commensurable in length to the line CF and the line EB to the line FD. Now the lines AE and EB are mediall wherfore by the 23. of the tenth the lines CF and FD are also mediall And for that as the line AE is to the line EB so is the line CF to the line FD. But the lines AE and EB are commensurable in power onely wherfore the lines CF and FD are also commensurable in power onely And it is proued that they are mediall Wherfore the lyne CD is a bimediall line I say also that it is of the selfe same order that the line AB is For for that as the line AE is to the line EB so is the line CF to the line FD but as the line CF is to FD so is the square of the lyne CF to the parallelogramme contained vnder the lynes CF and FD by the first of the sixt Therfore as the line AE is to the line EB so by the 11. of the fift is the square of the line CF to the parallelogramme contained vnder the lines CF and FD but as AE is to EB so by the 1. of the sixt is the square of the line AE to the parallelogramme contained vnder the lines AE and EB therfore by the 11. of the fift as the square of the line AE is to that which is contained vnder the lines AE and EB so is the square of the line CF to that which is contained vnder the lines CF and FD. Wherfore alternately by the 16. of the fift as the square of the line AE is to the square of the line CF so is that which is contained vnder the lines AE and EB to that which is contained vnder the lines CF FD. But the square of the line AE is commensurable to the square of the line CF because AE and CF are commensurable in length Wherfore that which is contained vnder the lines AE and EB in commensurable to that which is contained vnder the lines CF and FD. If therfore that which is contained vnder the lines AE and EB be rationall that is if the line AB be a first bimediall line that also which is contained vnder the lines CF and FD is rationall Wherfore also the line CD is a first bimediall line But if that which is contained vnder the lines AE and EB be mediall that is if the line AB be a second bimediall line that also which is contayned vnder the lines CF and FD is mediall wherfore also the line CD is a second bimediall line Wherfore the lines AB and CD are both of one and the selfe same order which was required to be proued ¶ A Corollary added by Flussates but first noted by P. Montaâ—reus A line commensurable in power onely to a bimediall line is also a bimediall line and of the selfe same order Suppose that AB be a bimediall line either a first or a second wherunto let the line GD be coÌ„mensurable in power onely Take also a rationall line EZ vpon which by the 45. of the first apply a rectangle parallelogramme equall to the square of the line AB which let be EZFC and let the rectangle parallelogramme CFIH be equall to the square of the line GD And forasmuch as vpon the rationall line EZ is applyed a rectangle parallelogramme EF equall to the square of a first bimediall line therefore the other side therof namely EC is a second binomiall line by the 61. of this booke And forasmuch as by supposition the squares of the lines AB GD are commensurable therefore the parallelogrammes EF and CI which are equall vnto them are also commensurable And therefore by the 1. of the sixt the lines EC and CH are commensurable in length But the line EC is a second binomiall line Wherefore the line CH is also a second binomiall line by the 66. of this booke And forasmuch as the superficies CI is contayned vnder a rationall line EZ or CF and a second binomiall line CH therefore the line which contayneth it in power namely the line GD is a first bimediall line by the 55. of this booke And so is the line GD in the selfe same order of bimediall lines that the line AB is The like demonstration also will serue if the line AB be supposed to bâ— a second bimediall line For so shall it make the breadth EC a third binomiall line whereunto the line CH shall be commensurable in length and therefore CH also shall be a third binomiall line by meanes whereof the line which contayneth in power the superficies CI namely the line GD shall also be a second bimediall line Wherefore a line commensurable either in length or in power onely to a bimediall line is also a bimediall line of the selfe same order But so is it not of necessitie in binomiall lines for if their powers onely be commensurable it followeth not of necessitie that they are binomialls of one and the selfe same order but they are eche binomialls eyther of the three first kindes or of the three last As for example Suppose that AB be a first binomiall line whose greater name let be AG and vnto AB let the DZ be coÌ„meÌ„surable in power onely Then I say that the line DZ is not of the selfe same order that the line AB is For if it be possible let the line DZ be of the selfe same order that the line AB is Wheâ—efore the line DZ may
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
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 in breadth the line HM Wherefore the line HM is rationall and incommensurable in length to the line EF. And forasmuch as the squares of the lines AC and CB are incommensurable to that which is contained vnder the lines AC and CB twise therefore the parallelogramme EG is incommensurable to the parallelogramme HG Wherefore the line EM is incommensurable in length to the line MH and they are both rationall Wherefore the lines EM and MH are rationall commensurable in power onely Wherefore EH is a residuall line And the line ioyned vnto it is HM And in like sort may we proue that the line EH is a residuall line and that the line HN is ioyned vnto it Wherefore vnto a residuall line is ioyned two sundry lines being eche commensurable in power onely so the whole lineâ— which by the 79. of the tenth is impossible Wherefore vnto the line AB can not be ioyned any other right line besides the line BC which shall be incommensurable in power to the whole line haue 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 incommensurable to that which is made of their squares added together Wherefore vnto a line making with a mediall superficies the whole superficies mediall 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 contained vnder theÌ„ which was required to be proued ¶ Third Definitions AS of binomiall lynes there are 6. diuers kindes so also of residuall lynes which are correspondent vnto them and depend of them for a residuall line is nothing els as was before said but that which remayneth wheÌ„ the lesse part of a binomiall line is taken from the greater part or name therof there are likewise sixe seuerall kindes All which are knowne and considered in comparison to a rationall line set forth appointed and these residual line haue the selfe same order of production that the binomials had For as the three first kindes of binomiall lines namely the first second and third were produced when the square of the greater part of the part of the binomiall excedeth the square of the lesse part thereof by the square of a line commensurable vnto it in length so in likewise the first three kindes of residuall lines namely the first second and
FG is commensurable to the square of the line GH But the square of the line FG is rationall wherfore the square also of the line GH is rationall wherfore the line GH is also rationall And for that the number Bâ— hath not to the number CD that proportion that a square number hath to a square number therfore the line FG is incommensurable in length to the line GH and they are both rationall Wherefore the lines FG and GH are rationall commensurable in power onely Wherfore the lyne FH is a residuall line I say moreouer that it is a sixt residuall line For for that as the number E is to the number BC so is the square of the line A to the square of the line FG and as the number BC is to the nuÌ„ber CD so is the square of the line FG to the square of the line GH therefore by equalitie of proportion as the number E is to the number CD so is the square of the line A to the square of the line GH But the number E hath not to the number CD that proportion that a square number hath to a square number Wherefore the line A is inâ—â—mmensurable in length to the line GH and neither of these lines FG nor Gâ— is commensurable in length to the rationall line A. And forasmuch as the square of the line FG is greater then the square of the line GH vnto the square of the line FG let the the squares of the lines GH and K be equall Now therfore for that as the number Bâ— is to the number CD so is the square of the line FG to the square of the line GH therefore by conuersion of proportion as the number BC is to the number BD so is the square of the line FG to the square of the line K. But the number BC hath not to the number BD that proportion that a square number hath to a square number therfore the line FG is incommensurable in length to the line K. Wherfore the line FG is in power more then the lyne GH by the square of a line incommensurable in length to the line FG and neither of the lines FG nor GH is commensurable in length to the rationall line A. Wherfore the line FH is a sixt residual line Wherfore there is found out a sixt residuall line which was required to be done There is also a certayne other redier way to finde out euery one of the forsayd sixe residuall lines which is after this maner Suppose that it were required to finde out a first residuall line Take a first binomiall line AC let the greater name thereof be AB And vnto the line BC let the line BD be equall Wherefore the lines AB and BC that is the lines AB and BD are rationall commensurable in power onely and the line AB is in power more then the line BC that is then the line BD by the square of a line commensurable in length to the line AB And the line AB is commensurable in length to the rationall line geuen For the line AC is put to be a first binomiall line Wherefore the line AD is a first residual line And in like maner may ye finde out a second a third a fourth a fift and a sixt residuall line if ye take for eche a binomiall line of the same order ¶ The 67. Theoreme The 91. Proposition If a superficies be contayned vnder a rationall line a first residuall line the line which contayneth in power that superficies is a residuall line SVppose that there be a rectangle superficies AB contayned vnder a rationall line AC and a first residuall line AD. Then I say that the line which contayneth in power the superficies AB is a residuall line For forasmuch as AD is a first residuall line let the line ioyned vnto it be DG by the line ioyned vnto it vnderstand such a line as was spoken of in the end of the 79. proposition Wherefore the lines AG and GD are rationall coÌ„meÌ„surable in power only the whole line AG is coÌ„mensurable in length to the rationall line AC and the line AG is in power more then the line GD by the square of a line commensurable in length vnto AG by the definition of a first residuall line Deuide the line GD into two equall partes in the poynt E. And vpon the line AG apply a parallelogramme equall to the square of the line EG and wanting in figure by a square and let the sayd parallelogramme be that which is coÌ„tayned vnder the lines AF and FG. Wherefore the line AF is commensurable in length to the line FG by the 17. of the tenth â— And by the poyntes E F and G draw vnto the line AC these parallel lines EH FI and GK And make perfect the parallelograme AK And forâ—as much as the line AF is commensurable in length to the line FG therefore also the whole line AG is commensurable iâ— length to either of the lines AF and FG by the 15. of the tenth But the line AG is commensurable in length to the line AC Wherefore either of the lines AF and FG is commensurable in length to the line AC But the line AC is rationall wherefore either of the lines AF and FG is also rationall Wherefore by the 19. of the tenth either of the parallelogrammes AI and FK is also rationall And forasmuch as the line DE is commeÌ„surable in length to the line EG therfore also by the 15. of the tenth the line DG is commensurable in length to either of the lines DE and EG But the line DG is rationall wherefore either of the lines DE and EG is rationall and the selfe same line DG is incommensurable in length to the line AC by the definition of a first residuall line or by the 13. of the tenthâ— For the line DG is incommensurable in length to the line AG which line AG is coÌ„mensurable in length to the line AC wherfore either of the lines DE and EG is rationall and incommensurable in length to the line AC â— Wherefore by the 21. of the tenth either of these parallelogrammes DH and EK is mediall Vnto the parallelogramme AI let the square LM be equall and vnto the parallelogramme FK let the square NX be equall being taken away from the square LM â— and haâ—ing the angle LOM common to them both And to doo this there must be founde out the meane proportionall betwene the lines FI and FG. For the square of the meane proportionall is equall to the parallelogramme contayned vnder the lines FI and FG. And from the line LO cut of a line equall to the meane proportionall so founde out and descriâ—e the square thereof Wherefore both the squares LM and NX are about one and the selfe same diameter by the 20. of the sixth let their diameter be OR and describe the figure as
it is hâ—â—e sâ—t forthâ— Now then forasmuch as the parallelogramme contayned vnder the lines AF FG is equal to the square of the line EG therefore by the 17. of the sixth as the line AF is to the line EG so is the line EG to the line FG. But as the line AF is to the line EG so is thâ— parallelogramme AI to the parallelogramme EK And as the line EG is to the line FG so is the parallelogramme EK to the parallelogramme FK Wherfore betwene the pagrammes AI and FK the parallelogramme EK is the meane proportionall But by the second part of the assumpt going before the 54. of the tenth beâ—wene the squares LM and NX the parallelogramme MN is the meane proportionall And vnto the parallelogramme AI is equall the square LM and vnto the parallelogramme FK is equall the square NX by construction Wherefore the parallelogramme MN is equall to the parallelogramme EK by the 2. assumpt going before the 54. of the tenth But the parallelogramme EK is by the first of the sixth equall to the parallelogramme DH and the parallelogramme MN is by the 43. of the first equall to the parallelogramme LX. Wherefore the whole parallelogramme DK is equall to the gnomon VTZ which gnomon consisteth of those parallelogrammes by which ye see in the figure passeth a portion of a circle greater then a semicircle and moreouer to the square NX and the parallelogramme AK is equall to the squares LM and NX by construction and it is now proued that the parallelogramme DK is equall to the gnomoÌ„ VTZ and moreouer to the square NX Wherfore the residue namely the parallelogramme AB is equall to the square SQ which is the square of the line LN Wherefore the square of the line LN is equall to the parallelogramme AB Wherefore the line LN contayneth in power the parallelogramme AB I say moreouer that the line LN is a residuall line For forasmuch as either of these parallelogrammes AI and FK is rationallâ— as it is before sayd therefore the squares LM and NX which are equall vnto them that is the squares of the lines LO and ON are rationall Wherefore the lines LO and ON are also rationall Agayne forasmuch as the parallelogramme DH that is LX is mediall therefore the parallelogramme LX is incommensurable to the square NX Wherefore by the 1. of the sixth and 10. of the tenth the line LO is incoÌ„mensurable in length to the line ON â— and they are both rationall Wherefore they are lines rationall commensurable in power onely Wherefore LN is a residuall line by the definition and it contayneth in power the paralleloparallelogramme AB If therefore a superficies be contayned vnder a rationall line and a first residual line the line which contayneth in power that superficiesâ— is a residuall line which was required to be demonstrated ¶ The 68. Theoreme The 92. Proposition If a superficies be contained vnder a rationall line and a second residuall line the line which containeth in power that superficies is a first mediall residuall line SVppose that AB be a superficies contained vnder â—raâ—â—onall line AC and a second residuall liâ—â— AD. Then I sayâ— that the line thâ—â— containeth in power the superficies AB is a first mediall residuall line For let the line ioyned to the line AD be DG Wherefore the lines AG and GD are rationall commensurable in power onely and the line that is ioyned to the residual line namely the line DG is coÌ„mensurable in length to the rationall line AC and the line AG is in power more then the line DG by the square of a line coÌ„mensurable in length to the line AG. Deuide the line DG into two equall partes in the point E. And vnto the line AG apply a parallelogramme equall to the fourth part of the square of the line DG that is equall to the square of the line EG and wanting in figure by a square and let that parallelogramme be that which is contained vnder the lines AF and FG. Wherefore by the 1â—â— of the tenth the line AF is commensurable in length to the line FG. And by the pointes E F and G draw vnto the line AC these parallell lines EH FI and GK and forasmuch as the line AF is coÌ„mensurable in length to the line FG therefore the whole line AG is commensurable in length to either of these lines AF and FG. But the line AG is rationall and incommensurable in length to the line AC Wherefore either of these lines AF and FG are rationall and incommensurable in length to the line AC Wherefore either of thâ—se parallelogrammes AI and FK is by the 21. of the tenth mediall Againe forasmuch as the line DE is commensurable in length to the line EG therefore the line DG is commensurable in length to either of these lines DE and EG But the line DG is commensurable in length to the rationall line AC Wherefore either of these lines DE and EG is rationall and commensurable in length to the line AC Wherefore by the 19. of the tenth either of these parallelogrammes DH and EK is rationall Vnto the parallelogramme AI describe an equall square LM and vnto the parallelogramme FK let the square NX be equall as in the Proposition going before Wherefore the squares LM and NX are both about one and the same diameter Let the diameter be OR and describe the figure as is in the former Proposition expressed Now therefore forasmuch as the parallelogrammes AI and FK are mediall are commensurable the one to the other and the squares of the lines LO ON which are equall to those parallelograÌ„mes are mediall therefore the lines LO and ON are also mediall commensurable in power And it is manifest that the lines LO and ON are coÌ„mensurable in power for their squares are commensurable and those squares namely the squares of the lines LO ON are commensurable for they are equall to the parallelogrammes AI and FK which are commensurable the one to the other and that those parallelogrammes AI and FK are commensurable the one to the other hereby it is manifest for that it was before proued that the lines AF and FG are commensurable in length Wherefore by the 1. of the sixt and 10. of the tenth the parallelogrammes AI and FK are commensurable the one to the other Wherefore it is now manifest by the way of resolution that the lines LO ON are coÌ„mensurable in power â— And forasmuch as the parallelogramme contained vnder the lines AF and FG is equall to the square of the line EG therefore as the line AF is to the line EG so is the line EG to the line FG. But as the line AF is to the line EG so is the parallelogramme AI to the parallelogramme EK and as the line EG is to the line FG so is the parallelogramme EK to the parallelogramme FK Wherefore
the parallelogramme â—K is the meanâ— proportionall betwene the parallelogrammes AI and FK and the parallelogramme MN iâ— also the meane proportionall betwene the squares LM NX and the parallelogramme AI is equall to the square LM and the parallelograÌ„me FK is equall to the square NX Wherefore the parallelogramme MN is equall to the parallelogramme EK But the parallelogramme EK is equall to the parallelogramme DH and the parallelogramme LX is equall to the parallelogramme MN Wherefore the whole parallelogramme DK is equall to the Gnomon VTZ and to the square NX Wherefore the residuâ— namely the parallelogramme AB is equall to the square SQ that is to the square of the line LN Wherefore the line LN containeth in power the superficies AB I say moreouer that the line LN is a first mediall residuall line For forasmuch as the parallelogramme EK is rationall and is equall to the parallelogramme MN that is LX therefore LX that is the parallelogramme contained vnder the lines LO and ON is rationall But the square NX is mediall for it is already proued that the parallelogramme FK which is equall to the square NX is mediall Wherefore the parallelogramme LX is incommensurable to the square NX But as the parallelogramme LX is to the square NX so is the line LO to the line ON by the 1. of the sixt Wherefore by the 10. of the tenth the lines LO and ON are incommensurable in length And it is already proued that they are mediall commensurable in power Wherefore the lines LO ON are mediall commensurable in power onely containing a rationall superficies Wherefore the line LN is a first mediall residuall line and containeth in power the superficies AB which is contained vnder a rationall line and a second residuall line If therefore a superficies be contained vnder a rationall line and a second residuall line the line which containeth in power that superficies is a first mediall residuall line which was required to be demonstrated ¶ The 69. Theoreme The 93. Proposition If a superficies be contained vnder a rationall line and a third residuall line the line that containeth in power that superficies is a second mediall residuall line SVppose that AB be a superficies contained vnder a rationall line AC a third residuall line AD. Then I say that the line which containeth in power the superficies AB is a second mediall residuall line Let the line ioyned vnto AD be DG Wherefore the lines AG and GD are rationall commensurable in power onely and neither of the lines AG nor GD is commensurable in length to the rationall line AC and the whole line AC is in power more then the line GD by the square of a line commensurable in length to the line AG. Let the rest of the construction be as it was in the former Propositions Wherefore the lines AF and FG are commensurable in length and the parallelogramme AI is commensurable to the parallelogramme FK And forasmuch as the lines AF and FG are commensurable in length therefore the whole line AG is commensurable in length to either of these lines AF and FG. Bâ—â— the linâ— AG is rationall and incommensurable in length to the line AG. Wherefore either of these lines AF and FG is rationall and incommensurable in length to the line AG. Wherefore by the 21. of the tenth either of these parallelogrammes AI and FK is mediall Againâ— forasmuch as the line DB is commensurable in length to the line EG therefore also the whole line DG is commensurable in length to either of these lines DE and EG But the line DG is rationall commensurable in power only to the line AC Wherfore also either of the lines DE and EG is rationall and commensurable in power only to the line AC Wherefore either of these parallelogrammes DH and EK is mediall Againe forasmuch as the lines AG and DG are commensurable in power onely therefore they are incommensurable in length But the line AG is commensurable in length to the line AF and the line DG is commensurable in length to the line GE. Wherefore the line AF is incommensurable in length to the line EG But as the line AF is to the line EG so is the parallelogramme AI to the parallelogramme EK Wherefore the parallelogramme AI is incommensurable to the parallelogramme EK Vnto the parallelogramme AI describe an equall square LM and vnto the parallelogramme FK describe an equall square NX and describe the figure as you did in the former Proposition Now forasmuch as the parallelogramme contained vnder the lines AF and FG is equall to the square of the line EG therefore as the line AF is to the line EG so is the line EG to the line FG. But as the line AF is to the line EG so is the parallelogramme AI to the parallelogramme EK and as the line EG is to the line FG so is the parallelogramme EK to the parallelogramme FK Wherefore as the parallelograÌ„me AI is to the parallelogramme EK so is the parallelogramme EK to the parallelogramme FK Wherefore the parallelogramme EK is the meane proportionall betwene the parallelograÌ„mes AI and FK But the parallelogramme MN is the meane proportionall betwene the squares LM and NX Wherefore the parallelogramme EK is equall to the parallelogramme MN Wherefore the whole parallelogramme DK is equall to the Gnomon VTZ to the square NX And the parallelogramme AK is equall to the squares LM and NX Wherefore the residue namely the parallelogramme AB is equall to the square QS that is to the square of the line LN Wherefore the line LN containeth in power the superficies AB I say moreouer that the line LN is a second mediall residuall line For for that as it is proued the parallelogrammeâ— AI and FK are mediall therefore the squares that are equall vnto them namely the squares of the lines LO and ON are also mediall Wherefore either of these lines LO and ON is mediall And forasmuch as the parallelogramme AI is commensurable to the parallelogramme FK therefore the squares that are equall to them namely the squares of the lines LO and ON are also commensurable Againe forasmuch as it is proued that the parallelogramme AI is incommensurable to the parallelogramme EK therfore the square LM is incommensurable to the parallelogramme MN that is the square of the line LO to the parallelogramme contained vnder the lines LO ON Wherfore also the line LO is incommensurable in length to the line ON Wherefore the lines LO and ON are mediall commensurable in power onely I say moreouer that they containe a mediall superficies For forasmuch as it is proued that the parallelogramme EK is mediall therefore the parallelogramme which is equall vnto it namely the parallelogramme contained vnder the lines LO and ON is also mediall Wherefore the line LN is a second mediall residuall line and containeth in power the
superficies AB Wherefore the line that containeth in power the superficies AB is a second mediall residuall line If therefore a superficies be contained vnder a rationall line and a third residuall line the line that containeth iâ— power that superficies is a second mediall residuall line which was required to be demonstrated The 70. Theoreme The 94. Proposition If a superficies be contayned vnder a rationall lyne and a fourth residuall lyne the lyne which contayneth in power that superficies is a lesse lyne SVppose that there be a superficies AB contained vnder a rationall line AC and a â—ourth residuall line AD. Then I say that the line which containeth in power the superficies AB is a lesse line For let the line ioyned vnto it be DG Wherefore the lines AG and DG are rationall commensurable in power only and the line AG is in power more then the line DG by the square of a line incoÌ„mensurable in length to the line AG and the line AG is commensurable in length to the line AC Deuide the line DG into two equall partes in the point E. And vnto the line AG apply a parallelogramme equal to the square of the line EG and wanting in figure by a square and let that parallelograme be that which is contayned vnder the lines AF and FG. Wherfore by the 18. of the tenth the line AF is incoÌ„mensurable in length to the line FG. Draw by the pointes E F G vnto the lines AC and DB these parallel lines EH FI and GK Now forasmuch as the line AG is rational and commensurable in length to the line AC therfore the whole parallelogramme AK is by the 19. of the tenth rational Againe forasmuch as the line DG is incommensurable in length to the line AC for if the line DG were commensurable in length to the line AC then forasmuch as the line AG is commensurable in length to the same line AC the lines AG and DG should be commensurable in length the one to the other when yet they are put to be commensurable in power onely and both these lines AC and DG are rationall Wherfore the parallelograme DK is mediall Againe forasmuch as the line AF is incommensurable in length to the line FG therfore the parallelogramme AI is incommensurable to the parallelogramme FK Vnto the parallelogramme AI describe an equal square LM and vnto the parallelograme FK describe an equall square NX and let the angle LOM be common to both those squares Wherefore the squares LM and NX are about one and the selfe same diameter Let their diameter be OR and describe the figure And forasmuch as the parallelograme contained vnder the lines AF and FG is equall to the square of the line EG therfore proportionally as the line AF is to the line EG so is the line EG to the line FG but as the line AF is to the line EG so is the parallelogramme AI to the parallelogramme EK by the 1. of the sixt And as the line EG is to the line FG so is the parallelogramme EK to the parallelogramme FK Wherfore the parallelogramme EK is the meane proportionall betwene the parallelogrammes AI and FK wherfore as iâ— was said in the former propositions the parallelogramme MN is equal to the parallelogramme EK but the parallelogramme DH is equall to the parallelograme EK and the parallelogramme MN to the parallelogramme LX. Wherfore the whole parallelogramme DK is equâ—ll to the gnomon VTZ and to the square NX Wherfore the residue namely the parallelogramme AB is equall to the residue namely to the square SQ that is to the square of the line LN I say moreouer that LN is that irrationall line which is called a lesse line For forasmuch as the parallelogramme AK is rationall and is equall to the squares of the lines LO and ON therfore that which is made of the squares of the lines LO and ON added together is rationall Againe forasmuch as the parallelogramme DK is mediall and is equall to that which is contained vnder the lines LO and ON twise therefore that which is contained vnder the lynes LO and ON twise is also mediall And forasmuch as the parallelogramme AI is incommensurable to the parallelogramme FK therefore the squares which are equall vnto them namely the squares of the lines LO and ON are incommensurable the one to the other Wherfore the lines LO and ON are incommensurable in power hauyng that which is made of their squares added together rationall and that which is contained vnder them twise mediall which is commensurable to that which is contayned vnder them once Wherfore that which is contained vnder them once is also mediall Wherfore LN is that irrationall line which is called a lesse line and it containeth in power the superficies AB If therfore a superficies be contained vnder a rationall line and a fourth residuall line the line which containeth in power that superficies is a lesse line which was required to be demonstrated ¶ The 71. Theoreme The 95. Proposition If a superficies be contained vnder a rationall line and a fift residual line the line that coÌ„tayneth in power the same superficies is a line making with a rationall superficies the whole superficies mediall SVppose that there be a superficies AB contained vnder a rationall line AG and a fift residuall line AD. TheÌ„ I say that the line that coÌ„taineth in power the supâ—rficies AB is a line making with a rationall superficies the whole superficies mediall For vnto the line AD let the line DG be ioyned which shal be coÌ„meÌ„sâ—â—able in leÌ„gth to the rational line AC And let the rest of the constructioÌ„ be as in the propositioâ— next going before And forasmuch as the line AG is incoÌ„mensurable in leÌ„gth to the line AC and they are both rationall therfore the parallelograÌ„me AK is medial Againe forasmuch as the line DG is rationall and commensurable in length to the lyne AC therefâ—re the parallelogramme DK is rationall Vnto the parallelâ—gramme AI describe an equall square LM and vnto the parallelograme â—â— describe an equall square Nâ— and as in 〈◊〉 proposition next going before so also in this may we proue that the line LN containeth in power the superficies AB I say moreouer that that line LN is a line making with a rationall superficies the whole superficies mediall For forasmuch as the parallelogramme AK is mediall therefore that which is equall vnto it namely that which is made of the squares of the lines LO and ON added together is also mediall Againe forasmuch as the parallelogramme DK is rationall therfore that which is equall vnto it namely that which is contained vnder the lines LO and ON twise is also rationall And forasmuch as the line AF is incommensurable in length to the line FG therfore by the 1. of the sixt 10. of the tenth the parallelograÌ„me Aâ— is incommensurable to the
parallelogramme FK wherfore also the square of the lyne LO is incommensurable to the square of the line ON Wherfore the lines LO and ON are incommensurable in power hauing that which is made of their squares added together mediall and that which is contayned vnder them twise rationall Wherfore the line LN is that irrationall line which is called a lyne making with a rationall superâ—icies the whole superficies mediall and it contayneth in power the superficies AB Wherfore the line contayning in power the superâ—icies AB is a line making with a rationall superficies the whole superficies mediall If therfore a superficies be contayned vnder a rationall lyne a fift residuall line the line that contayneth in power the same superficies is a line makyng with a rationall superficies the whole superâ—icies mediall which was required to be proued ¶ The 72. Theoreme The 96. Proposition If a superficies be contayned vnder a rationall line and a sixth residuall line the line which contayneth in power the same superficies is a line making with a mediall superficies the whole superficies mediall SVppose that AB be a superficies contayned vnder a rationall line AC a sixt residuall line AD. Then I say shal the line which contayneth in power the superficies AB is a line making with a medial superficies the whole superficies mediall For vnto the line AD let the line DG be ioyned And let the rest be as in the propositions going before And forasmuch as the line AF is incommensurable in length to the line FG therefore the parallelogramme AI is incoÌ„mensurable to the parallelogramme FKâ— And forasmuch as the lines AG and AC are rationall commensurable in power onely therefore the parallelogramme AK is mediall and in like maner the parallelogramme DK is mediall Now forasmuch as the lines AG and GD are commensurable in power onely therefore they are incommensurable in length the one to the other But as the line AG is to the line GD so is the parallelogramme AK to the parallelogramme DK therefore the parallelogramme AK is incommensurable to the parallelogramme DK Describe the like figure that was described in the former propositions and we may in like sort proue that the line LN contayneth in power the superficies AB I say moreouer that it is a line making wâ—th a mediall superficies the whole superficies mediall For the parallelograme AK is mediall wherefore that which is equal vnâ—â— it namely that which is made of the sqâ—ares of the lines LO and ON added together is also mediall And forasmuch as the parallelogramme â—K is mediall therefore that which is equall vnto â—t namely that which is contayned vnder the lines LO and ON twise is also mediall And forasmuch as the parallogramme AK is incommensurable to the parallelogramme DK therefore the squares of the lines LO and ON are incommensurable to that which is contained vnder the lines LO and ON twise And forasmuch as the parallelogramme AI is incommensurable to the parallelogramme FK therefore also the square of the line LO is incommensurable to the square of the line ON Wherefore the lines LO and ON are incommensurable in power hauing that which is made of the squares of the lines LO and ON mediall and that which is contayned vnder them twise mediall and moreouer that which is made of the squares of them is incommensurable to thatâ— which is contayned vnder them twise Wherfore the line LN is that irrationall line which is called a line making with a mediall superficies the whole superficies mediall and it contayneth in power the superficies AB Wherefore the line which contayneth in power the superficies AB is a line making with a mediall superficies the whole superficies mediall If therefore a superficies be contayned vnder a rationall line and a sixth residuall line the line which contayneth in power the same superficies is a line making which a mediall superficies the whole superficies mediallâ— which was required to be demonstrated ¶ The 73. Theoreme The 97. Proposition The square of a residuall line applyed vnto a rationall line maketh the breadth or other side a first reâ—iduall line SVppose that AB be a residuall line and let CD be a rationall line And vnto the line CD apply the parallelogramme CE equall to the square of the line AB and making the breadth the line CF. Then I say that the line CF is a first residuall lineâ— For vnto the line AB let the line coÌ„ueniently ioyned be supposed to be Bâ— which selfe line is also called a line ioyned as we declared in the end of the 79. proposition Wherefore the â—ines AG and GB are rationall â—â—â—mensurable in power onely And vnto the line CD apply the parallelogramme CH equall to the square of the line AG and vnto the line KH which is equall to the line CD apply the parallelogramme KL equall to the square of the line BG Wherfore the whole parallelogramme CL is equall to the squares of the linâ—s Aâ— and GBâ— And the parallelogrâ—mmâ— CE is equall to the square of the line AB wherefore the parallelogramme remayning namely the parallelogramme FL is equall to that which is contayned vnder the lines AG and GB twise For by the 7. of the second the squares of the lines AG and GB are equall to that which is contayned vnder the lines AG GB twise and to the square of the line AB Deuide the line FM into two equall partes in the point N. And by the poynt N draw vnto the line CD a parallell line NX Wherfore either of the parallelogrammes FX and NL is equall to that which is contayned vnder the lines AG and GB once And forasmuch as the squares of the lines AG and GB are rationall vnto which squares the parallelogramme CL is equall therefore the parallelogramme CL also is rationall wherefore the line GM is rationall and commensurable in length to the line CD Agayne forasmuch as that which is contayned vnder the lines AG and GB twise is mediall therefore the parallelogramme equall vnto it namely the paralâ—elogramme FL is also mediall Wherefore the line FM is rationall and incommensurable in length to the line CD And forasmuch as the squares of the lines AG and GB are rationall and that which is contayned vnder the lines AG and GB twise is mediall therefore the squares of the lines AG and GB are incoÌ„mensurable to that which is contayned vnder the lines AG and GB twise But vnto the squares of the lines AG and GB is equall the parallelogramme CL and to that which is contayned vnder the lines AG and GB twise is equall the parallelogramme FL wherefore the parallelogramme CL is incommensurable to the parallelogramme FL. Wherefore also the line CM is incommensurable in length to the line FM and they are both rationall Wherefore the lines CM and FM are rationall commensurable in power onely and therefore the line CF is
a residuall line by the 73. of the tenth I say moreouer that it is a first residuall line For forasmuch as that which is contayned vnder the lines AG and GB is the meane proportionall betwene the squares of the lines AG and GB by the assumpt going before the 54. of the tenth And vnto the square of the line AG is equall the parallelogramme CH and vnto that which is contayned vnder the lines AG and GB is equall the parallelogramme NL and vnto the square of the line GB is equall the parallelograme KL Wherefore the parallelogramme NL is the meane proportionall betwene the parallelogrammes CH and KL Wherefore as CH is to NL so iâ— NL to KL But as CH is to NL so is the line CK to the line NM as NL is to KL so is the line NM to the line KM Wherfore as the line CK is to the line NM so is the line NM to the line KM Wherfore the parallelogramme contayned vnder the lines CK and KM is equall to the square of the line NM that is to the fourth part of the square of the line FM And forasmuch as the square of the line AG is commensurable to the square of the line GB therefore the parallelogramme CH is coÌ„mensurable to the parallelogramme KL But as CH is to KL so is the line CK to the line KM wherefore the line CK is commensurable in length to the line KM Wherefore by the 17. of the tenth the line CM is in power more then the line FM by the square of a line commensurable in length to the line CM But the line CM is commensurable in length to the rationall line CD Wherefore the line CF is a fâ—irst residuall line Wherefore the square of a residuall line applyed vnto a rationall line maketh the breadth or other side a first residuall line which was required to be demonstrated ¶ The â—4 Theoreme The 98. Proposition The square of a first mediall residuall line applied to a rationall line maketh the breadth or other side a second residuall line SVppose that AB be a first mediall residual line and let CD be a rationall line And vnto the line CD apply the parallelogramme CE equall to the square of the lyne AB and making in breadth the line CF. Then I say that the line CF is a second residuall line For vnto the line AB let the lyne coÌ„ueniently ioyned be supposed to be BG Wherefore the lines AG and BG are mediall commensurable in power onely comprehending a rationall superâ—icies And vnto the line CD apply the parallelogramme CH equal to the square of the line AG and making in bredth the line CK and vn-the line KH which is equall to the line CD apply the parallelogramme KL equall to the square of the line GB and making in breadth the line KM Wherfore the whole parallelogramme CL is equall to both the squares of the lines AG and GB which are mediall commensurable the one to the other Wherfore the parallelogrammes CH and KL are mediall and commensurable the one to the other Wherfore by the 15. of the tenth the whole parallelogramme CL is commensurable to either of these parallelogrammes CH and KL Wherfore by the corollary of the 23. of the tenth the whole parallelogramme CL is also mediall Wherefore by the 22. of the tenth the line CM is rationall and incommensurable in length to the line CD And forasmuch as the parallelogramme CL is equal to the squares of the lines AG and GBâ— and the squares of the lines AG and GB are equall to that which is coÌ„tained vnder the lines AG and GB twise together with the square of the line AB by the 7. of the second and vnto the square of the line AB is equall the parallelogramme CE. Wherfore the residue namely that which is contained vnder the lines AG and GB twise is equal to the residue namely to the parallelograÌ„me FL. But that which is contained vnder the lines AG GB twise is rational Wherfore the parallelogramme FL is also rationall Wherfore the line FM is rationall and coÌ„mensurable in length to the line CD by the 20. of the tenth Now forasmuch as the parallelogramme CL is mediall and the parallelogramme FL is rational therfore they are incoÌ„mensurable the one to the other Wherfore also the lyne CM is incommensurable in length to the lyne FM and they are both rationall Wherfore the line CF is a residuall line I say moreouer that it is a second residuall lyne For deuide the line FM into two equall partes in the point N from which point draw vnto the line CD a parallel line NX Wherfore either of these parallelogrammes FX and NL is equall to the parallelogramme contained vnder the lines AG and GB And forasmuch as the parallelogramme contained vnder the lines AG and GB is the meane proportionall betwene the squares of the lines AG and BG Therefore the parallelogramme NL is the meane proportionall betwene the parallelogrammes CH and KL But as CH is to NL so is the line CK to the line NM and as NL is to KL so is the line NM to the line KM Wherfore as the line CK is to the lyne NM so is the lyne NM to the line KM Wherfore the parallelograÌ„me contayned vnder the lines CK and KM is equall to the square of the line NM that is to the fourth part of the square of the lyne FM But the parallelogramme CH is commensurable to the parallelogramme KL Wherfore also the lyne CK is commensurable in length to the lyne KM Wherfore by the 17. of the tenth the line CM is in power more then the line FM by the square of a line commensurable in length to the line CM And the line FM which is the line conueniently ioyned is commensurable in length to the rationall lyne CD Wherfore the linâ— CF is a second residuall lyne Wherfore the square of a first mediall residual line applied to a rationall line maketh the breadth or other side a second residuall lyne which was required to be proued ¶ The 75. Theoreme The 99. Proposition The square of a second mediall residuall line applied vnto a rationall line maketh the breadth or other side a third residuall line SVppose that AB be a second mediall residuall line and let CD be a rationall line And vnto the line CD apply the parallelogramme CE equall to the square of the line AB and making in breadth the line CF. Then I say that the line CF is a third residuall line For vnto the line AB let the line conueniently ioyned be supposed to be BG Wherefore the lines AG GB are mediall commensurable in power onely containing a mediall superâ—icies And let the rest of the construction be as in the Proposition next going before Wherefore the line CM is rationall and incommensurable in length to the rationall line CD And either of the parallelogrammes FX
line incommensurable in length to the line CM And the line conueniently ioyned namely the line FM is commensurable in length to the rationall line CD Wherfore the line CF is a â—iâ—t residuall line Wherfore the line CF is a fift residuall line Wherfore the square of a line making with a rationall superficies the whole superficies medial applied vnto a rational line maketh the breadth or other side a fift residuall lyne which was required to be demonstrated ¶ The 78. Theoreme The 102. Proposition The square of a lyne making with a mediall superficies the whole superficies mediall applied to a rationall line maketh the breadth or other side a sixt residuall line SVppose that AB be a line making with a mediall superficies the whole superficies mediall and let CD be a rationall line And vnto the line CD apply the parallelogramme CE equall to the square of the line AB and making in breadth the line CF. Then I say that the line CF is a sixt residual line For vnto the line AB let the line conueniently ioyned be BG Wherfore the lines AG and BG are incommensurable in power hauing that which is made of their squares added together mediall that which is contained vnder them mediall and moreouer that which is made of their squares added together is incommensurable to that which is contained vnder them Let the rest of the construction be in this as it was in the propositioÌ„s going before Wherfore the whole parallelogramme CL is mediall for it is equall to that which is made of the squares of the lines AG GB added together which is supposed to be mediall Wherefore the line CM is rationall and incommensurable in length to the line CD and in like manner the parallelogramme FL is mediall Wherfore also the line FM is rationall and incommensurable â—n length to the line CD And forasmuch as that which is made of the squares of the lines AG and GB added together is incommensurable to that which is contained vnder the lines AG and GB twise therefore the parallelograÌ„mes equall to them namely the parallelogrammes CL and FL are incommensurable the one to the other Wherfore also the lines GM and FM are incommensurable in length and they are both rational Wherfore they are rationall coÌ„mensurable in power only Wherfore the line CF is a residuall line I say moreouer that it is a sixt residuall line Let the rest of the demonstration be as it was in the former propositions And forasmuch as the lines AG and BG are incommensurable in power therfore their squares that is the parallelogrammes which are equall vnto them namely the parallelogrammes CH and KL are incommensurable the one to the other Wherfore also the line CK is incommensurable in length to the line KM Wherfore by the 18. of the tenth the line CM is in power more then the line FM by the square of a line incommensurable in length to the line CM And neither of the lines CM nor FM is commensurable in length to the rationall line CD Wherfore the line CF is a sixt residuall line Wherfore the square of a line making with a medial superficies the whole superficies mediall applied to a rationall line maketh the breadth or other side a sixt residuall line which was required to be demonstrated ¶ The 79. Theoreme The 103. Proposition A line commensurable in length to a residuall line is it selfe also a residuall line of the selfe same order SVppose that AB be a residuall line vnto which let the line CD be commensurable in length Then I say that the line CD is also a residuall line and of the selfe same order of residuall lines that the line AB is For forasmuch as the line AB is a residuall line let the line conueniently ioyned vnto it be supposed to be BE. Wherefore the lines AE and BE are rationall commensurable in power onely As the line AB is to the line CD so by the 12. of the sixt let the line BE be to the line DF. Wherefore by the 12. of the fift as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes Wherefore as the line AB is to the line CD so is the whole line AE to the whole line CF and the line BE to the line DF. Wherefore by the 10. of the tenth the line AE is commensurable in length to the line CF and the line BE to the line DF. But the line AE is rationall Wherefore the line CF is also rationall And in like sort the line DF is rationall for that the line BE to whom it is commensurable is also rationall And for that as the line BE is to the line AE so is the line DF to the line CF. But the lines BE and AE are commensurable in power onely Wherefore the lines CD and DF are commensurable in power onely Wherefore the line CD is a residuall line I say moreouerâ— that it is a residuall line of the selfe same order that the line AB is For for that as we haue before said as the line Aâ— is to the line CF so is the line BE to the line DFâ— therefore alternately as the line AE is to the line BE so is the line CF to the line DF. But the line AE is in power more then the line EB either by the square of a line commensurable in length to the line AE or by the square of a line incommensurable in length to the line AE If AE be in power 〈◊〉 then BE by the square of a line commensurable in length to AE then the line â—Fâ— shall also by the 14. of the tenth be in power more then the line DF by the square of a line commensurable in length to the line CF and so if the line AE be commensurable in length to the rationall line put forasmuch as the line AE is commensâ—rable in length to the line CF therefore by the 12. of the tenth the line CF shall also be commensurable in length to the same rationall line Wherefore either of the lines AB and CD is a first residuall line And if the line BE be commensurable in length to the rationall line put forasmuch as the line BE is commensurable in length to the line DF therefore the line DF shall also be commensurable in length to the rationall line putâ— and then either of the lines AB and CD is a second residuall line And if neither of the lines AE nor BE be commensurable in length to the rationall line put then neither of the lines CF nor DF shall be commensurable in length to the same rationall line by the 13. of the teÌ„th And so either of the lines AB CD is a third residuall line But if the line AE be in power more theÌ„ the line BE by the square of a line incoÌ„mensurable in length to the line AE the line CF
medial residuall line or a second mediall residuall line according as the line A is supposed to be which was required to be proued ¶ The 81. Theoreme The 105. Proposition A line commensurable to a lesse line is it selfe also a lesse line SVppose that AB be a lesse line vnto whom let the line CD be commensurable Then I say that the line CD is also a lesse line For let the same construction be in this that was in the former Propositions And forasmuch as the lines AE and EB are incommensurable in power therefore by the 22. of the sixt and 10. of the tenth the lines CF FD are incommensurable in power Againe by the 22. of the sixt as the square of the line AE is to the square of the line BE so is the square of the line CF to the square of the line DF. Wherfore by coÌ„position as the squares of the lines AE and BE are to the square of the line BE so are the squares of the lines CF and DF to the square of the line DF and alternately as the squares of the lines AE and BE are to the squares of the lines CF and DF so is the square of the line BE to the square of the line DF. But the square of the line BE is commensurable to the square of the line DF for the lines BE and DF are commensurable Wherefore that which is made of the squares of the lines AE and BE added together is commensurable to that which is made of the squares of the lines CF and DF added together But that which is made of the squares of the lines AE and BE added together is rationall Wherefore that which is made of the squares of the lines CF and DF added together is also rationall Againe for that as the square of the line AE is to the parallelogramme contained vnder the lines AE and BE so is the square of the line CF to the parallelogramme contained vnder the lines CF and DF as we declared in the Proposition next going before therefore alternately as the square of the line AE is to the square of the line CF so is the parallelogramme contained vnder the lines AE and BE to the parallelogramme contained vnder the lines CF and DF. But the square of the line AE is commensurable to the square of the line CF for the lines AE CF are commensurable Wherefore the parallelogramme contained vnder the lines AE and BE is commensurable to the parallelogramme contained vnder the lines CF and DF. But the parallelogramme contained vnder the lines AE and BE is mediall Wherefore the parallelogramme contained vnder the lines CF and DF is also mediall Wherefore the lines CF and DF are incommensurable in power hauing that which is made of their squares added together rationall and the parallelogramme contained vnder them mediall Wherefore the line CD is a lesse line A line therefore commensurable to a lesse line is it selfe also a lesse line which was required to be proued An other demonstration Suppose that A be a lesse line and vnto A let the line B be commensurable whether in length and power or in power onely Then I say that B is a lesse line Take a rationall line CD And vnto the line CD apply by the 44 of the first the parallelogramme CE equall to the square of the line A and making in bredth the line CF. Wherefore by the 100. proposition the line CF is a fourth residuall line Vnto the line FE apply by the same the parallelogramme EH equall to the square of the line B and making in breadth the line FH Now forasmuch as the line A is commensurable to the line B therefore also the square of the line A is coÌ„mensurable to the square of the line B. But vnto the square of the line A is equall the parallelogramme CE vnto the square of the line B is equal the parallelogramme EH Wherfore the parallelogramme CE is commensurable to the parallelogramme EH But as the parallelogramme CE is to the parallelogramme EH so is the line CF to the line FH Wherfore the line CF is commensurable in length to the line FH But the line CF is a fourth residuall line Wherfore the line FH is also a fourth residuall line by the 103. of the tenth and the line FE is rationall But if a superficies be contained vnder a rationall line and a fourth residuall lyne the line that containeth in power that superficies is by the 94. of the tenth a lesse lyne But the line B containeth in power the superficies EH Wherfore the line B is a lesse line which was required to be proued ¶ The 82. Theoreme The 106. Proposition A line commensurable to a lyne making with a rationall superficies the whole superficies mediall is it selfe also a lyne making with a rationall superficies the whole superficies mediall SVppose that AB be a line making with a rationall superficies the whole superficies mediall vnto whom let the line CD be commensurable Then I say that the line CD is a line making with a rationall superficies the whole superficies mediall Vnto the line AB let the line conueniently ioyned be BE. Wherefore the lines AE and EB are incommensurable in power hauing that which is made of their squares added together mediall and the parallelogramme contained vnder them rationall Let the construction be in this as it was in the former propositions And in like sort may we proue that as the line AE is to the line BE so is the line CF to the line DF and that that which is made of the squares of the lines AE and BE added together is commensurable to that which is made of the squares of the lines CF and DF added together and that that which is contained vnder the lynes AE and EB is in like sort commensurable to that which is contained vnder the lines CF and DF. Wherfore also the lines CF and DF are commensurable in power hauing that which is made of their squares added together mediall and that which is contained vnder them rationall Wherfore the line CD is a lyne making with a rationall superficies the whole superficies mediall Wherfore a line commensurable to a line making with a rationall superficies the whole superficies mediall is it selfe also a lyne making with a rational superficies the whole superficies mediall which was required to be demonstrated An other demonstration Suppose that A be a line making with a rationall superficies the whole superficies mediall and vnto it let the lyne B be commensurable either in length and in power or in power onâ—ly Then I say that B is a lyne making with a rationall superficies the whole superficies mediall Take a rational line CD and vnto the line CD apply the parallelogramme CE equal to the square of the line A and makyng in breadth the lyne GF Wherfore by the 101. proposition the lyne CF is a fift residuall lyne
Againe vnto the line FE apply the parallelogramme FG equall to the square of the line B and makyng in breadth the lyne FH Now forasmuch as the line A is coÌ„mensurable to the lyne B therfore the square of the lyne A is commeÌ„surable to the square of the line B. But vnto the square of the lyne A is equall the parallelogramme CE and vnto the square of the line B is equall the parallelogramme FG. Wherfore the parallelogramme CE is commensurable to the parallelogramme FG. Wherefore the line CF is also commensurable in length to the line FH But the line CF is a fift residuall line Wherfore also the line FH is a fift residual line And the line FE is rationall But if a supersicies be contayned vnder a rationall line and a â—ift residuall lyne the lyne that contayneth in power that superficies is by the 95. of the tenth a lyne making with a rationall superficies the whole superâ—icies mediall But the lyne B containeth in power the parallelogramme FG. Wherfore the lyne B is a lyne making with a rationall superficies the whole superficies mediallâ— which was required to be demonstrated ¶ The 83. Theoreme The 107. Proposition A line coÌ„mensurable to a line making with a mediall superficies the whole superficies mediall is it selfe also a line making with a mediall superficies the whole superficies mediall SVppose that AB be a line making with a mediall superficies the whole superficies mediall vnto whome let the line CD be commensurable Then I say that the line CD is also a line making with a mediall superficies the whole superficies mediall For vnto the line AB let the line coueniently ioyned be BE. And let the rest of the construction be in this as it was in the former propositions Wherefore the lines AE and BE are incoÌ„mensurable in power hauing that which is made of their squares added together mediall and that which is contained vnder them also mediall and moreouer that which is made of their squares added together is incommensurable to that which is contained vnder them But the lines AE and BE as we haue before proued are commensurable to the lines CF DF and that which is made of the squares of the lines AE and BE added together is commensurable to that which is made of the squares of the lines CF and FD added together and the parallelogramme contained vnder the lines AE and BE is commensurable to the parallelogramme contained vnder the lines CF and DF. Wherfore the lines CF and DF are incommensurable in power hauing that which is made of their squares added together mediall and that which is contained vnder them also mediall and moreouer that which is made of their squares added together is incommensurable to that which is contained vnder them Wherfore the line CD is a line making with a mediall superficies the whole superficies mediall A line therefore commensurable to a lyne making with a mediall superficies the whole superficies mediall is it selfe also a line makyng with a mediall superficies the whole superficies mediall which was required to be proued This proposition may also be an other way demonstrated as the three former propositions were If vpon a rationall line you apply parallelogrammes equall to the squares of the lines AB and CD the breadthes of which parallelogrames shall be eche a sixth residual line aud therfore the lines which coÌ„tayne them in power namely the lines AB and CD shall be both such lines as is required in the proposition which is easy to concludâ— marking the ordeâ— of the demonstration in the three former propositions ¶ The 84. Theoreme The 108. Proposition If from a rationall superficies be taken away a medialt superficies the line which containeth in power the superficies remayning is one of these two irrationall lines namely either a residuall line or a lesse line SVppose that BC be a rationall superficies and from it take away a mediall superficies namely BD. Then I say that the line which containeth in power the superficies remayning namely the superficies EC is one of â—hese two irrationall lines namely either a residuall line or a lesse line Take a rationall line FG. And vpon FG describe by the 44. of the first a rectangle parallelogramme GH equall to the superâ—icies BC. And from the parallelogramme GH take away the parallelogramme GK equall to the superficies BD. Wherefore by the third common sentence the superâ—icies remayning namely EC is equall to the parallelogramme remayning namely to LH And forasmuch as BC is rationall and BD is mediall and BC is equall to the parallelogramme GH and BD to the parallelogramme GK therefore GH is rationall and GK is mediall and the parallelogramme GH is applied vnto the rationall line FG. Wherefore by the â—0 of the tenth the line FH is rationall and commensurable in length to the line FG. And the parallelogramme GK is also applied vnto the raâ—ionall line FG. Wherefore by the 22. of the tenth the line FK is rationall and incommensurable in length to the line FG. Wherefore by the Assumpt of the 12. of the tenth the line FH is incommensurable in length to the line FK And they are both rationall Wherefore the lines FH and FK are rationall commensurable in power onely Wherefore the line KH is a residuall line and the line conueniently â—oyâ—ed vnto it is KF Now the line FH is in power more thâ—n the line KF either by the square of a line commensurable in lengtâ— to the line FH or by the square of a line incommensurable in length to the line FH First let it be in power more then the line FK by the square of a line commensurable in length to the line FH and the whole line FH is commensurable in length to the rationall line put namely to FG. Wherefore the line KH is a first residuall line But if a superficies be contained vnder a rationall line and a first residuall line the line that containeth in power that superficies is by the 91. of the tenth a residuall line Wherefore the line which containeth in power LH that is the superficies EC is a residuall line But if the line HF be in power more then the line FK by the square of a line incommensurable in length to the line FH and the whole line FH is commensurable in length to the rationall line geuen FG. Wherefore the line KH is a fourth residuall line But a line containing in power a superficies contained vnder a rationall line and a fourth residuall line as a lesse line by the 94. of the tenth Wherefore the line that containeth in power the superficies LH that is the superficies EC is a lesse line If therefore from a rationall superficies be taken away a mediall superâ—icies the line which containeth in power the superficies remayning is one of these two irrationall lines namely either a residuall line or a lesse line which was required to be proued ¶ The
85. Theoreme The 109. Proposition If from a mediall superficies be taken away a rationall superficies the line which contayneth in power the superficies remayning is one of these two irrationall lines namely either a first mediall residuall line or a line making with a rationall superficies the whole superficies mediall SVppose that BC be a mediall superâ—icies and from it take away a rational superâ—icies namely BD. Then I say that the line which contayneth in power the superficies remayning namely the superficies EC is one of these two irrationall lines either a first mediall residuall line or a line making with a rational superficies the whole superficies mediall Take a rationall line FG and let the rest of the construction be in this as it was in the former proposition Wherefore it followeth that the line Fâ— is rationall and incommensurable in length to the line FG by the 22. of the tenth And that the line KF is by the 20. of the tenth rationall and commensurable in length to the line FG. Wherefore the lines FH and FK are rationall commensurable in power onely Wherefore KH is a residuall line And the line conueniently ioyned vnto it is FK Now the line FH is in power more then the line FK either by the square of a line commensurable in length to the line FH or by the square of a line incommensurable in length vnto it If the line FH be in power more then the line FK by the square of a line commensurable in length to the line FH and the line coÌ„ueniently ioyned vnto it namely FK is coÌ„mensurable in leÌ„gth to the rationall line FG. Wherefore the line KH is a second residuall line And the line FG is a rationall line But a line contayning in power a superficies comprehended vnder a rationall line and a second residuall line is by the 92. of the tenth a first mediall residuall line Wherefore the line thus contayneth in power the superficies LH that is the superficies CE is a first mediall residuall lne But if the line HF be in power more then the line FK by the square of a line incommensurable in length to the line FH and the line conueniently ioyned namely the line FK is commensurable in length to the rationall line put namely to FG wherefore the line KH is a â—ift residuall line Wherefore by the 95. of the tenth the line that contayneth in power the superficies LH that is the superficies EC is a line making with a rationall superâ—icies the whole superficies mediall which was required to be proued ¶ The 86. Theoreme The 110. Proposition If from a mediall superficies be taken away a mediall superficies incommensurable to the whole superficies the line which containeth in power the superficies which remaineth is one of these two irrationall lines namely either a second mediall residuall line or a line making with a mediall superficies the whole superficies mediall AS in the former descriptions s◠〈◊〉 also â—ake away froÌ„ the mediall superficiâ—s BCâ— mediall superficies BDâ— and let â—D be incommâ—nsurâ—ble to the whole superficies BC. Then I say that the line which containeth in power the superâ—icies EC is one of thâ—se two irrationall lines namely either a second mediall residuall line or a liâ—e making with a mediall superfiâ—â—es the whole superficies mediall For forasmuch as either of these superficieâ—â—s BC and BD is mediall and BC is incommensurable to BD it followeth by the 22. of the tenth thaâ— either of these lines FH and FK is rationall and incommâ—â—surable in length to the line FG. And forasmuch as the superficies BC is incommensurable to the superficies BD that is the superficies GH to the superâ—icies GK therefore by the first of the sixt 10â— of the tenth the line FH is incommensurable in length to the line FK Wherfore the lines HF and FK are rational coÌ„mensurable in power onely Wherefore by the 73. of the tenth the line KH is a residuall line and the line coÌ„ueniently ioyned vnto it is FK Now the line HF is in power more then the line FK either by the square of a line coÌ„mensurable in length to the line HF or by the square of a line incommensurable in length vnto it If the line HF be in power more then the line FK by the square of a line coÌ„mensurable in length to the line FH and neither of the lines HF nor FK iâ— commensurâ—ble to the rationall line put FG. Wherfore the line KH is a third residuall But the line CF that is the line KL is rationall And a rectangle superâ—icies contained vnder a rationall line and a third residuall line is irrationall and the line which containeth in power that superficies is by the 93. of the tenth a second mediall residuall line Wherefore the line that containeth in power the superficies LH that is the superficies EC is a second mediall residuall line But if the line HF be in power more then the line FK by the square of a line incommeâ—surable in length to the line FH and neither of the lines HF nor FK is commensurable in length to the line FG. Wherefore the line HK is a sixt residuall line But a line containing in power a superâ—icies contained vnder a rationall line and a sixt residuall line is by the 96. of the tenth a line making with a mediall superficies the whole superficies mediall Wherefore the line that containeth in power the superficies LH that is the superficies EC is a line making with a mediall superficies the whole superficies mediall If therefore from a mediall superâ—icies be taken away a mediall superficies incommensurable to the whole superficies the line that containeth in power the superficies which remaineth is one of the two irrationall lines remaining namely either a second mediall residuall line or a line making with a mediall superficies the whole superficies mediall which was required to be proued ¶ The 87. Theoreme The 111. Proposition A residuall line is â—ot one and the same with a binomiall lyne SVppose that AB be a residuall line Then I say that AB is not one and the same with a binomiall line For if it be possible let it be a binomiall line And take a rationall line DC And by the 44. of the first vnto the line CD apply a rectangle parallelogramme CE equall to the square of the line AB and making in bredth the line DE. And forasmuch as AB is a residuall line therfore by the 97. of the tenth the line DE is a first residuall line Let the line coâ—â—â—niently ioyned vnto it be Eâ— Wherfore the lines D F and FE are rationall commensurable in power â—nely and the line DF is in power more then the line FE by the square of a line commensurable in length to the line DE the line DF is coÌ„mensurable in leÌ„gth to the rational line put DC Again forasmuch as AB is by position a binomiall line therefore by the 60.
of the tenth the line DE is a first binomiall line Deuide it into his names in the point G. And let DG be the greater name Wherfore the lines DG and â—E are rationall commensurable in power onely And the line DG is in power more then the line GE by the square of a line commensurable in length to the lyne DG and the line DG is commensurable in length to the rationall line put DC Wherefore the line DF is commensurable in length to the line DG Wherfore by the 13. of the tenth the whole line DF is commensurable in leÌ„gth to the line remaining namely â—o the line GF And forasmuch as the line DF is coÌ„meÌ„surable to the line FG but the line FD is rationall Wherfore the line FG is also rationall And forasmuch as the line FD is commensurable in length to the line FG but the line DF is incommensurable in length to the line FE Wherfore the line FG is incommensurable in length to the line FE by the 13. of the tenth and they are both rationall lines Wherfore the lines GF and FE are rationall commensurable in power onely Wherfore by the 73. of the tenth the line EG is a residuall line but it is also rationall as before hath bene proued which is impossible namely that one the same line should be both rationall and irrationall Wherfore a residuall line is not one and the same with a binomiall line that is is not a binomiall line which was required to be demonstrated ¶ A Corollary A residuall lyne and the other fiue irrationall lynes following it are neither mediall lines nor one and the same betwene themseluesâ— that is one is vtterly of a diuers kinde froÌ„ an other For the square of a mediall line applied to a rationall line maketh the breadth rationall and incommensurable in length to the rationall lyne whereunto it is applied by the 22. of the tenth The square of a residuall line applied to a rationall line maketh the breadth a first residuall line by the 97. of the tenth The square of a first mediall residuall line applied to a rationall line maketh the breadth a second residuall lyne by the 98. of the tenth The square of a second mediall residuall line applied vnto a rationall line maketh the breadth a third residuall line by the 99. of the tenth The square of a lesse line applied to a rationall line maketh the breadth a fourth residuall line by the 100. of the tenth The square of a line making with a rationall superficies the whole superficies mediall applied to a rationall line maketh the breadth a â—ift residuall line by the 101. of the tenth And the square of a line making with a mediall superâ—icies the whole superficies mediall applied to a rationall line maketh the breadth a sixt residuall line by the 102. of the teÌ„th Now forasmuch as these foresaid sides which are the breadthes differ both from the first breadth sor that it is rational and differ also the one froÌ„ the other for that they are residuals of diuers orders and kindes it is manifest that those irrationall lines differ also the one from the other And forasmuch as it hath bene proued in the 111. proposition that 〈◊〉 residual 〈◊〉 is not one and the same with a binomiall line and it hath also bene proued that the 〈…〉 of a residuall line and of the fiue irrationall lines that follow it being applied to a rational line do make their breadthes one of the residuals of that order of which they were whose squareâ— were applied to the rationall line likewise also the squares of a binomiall line and of the fiue irrationall lines which follow it being applied to a rationall line do make the breadthes one of the binomials of that order of which they were whose squares were applied to the rationall line Wherfore the irrationall lines which follow the binomiall line and the irrationall lines which follow the residuall line differ the one from the other so that all the irrationall lynes are 13. in number namely these 1 A mediall line 2 A binomiall line 3 A first bimediall line 4 A second bimediall line 5 A greater line 6 A line containing in power a rationall superficies and a mediall superficies 7 A line contayning in power two mediall superficieces 8 A residuall line 9 A first mediall residuall line 10 A second mediall residuall line 11 A lesse line 12 A line making with a rationall superficies the whole superficies mediall 13 A line making with a mediall superficies the whole superficies mediall ¶ The 88. Theoreme The 112. Proposition The square of a rationall line applyed vnto a binomiall line maketh the breadth or other side a residuall line whose names are commensurable to the names of the binomiall line in the selfe same proportioÌ„ moreouer that residuall line is in the selfe same order of residuall lines that the binomiall line is of binomiall lines SVppose that A be a rationall line and BC a binomiall line whose greater name let be CD And vnto the square of the line A let the parallelogramme contayned vnder the lines BC and EF so that EF be the breadth be equall Then I say that EF is a residuall line whose names are commensurable to the names of the binomiall line BC which names let be CD and DB and are in the same proportion with them and moreouer the line EF is in the selfe same order of residual lines that the line BC is of binomiall lines Vnto the square of the line A let the parallelogramme contayned vnder the lines BD and G be equall Now forasmuch as that which is coÌ„tayned vnder the lines BC EF is equal to that which is contayned vnder the lines BD and G therfore reciprocally by the 14. of the sixth as the line CB is to the BD so is the line G to the line EF. But the line BC is greater then the line BD wherefore the line G is greater then the line EF. Vnto the line G let the line EH be equall Wherefore by the 11. of the fift as the line CB is to the line BD so is the line HE to the line FE Wherefore by deuision by the 17. of the fifth as the line CD is to the line BD so is the line HF to the line FE As the line HF is to the FE so let the line FK be to the line KE how this is to be done we will declare at the end of this demonstration Wherefore by the 12. of the fift the whole line HK is to the whole line KF as the line FK is to the line KE For as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes But as the line FK is the line KE so is the line CD to the line DB for FK is to EK as HF is to FE and HF is to FE as CD is DB Wherfore by the
that superficies is rationall SVppose that a parallelogramme be contained vnder a residuall line AB and a binomiall line CD and let the greater name of the binomiall line be CE and the lesse name be ED and let the names of the binomiall line namely CE and ED be commensurable to the names of the residuall line namely to AF and Fâ— and in the selfe same proportion And let the line which containeth in power that parallelograÌ„me be G. TheÌ„ I say that the line G is rational Take a rational line namely H. And vnto the line CD apply a parallelograÌ„me equal to the square of the line H and making in breadth the line KL Wherefore by the 112. of the tenth KL is a residuall line whose names let be KM and ML which are by the same coÌ„mensurable to the names of the binomiall line that is to CE and ED and are in the selfe same proportioÌ„ But by position the lines CE and ED are coÌ„mensurable to the lines AF and FB and are in the selfe same proportion Wherfore by the 12. of the tenth as the line AF is to the line FBâ— so is the line KM to the line ML Wherfore alternately by the 16. of the fift as the line AF is to the line KM so is the line BF to the line LM Wherfore the residue AB is to the residue KL as the whole AF is to the whole KM But the line AF is commensurable to the line KM for either of the lines AF and KM is commensurable to the line CE. Wherfore also the line AB is commensurable to the line KL And as the line AB is to the line KL so by the first of the sixt is the parallelogramme contained vnder the lines CD and AB to the parallelogramme contained vnder the lines CD and KL Wherfore the parallelogramme contained vnder the lines CD and AB is commensurable to the parallelogramme contained vnder the lines CD and KL But the parallelogramme contained vnder the lines CD and KL is equall to the square of the line H. Wherfore the parallelograÌ„me coÌ„tained vnder the lines CD AB is coÌ„mensurable to the square of the line H. But the parallelograÌ„me contained vnder the lines CD and AB is equall to the square of the line G. Wherfore the square of the line H is commensurable to the square of the line G. But the square of the line H is rationall Wherfore the square of the line G is also rationall Wherfore also the line G is rational and it containeth in power the parallelogramme contained vnder the lines AB and CD If therfore a parallelogramme be contained vnder a residuall line and a binomiall line whose names are commensurable to the names of the residuall line and in the selfe same proportion the line which containeth in power that superficies is rationall which was required to be proued ¶ Corollary Hereby it is manifest that a rationall parallelogramme may be contained vnder irrationall lines ¶ An otâ—â—r 〈…〉 Flussas 〈…〉 line â—D whosâ— names Aâ— and â—D let be commensurable in length vnto the names of the residuall line Aâ— which let be AF and FB And let the liâ—e AEâ— be to the line EDâ— in the same proportion that the line AF is to the line Fâ— And let the right line â— contayne in power the superficies Dâ— Then I say thaâ— the liâ—e â— is a rationall lin◠〈…〉 lâ—ne which lâ—â— bâ—â— And vpon the line â—â— describe by the 4â— of the first a parallelogramme eqâ—all to the squarâ— of the line â—â— and making in breadth the line DC Wherefore by the â—12 of this booke CD is a residuâ—ll lineâ— whose names Which let be â—â— and OD shall be coâ—mensurablâ— in leâ—gth vnto the names Aâ— and â—D and the line C o shall be vnto the line OD in the same proporâ—ion that the line AE is to the line EDâ— But as the line Aâ— is to the line â—D so by supposition is the line AF to the line FE Wherfore as the line CO is to the line OD so is the line AF to the line Fâ—â— Wherefore the lines CO and OD are commensurable with the lines Aâ— and â—â— by the â—â— of this boke Wherfore the residue namely the line CD is to the residue namely to the line Aâ— as the line CO is to the line AF by the 19. of the fifth But it is proued that the line CO is coÌ„mensurable vnto the line AF. Wherefore the line CD is commensurable vnto the line AB Wherefore by the first of the sixth the parallelogramme CA is commensurable to the parallelogramme Dâ— But the parallelogramme â—â— iâ— by construction rationall for it is equall to the square of the rationall line â— Whâ—refore the parallelogramme â—D â—s also ratâ—â—nâ—llâ— Wherâ—fore the line â— which by supposition coÌ„tayneth in power the superficies â—Dâ— is also rationall If therfore a parallelograÌ„me be contayned c which was required to be proued ¶ The 91. Theoreme The 115. Proposition Of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before SVppose that A be a mediall line Then I say that of the line A may be produced infinite irrationall lines of which none shall be of the selfe same kinde with any of those that were before Take a rationall line B. And vnto that which is contained vnder the lines A and B let the square of the line C be equall by the 14. of the second â— Wherefore the line C is irrationall For a superficies contained vnder a rationall line and an irrationall line is by the Assumpt following the 38. of the tenth irrationall and the line which containeth in power an irrationall superficies is by the Assumpt going before the 21. of the tenth irrationall And it is not one and the selfe same with any of those thirtene that were before For none of the lines that were before applied to a rationall line maketh the breadth mediall Againe vnto that which is contained vnder the lines B and C let the square of D be equall Wherefore the square of D is irrationall Wherefore also the line D is irrationall and not of the self same kinde with any of those that were before For the square of none of the lines which were before applied to a rationall line maketh the breadth the line C. In like sort also shall it so followe if a man proceede infinitely Wherefore it is manifest that of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before which was required to be proued An other demonstratioâ— Suppose that AC be a mediall line Then I say that of the line AC may be produced infinite irrationall lines of which none shall be of the selfe same kinde with any of those irrationall lines before named Vnto the line AC and from the point A
or Tetrahedron ADC is to the pyramis AKG as 64. is to â—7 which is triple to the proportion of 4. to 3. And forasmuch as the line AC is vnto the line AG in length sesquitertia of what partes the line AC containeth in power 64 of the same partes doth the line AG containe in power 36. For by the 2. of the sixth the proportion of the powers or squares is duple to the porportion of the sides which are as 64. is to 48. Now then vpon the line RS which let be equall to the line AG let there be an equilater triangle QRS described by the first of the first And from the angle Q draw to the base RS a perpendicular line QT And extend the line RS to the poynt X. And as 27. is to 64. so by the corollary of the 6. of the tenth let the line RS be to the line RX And diuide â—he line RX into two equall partes in the poynt V and draw the line QV. And forasmuch as the line RS is equall to the line AG of what partes the line AC contayneth in power 64. of the same part the line RS contayneth in power 36. for it is proued that the line AG contayneth in power 36. of those partes And of what partes the line RS contayneth in power 36 of the same partes the 〈◊〉 QT contayneth in power â—7 by the corollary of the 12. of the thiâ—tenth Wherfore of what partes the line AZ contayneth in power 64. of the same parts the line QT contayneth in power 27. Wherefore the right line QT shall be equâ—ll to the right line LN by supposition Agayne forasmuch as the line RS is put equall to the line AG and of what partes the line RS contayneth in length 27. of the same parts is the line RX put to contayne in length 64. and of what partes the line RX contayneth in length 64. of the same the line AC which is in length sesquitertia to the line AG or RS contayneth 36. Wherefore the line RV which is the halfe of the line RX containeth in leÌ„gth of the same partes 32. of which the line AC contayned in length 36. Wherefore the line RV is to the line AC subsesquioctaua and therefore the line RV is equall to the line LM which is also subsesquioctaua to the same line AC And forasmuch as the line NL is equall to the line QT and the line LM to the line RV as before hath bene proued the rectangle parallelogramme contayned vnder the lines QT and RV shall be equall to the rectangle parallelogramme contayned vnder the line NL which is in power â—7 64 to the side AC and vnder the line LM which is in length subsesquioctaua to the same side AC But that which is contayned vnder the lines QT and RV is double to the triangle QVR by the 41. of the first and to the same triangle QVR is the triangle QXR duple by the first of the sixth Wherefore the whole triangle QXR is equall to that which is contayned vnder the lines QT and RV and therefore is equall to the parallelogramme MN And forasmuch as the line RX by supposition contayneth in length 64. of those partes of which the line RS contayneth 27 and the triangles QRX and QRS are by the first of the sixth in the proportion of their bases that is as 64. is to 27 but as 64. is to 27. so is the pyramis or tetrahedron ADC to the pyramis AKG wherefore as the parallelogramme NM or the triangle QRX is to the triangle QRS so iâ— the pyramis ADC to the pyramis AKG And forasmuch as the semidiameter AH is the altitude of the pyramis AKG and also of the two equall and like pyramidâ— of the octohedron which haue their common base in the square of the octohedron by the corollary of the 14. of the thirtenth therefore as the base of the pyramis AKG which is the triangle QRS is to two squares of the octohâ—dron that is to the square of the diameter AB which is equall to those squares by the 47. of the first so is the pyramis AKG to the octohedron AEB by the 6. of the twelfth And forasmuch as the parallelogramme MN is to the base QRS as the pyramis ADC is to the pyramis AKG and the base QRS is to the square of the line BE as the pyramis AKG is to the octohedron AEB therefore by pâ—oportion of â—quality taâ—ing away thâ— meaneâ— by the 22. of the fifth as the parallelogramme NM is to the square of the line BE so is the pyramis ADC to the octohedron AEB inscribed in one and the selfe same sphere But the parallelogramme NM is contayned vnder the line NL which by supposition is in power â—7 â—â— to AC the side of the tetrahedron ADC and vnder the line LM which is also by supposition in length subsesquioctaua to the same line AC Wherfore a tetrahedron an octohedron inscribed in one and the selfe same sphere are in proportioÌ„ as the rectangle parallelogramme contayned vnder the line which contayneth in power 27. sixty fower parts of the side of the Tetrahedron and vnder the line which is subsesquioctaua to the same side of the Tetrahedron is to the square of the diameter of the sphere which was required to be proued ¶ The 12. Proposition If a cube be contayned in a sphere the square of the diameter doubled is equall to all the superficieces of the cube taken together And a perpendicular line drawne from the centre of the sphere to any base of the cube is equall to halfe the side of the cube FOr forasmuch as by the 15. of the thirtenth the diameter of the sphere is in power triple to the side of thâ— cube therefore the square of the diameter doubled is sextuple to the base of the same cube But the sextuple of the power of one of the sides contayneth the whole superficies of the cube 〈◊〉 or the cube is composed of sixe square superficieces by the 2â— diffinition of thâ— eleuenth whose sides therefore are equall wherefore the square of the diâ—meter â—oubled is equall to the whole superficies of the cube And forasmuch as the diameter of the cube and the line which falleth perpendicularly vpoÌ„ the opposite bases of the cube do cut the one the other into two equall partes in the centre of the sphere which containeth the cube by the 2. corollary of the 15. of the thirtenth and the whole right line which coupleth the centres of the opposite bases is equall to the side of the cube by the 33. of the first for it coupleth the equall and parallel semidiameters of the bases therefore the halfe thereof shall be equall to the halfe of the side of the cube by the 15. of the fifth If therefore a cube be contayned in a sphere the square of the diameter doubled is equall to all the superficieces of the cube taken
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
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 square of the line A haue not vnto the square of the line B the same proportion that a square number hath to a square number Then I say that the lines A and B are incommensurable in length For if the lines A and B be commensurable in length then the square of the line A should haue vnto the square of the line B the same proportion that a square number hath to a square number by the first part of this proposition but by supposition it hath not wherfore the lines A and B are not commensurable in length Wherfore they are incomensurable in length Wherfore squares made of right lines commensura in length haue that proportion the one to the other that a square number hath to a square number And squares which haue that proportion the one to the other that a square number hath to a squaâ—e number shall also haue the sides commensurable in length But squares described of right lines incommensurable in length haue not that proportion the one to the other that a square number hath to a square number And squares which haue not that propoâ—tion the one to the other that a sâ—uare number hath to a square number haue not also their sides commâ—nsurable in length which was all that was required to be proued ¶ Corrollary Hereby it is manifest that right lines coÌ„mensurable in length are also euer commensurable in power But right lines commensurable in power are not alwayes commensurable in length And right lines incoÌ„mensurable in leÌ„gth are not alwayes incommensurable in power But right lines incommensurable in power are euer also incommensurable in length For forasmuâ—h as squares made of right lines commensurable in length haue that proportion the one to the other that a square number hath to a square number by the first part of this proposition but magnitudes which haue that proportion the one to the other that number simply hath to number are by the sixt of the tenth commensurable Wherfore right lines commensurable in length are commensurable not onely in length but also in power Againe forasmuch as there are certaine squares which haue not that proportion the one to the other that a square number hath to a square number but yet haue that proportion the one to the other which number simply hath to number their sides in dede are in power commensurable for that they describe squares which haue that proportion which number simply hath to number which squares are therfore commensurable by the 6. of this booke but the said sides are incommensurable in length by the latter part of this proposition Wherâ—fore it is tâ—ue that lines commensurable in power are not straight way commensurable in length also And by the selâ—e same reason is proued also that third part of the corollary that lines incommensurable in length are not alwayes incommensurable in power For they may be incommâ—nsurable in length but yet commensurable in power As in those squares which are in proportion the one to the other as number is to number but not as a square number is to a square number But right lines incommensurable in power are alwayes also incommensurable in length For iâ— they be commensurable in length they shal also be commensurable in power by the first part of this Corollary But they are supposed to be incommensurable in length which is absurde Wherâ—ore right lines incommensurable in power are euer incommensurable in lengthâ— For the better vnderstanding of this proposition and the other following I haue here added certayne annotacions taken out of Montaureus And first as touching the signiâ—ication oâ— wordes and termes herein vsed whâ—ch arâ— such that vnlesse they be well marked and peysed the matter will be obscure and hard and in a maner inexplicable First this ye must note that lines to be commensurable in length and lines to be in proportion the one to the other as number is to number is all one So that whatsoeuer lines are commensurable in length are also in proportion the one to the other as number is to number And conuersedly what so euer lynes are in proportion the one to the other as number is to number are also commeÌ„surable in length as it is manifest by the 5 and 6 of this booke Likewise lines to be incommensurable in length and not to be in proportion the one to the other as number is to number is all one as it is manifest by the 7. and 8. of this booke Wherfore that which is sayd in this Theoreme ought to be vnderstand of lines commensurable in length and incommensurable in length This moreouer is to be noted that it is not all one numbers to be square numbers and to be in proportioÌ„ the one to the other as a square number is to a square number For although square numbers be in proportion the one to the other as a square number is to a square number yet are not all those numbers which are in proportion the one to the other as a square number is to a square number square numbers For they may be like superficiall numbers and yet not square numbers which yet are in proportioÌ„ the one to the other as a square number is to a square number Although all square numbers are like superficiall numbers For betwene two square numbers there â—alleth one meane proportionall number by the 11. of the eight But if betwene two numbers there fall one meane proportionall number those two numbers are like superficiall numbers by the 20. of the eight So also if two numbers be in proportion the one to the other as a square number is to a square number they shall be like superficiall nuÌ„bers by the first corollary added after the last proposition of the eight booke And now to know whether two superficiall numbers geuen be like superficiall numbers or no it is thus found out First if betwene the two numbâ—rs geuen there fall no meane proportionall then are not these two numbers like superficiall numbers by the 18. of the eight But if there do fall betwene them a meane proportionall then are they like superâ—iciall numbers by the 20. of the eight Moreouer two like superficiall numbers multiplied the one into the other do produce a square number by the firsâ— of the ninth Wherfore if they do not produce a square number then are they not like superficiall numbers And if the one being multiplied into the other they produce a square number then are they like superficiall by the 2. of the ninth Moreouer if the said two superficial numbers be in superperticular or superbipartient proportion then are they not like superficiall numbers For if they should be like then should there be a meane proportionall betwene them by the 20. of the eight But that is contrary to the Corollary of the 20. of the eight And the easilier to conceiue the demonstrations following take this example of that which we haue sayd ¶
An Assumpt Forasmuch as in the eight booke in the 26. proposition it was proued that like playne numbers haue that proportion the one to the other that a square number hath to a square number and likewise in the 24. of the same booke it was proued that if two numbers haue that proportion the one to the other that a square number hath to a square number those numbers are like plaine numbers Hereby it is manifest that vnlike plaine numbers that is whose sides are not proportionall haue not that proportion the one to the other that a square number hath to a square number For if they haue then should they be like plaine numbers which is contrary to the supposition Wherfore vnlike plaine numbers haue not that proportion the one to the other that a square number hath to a square nuÌ„ber And therfore squares which haue that proportion the one to the other that vnlike plaine numbers haue shall haue their sides incommensurable in length by the last part of the former proposition for that those squares haue not that proportion the one to the other that a square number hath to a square number ¶ The 8. Theoreme The 10. Proposition If foure magnitudes be proportionall and if the first be commensurable vnto the second the third also shal be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth SVppose that these foure magnitudes A B C D be proportionall As A is to B so let C be to D and let A be commensurable vnto B. Then I say that C is also commensurable vnto D. For forasmuch as A is commensurable vnto B it hath by the fift of the tenth that proportion that number hath to number But as A is to B so is C to D. Wherfore C also hath vnto D that proportion that number hath to number Wherfore C is commensurable vnto D by the 6. of the tenth But now suppose that the magnitude A be incommensurable vnto the magnitude B. Then I say that the magnitude C also is incommensurable vnto the magnitude D. For forasmuch as A is incommensurable vnto B therfore by the 7. of this booke A hath not vnto B such proportion as number hath to number But as A is to B so is C to D. Wherefore C hath not vnto D such proportion as number hath to number Wherfore by the 8. of the tenth C is incommensurable vnto D. If therefore there be foure magnitudes proportionall and if the first be commensurable vnto the second the third also shall be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth which was required to be proued ¶ A Corollary added by Montaureus If there be foure lines proportionall and if the two first or the two last be commensurable in power onely the other two also shall be commensurable in power onely This is proued by the 22. of the sixt and by this tenth proposition And this Corollary Euclide vseth in the 27. and 28. propositions of this booke and in other propositions also ¶ The 3. Probleme The 11. Proposition Vnto a right line first set and geuen which is called a rationall line to finde out two right lines incommensurable the one in length onely and the other in length and also in power SVppose that the right line first set and geuen which is called a rationall line of purpose be A. It is required vnto the said line A to finde out two right lines incommensurable the one in length onely the other both in length and in power Take by that which was added after the 9. proposition of this booke two numbers B and C not hauing that proportion the one to the other that a square number hath to a square number that is let them not be like plaine numbers for like plaine numbers by the 26. of the eight haue that proportion the one to the other that a square number hath to a square number And as the number B is to the number C so let the square of the line A be vnto the square of an other line namely of D how to do this was taught in the assumpt put before the 6. proposition of this booke Wherfore the square of the line A is vnto the square of the line D commensurable by the sixt of the tenth And forasmuch as the number B hath not vnto the number C that proportion that a square number hath to a square nuÌ„ber therfore the square of the line A hath not vnto the square of the line D that proportioÌ„ that a square number hath to a nuÌ„ber Wherfore by the 9. of the tenth the line A is vnto the line D incommensurable in length onely And so is found out the first line namely D incommensurable in length onely to the line geuen A. Agayne take by the 13. of the sixt the meane proportionall betwene the lines A and D and let the same be E. Wherfore as the line A is to the line D so is the square of the line A to the square of the line E by the Corollary of the 20. of the sixt But the line A is vnto the line D incommensurable in length Wherfore also the square of the line A is vnto the square of the line E incommensurable by the second part of the former proposition Now forasmuch as the square of the line A is incoÌ„meÌ„surable to the square of the line E it followeth by the definition of incommensurable lynes that the line A is incommensurable in power to the line E. Wherfore vnto the right line geuen and first set A which is a rationall line and which is supposed to haue such diuisions and so many partes as ye list to conceyue in minde as in this example 11 whereunto as was declared in the 5. definition of this booke may be compared infinite other lines either commensurable or incommensurable is found out the line D incommensurable in length onely Wherfore the line D is rationall by the sixt definitioÌ„ of this booke for that it is incommensurable in length onely to the line A which is the first line set and is by suppositioÌ„ rational There is also found out the line E which is vnto the same line A incommensurable not onely in length but also in power which line E compared to the rationall line A is by the definition irrationall For Euclide alwayes calleth those lines irrationall which are incommensurable both in length and in power to the line first set and by supposition rationall ¶ The 9. Theoreme The 12. Proposition Magnitudes commensurable to one and the selfe same magnitude are also commensurable the one to the other SVppose that either of these magnitudes A and B be commensurable vnto the magnitude C Then I say that the magnitude A is commensurable vnto the magnitude B. For â—orasmuch as the
magnitude A is commensurable vnto the magnitude C therefore by the 5. of the tenth A hath vnto C such proportion as number hath to number Let A haue vnto C that proportion that the number D hath to the number E. Againe forasmuch as B is commensuâ—able vnto C therefore by the selfe same C hath vnto B that proportion that number hath to number Let C haue vnto B that proportion that the number F hath vnto the number G. Now then take the least numbers in continuall proportion and in these proportions geuen namely that the number D hath to the number E and that the number F hath to the number G by the 4. of the eight which let be the numbers â— K L. So that as the number D is to the number E so let the number H be to the number K and as the nuÌ„ber F is to the nuÌ„ber G so let the nuÌ„ber K be to the nuÌ„ber L. Now for that as A is to C so is D to E but as D is to E so is H to K therfore as A is to C so is H to K. Againe for that as C is to B so is F to G but as F is to G so is K to L therefore as C is to B so is K to L. But it is now proued that as A is to C so is H to K. Wherefore of equalitie by the 22. of the fift as A is to B so is the number H to the number L. Wherefore A hath vnto B such proportion as number hath to number Wherefore by the sixt of the tenth the magnitude A is commensurable vnto the magnitude B. Magnitudes therefore commensurable to one and the selfe same magnitude are also commensurable the one to the other which was required to be proued ¶ An Assumpt If there be two magnitudes compared to one and the selfe same magnitude and if the one of them be commensurable vnto it and the other incommensurable those magnitudes are incommensurable the one to the other SVppose that there be two magnitudes namely A and B and let C be a certayne other magnitude And let A â—e commensurable vnto C and let B be commeÌ„surable vnto the selfe same C. Then I say that the magnitude A is incommensurable vnto B. For if A be commensurable vnto B forasmuch as A is also commeÌ„surable vnto C therefore by the 12. of the tenth B is also commeÌ„surable vnto C which is contrary to the supposition ¶ The 10. Theoreme The 13. Proposition If there be two magnitudes commensurable and if the one of them be incommensurable to any other magnitude the other also shall be incommensurable vnto the same SVppose that these two magnitudes A B be commensurable the one to the other and let the one of them namely A be incommensurable vnto an other magnitude namely vnto C. Then I say that the other magnitude also namely B is incommensurable vnto C. For if B be commensurable vnto C then forasmuch as A is commensurable vnto B therefore by the 12. of the tenth the magnitude A also is commensurable vnto the magnitude C. But it is supposed to be incommensurable vnto it which is impossible Wherefore the magnitudes B and C are not commensurable Wherefore they are incommensurable If therefore there be two magnitudes commensurable and if the one of them be incommensurable to any other magnitude the other also shal be incommensurable vnto the same which was required to be proued ¶ A Corollary added by Montaureus Magnitudes commensurable to magnitudes incommeÌ„surable are also incommensurable the one to the other Suppose that the magnitudes A and B be incommensurable the one to the other and let the magnitudâ— C be coÌ„mensurable to A and let the magnitude D be coÌ„mensurable vnto B. Then I say that the magnituâ—â—s C and D are incommensurable the one to the other For A and C are commensurable of which the magnitude A is incommensurable vnto B wherefore by this 13. proposition the magnitudes C and B are also incommensurable but the magnitudeâ— B and D are coÌ„mensurable wherefore by the same or by the former assumpt the magnitudes C and D are incommensurable the one to the other This corollary Theon vseth often times as in the 22. 26. and 36 propositions of this booke and in other propositions also ¶ An Assumpt Two vnequall right lines being geuen to fiâ—de out how much the greater is in power more then the lesse And like in sorte two right lines being geuen by this meanes may be founde out a right lyne which contayneth them both in power Suppose that the two right lines geuen be AD and DB. It is required to â—inde out a right lyne that contayneth them both in power Let the lines AB and DB be so put that they comprehend a right angle ADB and draw a right line from A to B. Now agayne it is manifest by the 47. of the â—irst that the line AB contayneth in power the lines AD and DB. ¶ The 11. Theoreme The 14. Proposition If there be sower right lines proportionall and if the first be in power more then the second by the square of a right line commensurable in length vnto the first the third also shal be in power more then the fourth by the square of a right line commensurable vnto the third And if the first be in power more then the second by the square of a right line incommensurable in length vnto the first the third also shall be in power more then the fourth by the square of a right line incommensurable in length to the third SVppose that these foure right lines A B C D be proportionall As A is to B so let C be to D. And let A be in power more then B by the square of the line E. And likewise let C be in power more then D by the square of the line F. Then I say that if A be commensurable in length vnto the line E C also shall be commensurable in length vnto the line F. And if A be incommensurable in length to the line E C also shall be incommensurable in length to the line F. For for that as A is to B so is C to D therefore as the square of the line A is to the square of the line B so is the square of the line C to the square of the line D by the 22. of the sixt But by supposition vnto the square of the line A are equall the squares oâ— the lines E and B and vnto the square of the line C are equall the squares of the of the lines D and F Wherefore as the squares of the lines E and B which are equall to the square of the line A are to the square of the line B so are the squares of the lines D and F which are equall to the square of the line C to the square of the line D by the seuenth of the fift Wherfore
by the 17. of the fift as the square of the line â— iâ— to the square of the line B so is the square of the line F to the square of the line D. Wherfore also as the line E is to the line â— so is the line F to the line D by the second parte of the 22. of the sixt wherefore contrariwise by the Corollary of the fourth of the fift as B is to E so is D to F. But by supposition â—s A is to B so is C to D Wherfore of equallitie by the 22. of the fift aâ— A is to â— so is C is F. If therfore A be commensurable in length vnto E C also shall be commâ—nsurable in leÌ„gth vnto F and if it ba incommensurable in length vnto E C also shal be incommensurablel in length vnto F by the 10. of this booke If therfore there be foure right tlines proportionall and if the first be in power more then the secondby the square of a right line commensurable in length vnto the first the third also shall be in power more then the fourth by the square of a right line commensurable in length vnto the third and if the first be in power more theÌ„ the second by the square of a right line incoÌ„mensurable in length vnto the first the third also shall be in power more the the fourth by the square of a right line incommensurable in length to the third which was required to be proued Note that the line A may be proued to be in proportion to the line E as the line C is to the line F by an other way namely by conuersion of proportion of someâ— as we haue before noted called inuerse proportion by the 19. of the fift For forasmuch as the foure lines A B C D are proportionall therfore by the 22. of the sixt their squares also are proportionall And forasmuch as the antecedent namely the square of the line A excedeth the consequent namely the square of the line B by the square of the line E and the other antecedent namely the square of the line C excedeth the other consequent namely the square of the line D by the square of the line F therefore as the square of the line A is to the excesse namely to the square of the line E so is â—he square of the line C to the excesse namely to the square of the line F. Wherefore by the second part of the 22. of the sixt as the line A is to the line E so is the line C to the line F. ¶ The 12. Theoreme The 15. Proposition If two magnitudes commensurable be composed the whole magnitude composed also shall be commensurable to either of the two partes And if the whole magnitude composed be commensurable to any one of the two partes those two partes shall also be commensurable LEt these two commensurable magnitudes AB and BC be composed or added together Then I say that the whole magnitude AC is coÌ„mensurable to either of these partes AB and BC For forasmuch as AB and BC are commensurable therfore by the first definition of the tenth some one magnitude measureth them both Let there be a magnitude that measureth them and let the same be D. Now forasmuch as D measureth AB and BC it shall also measure the whole magnitude composed AC by this common sentence what soeuer magnitude measureth two other magnitudes shall also measure the magnitude composed of them But the same D measureth AB and BC by supposition Wherefore D measureth AB BC and AC Wherefore AC is commensurable to either of these magnitudes AB and BC. But now suppose that the whole composed magnitude AC be commensurable to any one of these two magnitudes AB or BC let it be commensurable I say vnto AB Then I say that the two magnitudes AB and BC are commensurable For forasmuch as AB and AC are commonsurable some one magnitude measureth them by the first definition of the tenth Let some magnitude measure them and let the same be D. Now forasmuch as D measureth AB and AC it also measureth the residue BC by this common sentence what soeuer measureth the whole and the part taken away shall also measure the residuâ— But the same D measureth the magnitude AB by supposition Wherefore D measureth either of these magnitudes AB and BC. Wherefore the magnitudes AB and BC are commensurable If therfore two magnitudes commensurable be composed the whole magnitude composed also shall be commensurable to either of the two partes And if the whole magnitude composed be commensurable to any one of the two partes those two partes shall also be commensurable which was required to be demonstrated ¶ A Corollary added by Montaureus If an whole magnitude be commensurable to one of the two magnitudes which make the whole magnitude it shall also be commensurable to the other of the two magnitudes For if the whole magnitude AC be commensurable vnto the magnitude BC then by the 2. part of thys 15 Proposition the magnitudes AB and BC are commensurable Wherefore by the first part of the same the magnitude AC shall be commensurable to either of these magnitudes AB and BC. This Corollary Theon vseth in the demonstration of the 17. Proposition and also of other Propositions Howbeit Euclide left it out for that it seemed easie as in a maner do all other Corollaryes ¶ The 13. Theoreme The 16. Proposition If two magnitudes incommensurable be composed the whole magnitude also shall be incommensurable vnto either of the two partes coÌ„ponentes And if the whole be incommensurable to one of the partes componentes those first magnitudes also shall be incommensurable LEt these two incommensurable magnitudes AB BC be composed or added together Then I say that the whole magnitude AC is incommensurable to either of these magnitudes AB and BC. For if AC and AB be not incommensurable then some one magnitude measureth them by the first definition of the tenth Let there be such a magnitude if it be possible and let the same be D. Now forasmuch as D measureth CA and AB it also measureth the residue BC it likewise measureth AB Wherefore D measureth AB and BC. Wherfore by the first definitioÌ„ of the tenth the magnitudes AB and BC are commensurable But it is supposed that they are incommensurable which is impossible Wherefore no magnitude doth measure the magnitudes AB and AC Wherefore the magnitudes CA and AB are incommensurable In like sort also may we proue that the magnitudes AC and CB are incommensurable But now suppose that the magnitude AC be incommensurable to one of these magnitudes AB or BC and first let it be incommensurable vnto AB Then I say that the magnitudes AB and BC are incommensurable For if they be commensurable some one magnitude measureth them Let some one magnitude measure them let the same be D.
Now forasmuch as D measureth AB and BC it also measureth the whole magnitude AC And it measureth AB Wherefore D measureth these magnitudes CA and AB Wherefore CA AB are commensurable And they are supposed to be incoÌ„mensurableâ— which is impossible Wherfore no magnitude measureth AB and BC. Wherefore the magnitudes AB and BC are incommensurable And in like sort may they be proued to be incommensurable if the magnitude AC be supposed to be incommensurable vnto BC. If therefore there be two magnitudes incommensurable composed the whole also shall be incommensurable vnto either of the two partes component and if the whole be incommensurable to one of the partes component those first magnitudes shall be incommensurable which was required to be proued ¶ A Corollary added by Montaureus If an whole magnitude bee incommensurable to one of the two magnitudes which make the whole magnitude it shall also be incommensurable to the other of the two magnitudes For if the whole magnitude AC be incoÌ„mensurable vnto the magnitude BC then by the 2 part of this 16. Theorâ—me the magnitudes AB and BC shall be incommensurable Wherefore by the first part of the same Theoreme the magnitude AC shall be incommensurable to either of these magnitudes AB and BC. This Corollary 〈◊〉 vseth in the demonstration of the â—3 Theoreme also of other Propositions ¶ An Assumpt If vpon a right line be applied a parallelogramme wanting in figure by a square the parallelogramme so applied is equall to that parallelogramme which is contayned vnder the segmentes of the right line which segmentes are made by reason of that application This Assumpt I before added as a Corollary out of Flussates after the 28. Proposition of the sixt booke ¶ The 14. Theoreme The 17. Proposition If there be two right lines vnequall and if vpon the greater be applied a parallelogramme equall vnto the fourth part of the square of the lesse line and wanting in figure by a square if also the parallelogramme thus applied deuide the line where vpon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a right line commensurable in length vnto the greater and if also vpon the greater be applied a parallelograÌ„me equall vnto the fourth part of the square of the lesse line and wanting in figure by a square then shall it deuide the greater line into partes commensurable But now suppose that the line BC be in power more then the line A by the square of a line commensurable in length vnto the line BC. And vpon the line BC let there be applied a rectangle parallelograme equall vnto the fourth part of the square of the line A and wanting in figure by a square and let the sayd parallelograme be that which is contained vnder the lines BD and DC Then must we proue that the line BD is vnto the line DC commensurable in length The same constructions and suppositions that were before remayning we may in like sort proue that the line BC is in power more then the line A by the square of the line FD. But by suppositioÌ„ the line BC is in power more theÌ„ the line A by the square of a line coÌ„mensurable vnto it in length Wherfore the line BC is vnto the line FD coÌ„mensurable in length Wherefore the line composed of the two lines BF and DC is coÌ„mensurable in length vnto the line FD by the second part of the 15. of the tenth Wherefore by the 12. of the tenth or by the first part of the 15. of the tenth the line BC is commensurable in length to the line composed of BF and DC But the whole line conposed BF and DC is commensurable in length vnto DC For BF as before hath bene proued is equall to DC Wherefore the line BC is commensurable in length vnto the line DC by the 12. of the tenth Whâ—â—â—fore also the line BD is commensurable in length vnto the line DC by the second part of thâ— 15. of the teâ—th If therfore there be two right lines vnequall and if vpon the greater be appliâ—d a parallelograme equall vnto the fourth part of the square of the lesse and wanting in figure by a square if also the parallelograme thus applied deuide the line whereupon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a line commeÌ„surable in length vnto the greater and if also vpon the greater be applied a parallelograme equall vnto the fourth part of the square made of the lesse and wanting in figure by a square then shall it deuide the greater line into partes commensurable in length which was required to be proued Campanâ— after this proposition reacheth how we may redily apply vpon the line BC a parallelograme equall to the fourth part of the square of halfe of the line A and wanting in figure by a square after this maner Deuide the line BC into two lines in such sort that halfe of the line A shal be the meane proportionall betwene those two lines which is possible when as the line BC is supposed to be greater then the line A and may thus be done Deuide the line BC into two equal partes in the point E and describe vpon the line BC a semicircle BHC And vnto the line BC and from the point C erect a perpâ—dicular line CK and put the line CK equall to halfe of the line Aâ— And by the point K draw vnto the line EC a parallel line KH cutting the semicircle in the point H which it must needes cut foâ—asmuch as the line BC is greater then the line A And froÌ„ the point H draw vnto the line BC a perpendicular liâ—e HD which line HDâ— forasmuch as by the 34 of the first it is equall vnto the line KC shall also be equall to halfe of the line A draw the lines BH and HC Now then by the â—â— of the third the angle BHC is a right aâ—gle Wherefore by the corollary of the eight of the sixt booke the line HD is the meane proportionall betwene the lines BD and DC Wherefore the halfe of the line A which is equall vnto the line HD is the meane proportionall betwene the lines BD and DC Wherefore that which is contained vnder the lines BD and DC is equall to the fourth part of the square of the line A. And so if vpon the line BD be described a rectangle parallelograme hauing his other side equall to the line DC there shal be applied vpon the line BC a rectangle parallelograme equall vnto the square of halfe of the line A and wanting in figure by
a square which was required to be done ¶ The 15. Theoreme The 18. Proposition If there be two right lines vnequall and if vpon the greater be applied a parallelograme equall vnto the fourth part of the square of lesse and wanting in figure by a square if also the parallelograme thus applied deuide the line whereupon it is applied into partes incommensurable in length the greater line shal be in power more then the lesse line by the square of a line incommensurable in length vnto the greater line And if the greater line be in power more then the lesse line by the square of a line incommeÌ„surable in length vnto the greater and if also vpon the greater be applied a parallelograme equall vnto the fourth part of the square of the lesse and wanting in figure by a square then shall it deuide the greater line into partes incommensurable in length But now suppose that the line BC be in power more then the line A by the square of a line incommensurable in length vnto BC. And vpon the line BC let there be applied a parallelogramme equall vnto the fourth part of the square of the line A and wanting in figure by a square and let the said parallelogramme be that which is contained vnder the lines BD DC Then must we proue that the line BD is vnto the line DC incommensurable in leÌ„gth The same order of construction and demonstration being kept we may in like sort proue that the line BC is in power more then the line A by the square of the line FD. But now by supposition the line BC is in power more then the line A by the square of a line incommensurable in length vnto BC. Wherefore the line BC is vnto the line FD incommensurable in length Wherfore the line composed of BF and DC taken as one line shall be incommensurable in length to the line FD by the second part of the 16. of the tenth wherefore also by the first part of the same the line BC shall be incommensurable in length to the line composed of the lines BF and DC But the line composed of the lines BF and DC is commensurable in length to the line DC for that BF as before hath bene proued is equall to DC Wherfore the line BC is incoÌ„mensurable in length to the line DC by the 13. of the tenth Wherefore by the second part of the 16. of the tenth the line BD is incommeÌ„surable in leÌ„gth vnto the line DC If therfore there be two right lines vnequall and if vpon the greater be applied a parallelogramme equall vnto the fourth part of the square of the lesse line wanting in â—igure by a square if also the parallelogramme thus applied deuide the line wherupon it is applied into partes incommensurable in length the greater line shall be in power more then the lesse line by the square of a line incommensurable in length vnto the greater And if the greater line be in power more then the lesse by the square of a line incommensurable in length vnto the greater and if also vpon the greater be applied a parallelogramme equall vnto the fourth part of the square of the lesse line and wanting in figure by a square then shall it deuide the greater line into partes incommensurable in length which was required to be demonstrated This Proposition may also be demonstrated by the former proposition namely the first part of this by the second part of the former and the second part of this by the first part of the former by an argument leading to an absurditie For as touching the first part of this proposition the line BC contayning in power more then the line A by the square of the line FD if the line BG be not incommensurable vnto the line FD then is it commensurable vnto it Wherfore by the second part of the 17. proposition the lines BD and DC also are commensurable which is impossible for they are supposed to be incommensurable So likewise as touching the second parte of the same the line BC contayning in power more then the line A by the square of the line FD if the line DB be not incommensurable to the lyne DC then is it commensurable vnto it wherfore by the first part of the â—â— proposition the lines BC and FD are also commensurable which were absuâ—de For the lines BC and FD are supposed to be incommensuâ—able which was required to be proued ¶ An assumpt Forasmuch as it hath bene proued that lines commensurable in length are alwayes also commensurable in power but lines commensurable in power are not alwayes commensurable in length but may be in length both commensurable and also incommensurable it is manifest that if vnto the line propounded which is called rationall of purpose a certayne line be coÌ„mensurable in length it ought to be called rationall and coÌ„mensurable vnto it not only in length but also in power for lines commensurable in length are also alwayes commensurable in power But if vnto the line propounded which is called rationall of purpose a certayne line be commensurable in power then if it be also commensurable vnto it in length it is called rationall and commensurable vnto it both in length and in power But againe if vnto the said line geuen which is called rationall a certayne line be commensurable in power and incommensurable in length that also is called rationall commensurable in power onely An annotacion of Proclus He calleth those lines rationall which are vnto the rationall line first set coÌ„mensurable in length in power or in power only And there are also other right lines which are vnto the rationall line first set incommensurable in length and are vnto it commensurable in power only and therfore they are called rationall commeÌ„surable the one to the otherâ— for which cause they are rationall But euen these lines may be commensurable the one to the other either in length and therefore in power or els in power onely Now if they be commensurable in length then are those lines called rationall commensurable in length but yet so that they be vnderstand to be in power commensurable but if they be commensurable the one to the other in power onely they also are called rationall commensurable in power onely ¶ A Corollary And that two lines or more being rationall and commensurable in length to the rational line first set are also commensurable the one to the other in length hereby it is manifest for forasmuch as they are rationall and commensurable in length to the rationall line first set but those magnitudes whiche are commensurable to one and the selfe same magnitude are also commensurable the one to the other by the 12. of the tenth wherfore the rationall lines commensurable in length to the rational lyne first set are also commensurable in length the one to the other And as touching those which are rationall commensurable in power onely to the rationall line
first set they also must needes be at the least commensurable in power the one to the other For forasmuch as their squares are rationall they shall bee commensurable to the square of the rationall line first set Wherfore by the 12. of this booke they are also commensurable the one to the other Wherefore their lines are at the least commensurable in power the one to the other And it is possible also that they may be commeÌ„surable in leÌ„gth the one to the other For suppose that A be a rationall liâ—e first set and let the line B be vnto the same rationall line A commensurable in power onely that is incommensurable in length vnto it Let there be also an other line C commensurable in length to the lyne B which is possible by the principles of this booke Now by the 13. of the tenth it is manifest that the line C is incommensurable in length vnto the line A. But the square of the line A is coÌ„meÌ„surable to the square of the line B by supposition and the square of the line C is also commensurable to the square of the line B by supposition Wherefore by the 12. of this booke the square of the line C is commensurable to the square of the line A. Wherfore by the definition the line C shall be rationall commensurable in power onely to the line A as also is the line B. Wherefore there are geuen two rationall lines commensurable in power onely to the rationall line first set and commeÌ„surable in length the one to the other Here is to be noted which thing also we before noted in the definitions that Campane and others which followed him brought in these phrases of speaches to call some lynes rationall in power onely and other some rationall in length and in power which we cannot finde that Euclide euer vsed For these wordes in length and in power are neuer referred to rationalitie or irrationalitie but alwayes to the commensurabilitie or incommensurablitie of lines Which peruerting of wordes as was there declared hath much increased the difficulty and obscurenes of this booke And now I thinke it good agayne to put you in minde that in these propositions which follow we must euer haue before our eyes the rationall line first set vnto which other lines compared are either rationall or irrationall according to their commensurability or incommensurabilitie ¶ The 16. Theoreme The 19. Proposition A rectangle figure comprehended vnder right lines commensurable in lengthe being rationall according to one of the foresaide wayes is rationall SVppose that this rectangle figure AC be comprehended vnder these right lines AB and BC being commensurable in length and rationall according to one of the foresaid wayes Then I say that the superficies AC is rationall describe by the 46. oâ— the first vpon the line AB a square AD. Wherfore that square AD is rationall by the definition And forasmuch as the line AB is commensurable in length vnto the line BC and the line AB is equall vnto the lyne BD therefore the lyne BD is commensurable in length vnto the line BC. And as the line BD is to the line BC so is the square DA to the superficies AC by the first of the sixt but it is proued that the line BD is commensurable vnto the line BC wherfore by the 10. of the tenth the square DA is commensurable vnto the rectangle superficies AC But the square DA is rationall wherfore the rectangle superficies AC also is rationall by the definition A rectangle figure therfore comprehended vnder right lines commensurable in length beyng rationall accordyng to one of the foresayd wayes is rationall which was required to be proued Where as in the former demonstration the square was described vpon the lesse line we may also demonstrate the Proposition if we describe the square vpon the greater line and that after thys maner Suppose that the rectangle superficies BC be contayned of these vnequall lines AB and AC which let be rationall commensurable the one to the other in length And let the line AC be the greater And vpon the line AC describe the square DC Then I say that the parallelogramme BC is rationall For the line AC is commensurable in length vnto the line AB by supposition and the line DA is equall to the line AC Wherefore the line DA is commensurable in length to the line AB But what proportion the line DA hath to the line AB the same hath the square DC to the paraâ—lelogramme Câ— by the first of the sixt Wherefore by the 10. of this booke the square DC is commensurable to the parallelogramme CB. But it is manifest that the square DC is rationall for that it is the square of a rationall line namely AC Wherefore by the definition the parallelograÌ„me also CB is rationall Moreouer forasmuch as those two former demonstrations seeme to speake of that parallelograÌ„me which is made of two lines of which any one may be the liâ—e first set which is called the first rationall line from which we sayd ought to be taken the measures of the other lines compared vnto it and the other is commeÌ„surable in length to the same first rationall line which is the first kinde of rationall lines coÌ„mensurable in length I thinke it good here to set an other case of the other kinde of rationall lines of lines I say rationall coÌ„mensurable in length compared to an other rationall line first set to declare the generall truth of this Theoreme and that we might see that this particle according to any of the foresayd wayes was not here in vaine put Now then suppose first a rationall line AB Let there be also a parallelograÌ„me CD contayned vnder the lines CE and ED which lines let be rationall that is commensurable in length to the â—irst rationall line propounded AB Howbeit let those two lines CE and ED be diuers and vnequall lines vnto the first rationall line AB Then I say that the parallelogramme CD is rationall Describe the square of the line DE which let be DF. First it is manifest by the 12. of this booke that the lines CE ED are commensurable in leÌ„gth the one to the other For either of them is supposed to be commensurable in length vnto the line AB But the line ED is equall to the line EF. Wherefore the line CE is commensurable in length to the line BF But 〈◊〉 the line CE is â—o the line â— F â—o is the parallelogramme CD to the square DF by the first of the sixt Whâ—refore by the 10. of this booke the parallelogramme CD shall be commensurable to the square DF. But the square DF is commensurable to the square of the line AB which is the first rationall line propounded Wherfore by the 12. of this booke the parallelogramme CD is commensurable to the square of the line AB But the square of
of BE. But that which is produced of AB into BC together with the square number of CE is supposed to be equal to the square number of BE wherfore that which is produced of GB into BC together with the square number of CE is equall to that which is produced of AB into BC together with the square number of CE. Wherefore taking away the square number of CE which is common to them both the number AB shall be equall to the number GB namely the greater to the lesse which is impossible Wherfore that which is produced of AB into BC together with the square number of CE is not equall to the square number of BE I say also that that which is produced of AB into BC together with the square number of CE is not lesse then the square number of BE. For if it be possible theÌ„ shall it be equâ—l to some square number lesse then the square number of BE. Wherfore let the number produced of AB into BC together with the square of the number CE be equal to the square number of BF And let the number HA be double to the number DF. TheÌ„ also it followeth that the number HC is double to the number CF so that HC also is deuided into two equall partes in F and therfore also the number which is produced of HD into BC together with the square number of FC is equall to the square number of the number BF But by supposition the number which is produced of AB into BC together with the square number of CE is equall to the square number of BF Wherfore it followeth that the number produced of AB into BC together with the square number of CE is equall to that which is produced of HB into BC together with the square number CF which is impossible For if it should be equall then forasmuch as the square of CF is lesse then the square of CE the number produced of HB into BC should be greater then thâ— number produced of AB into BC. And so also should the number HB be greater then the number AB when yet it is lesse then it Wherfore the number produced of AB into BC together with the square number of CE is not lesse then the square nuÌ„ber of â— E. And it is also proued that it cannot be equall to the square number of BE neither greater then it Wherfore that which is produced of AB into BC added to the square number of CE maketh not a square number And although it be possible to demonstrate this many other wayes yet this semeth to vs sufficiâ—nâ— least the matter beyng ouer long should seeme to much tedious ¶ The 6. Probleme The 29. Proposition To finde out two such rationall right lynes commensurable in power only that the greater shall be in power more then the lesse by the square of a right line commensurable in length vnto the greater LEt there be put a rational line AB and take also two such square numbers CD and DE that their excesse CE be not a square number by the corolary of the first assumpt of the 28. of the tenth And vpon the line AB describe a semicircle AFB And by the corollary of the 6. of the tenth as the number DC is to the number CE so let the square of the lyne BA be to the square of the line AF. And draw a line from F to B. Now for that as the square of the line BA is to the square of the line AF so is the number CD to the number CE therfore the square of the line BA hath to the square of the line AF that proportion that the nuÌ„ber CD hath to the number CE. Wherfore the square of the line BA is coÌ„meÌ„surable to the square of the line AF by the 6. of the teÌ„th But the square of the line AB is rational Wherfore also the square of the line AF is rational Wherfore also the line AF is rationall And forasmuch as the number CD hath not vnto the number CE that proportion that a square number hath to a square number therfore neither also hath the square of the line AB to the square of the line AF that proportion that a square number hath to a square number Wherfore by the 9. of the teÌ„th the line AB is vnto the line AF incommensurable in length Wherfore the lines AF and AB are rationall commensurable in power onely And for that as the number DC is to the number CE so is the square of the line AB to the square of the line AF therfore by conuersion or euerse proportioÌ„ which is demonstrated by the corollary of the 19. of the fifth as the number CD is to the number DE so is the square of the line AB to the square of the line BF which is the excesse of the square of the line AB aboue the square of the line AF by the assumpt put before the 14. of this booke But the number CD hath to the number DE that proportion that a square number hath to a square number wherfore the square of the line AB hath to the square of the line BF that proportion that a square numâ—er hath to a square number Wherefore by the 9. of the tenth the line AB is commensurable in length vnto the line BF And by the 47. of the first the square of the line AB is equall to the squares of the lines AF and FB Wherfore the line AB is in power more then the line AF by the square of the line BF which is commensurable in length vnto the line AB Wherefore there are found out two such rationall lines commensurable in power onely namely AB and AF so that the greater line AB is in power more then the lesse line AF by the square of the line FB which is commensurable in length vnto the line AB which was required to be done ¶ The 7. Theoreme The 30. Proposition To finde out two such rationall lines commensurable in power onely that the greater shal be in power more then the lesse by the square of a right line incommensurable in length to the greater LEt there be put a rationall line AB and take also by the 2. assumpt of the 28. of the tenth two square numbers CE and ED which being added together make not a square number and let the numbers CE and ED added together make the number CD And vpon the line AB describe a sencircle AFB And by the corollary of the 6. of the tenth as the number DC is to the number CE so let the square of the line AB be to the square of the line AF and draw a line from F to B. And we may in like sort as we did in the former proposition proue that the lines BA and AF are rationall commensurable in power onely And for that as the number DC is
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
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
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
is equall to that which is contained vnder the lines AC and CB twise Agayne from the the parallelograme EK take away the parallelograme EL equall to the squares of the lines AD and DB which are lesse then the squares of the lines AC and CB. Wherefore the residue namely the parallelograme MK is equall to that which is contayned vnder the lines AD and DB twise And forasmuch as the squares of the lines AC and CB are mediall therefore the parallelograme EG also is mediall And it is applyed vpon the rationall line EF â— wherefore the line EH is rationall and incommensurable in length to the line EF. And by the same reason the parallelograme HK is mediall for that which is equall vnto it namely that which is contayned vnder the lines AC and CB twise is mediall therefore the line HN is also rationall and incommensurable in length vnto the line EF. And forasmuch as the lines AC and CB are mediall lines commensurable in power onely therefore the line AC is incommensurable in length vnto the line CB. But as the line AC is to the line CB so is the square of the line AC to that which is contayned vnder the lines AC and CB by the 1. of the sixt Wherefore the square of the line AC is incommensurable to that which is contayned vnder the lines AC and CB. But by the 16. of the tenth vnto the square of the line AC are commensurable the squares of the lines AC and CB added together for the lines AC and CB are commensurable in power onely And vnto that which is contayned vnder the lines AC and CB is commensurable that which is contayned vnder the lines AC and CB twise Wherefore 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 But to the squares of the lines AC and CB is equall the parallelograme EG and to that which is contained vnder the lines AC and CB twise is equall the paralelograme HK Wherfore the parallelograme EG is incommensurable to the parallelograme HK Wherefore also the line EH is incommensurable in length to the line HN. And the lines EH and HN are rationall Wherefore they are rationall commensurable in power onely but if two rationall lines commensurable in power onely be added together the whole line is irrationall and is called a binomiall line by the 36. of the tenth Wherefore the binomiall line EN is in the poynt H deuided into his names And by the same reason also may it be proued that the lines EM and MN are rationall lines commensurable in power onely Wherfore EN being a binomiall line is deuided into his names in sundry poyntes namely in H and M neither is the line EH one and the same that is equal with MN For the squares of the lines AC and CB are greater then the squares of the lines BD and AD by the 1. assumpt put after the 41. of the tenth But the squares of the lines AD and DB are greater then that which is contayned vnder the lines AD and DB twise by the assumpt put after the 39. of the tenth Wherefore the squares of the lines AC and CB that is the parallelograme EG is much greater then that which is contained vnder the lines AD and DB twise that is then the parallelograme MK Wherfore by the first of the sixt the line EH is greater then the line MN Wherefore EH is not one and the same with MN Wherefore a binomiall line is in two sundry poyntes deuided into his names Which is impossible The selfe same absurditie also will follow if the line AC be supposed to be lesse then the line DB. A second binomiall line therefore is not deuided into his names in sundry poyntes Wherefore it is deuided in one onely which was required to be demonstrated ¶ The 33. Theoreme The 45. Proposition A greater line is in one poynt onely deuided into his names LEt AB being a greater line be deuided into his names in the poynt C so that let the lines AC and CB be rationall incommensurable in power hauing that which is composed of the squares of the lines AC and CB rationall and that which is contained vnder the lines AC and CB mediall Then I say that the line AB can not in any other poynt then in C be deuided into his names For if it be possible let it be deuided into his names in the poynt D so that let AD and DB be lines incommeÌ„surable in power hauing that which is composed of the squares of the lines AD and DB rationall and that which is contayned vnder the lines AD and DB mediall Now forasmuch as how much the squares of the lines AC and CB diâ—â—er from the squares of the lines AD and DB so much differeth that which is contained vnder the lines AD and DB twise from that which is contained vnder the lines AC and CB twise by those thinges which haue beâ—e sayd in the demonstration of the 42. proposition But the squares of the lines AC and CB exceede the squares of the lines AD and DB by a rationall superâ—icies for they are either of them rationall Wherfore that which is containeâ—â—â—der the lines AD and DB twise exceedeth that which is contained vnder the lines AC and CB twise by a rationall superâ—icies when as either of them is a mediall supersâ—â—â—es Which is impossible by the 26. of the tenth Wherefore a greater line is in one poynt onâ—ly deuided into his names which was required to be proued ¶ The 34. Theoreme The 46. Proposition A line contayning in power a rationall and a mediall is in one point onely deuided into his names LEt AB being a line containing in power a rationall and a mediall be deuided into his names in the point C so that let the lines AC CB be incommensurable in power hauing that which is composed of the squares of the lines AC and CB mediall and that which is contained vnder the lines AC and CB rationall Then I say that the line AB can not in any 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 the lines AD and DB be incommensurable in power hauing that which is composed of the squares of the lines AD and DB mediall and that which is contayned vnder the lines AD and DB rationall Now forasmuch as how much that which is contained vnder the lines AD DB twise differeth from that which is contained vnder the lines AC and CB twise so much differ the squares of the lines AC CB added together from the squares of the lines AD and DB added together But that which is contayned vnder the lines AC and CB twise excedeth that
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
line is when the square of the greater part excedeth the square of the lesse part by the square of a line coÌ„mensurable in length vnto it And neither part is commensurable in length to the rationall line geueÌ„ As suppose the line CE to be a binomiall line whose partes are ioyned together in the poynt D and let the square of the line CD the greater part exceede the square of the lesse part DE by the square of the line FG and let the line FG be commensurable in length to the line CD the greater part of the binomiall Moreouer let neither the greater part CD nor the lesse part DE be commensurable in length to the rationall line AB then is the line CE by this definition a third binomiall line A fourth binomiall line is when the square of the greater part exceedeth the square of the lesse by the square of a line incommensurable in length vnto the greater part And the greater is also commensurable in length to the rationall line As let the line CE be a binomiall line whose partes let be CD DE let the square of the line CD the greater part exceede the square of the line DE the lesse by the square of the line FG. And let the line FG be incommensurable in length to the line CD the greater Let also the line CD the greater part be commensurable in length vnto the ratâ—onall line AB Then by this definition the line CE is a fourth binomiall line A fift binomiall line is when the square of the greater part exceedeth the square of the lesse part by the square of a line incommensurable vnto it in length And the lesse part also is commensurable in length to the rationall line geuen As suppose that CE be a binomiall line whose greater part let be CD and let the lesse part be DE. And let the square of the line CD excede the square of the line DE by the square of the line FG which let be incoÌ„mensurable in length vnto the line CD the greater part of the binomiall line And let the line DE the second part of the binomiall line be commensurable in length vnto the rationall line AB So is the line CE by this definition a fift binomiall line A sixt binomiall line is when the square of the greater part exceedeth the square of the lesse by the square of a line incommensurable in length vnto it And neither part is commensurable in length to the rationall line geuen As let the line CE be a binomiall line deuided into his names in the point D. The square of whose greater part CD let exceede the square of the lesse part DE by the square of the line FG and let the line FG be incommensurable in length to the line CD the greater part of the binomiall line Let also nâ—ither CD the greater part nor DE the lesse part be commensurable in length to the rationall line AB And so by this definition the line CE is a sixt binomiall line So ye see that by these definitions their examples and declarations all the kindes of binomiall lines are made very playne This is to be noted that here is nothing spoken of those lines both whose portions aâ—e comâ—ensurable in length vnto the rationall line first set for that such lines cannot be binomiall lines â—or binomiall lines are composed of two rationall lines commensurable in Power onely by the 36. of this booke But lines both whose portions are commâ—nsurable in length to the rationall line first set are not binomiall lines For that the partes of such lines should by the 12. of this booke be commensurable in length the one to the other And so should they not be such lines as are required to the composition of a binomiall line Moreouer such lines should not be irrationall but rationall for that they are commensurable tâ—â—ch of the parts whereof they are coÌ„posed by the 15. oâ— this booke And therefore they should be rationall for that the lines which composâ— them are rationall ¶ The 13. Probleme The 48. proposition To finde out a first binomiall line TAke two numbers AC and CB let them be such that the number which is made of them both added together namely AB haue vnto one of them 〈◊〉 vnto BC that proportion that a square number hath to a square numbâ—r â—ut vnto the other namely vnto CA let it not haue that proportion that a square number hath to a square number such as is euery square number which may be deuided into a square number and into a number not square Take also a certayne rationall line and let the same be D. And vnto the line D let the line EF be commensurable in length Wherefore the line EF is rationall And as the number AB is to the nuÌ„ber AC so let the square of the line EF be to the square of an other â—iâ—e namely of FG by the corollary of the sixt of the teÌ„th Wherfore the square of the line EF hath to the square of the line FG that proportion that number hath to number Wherefore the square of the line EF is commensurable to the square of the line FG by the 6. of this booke And the line EF is rationall Wherefore the line FG also is rationall And forasmuch as the number AB hath not to the number AC that proportion that a square number hath to a square number neither shal the square of the line EF haue to the square of the line FG that proportion that a square number hath to a square number Wherefore the line EF is incommensurable in length to the line FG by the 9. of this booke Wherefore the lines EF and FG are rationall commensurable in power onely Wherefore the whole line EG is a binomiall line by the 36. of the tenth I say also that it is a â—irst binomiall line For for that as the 〈◊〉 BA is to the number AC so is the square of the line EF to the square of the line â—G but the number BA is greater then the number AC wherefore the square of the line â—F is also greater then the square oâ— the line FG. Vnto the square of the line EF let the squares of the lines FG and H be equall which how to finde out is taught in the assumpt put â—ftâ—r the 13. of the tâ—nth And fâ—r that as thâ— number BA is to the number AC so is the square of the line EF to the square of the line FG therefore by coâ—uersion or eâ—ersion of proportion by the corollary of the 19. of the fift as the number AB is to the number BC so is the square of the line EF to the square of the line H. But the number AB hath to the number BC that proportion that a square number hath to a square number Wherefore also the square of the line EF hath to the square of the line H
that proportion that a square number hath to a square number Wherefore the line EF is commensurable in length to the line H by the 9. of this booke Wherefore the line EF is in power more then the line FG by the square of a line commensurable in length to the line EF. And the lines EF and FG are rationall commensurable in power onely And the line EF is commensurable in length to the rationall line D. Wherefore the line EG is a first binomiall line which was required to be doone The 14. Probleme The 49. Proposition To finde out a second binomiall line TAke two numbers AC and CB and let them be such that the number made of them both added together namely AB haue vnto BC that proportioÌ„ that a square number hath to a square number and vnto the number CA let it not haue that proportion that a square number hath to a square number as it was declared in the former proposition Take also a raâ—ionall line and let the same be D and vnto the line D let the line FG be commensurable in length Wherefore FG is a rationall line And as the number CA is to the number AB so let the square of the line GF be to the square of the line FE by the 6. of the tenth Wherefore the square of the linâ— GF is commensurable to the square of the line FE Wherfore also FE is a rationall line And forasmuch as the nuÌ„ber CA hath not vnto the number AB that proportioÌ„ that a square number hath to a square nuÌ„ber therfore neither also the squarâ— of the line GF hath to the square of the line FE that proportion that a square number hath to a square number Wherefore the line GF is incommensurable in length vnto the line FE by the 9. of the tenth wherefore the lines FG and FE are rationall commensurable in power onely Wherefore the whole line EG is a binomiall line I say moreouer that the linâ— EG is a second binomiall line For for that by contrary proportion as the number BA is to the number AC so is the square of the line EF to the square of the line FG. But the number BA is greater then the number AC wherefore also the square of the line EF is greater then the square of the line FG. Vnto the square of the line EF let the squares of the linâ—s GF and H be equall Now by conuersion by the corollary of the 19. of of the fift as the number AB is to the number BC so is the square of the line EF to the square of the line H. But the number AB hath to the number BC that proportion that a square number hath to a square number Wherefore the square of the line EF hath to the square of the line H that proportion that a square number hath to a square number Wherefore by the 9. of the tenth the line EF is commensurable in length vnto the line H. Wherefore the line EF is in power more then the line FG by the square of a line commensurable in length vnto the line EF and the lines EF and FG are rationall commensurable in power onely and FG being the lesse name â—s commensurable in length vnto the rationall line geuen namely to D. Wherefore EG is a second binomiall line which was required to be done ¶ The 15. Probleme The 50. Proposition To finde out a third binomiall line TAke two numbers AC and CB and let them be such that the numbâ—r made of them both added together namely AB haue to the number BC that proportion that a square number hath to a square number But to the number AC let it not haue that proportion that a square number hath to a square number as it was declared in the two former And take also some other number that is not a square number and let the same be D and let not the number D haue either to the number BA or to the number AC that proportion that a square number hath to a square number And take a rational line and let the the same be E. And as the number D is to the number AB so let the square of the line E be to the square of the line FG. Wherfore the square of the line E is coÌ„mensurable to the square of the line FG but the line E is rational wherfore the line FG also is rational And for that the nuÌ„ber D hath not to the nuÌ„ber AB that proportion that a square number hath to a square number neither also shall the square of the line E haue to the square of the line FG that proportion that a square number hath to a square nuÌ„ber Wherfore the line E is incommensurable in length to the line FG by the 9. of the tenth Now againe as the nuÌ„ber AB is to the nuÌ„ber AC so let the square of the line FG be to the square of the line GH Wherfore the square of the line FG is commensurable to the square of the line GH And the line FG is rationall Wherfore also the line GH is rationall And for that the number BA hath not to the nuÌ„ber AC that proportion that a square number hath to a square number therefore neither also hath the square of the line FG to the square of the line GH that proportion that a square number hath to a square number Wherfore the line FG is incoÌ„mensurable in leÌ„gth to the line GH Wherfore the lines FG GH are rational coÌ„mensurable in power only Wherfore the whole line FH is a binomial line I say moreouer that it is a third binomial line For for that as the nuÌ„ber D is to the nuÌ„ber AB so is the square of the line E to the square of the line FG but as the nuÌ„ber AB is to the number AC so is the square of the line FG to the square of the line GHâ— therfore oâ— equalitie by the 22. of the fift as the number D is to the number AC so is the square of the line E to the square of the line GH But the nuÌ„ber D hath not to the nuÌ„ber A. C that proportioÌ„ that a square nuÌ„ber hath to a square nuÌ„ber Wherfore neither also hath the square of the line E to the square of the line GH that proportioÌ„ thaâ— a square number hath so a square number Wherfore the line E is incommensurable in leÌ„gth to the line GH And for that as the number AB is to the number AC so is the square of the line FG to the square oâ— the line GH therfore the square of the line FG is greater then the square of the line GH Vnto the square of the line FG let the squares of the lines GH and K be equal Wherfore by â—uerse proportioÌ„ by the corollary of the 19. of the â—ift as the nuÌ„ber AB is to the number BC so is the square of the line FG to
the square of the line K. But the nuÌ„ber AB hath to the nuÌ„ber BC that proportioÌ„ that a square number hath to a square number Wherfore also the square of the line FG hath to the square of the line K that proportion that a square number hath to a square number Wherfore the line FG is commensurable in length to the line K. Wherfore the line FG is in power more then the line GH by the square of a line commensurable in length vnto it And the lines FG and GH are rationall commensurable in power onely and neither of them is commensurable in length vnto the rationall line E wherfore the lyne FH is a third binomiall line which was required to be done ¶ The 16. Probleme The 51. Proposition To finde out a fourth binomiall line TAke two numbers AC and CB let theÌ„ be such that the nuÌ„ber made of theÌ„ both added together namely AB haue to neither of the nuÌ„bers AC and CB that proportioÌ„ that a square number hath to a square nuÌ„ber such as is euery square nuÌ„ber to two nuÌ„bers not square which are lesse theÌ„ it make the said square nuÌ„ber And take a rationall line and let the same be D. And vnto the line D let the line EF be coÌ„mensurable in length Wherfore EF is a rationall line and as the number BA is to the number AC so let the square of the line EF be to the square of the line FG. Wherefore the square of the line EF is commensurable to the square of the line FG and the line BF is a rationall line Wherfore also the line FG is a rationall line And for that the number BA hath not to the number AC that proportion that a square number hath to a square number neither also shall the square of the line EF haue to the square of the line FG that proportion that a square number hath to a square number Wherfore the line EF is incommensurable in leÌ„gth to the line FG. Wherfore the lines EF and FG are rationall commensurable in power onely Wherefore the whole line EG is a binomiall line I say moreouer that it is a fourth binomiall lyne For for that as the number BA is to the number AC so is the square of the line EF to the square of the line FG. But the number BA is greater then the number AC Wherefore also the square of the line EF is greater then the square of the line FG. Vnto the square of the line EF let the squares of the lines FG and H be equall Wherfore by conuersion by the corollary of the 19. of the fift as the number AB is to the number BC so is the square of the line EF to the square of the line H. But the number AB hath not to the number BC that proportion that a square number hath to a square number therfore neither also hath the square of the line EF to the square of the line H that proportioÌ„ that a square nuÌ„ber hath to a square nuÌ„ber Wherfore by the 9. of the teÌ„th the line EF is incoÌ„mensurable in length vnto the line H. Wherfore the line EF is in power more theÌ„ the line FG by the square of a line incommensurable in length vnto it And the lynes EF and FG are rationall commensurable in power onely and the line EF is commensurable in length to the rationall line D. Wherfore the line EG is a fourth binomiall line which was required to be found out ¶ The 17. Probleme The 52. Proposition To finde out a fift binomiall lyne TAke two numbers AC and CB and let them be such that the number AB haue to neither of the numbers AC or CB that proportion that a square number hath to a square number as in the former proposition And take a rationall line and let the same be D. And vnto the line D let the line FG be commensurable in length Wherfore the line FG is rationall And as the number CA is to the number AB so let the square of the line GF be to the square of the line EF. Wherefore the square of the line GF is commensurable to the square of the line FE Wherefore also the line FE is rationall Anâ— for that the number CA hath not to the number AB that proportioÌ„ that a square number hath to a square number therâ—ore neither also hath the square of the line GF to the square of the line FE that proportion that a square number hath to a square number Wherfore by the 9. of the tenth the line GF is incommensurable in length to the line FE Wherfore the lines EF and FG are rationall commensurable in power only Wherfore the whole line EG is a binomiall line I say moreouer that it is a fift binomiall line For for that as the number CA is to the number AB so is the square of the line GF to the square of the line FE therfore contrariwise as the number BA is to the number AC so is the square of the line EF to the square of the line FG. but the number BA is greater then the number AC Wherfore also the square of the line EF is greater then the square of the line FG. Vnto the square of the line EF let the squares of the lines FG and H be equall Wherfore by conuersioÌ„ by the corollary of the 19. of the fift as the nuÌ„ber AB is to the number BCâ— so is the square of the line EF to the square of the line H. But the nuÌ„ber AB hath not to the number BC that proportioÌ„ that a square number hath to a square number Wherefore neither also hath the square of the line EF to the square of the line H that proportion that a square number hath to a square number Wherfore by the 9. of the tenth the line EF is incommensurable in length to the line H. Wherfore the line EF is in power more then the line FG by the square of a line incommensurable in length vnto it And the lines EF and FG are rationall commensurable in power onely And the line FG being the lesse name is commensurable in length to the rationall line geuen namely to D. Wherfore the whole line EG is a fift binomiall line which was required to be found out ¶ The 18. Probleme The 53. Proposition To finde out a sixt binomiall line TAke two numbers AC CB and let theÌ„ be such that the number which is made of them both added together namely AB haue to neither of the numbers AC or CB that proportion that a square nuÌ„ber hath to a square number Take also any other number which is not a square number and let the same be D. And let not the number D haue to any one of these numbers AB and AC that proportion that a square number hath to a square nuÌ„ber Let there be put moreouer a rationall line and
let the same be E. And as the number D is to the number AB so let the square of the line E be to the square of FG. Wherefore by the 6. of the tenth the line E is commensurable in power to the line FG the line E is rationall Wherfore also the line FG is rationall And for that the number D hath not â—o the number AB that proportion that a square nuÌ„ber hath to a square number therefore neither also shall the square of the line E haue to the square of the line FG that proportion that a square number hath to a square number Wherefore the line FG is incommensurable in length to the line E. Againe as the number BA is to the number AC so let the square of the line FG be to the square of the line GH Wherefore by the 6. of the tenth the square of the line FG is commensurable to the square of the line GH And the square of the line FG is rationall Wherefore the square of the line GH is also rationall Wherefore also the line GH is rationall And for that the number AB hath not to the number AC that proportion that a square number hath to a square number thereâ—ore neither also hath the square of the line FG to the square of the line GH that proportion that a square number hath to a square number Wherefore the line FG is incommensurable in length to the line GH Wherefore the lines FG and GH are rationall commensurable in power onely Wherefore the whole line FH is a binomiall line I say moreouer that it is a sixt binomiall line For for that as the number D is to the number AB so is the square of the line E to the square of the line FG. And as the number BA is to the number AC so is the square of the line FG to the square of the line GH Wherefore of equalitie by the 22. of the fift as the number D is to the number AC so is the square of the line E to the square of the line GH But the number D hath not to the nuÌ„ber AC that proportion that a square number hath to a square number Wherefore neither also hath the square of the line E to the square of the line GH that proportion that a square number hath to a square number Wherfore the line E is incommensurable in length to the line GH And it is already proued that the line FG is also incommensurable in length to the line E. Wherefore either of these lines FG and GH is incommensurable in length to the line E. And for that as the number â—A is to the number AC so is the square of the line FG to the square of the line GH therfore the square of the line FG is greater then the square of the line GH Vnto the square of the line FG let the squares of the lines GH and K be equall Wherefore by euersion of proportion as the number AB is to the number BC so is the square of the line FG to the square of the line K. But the number AB hath not to the number BC that proportion that a square number hath to a square number Wherefore neither also hath the square of the line FG to the square of the line K that proportion that a square number hath to a square number Wherefore the line FG is incommensurable in length vnto the line K. Wherefore the line FG is in power more then the line GH by the square of a line incommensurable in length to it And the lines FG and GH are rationall commensurable in power onely And neither of the lines FG GH is commensurable in length to the rationall line geuen namely to E. Wherefore the line FH is a sixt binomiall line which was required to be found out ¶ A Corollary added out of Flussates By the 6. formâ—r Proposiâ—â—â—â—â— it iâ— manifest ho◠〈◊〉 divide any right line geuen into the names of euery one of the sixâ— foresayd binomiall lines For if it be required to deuide a right line geuen into a first binomiall line then by the 48â— of this booke finde out a first binomiall line And this right line being so found out deuided into his names you may by the 10. of the sixt deuide the right line geuen in like sort And so in the other fiue following Although I here note vnto you this Corollary out of 〈…〉 in very conscience and of gratefull â—indeâ— I am enforced to certifie you that iâ— any yeareâ— before the trauailes of Flussas vpoÌ„ Euâ—liâ—â—â— Geometricall Elementes were published the order how to deuide not onely the 6. Binomiall lines into their names but also to adde to the 6. Residâ—â—ls their due partes â—nd fâ—rthermore to deuide all the other irrationalâ— lines of this tenth booke into the partes distinct of which they are composed with many other straunge conclusions Mathematicall to the better vnderstanding of this tenth booke and other Mathematicall bookes most necessary were by M. Iohn Dee inuented and demonstrated as in his booke whose title is Tyrocinium Mathematicum dedicated to Petruâ— Nonnius An. 1559. may at large appeare Where also is one new arte with sundry particular pointes whereby the Mathematicall Sciences greatly may be enriched Which his booke I hope God will one day allowe him opportunitie to publishe with diuers other his Mathematicall and Metaphysicall labours and inuentions ¶ An Assumpt Is a right line be deuided into two partes how soeuer the rectangle parallelogramme contayned vnder both the partes is the meane proportionall betwene the squares of the same parts And the rectangle parallelogramme contained vnder the whole line and one of the partes is the meane proportionall betwene the square of the whole line and the square of the sayd part Suppose that there be two squares AB and BC and let the lines DB and BE so be put that they both make one right line Wherefore by the 14. of the first the lines FB and BG make also both one right line And make perfect the parallelogramme AC Then I say that the rectangle parallelogramme DG is the meane proportionall betwene the squares AB and BC and moreouer that the parallelogramme DC is the meane proportionall betwene the squares AC and CB. First the parallelogramme AG is a square For forasmuch as the line DB is equall to the line BF and the line BE vnto the line BG therfore the whole line DE is equall to the whole line FG. But the line DE is equall to either of these lines AH KC and the line FG is equall to either of these lines AK and HC by the 34. of the first Wherfore the parallelograÌ„me AC is equilater it is also rectangle by the 29. of the first Wherefore by the 46. of the first the parallelograÌ„me AC is a square Now for that as the line FB is to the line BG so is the line DB
therfore be deuided into two vnequall partes the squares which are made of the vnequall partes are greater theÌ„ the rectangle parallelogramme contained vnder the vnequall partes twise which was required to be demonstrated In numbers I neede not to haue so alleaged for the 17. of the seuenth had confirmed the doubles to be one to the other as their singles were but in our magnitudes it likewise is true and euident by alternate proportion thus As the parallelogramme of the lines AC and CB is to his double so is the square of the line AD to his double eche being halfe Wherfore alternately as the parallelogramme is to the square so is the parallelograme his double to the double of the square But the parallelograme was proued lesse then thâ— square wherfore his double is lesse then the square his double by the 14. of the fifth This Assumpt is in some bookes not read for that in maner it semeth to be all one with that which was put after the 39. of this booke but for the diners maner of demonstrating it is necessary For the feare of inuentioÌ„ is therby furthered And though Zambert did in the demonstration hereof omitte that which P. Montaureus could not supplyâ— but plainly doubted of the sufficiencie of this proofe yet M. d ee by onely allegation of the due places of credite whose pithe forceâ— Theon his wordes do containe hath restored to the demonstration sufficiently both light and authoritie as you may perceiue and chiefly such may iudge who can compare this demonstration here thus furnished with the Greeke of Theon or latine translation of Zambert ¶ The 42. Theoreme The 60. Proposition The square of a binomiall line applyed vnto a rationall line maketh the breadth or other side a first binomiall line SVppose that the line AB be a binomiall line and let it be supposed to be deuided into his names in the poynt C so that let AC be the greater name And take a rationall line DE â— And by the â—4 of the first vnto the line DE apply a rectangle parallelograme DEFG equall to the square of theâ—line AB and making in breadth the line DG Then I say that the line DG is a first binomiall line Vnto the line DE apply the parallelograme DH equall to the square of the line AC and vnto the line KH which is equall to the line DE apply the parallelograÌ„me KL equall to the squarâ— of the line BC. Wherefore the residue namely that which is contayned vnder the lines AC CB twise is equall to the residue namely to the parallelograme MF by 〈◊〉 4â— of the second Deuide by the 1â— of the first the line MG into two equall partes in the poynt N. And by the 31. of the first draw the line NX parallel to either of these lines ML and GF Wherefore either of these parallelogrammes MX and NF is equall to that which is contayned vnder the lines AC and CB once by the 15. of the fifth And forasmuch as the line AB is a binomiall line and is deuided into his names in the poynt C therefore the â—ines AC and CB are rationall commensurable in power onely Wherefore the squares of the lines AC and CB are rationall and therefore commensurable the one to the other Wherefore by the 15. of the tenth that which is made of the squares of the lines AC and CB added together is commensurable to either of the squares of the lines AC or CB wherefore that which is made of the squares of the lineâ— AC and CB added together is rationall And it is equall to the parallelograme DL by construction Wherefore the parallelograme DL is rationall And it is applyed vnto the rational line DE wherefore by the 20. of the tenth the line DM is rationall and commensurable in length to the line DE. Agayne forasmuch as the lines AC and CB are rationall commensurable in power onely therefore that which is contayned vnder the lines AC and CB twise that is the parallelograme MF is mediall by the 21 of the tenth and it is applyed vnto the rationall line ML Wherefore the line MG is rationall and incommensurable in length to the line ML by the 22. of this tenth that is to the line DE. But the line MD is proued rationall and commensurable 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 Mâ— are rationall commensurable in power onely Wherefore by the 36. of the tenth the whole line DG is a binomiall line Now resteth to proue that it is a first binomiall line Forasmuch as by the the assumpt going before the 54. of the tenth that which is contayned vnder the lines AC and CB is the meane proportionall betwene the squares of the lines AC and CB therefore the parallelograme MX is the meane proportionall betwene the parallelogrammes DH and KL Wherefore as the parallelogame DH â— is to the parallelograme MX so is the parallelograme MX to the parallelograme KL that is as the line DK is to the line MN so is the line MN to the line MK Wherefore that which is contayned vnder the lines DK and KM is equall to the square of the line MN And forasmuch as the square of the line AC is commensurable to the square of the line CB the parallelograme DH is commensurable to the prarallelograme KL Wherefore by the 1. of the sixt and 10. of the tenth the line DE is coÌ„meÌ„sâ—rable in leÌ„gth to the line KM And 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 going before this proposition or by the assumpt after this 39. of the tenth therefore the parallelograme DL is greater then the parallelograme MF Wherefore by the first of the sixt the line DM is greater then the line MG 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 But by the 17. of the tenth if there be two vnequall right lines and if vppon the greater be applyed a parallelograme equall to the fourth part of the square made of the lesse line and wanting in figure by a square if also the parallelograme thus applyed deâ—ide the line whereupon it is applyed into parts commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length to the greater Wherefore 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 lines DM and MG are proued rationall commensurable in power onely And the line DM is proued the greater name and commensurable in length to the rationall line geuen DE. Wherefore by
to the line KM Wherefore by the 18. of the tenth the line DM is in power more theÌ„ the line MG by the square of a line incommensurable in length to the line DM And the lines DM and MG are rationall commensurable in power onely and the lesse line namely MG is commensurable in length to the rationall line geuen DE. Wherefore the line DG is a fift binomiall line which was required to be demonstrated ¶ The 47. Theoreme The 65. Proposition The square of a line contayning in power two medialls applyed vnto a rationall line maketh the breadth or other side a sixt binomiall line SVppose that the line AB be a line contayning in power two medialls and let it be supposed to be deuided into his partes in the poynt C. And take a rationall line DE. And by the 44. of the first vnto the rationall line DE apply the parallelograÌ„me 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 sixt binomiall line Let the selfe same construction be in thys that was in the former And forasmuch as the line AB is a line contayning in power two medialls and is diuided into his partes in the poynt C therefore the lines AC CB are incommensurable in power hauing that which is made of the squares of them added together mediall and that which is contayned vnder them mediall and moreouer incommensurable to that which is made of the squares of them added together Wherefore by those thinges which haue bene before proued either of these parallelograÌ„mes DL and MF is mediall and either of them is applyed vpon the rationall line DE. Wherefore by the 22. of the tenth either of these lines DM and MG is rationall and incommensurable in length to the line DE. And forasmuch as â—hat which is madâ— of the squares of the lines AC and CB added together is incommensurable to that which is contayned vnder the lines AC and CB twise therefore the parallelograÌ„me DL is incommensurable to the parallelograÌ„me MF Wherefore by the 1. of the sixt and 10. 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 also that is a sixt binomiall line For euen as in the other propositionâ— it hath bene proued 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 the line MN and that the line DK is incommensurable in length to the line KM and therfore by the 18. of the tenth the line DM is in power more then the line MG by the square of a line incommensurable in length to the line DM And neither of these lines DM nor MG is commensurable in length to ther anâ—nall line geuen DE. Wherefore the line DG is a sixt â—â—nomiall line which was required to be demonstrated ¶ The 48. Theoreme The 66. Proposition A line commensurable in length to a binomiall line is also a binomiall line of the selfe same order SVppose that the line AB be a binomiall line and vnto the line AB let the line CD be commensurable in length Then I say that the line CD is a binomiall line and of the selfe same order that the line AB is For forasmuch as AB is â— binomiall line let it be deuided into his names in the poynt E and let AE be the greater name Wherefore the lines AE and EB are rationall commensurable in power onely And as the line AB is to the line CD so by the 12. of the sixt let the line AE be to the line CF. Wherefore by the 19. of the fift the residue namely the line EB is to the residue namely to the line FD as the line AB is to the line CD But by supposition the line AB is commensurable in length to the line CD Wherefore by the 10. of the tenth the line AE is commensurable in length to the line CF and the line EB to the line FD. And the lines AE and EB are rationall Wherefore the lines CF and FD are also rationall And for that as the line AE is to the line CF so is the line EB to the line FD therefore alternately by the 16. of the fift as the line Aâ—E is to the line EB so is the line CF to the line FD. But the lines AE and EB are commensurable in power onely wherefore the lines CF and FD are also commensurable in power onely and they are rationall Wherefore the whole line CD is a binomiall line I say also that it is of the selfe same order of binomiall lines that the line AB is For the line AE is in power more then the line EB either by the square of a line commensurable in length to the line AE or by the square of a line incommensurable in length to the line AE If the line AE be in power more then the line EB by the square of a line commensurable in length to the line AE the line also CF by the 14. of the tenth shal be in power more then the line FD by the square of a line commensurable in length to CF. And if the lineâ— AE be commensurable in length to a rationall line geuen the line CF also shal be commensurable in length to the same by the 12. of the tenth And so either of these lines AB and CD is a first binomiall line that is they are both of one and the selfe same order But if the line EB be commensurable in length to the rationall line put the line FD also shal be commensurable in length to the same And by that meanes agayne the lines AB and CD are both of one and the selfe same orderâ— for either of them is a second binomiall line But if neither of the lines AE nor Eâ— be commensurable in length to the rationall line put neither also of these lines CF nor FD shal be commensurable in length to the same And so neither of the lines AB and CD is a third binomiall line But if the line AE be in power more then the line EB by the square of a line incommensurable in length to the line AE the line also CF shal be in power more then the line FD by the square of a line incommensurable in length to the line CF by the 14. of the tenth And then if the line AE be commensurable in length to the rationall line put the line CF also shal be commensurable in length to the same and so either of the lines AB and CD shal be a fourth binomiall line And if the line EB be commensurable in length to the rationall line geuen the line FD also shal be commensurable in length to the same And so either of the lines AB and CD shal be a
in like sort be deuided as the line AB is by that which hath bene demonstrated in the 66. Proposition of this bookeâ— let it be so deuided in the poynt E. Wherefore it can not be so deuided in any other poynt by the 42â— of this booke And for that the line AB â—â— to the line DZ as the line AG is to the line DE but the lines AG DE namely the greater names are commensurable in length the one to the other by the 10. of this booke for that they are commensurable in length to 〈◊〉 and the selfe same rationall line by the first definition of binomiall lines Wherefore the lines AB and DZ are commensurable in length by the 13. of this booke But by supposition they are commensurable in power onely which is impossible The selfe same demonstration also will serue if we suppose the line AB to be a second binomial line for the lesse names GB and EZ being commensurable in length to one and the selfe same rationall line shall also be commensurable in length the one to the other And therefore the lines AB and DZ which are in the selfe same proportion with them shall also be commensurable in length the one to the other which is contrary to the supposition Farther if the squares of the lines AB and DZ be applyed vnto the rationall line CF namely the parallelogrammes CT and HL they shall make the breadthes CH and HK first binomiall lines of what order soeuer the lines AB DZ whose squares were applyed vnto the rational line are by the 60. of this booke Wherefore it is manifest that vnder a rationall line and a first binomiall line are confusedly contayned all the powers of binomiall lines by the 54. of this booke Wherfore the onely commensuration of the powers doth not of necessitie bryng forth one and the selfe same order of binomiall lines The selfe same thyng also may be proued if the lines AB and DZ be supposed to be a fourth or fifth binomiall line whose powers onely are conmmensurable namely that they shall as the first bring forth binomiall lines of diuers orders Now forasmuch as the powers of the lines AG and GB and DE and EZ are commensurable proportionall it is manifest that if the line AG be in power more then the line GB by the square of a line commensurable in length vnto AG the line DE also shall be in power more then the line EZ by the square of a line commensurable in length vnto the line DE by the 16. of this booke And so shall the two lines AB and DZ be eche of the three first binomiall lines But if the line AG be in power more then the line GB by the square of a line incommensurable in length vnto the line AG the line DE shall also be in powâ—r 〈◊〉 then the line EZ by the square of a line incomensurable in length vnto the line DE by the selfâ— same Pâ—oposition And so shall eche of the lines AB and DZ be of the three last binomiall lines But why it is not so in the third and sixt binomiall lines the reason is For that in them neither of the nameâ— is commensurable in length to the rationall line put FC ¶ The 50. Theoreme The 68. Proposition A line commensurable to a greater line is also a greater line SVppose that the line AB be a greater line And vnto the line AB let the line CD be commensurable Then I say that the line CD also is a greater line Deuide the line AB into his partes in the point E. Wherfore by the 39. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ„ mediall 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 line CF thâ— line EB to the line FD but the line AB is commensurable to the line CD by suppositioÌ„ Wherfore the line AE is commensurable to the line CF and the line EB to the line FD. And for that as the line AE is to the line CF so is the line EB to the line FD. Therfore alternately by the 16. of the fift as the line AE is to the line EB so is the line CF to the line FD. Wherfore by composition also by the 18. of the fift as the line AB is to the line EB so is the line CD to the line FD. Wherefore by the 22. of the sixt as the square of the line AB is to the square of the line EB so is the square of the line CD to the square of the line FD. And in like sort may we proue that as the square of the line AB is to the square of the line AE so is the square of the line CD to the square of the line CF. Wherfore by the 11. of the fift as the square of the lyne AB is to the squares of the lines AE and EB so is the square of the line CD to the squares of the lines CF and FD. Wherfore alternately by the 16. of the fift as the square of the line AB is to the square of the line CD so are the squares of the lines AE and EB to the squares of the lines CF and FD. But the square of the line AB is commensurable to the square of the line CD for the line AB is commensurable to the line CD by suppositioÌ„ Wherfore also the squares of the lines AE and EB are commensurable to the squares of the lines CF and FD. But the squares of the lines AE and EB are incommensurable and being added together are rationall Wherfore the squares of the lines CF and FD are incommensurable being added together are also rationall And in like sort may we proue that that which is contained vnder the lines AE and EB twise is commensurable to that which is contained vnder the lines CF and FD twise But that which is contained vnder the lines AE and EB twise is mediall wherfore also that which is contained vnder the lines CF and FD twise is medial Wherfore the lines CF and FD are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ„ mediall Wherfore by the 39. of the tenth the whole line CD is irrationall is called a greater line A line therfore commensurable to a greater line is also a greater line An other demonstration of Peter Montaureus to proue the same Suppose that the line AB be a greater line and vnto it let the line CD be commensurable any way that is either both in length and in power or els in power onely Then I say that the line CD also is a greater
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
third are produced when the square of the whole namely of that which is made of the residuall line and the line ioyned vnto it added together excedeth the square of the line ioyned to the residuall by the square of a lyne which is commensurable vnto it in length And as the three last kindes of binomials namely the fourth fifth and sixth were produced when the square of the greater part excedeth the square of the lesse by the square of a line incommenâ—â—rable in length vnto it euen so the three last kyndes of residuall lynes are produced wheÌ„ the square of the whole excedeth the square of the lyne adioined by the square of a line incommensurable vnto it in length As ye may perceiue by their definitions following A first residuall line is when the square of the whole excedeth the square of the lyne adioyned by the square of a lyne commensurable vnto it in leÌ„gth and also the whole is commensurable in length to the rationall line first set As let AB be a rationall line whose partes are certaine distinct to be expressed by number And let the residual lyne be CD and let the line ioyned vnto it be ECâ— and let the whole being composed of the residuall CD and the line adioyned EC be the line ED let moreouer the square of the whole line ED excede the square of the line adioyned EC by the square of the line F which line F let be commensurable in length to the whole lyne ED and let the whole line ED be also commensurable in length to the rationall line AB then is the residuall lyne CD by this definition a first residuall line A second residual line is when the square of the whole excedeth the square of the line adioyned by the square of a line commensurable vnto it in leÌ„gth and also the line adioyned is commensurable in length to the rationall lyne As suppose the line CD to be a residuall and let the line adioyned vnto it be EC and the whole made of them both let be the line ED let the square of ED the whole line excede the square of the lyne adioyned EC by the square of the lyne F and let the lyne F be commensurable in length to the whole lyne ED moreouer let the line adioyned EC be commensurable in length to the rationall line AB then by this definition the residuall line CD is a second residuall line A third residuall line is when the square of the whole excedeth the square of the lyne adioyned by the square of a line commensurable vnto it in leÌ„gth and neither the whole line nor the line adioyned is coÌ„mensurable in length to the rationall lyne As the former supposition standing suppose that the square of the whole lyne ED exceede the square of the lyne adioyned EC by the square of the lyne F and let the lyne F be commensurable in length to the whole lyne EC and let neither the whole lyne ED nor the line adioyned EC be commensurable in length to the rationall lyne AB then by this definition the residuall lyne CD is a third residuall lyne A fourth residuall line is when the square of the whole lyne excedeth the square of the lyne adioyned by the square of a lyne incommensurable vnto it in length and the whole lyne is also commensurable in length to the rationall lyne As the residuall lyne beyng as before CD the lyne adioyned EC and the whole ED let the square of the whole lyne ED exceede the square of the line adioyned EC by the square of the lyne F and let the lyne F be incommensurable in length to the whole line ED and let ED the whole line be commensurable in length to the rationall lyne AB then is the residuall line CD by this declaration a fourth residuall lyne A fiueth residuall line is when the square of the whole lyne exceedeth the square of the lyne adioyned by the square of a lyne incommensurable vnto it in length and the lyne adioyned is commensurable in length to the rationall lyne As the residuall line beyng CD the lyne adioyned EC and the whole lyne ED let the square of the whole lyne ED exceede the square of the line adioyned EC by the square of the lyne F and let the line F be incommensurable in length to the whole lyne ED and let also EC the lyne adioyâ—ed be commensurable in length to the rationall line AE then shall the residuall CD be by this definition a fifth residuall lyne A sixth residuall line is when the square of the whole line exceedeth the square of the line adioyned by the square of a line incommensurable vnto it in length and neither the whole line nor the line adioyned is commensurable in length to the rationall line As suppose the residuall line to be CD and the lyne adioyned to be â—C and the whole lyne composed of them let be ED and let the square of the whole lyne â—D exceede the square of the line adioyned by the square of the lyne F which line F let be incommensurable in length to the whole lyne ED moreouer let neither the whole lyne CD nor the line adioyned EC be commâ—nsurable in length to the rationall line Aâ— then shall the residuall lyne CD be by this explication a sixt residuall lyne and the last ¶ The 19. Probleme The 85. Proposition To finde out a first residuall line TAke a rationall line and let the same be A and vnto it let the line BG be commensurable in length Wherefore the line BG also is rationall And take two square numbers DE and EF which let be such that the excesse of the greater namely of DE aboue the lesse EF which excesse let be the number DF be no square number by the corollary of the first assumpt of the 28. of the tenth Wherfore the number ED hath not to the number DF that proportion that a square number hath to a square number by the 24. of the eight And as the number ED is to the number DF so let the square of the line BG be to the square of the line GC by the corrollary of the sixt of the tenth Wherefore the square of the line BG is commeÌ„surable to the square of the line GC But the square of the line BG is rationall wherefore also the square of the line GC is rationall Wherefore the line GC is also rationall And forasmuch as the number ED hath not to the number DF that proportioÌ„ that a square nuÌ„ber hath to a square nuÌ„ber therfore neither also hath the square oâ— the line BG to the square of the line GC that proportion that a square number hath to a square number Wherfore by the 9. of the tenth the line BG is incommensurable in length to the line GC And they are both rationall Wherefore the lines BG and GC are rational commensurable in power onely
Wherefore the line BC is a residuall line I say moreouer that it is a first residuall line For forasmuch as the square of the line BG is greater then the square of the line GC that it is greater it is manifest for by supposition the square of the line BG is to the square of the line GC as the greater number namely ED is to the number DF vnto the square of the line BG let the squares of the lines GC and H be equall And for that as the number DE is to the number DF so is the square of the line BG to the square of the line GC therefore by conuersion of proportion by the corollary of the 9. of the fifth as the number DE is to the number EF so is the square of the line BG to the square of the line H. But the number DE hath to the number EF that proportion that a square number hath to a square number for either of them is a square number wherefore also the square of the line BG hath to the square of the line H that proportion that a square numbeâ— hath to a square number Wherefore the line GB is commensurable in length to the line H. Wherefore the line GB is in power more then the line GC by the square of a line commensurable in length to the line GB and the whole line namely GB is commensurable in length to the rationall line A. Wherefore the line BC is a first residuall line Wherefore there is founde out a first residuall line which was required to be done ¶ The 20. Probleme The 86. Proposition To finde out a second residuall line TAke a rationall line and let the same be A and vnto it let the line GC be commensurable in length And take two square numbers DE and EF and let them be such that the excesse of the greater namely DF be no square number And as the number DF is to the number DE so let the square of the line GC be to the square of the line GB Wherefore both the squares are commensurable And forasmuch as the square of the line GC is rationall therefore the square of the line BG is also rationall Wherefore also the line BG is rationall And forasmuch as the squares of the lines BG GC haue not that proportion the one to the other that a square number hath to a square number therefore the lines BG and GC are incommensurable in length and they are both rationall Wherefore the lines BG and GC are rationall commensurable in power onely Wherefore the line BC is a residuall line I say moreouer that it is a second residual line For forasmuch as the square of the line BG is greater theÌ„ the square of the line GC vnto the square of the line BG let the squares of the lines Gâ— H be equall And for that as the number DE is to the nuÌ„ber DF so is the square of the line GB to the square of the line GC therefore by conuersion of proportion as the number DE is to the number EF so is the square of the line BG to the square of the line H. But either of these numbers DE and EF is a square number Wherefore the line GB is commensurable in length to the line H. Wherefore the line BG is in power more then the line GC by the square of a line commensurable in length to the line BG and the line GC that is ioyned to the residuall line is commensurable in length to the rationall line A. Wherefore the line BC is a second residuall line Wherefore there is found out a second residuall line which was required to be done ¶ The 21. Probleme The 87. Proposition To finde out a third residuall line TAke rationall line let the same be A and take three numbers E B C and CD not hauing the one to the other that proportion that a square nuÌ„ber hath to a square number and let the number BC haue to the number BD that proportion that a square number hath to a square number And let the number BC be greater then the number CD And as the number E is to the number BC so let the square of the line A be to the square of the line FG and as the number BC is to the number CD so let the square of the line FG be to the square of the line HG Wherefore the square of the line A is commensurable to the square of the line FG. But the square of the line A is rationall Wherefore also the square of the line FG is rationall wherefore the line FG is also rationall And forasmuch as the number E hath not to the number BC that proportion that a square numberâ— hath to a square number therefore neither also hath the square of the line A to the square of the line FG that proportion that a square number hath to a square number Wherefore the line A is incommensurable in length to the line FG. Againe for that as the number BC is to the number CD so is the square of the line FG to the square of the line HG therefore the square of the line FG is commensurable to the square of the line HG But the square of the line FG is rationall Wherefore also the square of the line HG is rationall Wherefore also the line HG is rationall And for that the number BC hath not to the number CD that proportion that a square number hath to a square number therefore neither also hath the square of the line â—G to the square of the line HG that proportion that a square nuÌ„ber hath to a square number Wherefore the line FG is incommensurable in length to the line HG and they are both rationall Wherefore the lines FG HG are rationall coÌ„mensurable in power onely Wherfore the line FH is a residuall line I say moreouer that it is a third residuall line For for that as the number E is to the number BC so is the square of the line A to the square of the line FG and as the number â—C is to the number CD so is the square of the line FG to the square of the line HGâ— therefore by equalitie of proportion as the number E is to the number CD so is the square of the line A to the square of the line HGâ— but the number E hath â—â—â— to the number CD that proportioâ— that a square numâ—â—r hath to a square number therfore neither also hath the square of the line A to the square of the line HG that proportion that a square number hath to a square number therefore the line A is incommensurable in length to the line HG Wherefore neither of the lines FG and HG is commensurable in length to the rationall line A. And forasmuch as the square of the line FG is greater then the square of the line HG that the line FG is greater then the line HG it is
maniâ—est for by supposition the number BC is greater then the number CD vnto the square of the line FG let the squares of the lines HG Kâ— be equall And for that as the nuÌ„ber BC is to the number CD so is the square of the line FG to the square of the line Hâ—â—â—erfore by conuersion of proportion as the number BC is to the number BD so is the square of the line FG to the square of the line K. But the nuÌ„ber BC hath to the number BD that proportion that a square number hath to a square number Wherefore the square of the line FG hath to the square of the line K that proportion that a square number hath to a square number Wherefore the line FG is commensurable in length to the line K. Wherefore the line FG is in power more then the line HG by the square of a line commensurable in length to the line FG and neither of the lines FG and GH is commensurable in length to the rationall line A when yet notwithstanding either of the lines FG and GH is rationall Wherefore the line FH is a third residuall line Wherefore there is found out a third residuall line which was required to be done ¶ The 22. Probleme The 88. Proposition To finde out a fourth residuall line TAke a rationall line and let the same be A and vnto it let the line BG be commensurable in length Wherefore the line BG is rationall And take two numbers DF and FE and let them be such that the whole number namely DE haue to neither of the numbers DF and FE that proportion that a square number hath to a square number And as the number DE is to the number EF so let the square of the line BG be to the square of the line GC wherefore the square of the line BG is commensurable to the square of the line GC wherefore also the square of the line GC is rationall and the line GC is also rationall And for that the number DE hath not the number EF that proportion that a square number hath to a square number therefore the line BG is incommensurable in length to the line GC And they are both rationall wherefore the line BC is a residuall line I say moreouer that it is a fourth residuall line For forasmuch as the square of the line BG is greater then the square of the line GC vnto the square of the line BG let the squares of the lines CG and H be equall And for that as the number DE is to the number EF so is the square of the line BG to the square of the line GC therefore by conuersion of proportion as the number DE is to the number DF so is the square of the line BG to the square of the line H. But the numbers DE and DF haue not the one to the other that porportion that a square number hath to a square number Wherefore the line BG is incommensurable in length to the line H. Wherefore the line BG is in power more then the line GC by the square of a line incommensurable in length so the line BG and the whole line BG is commensurable in length to the rationall line A Wherefore the line BC is a fourth residuall line Wherefore there is founde out a fourth residuall line which was required to be doone ¶ The 23. Probleme The 89. Proposition To finde out a fift residuall lyne TAke a rational line and let the same be A and vnto it let the line CG be commensurable in length Wherefore the line CG is rationall And take two numbers DF and FE which let be such that the number DE haue to neither of these numbers DF nor FE that proportion that a square number hath to a square number And as the number FE is to the number DE so let the square of the line CG be to the square of the line BG Wherefore the square of the line CG is commensurable to the square of the line BGâ— Wherefore the square of the line BG is rationall and the line BG is also rational But the numbers DE and EF haue not that proportion the one to the other that a square number hath to a square nuÌ„ber Wherfore the lines BG and GC are rationall commensurable in power onely Wherâ—fore the line BC is a residuall line I say moreouer that it is a fift residuall line For forasmuch as the square of the line BG is greater then the square of the line GC vnto the square of the line BG let the squares of the lines GC and H be equal Now therfore for that as the number DE is to the number EF so is the square of the line BG to the square of the line GC therfore by conuersion of proportion at the number DE is to the nuÌ„ber DF so is the square of the line BG to the square of the line H. But the numbers DE DF haue not that proportion the one to the other that a square number hath to a square number Wherefore the line BG is incommensurable in length to the line H. Wherefore the line BG is in power more then the line CG by the square of a line incommensurable in leÌ„gth to the line BG and the line CG which is ioyned to the residual line is commeÌ„surable in leÌ„gth to the rationall line A. Wherefore the line BC is a fift residuall line Wherefore there is found out a fift residuall line which was required to be done ¶ The 24. Probleme The 90. Proposition To finde out a sixth residuall line TAke a rational line and let the same be A And take three numbers E BC and CD not hauing the one to the other that proportion that a square number hath to a square number And let not the number BC haue to the number BD that proportion that a square number hath to a square number And let the number BC be greater then the number CD as the number E is to the number BC so let the square of the line A be to the square of the lyne FG. And as the number BC is to the number CD so let the square of the line FG be to the square of the line GH Now therfore for that as the number E is to the nuÌ„ber BC so is the square of the line A to the square of the line FG therfore the square of the line A is commensurable to the square of the line F G. Wherfore the square of the line FG is rationall and the line FG is also rationall And for that the number E hath not to the number BC that proportion that a square number hath to a square number therfore the line A is incommensurable in length to the line FG. Againe for that as the number BC is to the number CD so is the square of the line FG to the square of the line GH therefore the square of the line
and NL is equall to that which is contained vnder the lines AG and GB But that which is contayned vnder the lines AG GB is mediall Wherfore that which is contained vnder the lines AG and GB twise is also mediall Wherfore the whole parallelogramme FL is also mediall Wherefore the line FM is rationall and incommensurable in length to the line CD And forasmuch as the lines AG and GB are incoÌ„mensurable in length therefore also the square of the line AG is incommensurable to the parallelogramme contayned vnder the lines AG and GB But vnto the square of the line AG are commensurable the squares of the lines AG and GB and vnto the parallelogramme contained vnder the lines AG and GB is commensurable that which is contained vnder the lines AG and GB twise Wherefore the squares of the lines AG and GB are incommensurable to that which is contained vnder the lines AG and GB twise Wherefore the parallelogrammes which are equall vnto them namely the parallelogrammes CL and FL are incommensurable the one to the other Wherefore also the line CM is incommensurable in length to the line FM and they are both rationall Wherefore the line CF is a residuall line I say moreouer that it is a third residuall line For forasmuch as the square of the line AG that is the parallelogramme CH is commensurable to the square of the line BG that is to the parallelogramme KL therefore the line CK is commensurable in length to the line KM And in like sort as in the former Proposition so also in this may we proue that the parallelogramme contayned vnder the lines CK and KM is equall to the square of the line NM that is to the fourth part of the square of the line FM Wherefore the line CM is in power more then the line FM by the square of a line commensurable in length to the line CMâ— and neither of the lines CM nor FM is commensurable in length to the rationall line â—D Wherefore the line CF is a third residual line Wherfore the square of a second mediall residual line applied vnto a rationall line maketh the breadth or other side a third residuall line which was required to be demonstrated ¶ The 76. Theoreme The 100. Proposition The square of a lesse line applied vnto a rationall line maketh the breadth or other side a fourth residuall line SVppose that AB be a lesse line and let CD be a rationall line And vnto the line CD apply the parallelogramme CE equall to the square of the line AB and making in breadth the line CF. Then I say that the line CF is a fourth residuall line For vnto the line AB let the line conueniently ioyned be supposed to be BG Wherefore the lines AG GB are incommensurable in power hauing that which is made of their squares added together rationall and that which is contained vnder them mediall And let the rest of the construction be as in the Propositions going before Wherefore the whole parallelogramme CL is rationall Wherefore the line CM is also rationall and commensurable in length to the line CD And forasmuch as that which is contained vnder the lines AG and GB twise is mediall therefore the parallelogramme which is equall vnto it namely the parallelogramme FL is also mediall Wherefore the line FM is rationall and incommensurable in length to the line CD But the line CM is commensurable in length to the line CD Wherefore by the 13. of the tenth the line CM is incommensurable in length to the line FM and they are both rationall Wherefore the lines CM and FM are rationall commensurable in power onely Wherefore the line CF is a residuall line I say moreouer that it is a fourth residuall line For forasmuch as the lines AG and GB are incommensurable in power therefore the squares of them that is the parallelogrammes which are equall vnto them namely the parallelogrammes CH and KL are incommensurable the one to the other Wherefore also the line CK is incommensurable in length to the line KM And in like sort may we proue that the parallelogramme contayned vnder the lines CK and KM is equall to the square of the line NM that is to the fourth part of the square of the line FM Wherefore by the 18. of the tenth the line CM is in power more then the line FM by the square of a line incommensurable in length to the line CM And the whole line CM is commensurable in length to the rationall line CD Wherefore the line CF is a fourth residuall line Wherefore the square of a lesse line applied vnto a rationall line maketh the breadth or other side a fourth residuall line which was required to be proued ¶ The 77. Theoreme The 101. Proposition The square of a lyne making with a rationall superficies the whole superficies mediall applied vnto a rational line maketh the breadth or other side a fift residuall lyne SVppose that AB be a line making with a rationall superficies the whole superficies mediall and let CD be a rationall line And vnto the line CD apply the parallelogramme CE equall to the square of the line AB and making in breadth the line CF. Then I say that the line CF is a fift residuall line For vnto the line AB let the line conueniently ioyned be supposed to â—e BG Wherfore the lines AG and GB are incommensurable in power hauing that which is made of their squares added together mediall and that which is contained vnder them rationall Let the rest of the construction be in this as it was in the former propositions Wherfore the whole parallelogramme CL is mediall Wherefore the line CM is rationall and incommensurable in length to the line CD And either of the parallelogramme FX NL is rationallâ— Wherforâ— the whole parallelogramme FL is also rationall Wherfore also the line FM is rationall and commensurable in leÌ„gth to the line CD And forasmuch as the parallelogramme CL is mediall and the parallelogramme FL is rationall therefore CL and FL are incommensurable the one to the other and the line CM is incommensurable in length to the line FM and they are both rationall Wherfore the lines CM and FM are rationall commensurable in power onely Wherfore the lyne GF is a residuall line I say moreouer that it is a â—iâ—t residual line For we may in like sort proue that the parallelograme contained vnder the lines CK and KM is equall to the square of the line NM that is to the fourth part of the square of the lyne FM And forasmuch as the square of the line AG that is the parallelogramme CH is incommensurable to the square of the line BG that is to the parallelogramme KL therfore the line CK is incommensurable in length to the line KM Wherfore by the 18. of the tenth the line CM is in power more then the line FM by the square of a
shall in like sort by the 14. of the tenth be in power more then the line DF by the square of a line incommensurable in length to the line CF and then if the line AE be commensurable in length to the rationall line the line CF shall also in like sort be commensurable in length to the same rationall line and so either of the lines AB and CD is a fourth residuall line And if the line BE be coÌ„meÌ„surable in leÌ„gth to the rationall line the line DF shall also be coÌ„mensurable in leÌ„gth to the same line and so either of the lines AB CD is a â—iâ—t residuall line And if neither of the lines AE nor BE be commensurable in length to the rationall line in like sort neither of the lines CF nor DF shall be coÌ„mensurable in leÌ„gth to the same rational line And so either of the lines AB CD is a sixt residual line Whereâ—ore the line CD is a residuall line of the selfe same order that the line AB is A line therfore commensurable in length to a residuall line is it selfe also a residuall line of the selfe same order which was required to be proued As before touching binomiall lines so also touching residuall lines this is to be noted that a line commensurable in length to a residuall line is alwayes a residuall line of the selfe same order that the residuall line is vnto whom it is coÌ„mensurable as hath before in this 103. propositioÌ„ bene proued But if a line be coÌ„mensurable in power only to a residuall lineâ— then followeth it not yea it is impossible that that line should be a residuall of the self same order that the residual line is vnto whom it is commensurable in power onely Howbeit those two lines shall of necessitie be both either of the three first orders of residâ—â—ll lines or of the three last orders which is not hard to proue if ye marke diligently the former demonstration and that which was spoken of binomiall lines as touching this matter ¶ The 80. Theoreme The 104. Proposition A line commensurable to a mediall residuall line is it selfe also a medial residuall line and of the selfe same order SVppose that AB be a mediall residuall line vnto whome let the line CD be commensurable in length and in power or in power onely Then I say that CD is also a mediall residuall line and of the selfe same order For forasmuch as the line AB is a mediall residuall line let the line conueniently ioyned vnto i◠〈◊〉 BE wherefore the lines AE and BE are mediall commensurable in power onely As AB is to CD so by the 22. of the sixth let BE be to DF. And in like sort as in the former so also in this may we proue that the line AE is commensurable in length and in power or in power onely vnto the line CF the line BE 〈◊〉 the line DF. Wherefore by the 23. of the tenth 〈◊〉 line CF is a mediall line and the line DF is also a mediall line for that it is commensurable to the mediall line BE. And in like sort the lines CF and DF are commensurable in power onely for that they haue the selfe same proportioÌ„ the one to the other that the lines AE and EB haue which are commensurable in power onely Wherefore the line CD is a mediall residuall line I say moreouer that it is of the selfe same order that the line AB is For for that as the line AE is to the line BE so is the line CF to the line DF. But as the line AE is to the line BE so is the square of the line AE to the parallelogramme contayned vnder the lines AE and BE by the first of the sixth and as the line CF is to the line DF so is the square of the line CF to the parallelogramme contayned vnder the lines CF and DF. Wherefore as the square of the line AE is to the parallelogramme contayned vnder the lines AE and BE so is the square of the line CF to the parallelogramme contayned vnder the lines CF and DF. Wherefore alternately as the square of the line AE is to the square of the line CF so is the parallelogramme contayned vnder the lines AE and BE to the parallelogramme contained vnder the â—ines CF and DF. But the square of the line AE is commensurable to the square of the line CF for the line AE is commensurable to the line CF Wherefore also the parallelogramme contayned vnder the lines AE and BE is commensurable to the parallelogramme contayned vnder the lines CF and DF. Wherefore if the parallelogramme contayned vnder the lines AE and EB be rationall the parallelogramme also contayned vnder the lines CF and FD shall be rationall And then either of the lines AB and CD is a first mediall residuall line But if the parallelogramme contayned vnder the lines AE and BE be mediall the parallelogramme also contayned vnder the lines CF and FD shall be also mediall by the corollary of the 23. of the teÌ„th and so either of the lines AB and CD is a second mediall residuall line Wherefore the line CD is a mediall residuall line of the selfe same order that the line AB is A line therefore commensurable to a mediall residuall line is it selfe also a mediall residuall line of the selfe same order which was required to be demonstrated This Theoreme is vnderstanded generally that whether a line be commensurable in length in power or in power onely to a mediall residuall line it is it selfe also a mediall residuall line and of the selfe same order which thing also is to be vnderstanded of the three Theoremes which follow An other demonstration after Campane Suppose that A be a mediall residuall line vnto whome let the line B be commensurable in length or in power onely And take a rationall line CD vnto which apply the parallelogramme CE equall to the square of the line A and vnto the line FE which is equall to the line CD apply the parallelogramme Fâ— equall to the square of the line B. Now then the parallelogrammes CE and FG shall be commensurable for that the lines A B are commensurable in power wherefore by the 1. of the sixth and 10. of this booke thâ— lines DE and FG are commensurable in length Now then if A be a first mediall residuall line then is the line DE a second residuall line by the 98. of this booke and if the line A be a sâ—cond mediall residuall line then is the line â— â— a third residuall line by the 99. of this booke But if DE be a second residuall line Gâ— also shall be a second residuall line by the â—03 of this boke And if DE be a third residuall line GE also shall by the same be also a third residuall line Wherefore it followeth by the 9â— and 93. of this booke that B is either a first
11. of the fift as the line HK is to the line KF so is the line CD to the line DB. But the square of the line CD is commensurable to the square of the line DB wherefore by the 10. of the tenth the square of the line HK is coÌ„mensurable to the square of the line FK But these three lines HK FK and EK are proportionall in coÌ„tinuall proportioÌ„ as it hath already bene proued Wherefore by the second corrollary of the 20. of the sixth the square of the line HK is to the square of the line FK as the line HK is to the line EK wherfore the line HK is commensurable in length to the line EK Wherefore by the 15. of the tenth the line HE is commensurable in length to the line FK And forasmuch as the square of the line A is equall to that which is contayned vnder the lines EH and BD but the square of the line A is rationall whereâ—ore that which is contayned vnder the lines EH and BD is rationall And it is applyed vnto the rationall line BD. Wherefore by the 20. of the tenth the line EH is rationall and commensurable in length to the line BD. Wherefore also the line EK which is commensurable in length to the line HE is rational and commensurable in length to the line BD. Now for that as the line CD is to the line DB so is tâ—e line FK to the line EK for it was before proued that as CD is to DB so is HF to FE and as HF is to FE so is FK to EK but the lines CD and DB are commensurable in power onely wherefore by the 10. of the tenth the lines FK and KE are also commensurable in power onely And for that as the line CD is to the line DB so is the line FK to the line EK therefore by contrary proportion as DB is to CD so is EK to FK and alternately as DB is to EK so is CD to FK but the lines BD and EK are commensurable in length as it hath already bene proued Wherfore also the lines CD and FK are commensurable in length But the line CD is rationall wherefore also the line FK is rationall Wherefore the lines FK and EK are rationall commensurable in power onely Wherefore the line FE is a residuall line whose names FK and KE are commensurable to the names CD and BD of the binomiall line BC and in the same proportion as is proued I say moreouer that it is a residuall line of the selfe same order that the binomiall line is For the line CD is in power more then the line BD either by the square of a line commensurable in length to the line CD or by the square of a line incommensurable in length Now if the line CD be in power more then the line BD by the square of a line commensurable in length vnto the line CD then by the 13. of the tenth the line FK is in power more then the line EK by the square of a line coÌ„mensurable in length to the line FK And so if the line CD be coÌ„mensurable in leÌ„gth to the rationall line put the line FK also shal be coÌ„mensurable in leÌ„gth to the same rationall line wherfore then the line BC is a first binomiall line the line FE is likewise a first residuall line And if the line BD be commensurable in length to the rationall line the line â—K is also commensurable in length to the same and then the line BC is a second binomiall line and the line FE a second residuall line And if neither of the lines CD nor DB be commensurable in length vnto the rationall line neither of the lines FK nor EK are commensurable in length vnto the same and then the line BC is a third binomiall line the line FE is a third residuall line And if the line CD be in power more then the line BD by the square of a line incommensurable in length to the line CD the line FK is also by the 14. of the tenth in power more then the line EK by the square of a line incommensurable in length to the line FK And so if the line CD be commensurable in length to a rationall line put the line FK also is commensurable in length to the same whereâ—ore the line BC is a fourth binomiall line and the line FE is a fourth residuall line And if the line BD be commensurable in length to the rationall line the line EK is likewise commensurable in length to the same and then the line BC is a fifth binomiall line and the line EF a fifth residuall line And if neither of the lines CD nor DB be commensurable in length to the rationall line neither also of the lines FK nor EK is commensurable in length to the same and then the line BC is a sixth binomiall line and the line FE a sixth residuall line Wherfore the line FE is a residuall line whose names namely FK and EK are commeÌ„surable to the names of the binomiall line namely to the names CD and DB and are in the selfe same proportion and the residuall line EF is in the selfe same order of residuall lines that the binomiall line BC is of binomiall lines Wherefore the square of a rationall line applyed vnto a binomiall line maketh the breadth or other side a residuall line whose names are commensurable to the names of the binomiall line and in the selfe same proportion and moreouer that residuall line is in the selfe same order of residuall lines that the binomiall line is of binomiall lines which was required to be demonstrated Here is the Assumpt of the foregoing Proposition confirmed Now let vs declare how as the line HF is to the line FE so to make the line FK to the line EK The line CD is greater then the line BD by supposition Wherefore also the line HF is greater then the line FE by alternate proportion and the 14. of the fifth From the line HF take away the line FL equall to the line FE Wherefore the line remayning namely HL is lesse then the line HF for the line HF is equall to the lines HL LF As HL is to HF so by the 12. of the sixt let FE be to FK Wherfore by contrary proportion by the Corollary of the 4. of the fifth as HF is to HL so is FK to FE Wherefore by conuersion of proportion by the Corollary of the 19. of the fifth as HF is to LF that is to the line equall vnto it namely to FE so is the line FK to the line EK M. d ee of this Assumpt maketh 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 that is Acquisiuely a Probleme vniuersall thus Two vnequall right lines being propounded to adioyne vnto the lesse a right line which takeÌ„ with the lesse as one right line shall haue the same proportion to the line adioyned which the greater of the two propounded hath to the lesse The
construction and demonstration hereof is worde for worde to be taken as it standeth here before after these wordes The line HF is greater then the line FE ¶ A Corollary also noted by I. Dee It is therefore euident that thus are three right lines in our handling in continuall proportion it is to weete the greater the lesse and the adioyned make the first the lesse with the adioyned make the second and the adioyned line is the third This is proued in the beginning of the demonstration after the Assumpt vsed An other demonstration after Flussas Take a rationall line A and let GB be a binomiall line whose greater line let be GD and vpon the line GB apply by the 45. of the first the parallelograÌ„me BZ equall to the square of the line A and making in breadth the line GZ Likewise vpon the line DB by the same apply the parallelogramme BI equall also to the square of the line A and making in breadth the line DI and put the line GZT equall to the line DI. Then I say that GZ is such a residuall line as is required in the Proposition Forasmuch as the parallelogrammes BZ BI are equall therefore by the 14. of the sixt reciprocally as the line GB is to the line BD so is the line DI or the line GT which is equall vnto it vnto the line GZ Wherefore by diuision as the line GD is to the line DB so is the line TZ to the line ZG by the 17. of the fifth Wherefore the line TZ is greater then the line ZG For the line GD is the greater name of the binomiall line GB Vnto the line ZG put the line ZC equall And as the line TC is to the line TZ so by the 11. of the sixth let the line ZG be to line ZK Wherefore contrary wise by the Corollary of the 4. of the fifth the line TZ is to the line TC as the line ZK is to the line ZG Wherefore by conuersion of proportion by the 19. of the fifth as the line TZ is to the line ZC that is to ZG which is equall vnto it so is the line ZK to the line KG But the line TZ is to the line ZG as the line GD is to the line DB. Wherfore by the 11. of the fifth the line ZK is to the line KG as the line GD is to the line DB. But the lines GD and DB are commensurable in power onely Wherefore also the lines ZK and KG are commensurable in power onely by the 10. of this booke Farther forasmuch as the line TZ is to the line ZG as the line ZK is to the line KG therefore by the 12. of the fifth all the antecedentes namely the whole line TK are to all the consequentes namely to the line KZ as one of the antecedentes namely the line ZK is to one of the consequentes namely to the line KG Wherefore the line ZK is the meane proportionall betwene the lines TK and KG And therefore by the Corollary of the 20. of the sixth as the first namely the line TK is to the third namely to the line KG so is the square of the line TK to the square of the second namely of the line KZ And forasmuch as the parallelograÌ„me BI which is equall to the square of the rationall line A is applied vpon the rationall line DB it maketh the breadth DI rationall and commensurable in length vnto the line DB by the 20. of the tenth And therfore the line GT which is equall vnto the line DI is commensurable in length to the same line DB. And for that as the line GD is to the line DB so is the line KZ to the line KG but as the line KZ is to the line KG so is the line TK to the line KZ therefore by the 11. of the fifth as the line GD is to the line DB so is the line T K to the line KZ Wherefore by the 22. of the sixth as the square of the line GD is to the square of the line DB so is the square of the line T K to the square of the line KZ But the square of the line GD is commensurable to the square of the line DB for the names GD and DB of the binomiall line GB are commensurable in power Wherefore the square of the line T K shall be commensurable to the square of the line KZ by the 10. of this booke But as the square of the line T K is to the square of the line KZ so is it proued that the right line T K is to the right line KG Wherfore the right line T K is commensurable in length to the right line KG Wherefore it is also commensurable in length to the line TG by the 15. of the tenth Which line TG is as it hath bene proued a â—ationall line and equall to the line DI. Wherefore the lines T K and KG are rationall commensurable in length And forasmuch as it hath bene proued that the line Z K is commensurable in power onely vnto the rationall line KG therefore the lines Z K and KG are rationall commensurable in power onely Wherefore the line GZ is a residuall line And forasmuch as the rationall line TG is commensurable in length to either of these lines DB and KG Wherefore the lines DB KG shall be commensurable in length by the 12. of the tenth But the line Z K is to the line KG as the line GD is to the line DB. Wherefore alternately by the 16. of the fifth the line KZ is to the line GD as the line KG is to the line DB. Wherefore the line Z K is commensurable in length vnto the line GD Wherefore the lines Z K and KG the names of the residuall line GZ are commensurable in length to the lines GD and DB which are the names of the binomiall line GB and the line Z K is to the line KG in the same proportion that the line GD is to the line DB. Wherefore if the whole line Z K be in power more then the line conueniently ioyned KG by the square of a line commensurable in length to the line Z K then the greater name G D shall be in power more then the lesse name DB by the square of a line commensurable in length to the line GD by the 14. of the tenth And if the line Z K be in power more then the line KG by the square of a line incommensurable in length to the line Z K the line also GD shall be in power more then the line DB by the square of a line incommensurable in length vnto the line GD by the same Proposition And if the greater or lesse name of the one be commensurable in length to the rationall line put the greater of lâ—sse name also of the other shall be commensurable in length to the same rationall line by the 12. of this booke But if neither name
of the one be commensurable in length to the rationall line put neither name of the other also shall be commensurable in length to the same rationall line put by the 13. of the same Wherefore the residuall line GZ shall be in the selfe same order of residuall lines that the binomiall line GB is of binomiall lines by the definitions of residuall and binomiall lines The square therefore of a rationall line applied to a binomiall lineâ— â— which was required 〈◊〉 be proued ¶ The 89. Theoreme The 113. Proposition The square of a rational line applied vnto a residuall maketh the breadth or other side a binomial line whose names are commensurable to the names of the residuall line and in the selfe same proportion and moreouer that binomiall line is in the selfe same order of binomiall lynes that the residual line is of residuall lynes SVppose that A be a rationall line and BD a residuall line And vnto the square of the line A let that which is contained vnder the lines BD and KH be equal Wherfore the square of the rationall line A applied vnto the residuall line BD maketh the breadth or other side KH Then I say that the line KH is a binomiall line whose names are commensurable to the names of the residuall line BD and in the selfe same proportion and that the line KH is in the selfe same order of binomiall lines that the line BD is of residuall lines Vnto the line BD let the line conueniently ioyned be DC Wherfore the lines BC and DC are rationall commensurable in power onely And vnto the square of the line A let the parallelogramme contained vnder the lines BC and G be equall But the square of the line A is rationall Wherfore the parallelogramme contained vnder the lines BC and G is also rationall Wherfore also the line G is rationall and commensurable in length to the line BC by the 20. of the tenth Now forasmuch as the parallelogramme contained vnder the lines BC and G is equall to that which is contained vnder the lines BD and KH therfore by the 16. of the sixt as the line BC is to the line BD so iâ— the line KH to the line G. But the line BC is greater then the line BD. Wherfore also the line KH is greater then the line G. Vnto the line G lâ—t the line KE be equall Wherfore the line KE is rationall and coÌ„mensurable in length to the line BC as also the line G was by the 12. of the tenth And for that as BC is to BD so is KH to KE Wherfore by â—duersion of proportioÌ„ by the corollary of the 19. of the fift as BC is to DC so is KH to EH â—KH into EH so let the line FH be to the line EF how this is to be done we will decare at the ende of this demonstration Wherfore the residue KF is to the residue FH as the whole KH is to the whole HE by the 19. of the fift that is as the line BC is to the line CD But the lines BC and CD are commensurable in power onely Wherefore also the lines KF and FH are commensurable in power only And for that as KH is to HE so is KF to FH but as KH is to HE so is also HF to FE therfore as KF is to FH so is FH to FE Wherfore by the corollary of the 19. of the sixt as the first is to the third so is the square of the first to the square of the second Wherefore as KF is to FE so is the square of the line KF to the square of the line FH but these squares are commensurable for the lines KF and FH are commensurable in power Wherfore the lines KF and FE are commensurable in length Wherfore by the second part of the 15. of the tenth the lines KE and EF are commensurable in length Wherfore by the same the lines KF and FE are commensurable in length But the line KE is rationall and commensurable in length to the line BC wherefore the line KF is also rationall and commensurable in leÌ„gth to the line BC. And for that as the line BC is to the line CD so it KF to EH therfore alternately by the 16. of the fift as BC is to KF so is CD to FH But the line BC is commensurable in length to the line KF Wherfore the line CD is commensurable in length to the line FH But the line CD is rationall Wherfore also the line FH is rationall And the lines BC and CD are rationall commensurable in power onely Wherfore the lines KF and FH are rationall commensurable in power onely Wherfore the line KH is a binomiall line whose names are commensurable to the names of the residuall line and in the same proportion I say moreouer that it is a binomiall of the selfe same order of binomial lines that the line BD is of residual lines For if the line BC be in power more then the line CD by the square of a line commensurable in length to the line BC the line KF is also in power more then the line FH by the square of a line commensurable in length to the line KF by the 14. of the tenth And if the line BC be commensurable in length to the rationall line put the line KF is also by the 12. of the tenth commensurable in length to the rationall line and so the lyne BD is a first residuall lyne and the line KH is in like sort a first binomiall line If the line CD be commensurable in length to the rational line the line FH is also commensurable in length to the same line and so the line BD is a second residuall line and the line KH a second binomiall line And if neither of the lines BC nor CD be commensurable in length to the rationall line neither also of the lines KF nor FH is commensurable in length to the same and so the line BD is a third residuall line and the line KH a third binomiall line But if the line BC be in power more then the line CD by the square of a line incommensurable in length to the line BC the line KF is in power more theÌ„ the line FH by the square of a line incommensurable in length to the line KF by the 14. of the tenth And if the line BC be commensurable in length to the rationall line put the line KF is also commensurable in length to the same line and so the line BD is a fourth residuall line and the line KH a fourth binomiall line And if the line CD be coÌ„meÌ„surable in leÌ„gth to the rational line the line FH is also coÌ„meÌ„surable in leÌ„gth to the same so the line BD is a fift residuall line the line KH a fift binomiall line And if neither of the lines BC nor CD be commensurable in length to the rationall line neither also of the lines KF nor FH is commensurable in length
to the same and so the line BD is a sixt residuall line and the line KH is a sixt binomiall line Wherfore KH is a binomiall line whose names KF and FH are commensurable to the names of the residuall line BD namely to BC and CD and in the selfe same proportion and the binomiall line KH is in the selfe same order of binomiall lines that the residuall BD is of residuall lines Wherefore the square of a rationall line applied vnto a residuall line maketh the breadth or other side a binomiall line whose names are commensurable to the names of the residuall line and in the selfe same proportion and moreouer the binomiall line is in the selfe same order of binomiall lines that the residuall line is of residuall lines which was required to be demonstrated The Assumpt confirmed Now let vs declare how as the line KH is to the line EH so to make the line HF to the line FE Adde vnto the line KH directly a line equall to HE and let the whole line be KL and by the tenth of the sixt let the line HE be deuided as the whole line KL is deuided in the point H let the line HE be so deuided in the point F. Wherfore as the line KH is to the line HL that is to the line HE so is the line HF to the line FE An other demonstration after Flussas Suppose that A be a rationall line and let BD be a residuall line And vpon the line BD apply the parallelogramme DT equall to the square of the line A by the 45. of the first making in breadth the line BT Then I say that BT is a binominall line such a one as is required in the proposition Forasmuch as BD is a residuall line let the line coÌ„ueniently ioyned vnto it be GD Wherfore the lines BG and GD are rationall commensurable in power onely Vpon the rationall line BG apply the parallelogramme BI equall to the square of the line A and making in breadth the line BE. Wherefore the line BE is rationall and commensurable in length to the line BG by the 20. of the tenth Now forasmuch as the parallelogrammes BI and TD are equall for that they are eche equall to the square of the line A therfore reciprokally by the 14. of the sixth as the line BT is to the line BE so is the line BG to the line BD. Wherefore by conuersion of proportion by the corrollary of the 19. of the fifth as the line BT is to the line TE so is the line BG to the line GD As the line BG is to the line GD so let the line TZ be to the line ZE by the corrollary of the 10. of the sixth Wherefore by the 11. of the fifth the line BT is to the line TE as the line TZ is to the line ZE. For either of them are as the line BG is to the line GD Wherefore the residue BZ is to the residue ZT as the whole BT is to the whole TE by the 19. of the fifth Wherefore by the 11. of the fifth the line BZ is to the line ZT as the line ZT is to the line ZE. Wherfore the line TZ is the meane proportionall betwene the lines BZ and ZE. Wherefore the square of the first namely of the line BZ is to the square of the second namely of the line ZT as the first namely the line BZ is to the third namely to the line ZE by the corollary of the 20. of the sixth And for that as the line BG is to the line GD so is the line TZ to the line ZE but as the line TZ is to the line ZE so is the line BZ to the line ZT Wherefore as the line BG is to the line GD so is the line BZ to the line ZT by the 11. of the fifth Wherfore the lines BZ and ZT are commensurable in power onely as also are the lines BG and GD which are the names of the residuall line BD by the 10. of this booke Wherfore the right lines BZ and ZE are coÌ„mensurable in length for we haue proued that they are in the same proportion that the squares of the lines BZ and ZT are And therefore by the corollary of the 15. of this booke the residue BE which is a rationall line is commensurable in length vnto the same line BZ Wherefore also the line BG which is commensurable in length vnto the line BE shall also be commensurable in length vnto the same line EZ by the 12. of the tenth And it is proued that the line RZ is to the line ZT commensurable in power onely Wherefore the right lines BZ and ZT are rationall commensurable in power onely Wherefore the whole line BT is a binomiall line by the 36. of this booke And for that as the line BG is to the line GD so is the line BZ to the line ZT therefore alternately by the 16. of the fifth the line BG is to the line BZ as the line GD is to the line ZT But the line BG is commensurable in length vnto the line BZ Wherefore by the 10. of this booke the line GD is commensurable in length vnto the line ZT Wherefore the names BG and GD of the residuall line BD are commensurable in length vnto the names BZ and ZT of the binomial line BT and the line BZ is to the line ZT in the same proportion that the line BG is to the line GD as before it was more manifest And that they are of one and the selfe same order is thus proued If the greater or lesse name of the residuall line namely the right lines BG or GD be coÌ„mensurable in length to any rationall line put the greater name also or lesse namely BZ or ZT shal be commensurable in length to the same rationall line put by the 12. of this booke And if neither of the names of the residuall line be commensurable in length vnto the rationall line put neither of the names of the binomiall line shal be commensurable in length vnto the same rationall line put by the 13. of the tenth And if the greater name BG be in power more then the lesse name by the square of a line commensurable in length vnto the line BG the greater name also BZ shal be in power more then the lesse by the square of a line commensurable in length vnto the line BZ And if the one be in power more by the square of a line incommensurable in length the other also shal be in power more by the square of a line incommensurable in length by the 14. of this booke The square therefore of a rationall line c. which was required to be proued ¶ The 90. Theoreme The 114. Proposition If a parallelograÌ„me be coÌ„tained vnder a residuall line a binomiall lyne whose names are commensurable to the names of the residuall line and in the selâ—e same proportion the lyne which contayneth in power
draw by the 11. of the firsâ— a perpeÌ„dicular line AB and let AB be a rationall line and make perfectâ— the parallelogramme BC. Wherefore BG is irrationall by that which was declared and proued in maner of an Assumpt in the end of the demonstration of the 38 and the line that containeth it iâ— power is also irrationall Let the line CD containe in power the superâ—icies BC. Wherefore CD is irrationall not of the selfe same kind with any of those that were before for the square of the line CD applied to a rationall line namely AB maketh the breadth a mediall line namely AC But the square of none of the foresaid lines applied to a rationall line maketh the breadth a mediall line Againe make perfecte the parallelogramme ED. Wherefore the parallelogramme ED is also irrationall by the sayd Assumpt in the end of the 98. his demonstration brieâ—ly proued and the line which containeth it in power is irrationallâ— let the line which containeth it in power be DF. Wherefore DF is irrationall and not of the selfe same kinde with any of the foresaid irrationall lines For the square of none of the foresayd irrationall lines applied vnto a rationall line maketh the breadth the line CD Wherefore of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before which was required to be demonstrated ¶ The 92. Theoreme The 116. Proposition Now let vs proue that in square figures the diameter is incommensurable in length to the side SVppose that ABCD be a square and let the diameter therof be AC Then I say that the diameter AC is incommensurable in length to the side AB For if it be possible let it be coÌ„mensurable in leÌ„gth I say that theÌ„ this will follow that one and the selfe same nuÌ„ber shall be both an euen number an odde number It is manifest by the 47. of the first that the square of the line AC is double to the square of the line AB And for that the line AC is commensurable in length to the line AB by supposition therfore the lyne AC hath vnto the line AB that proportion that a number hath to a number by the 5. of the tenth Let the lyne AC haue vnto the line AB that proportion that the number EF hath to the number G. And let EF and G be the least numbers that haue one and the same proportion with them Wherfore EF is not vnitie For if EF be vnitie and it hath to the number G that proportion that the line AC hath to the lyne AB and the line AC is greater then the lyne AB Wherfore vnitie EF is greater then the number G which is impossible Wherfore FE is not vnitie wherfore it is a number And for that as the square of the line AC is to the square of the lyne AB so is the square number of the number EF to the square number of the number G for in eche is the proportion of their sides doubled by the corollary of the 20 of the sixt and 11. of the eight and the proportion of the line AC to the line AB doubled is equal to the proportioÌ„ of the nuÌ„ber EF to the number G doubled for as the line AC is to the line AB so is the nuÌ„ber EF to the number G. But the square of the line AC is double to the square of the line AB Wherfore the square number produced of the number EF is double to the square number produced of the number G. Wherefore the square number produced of EF is an euen number Wherfore EF is also an euen number For if EF were an odde number the square number also produced of it should by the 23. and 29. of the ninth be an odde number For if odde numbers how many soeuer be added together and if the multitude of theÌ„ be odde the whole also shal be odde Wherfore EF is an euen number Deuide the number EF into two equall partes in H. And forasmuch as the numbers EF and G are the lest numbers in that proportion therfore by the 24. of the seuenth they are prime numbers the one to the other And EF is an euen number Wherfore G is an odde number For if G were an euen number the number two should measure both the number EF and the number G for euery euen nuÌ„ber hath an halfe part by the definition but these numbers EF G are prime the one to the other Wherfore it is impossible that they should be measured by two or by any other number besides vnitie Wherfore G is an odde number And forasmuch as the number EF is double to the number EH therfore the square number produced of EF is quadruple to the square number produced of EH And the square number produced of EF is double to the square number produced of G. Wherfore the square number produced of G is double to the square number produced of EH Wherfore the square number produced of G is an euen number Wherfore also by those thinges which haue bene before spoken the number G is an euen number but it is proued that it is an odde number which is impossible Wherefore the line AC is not commensurable in length to the line AB wherfore it is incommensurable An other demonstration We may by an other demonstration proue that the diameter of a square is incommensurable to the side thereof Suppose that there be a square whose diameter let be A and let the side thereof be B. Then I say that the line A is incommensurable in length to the line B. For if it be possible let it be commensurable in length And agayne as the line A is to the line B so let the number EF be to the number G and let them be the least that haue one and the same proportion with them wherefore the numbers EF and G are prime the one to the other First I say that G is not vnitie For if it be possible let it be vnitie And for that the square of the line A is to the square of the line B as the square number produced of EF is to the square number produced of G as it was proued in the â—ormer demonstration but the square of the line A is double to the square of the line B. Wherfore the square nuÌ„ber produced of EF is double to the square number produced of G. And by your supposition G is vnitie Wherefore the square number produced of EF is the number two which is impossible Wherefore G is not vnitie Wherefore it is a number And for that as the square of the line A is to the square of the line B so is the square number produced of EF to the square number produced of G. Wherefore the square number produced of EF is double to the square number produced of G. Wherefore the square number produced of G.
and meane proportion in the point C and let the greater segment thereof be AC Then I say that eyther of the lines AC and CB is an irrationall line of that kinde which is called a residuall Extend the line AB to the point D and let the line AD be equall to halfe of the line AB Now forasmuch as the right line AB is diuided by an extreme meane proportion in the point C and vnto the greater segmeÌ„t AC is added a line AD equall to the halfe of the right line AB therfore by the 1. of the thirtenth the square of the line CD is quintuple to the square of the line AD. Wherefore the square of the line CD hath to the square of the line AD that proportion that nuÌ„ber hath to nuÌ„ber Wherfore the square of the line CD is commeÌ„surable to the square of the line AD. But the square of the line DA is rationall for the line DA is rationall forasmuch as it is the halfe of the rationall line AB Wherefore the square of the line CD is rationall Wherefore also the linâ— CD is rationall And forasmuch as the square of the line CD hath not to the square of the line AD that proportion that a square number hath to a square number therfore by the 9. of the tenth the line CD is incommensurable in length to the line AD. Wherefore the lines CD and DA are rationall commensurable in power only Wherfore the line AC is a residuall line by the 73. of the tenth Againe forasmuch as the line AB is deuided by an extreme and meane proportion and the greater segment thereof is AC therfore that which is coÌ„tayned vnder the lines AB BC is equall to the square of the line AC Wherefore the square of the line AC applyed to the rationall line AB maketh the bredth BC. But the square of a residuall line applied to a rationall line maketh the bredth a first residuall line by the 97. of the tenth Wherefore the line CB is a first residuall line And it is proued that the line AC is also a residuall line If therefore a rationall right line be diuided by an extreme and meanâ— proportion either of the segments is an irrationall line of that kinde which is called a residuall line which was required to be demonstrated ¶ A Corollary added by Campane Hereby it is manifest that if the greater segment be a rationall line the lesse segment shal be a residuall line For if the greater segment AC of the right line ACB be diuided into two equall partes in the point D the square of the line DB shal be quintâ—plâ— to the square of the line DC by the 3. of this booke And forasmuch as the line CD beyng the â—alfe of the rationall line supposed AC is rationall by the 6. diffinition of the tenth And vnto the square of the line DC the square of the line DB is commensurable for it is quintuple vnto it wherfore the square of the line DB is rationall Wherfore also the line DB is rationall And forasmuch as the squares of the lines DB DC are not in proportion as a square nuÌ„ber is to a square number therefore the lines DB and DC are incommensurable in length by the 9. of the tenth Wherefore they are commensurable in power only Wherefore by the 73. of the tenth the line BC which is the lesse segment is a residuall line The 7. Theoreme The 7. Proposition If an equilater PeÌ„tagon haue three of his angles whether they follow in order or not in order equall the one to the other that Pentagon shal be equiangle SVppose that ABCDE be an equilater pentagon And let the angles of the sayd Pentagon namely first three angles folowing in order which are at the points A B C be equal the one to the other Then I say that the Pentagon ABCDE is equiangle Draw these right lines AC BE and FD. Now forasmuch as these two lines CB and BA are equall to these two lines BA and AE the one to the other and the angle CBA is equall to the angle BAE therefore by the 4. of the first the base AC is equall to the base BE and the triangle ABC is equall to the triangle ABE and the rest of the angles are equal to the rest of the angles vnder which are subtended equall sides Wherefore the angle BCA is equall to the angle BEA and the angle ABE to the angle CAB Wherefore also the side AF is equall to the side BF by the 6. of the first And it was proued that the whole line AC is equal to the whole line BE. Wherefore the residue CF is equall to the residue Fâ— And the line CD is equall to the line DE. Wherefore these two lines FC and CD are equall to these two lines FE ED and the base FD is common to them both Wherefore the angle FCD is equall to the angle FED by the 8. of the first And it is proued that the angle BCA is equal to the angle AEB Wherefore the whole angle BCD is equall to the whole angle AED But the angle BCD is supposed to be equall to the angles A and B. Wherefore the angle AED is equall to the angles A and B. In like sort also may we proue that the angle CDE is equall to the angles A and B. Wherefore the Pentagon ABCDE is equiangle But now suppâ—se that three angles which folow not in order be equall the one to the other namely let the angles A C D be equall Then I say that in this case also the Pentagon ABCDE is equiangle Draw a right line from the point B to the point D. Now forasmuch as these two lines BA and Aâ— are equall to these two lines BC and CD and they comprehende equall angles therefore by the 4. of the first the base BE is equall to the base BD. And the triangle ABE is equall to the triangle BDC and the rest of the angles are equall to the rest of the angles vnder which are subtended equall sides wherefore the angle AEB is equall to the angle CDB And the angle BED is equall to the angle BDE by the 5. of the first for the side BE is equall to the side BD. Wherefore the whole angle AED is equal to the whole angle CDE But the angle CDE is supposed to be equall to the angles A and C. Wherefore the AED is equall to the angles A and C. And by the same reason also the angle ABC is equall to the angles A C and D. Wherefore the Pentagon ABCDE is equiangle If therefore an equilater Pentagon haue three of his angles whither they follow in order or not in order equall the one to the other that Pentagon shal be equiangle which was required to be proued The 8. Probleme The 8. Proposition If in an equilater equiangle PeÌ„tagon two right lines do subtend two of
are right angles and that the line AC is double to the line CM Now forasmuch as the angle ALC is equall to the angle AMF for that they are both right angles and the angle LAC is common to both the triangles ALC and AMF wherefore the angle remayning namely ACL is equal to the angle remayning AFM by the corollary of the 32. of the first Wherefore the triangle ACL is equiangle to the triangle AMF Wherefore proportionally by the 4. of the sixth as the line LC is to the line CA so is the line MF to the line FA. And in the same proportion also are the doubles of the antecedents LC and MF by the 15. of the fifth Wherefore as the double of the line LC is to the line CA so is the double of the line MF to the line FA. But as the double of the line MF is to the line FA so is the line MF to the halfe of the line FA by the 15. of the fifth wherefore as the double of the line LC is to the line CA so is the line MF to the halfe of the line FA by the 11. of the fifth And in the same proportion by the 15. of the fifth are the halues of the consequents namely of CA and of the halue of the line AF. Wherefore as the double of the line LC is to the halfe of the line AC so is the line MF to the fourth part of the line FA. But the double of the line LC is the line DC and the halfe of the line CA is the line CM as hath before bene proued and the fourth part of the line FA is the line FK for the line FK is the fourth part of the line FH by construction Wherfore as the line DC is to the line CM so is the line MF to the line FK Wherfore by composition by the 18. of the fifth as both the lines DC and CM are to the line CM so is the whole line MK to the line FK Wherefore also by the 22. of the sixt as the squares of the lines DC and CM are to the square of the line CM so is the square of the line MK to the square of the line FK And forasmuch as by the 8. of the thirtenth a line which is subtended vnder two sides of a pentagon figure as is the line AC being diuided by an extreame meane proporâ—ion the greater segment is equall to the side of the pentagon figure that is vnto the line DC and by the 1. of the thirtenth the greater segment hauing added vnto it the halfe of the whole is in power quintuple to the square made of the halfe of the whole and the halfe of the whole line AC is the line CM Wherefore the square that is made of the lines DC and CM that is of the greater segment and of the halfe of the whole as of one line is quintuple to the square of the line CM that is of the halfe of the whole But as the square made of the lines DC and CM as of one line is to the square of the line CM so is it proued that the square of the line MK is to the square of the line FK Wherefore the square of the line MK is quintuple to the square of the line FK But the square of the line KF is rationall as hath before bene proued wherefore also the square of the line MK is rationall by the 9. diffinition of the tenth for the square of the line MK hath to the square of the line KF that proportion that number hath to number namely that 5. hath to 1. and therefore the sayd squares are commensurable by the 6. of the tenth Wherefore also the line MK is rationall And forasmuch as the line BF is quadruple to the line FK for the semidiameter BF is equal to the semidiameter FH therfore the line BK is quintuple to the line FK Wherefore the square of the line BK is 25. times so much as the square of the line KF by the corollary of the 20 of the sixt But the square of the line MK is quintuple to the square of the FK as is proued Wherfore the square of the line BK is quintuple to the square of the line KM Wherfore the square of the line BK hath not to the square of the line KM that proportioÌ„ that a square number hath to a square number by the corollary of the 25. of the eight Wherefore by the 9. of the tenth the line BK is incommensurable in length to the line KM and either of the lines is rationall Wherefore the lines BK and KM are rationall commensurable in power onely But if froÌ„ a rationall line be taken away a rationall line being commensurable in power onely to the whole that which remayneth is irrationall and is by the 73. of the tenth called a residuall line Wherefore the line MB is a residuall line And the line conueniently ioyned vnto it is the line MK Now I say that the line BM is a fourth residuall line Vnto the excesse of the square of the line BK aboue the square of the line KM let the square of the line N be equall which excesse how to finde out is taught in the assumpt put after the 13. proposition of the tenth Wherefore the line BK is in power more then the line KM by the square of the line N. And forasmuch as the line KF is commensurable in length to the line FB for it is the fourth part thereof therefore by the 16. of the tenth the whole line KB is commensurable in length to the line FB But the line FB is commensurable in length to the line BH namely the semidiameterâ— to the diameter wherefore the line BK is commensurable in length to the line BH by the 12. of the tenth And forasmuch as the square of the line BK is quintuple to the square of the line KM therefore the square of the line BK hath to the square of the line KM that proportion that fiue hath to one Wherefore by conuersion of proportion by the corollary of the 19. of the fifth the square of BK hath to the square of the line N that proportion that fiue hath to fower therfore it hath not that proportioÌ„ that a square number hath to a square number by the corollary of the 25. of the eight Wherfore the line BK is incommensurable in length to the line N by the 9. of the tenth Wherfore the line BK is in power more then the line KM by the square of a line incommensurable in length to the line BK Now then forasmuch as the whole line BK is in power more then the line conueniently ioyned namely then KM by the square of a line incoÌ„mensurable in length to the line BK and the whole line BK is commensurable in length to the rationall line geuen BH therefore the line MB is a fourth residuall line by the diffinition of a fourth residuall line But a rectangle parallelogramme
place from whence first it began to be moued it shal passe by the pointes F G H and the octohedron shall be comprehended in a sphere I say also that it is comprehended in the sphere geuen For forasmuch as the line LK is equal to the line KM by position and the line KE is common to them both and they contayne right angles by the 3. diffinition of the eleuenth therefore by the 4. of the first the base LE is equall to the base EM And forasmuch as the angle LEM is a right angle by the 31. of the third for it is in a semicircle as hath bene proued therefore the square of the line LM is double to the square of the line LE by the 47. of the first Againe forasmuch as the line AC is equall to the line BC therefore the line AB is double to the line BC by the diâ—finition of a circle But as the line AB is to the line BC so is the square of the line AB to the square of the line BD by the corollaries of the 8. and â—0 of the sixt Wherefore the square of the line AB is double to the square of the line BD. And it is proued that the square of the line LM is double to the square of the line LE. Wherefore the square of the line BD is equall to the square of the line LE. For the line EH which is equall to the line LF is put to be equall to the line DB. Wherefore the square of the line AB is equall to the square of the line LM Wherefore the line AB is equall to the line LM And the line AB is the diameter of the sphere geueÌ„ wherefore the line LM is equall to the diameter of the sphere geuen Wherefore the octoedron is contayned in the sphere geuen and it is also proued that the diameter of the sphere is in power double to the side of the octohedron Wherefore there is made an octohedron and it is comprehended in the sphere geuen wherein was comprehended the Pyramis and it is proued that the diameter of the sphere is in power double to the side of the octohedrn which was required to be doone and to be proued Certayne Corollaries added by Flussas First Corollary The side of a Pyramis is in power sesquitertia to the side of an octâ—hedron inscribed in the same Sphere For forasmch as the diameter is in power double to the side of the octohedron therefore of what partes the diameter contayneth in power 6. of the same the side of the octohedron coÌ„tayneth in power 3. but of what partes the diameter contayneth 6. of the same the side of the pyramis contayneth 4. by the 13. of this booke Wherefore of what partes the side of the pyramis contayneth 4. of the same the side of the octohedron contayneth 3. Second Corollary An octohedron is deuided into two equall and like Pyramids The common bases of these Pyramids are set vpon euery square contayned of the sides of the octohedron vpon which square are set the â—â— triangles of the octohedron which pyramids are by the â— diffinition of the eleuenth equall and like And the foresayd square common to those Pyramids is the halfe of the square of the diameter of the sphere for it is the square of the side of the octohedron Third Corollary The three diameters of the octohedron do cutte the one the other perpendicularly into two equall parts in the center of the sphere which contayneth the sayd octohedron As it is manifest by the three diameters EG FH and LM which cutte the one the other in the center K equally and perpendicularly ¶ The 3. Probleme The 15. Proposition To make a solide called a cube and to comprehend it in the sphere geuen namely that Sphere wherein the former two solides were comprehendâ—dâ— and to proue that the diameter of the sphere is in power treble to the side of the cube TAke the diameter of the sphere geuen namely AB and diuide it in the point Câ— So that let the line AC be double to the line BC by the 9. of the sixt And vpon the line AB describe a semicircle ADB And by the 11. of the first from the pâ—ynt C râ—yse vp vnto the line AB a perpeÌ„dicular line CD And draw a right linâ— DB. And describe a squarâ— EFGH hauing euery one of his sides equall to the line DB And from the pointes E F G H rayse vp by the 12. of the eleuenth vnto the playne superficies of the square EFGH perpendicular lines EK FL GM and HN and let euery one of the lines EK FL GM and HN be put equall to one of the lines EF FG GH or HE which are the sides of the square and draw these right lines KL LM MN and NK Wherfore there is made a cube namely FN which is contayned vnder six equall squares Now it is required to comprehend the same cube in the sphere geuen and to proue that the diameter of the sphere is in power ble to the side of the cube Draw these right lines KG and EG And forasmuch as the angle KEG is a right angle for that the line KE is erected perpendicularly to the playne superficies Eâ— and therefore also to the right line EG by the 2. diffinitioÌ„ of the eleuenth wherefore a semicircle described vpon the line KG shall passe by the poynt E. Agayne forasmuch as the line FG is erected perpendicularly to either of these lines FL and FE by the diffinition of a square by the 2. diffinition of the eleuenth therefore the line FG is erected perpendicularly to the playne superficies FK by the 4. of the eleuenth Wherefore if we draw a right line from the point F to the point K the line GF shall be erected perpendicularly to the line KF by the 2. diffinition of the eleuenth And by the same reason agayne a semicircle described vpon the line GK shall passe also by the point F. And likewise shall it passe by the rest of the pointes of the angles of that cube If now the diameter KG abiding fixed the semicircle be turned round about vntill it returne into the selfe same place from whence it began first to be moued the cube shal be compreheÌ„ded in a sphere I say also that it is comprehended in the sphere geuen For forasmuch as the line GF is equall to the linâ—â—E and the angle F is a right angle therefore the square of the line EG is by the 47. of the first double to the square of the line â—F But the line EF is equall to the line EK Wherefore the square of the line EG is double to the square of the line EK Wherfore the squares of EG and EK that is the square of the line GK by the 47. of the first are treble to the square of the line EK And forasmuch as the line AB is treble to the line BC but
22. of the sixt Wherefore by coÌ„position by the 18. of the fifth as both the lines AB BC added the one to the other together with the line AC that is as two such lines as AB is are to the line AC so are both the lines DE and EF added the one to the other together with the line DF that is two such lines as DE is to the line DF. And in the same proportion are the halues of the antecedents by the 15. of the fifth Wherefore as the line AB is to the line AC so is the line DE to the line DF. And therefore by the 19. of the fifth as the line AB is to the line BC so is the line DF to the line FE Wherefore also by diuision by the 17. of the fifth as the line AC is to the line CB so is the line DF to the line DE Now that we haue proued that any right line whatsoeuer being diuided by an extreame and meane proportion what proportion the line contayning in power the squares made of the whole line and of the greater segment added together hath to the line contayning in power the squares made of the whole line and of the lesse segment added together the same proportion hath the side of the cube to the side of the Icosahedron Now also that we haue proued that as the side of the cube is to the side of the Icosahedron so is the superficies of the Dodecahedron to the superficies of the Icosahedron being both described in one and the selfe same sphere and moreouer seing that we haue proued that as the superficies of the Dodecahedron is to the superficies of the IcosahedroÌ„ so is the DodecahedroÌ„ to the Icosahedron for that both the pentagon of the Dodecahedron and the triangle of the Icosahedron are comprehended in one and the selfe same circle All these thinges I say being proued it is manifest that if in one and the selfe same sphere be described a Dodecahedron and an Icosahedron they shall be in proportion the one to the other as a right line whatsoeuer being diuided by an extreame and meane proportion the line contayning in power the squares of the whole line and of the greater segment added together is to the line containing in power the squares of the whole line and of the lesse segment added together For for that as the Dodecahedron is to the Icosahedron so is the superficies of the Dodecahedron to the superficies of the Icosahedron that is the side of the cube to the side of the Icosahedron but as the side of the cube is to the side of the Icosahedron so any right line what so euer being diuided by an extreame and meane proportion is the line contayning in power the squares of the whole line and of the greater segment added together to the line contayning in power the squares of the whole line and of the lesse segment added together Wherefore as a Dodecahedron is to an Icosahedron described in one and the selfe same sphere so any right line what so euer being diuided by an extreame and meane proportion is the line contayning in power the squares of the whole line of the greater segment added together to the line contayning in power the squares of the whole line and of the lesse segment added together The ende of the fourtenth Booke of Euclides Elementes after Hypsicles ¶ The fourtenth booke of Euclides Elementes after Flussas FOr that the fouretenth Booke as it is set forth by Flussas containeth in it moe Propositions then are found in Hypsicles also some of those Propositions which Hypsicles hath are by him somewhat otherwise demonstrated I thought my labour well bestowed for the readers sake to turne it also all whole notwithstanding my trauaile before taken in turning the same booke after Hypsicles Where note ye that here in this 14. booke after Flussas and in the other bookes following namely the 15. and 16. I haue in alleadging of the Propositions of the same 14. booke followed the order and number of the Propositions as Flussas hath placed them ¶ The first Proposition A perpendicular line drawen from the centre of a circle to the side of a Pentagon inscribed in the same circle is the halfe of these two lines taken together namely of the side of the hexagon and of the side of the decagon inscribed in the same circle TAke a circle ABC and inscribe in it the side of a pentagon which let be BC and take the centre of the circle which let be the point D and froÌ„ it draw vnto the side BC a perpendicular line DE which produce to the point â— And vnto the line E F put the line EG equall And draw these right lines CG CD and CF. Then I say that the right line DE which is drawen from the centre to BC the side of the pentagon is the halfe of â—he sideâ— of the decagon and hexagon taken together Forasmuch as the line DE is a perpendicular â—nto the line BC therefore the sections BE and EC shall be equall by the 3. of the third and the line EF is common vnto them both and the angles FEC and FEB are right angles by supposition Wherefore the bases BF and FC are equall by the 4. of the first But the line BC is the side of a pentagon by construction Wherefore FC which subtendeth the halfe of the side of the pentagon is the side of the decagon inscribed in the circle ABC But vnto the line FC is by the 4. of the first equall the line CG for they subtend right angles CEG and CEF which are contained vnder equall sides Wherefore also the angles CGE and CFE of the triangle CFG are equall by the 5. of the first And forasmuch as the arke FC is subtended of the side of a decagon the arke CA shall be quadruple to the arke CF Wherefore also the angle CDA shall be quadruple to the angle CDF by the last of the sixâ— And forasmuch as the same angle CDA which is set at the center is double to the angle CFA which is set at the circumference by the 20. of the third therefore the angle CFA or CFD is double to the angle CDâ— namely the halfe of quadruple But vnto the angle CFD or CFG is proued equall the angle CGF Wherefore the outward angle CGF is double to the angle CDF Wherefore the angles CDG and DCG shall be equall For vnto those two angles the angle CGF is equall by the 32. of the first Wherefore the sides GC and GD are equall by the 6. of the first Wherefore also the line GD is equall to the line FC which is the side of the decagon But vnto the right line FE is equall the line EG by construction Wherefore the whole line DE is equall to the two lines Câ— and FE Wherefore those lines taken together namely the lines DF and FC shall
sesquitertia to the line AD by the Corollary of the 12. of the thirtenth of what partes the line AB containeth in power 12 of the same partes the line AD containeth in power 9 wherefore the residue BD containeth in power of the same partes 3. â—or the line AB coÌ„taineth in power the lines AD and BD by the 47. of the first Wherfore the lines AD and DB are rationall and commensurable to the rationall line set AB by the 6. of the tenth But forasmuch as the power of the line AD is to the power of the line DB in that proportion that 9. a square number is to 3. a number not square therfore they are not in the proportion of square numbers by the Corollary of the 25. of the eight And therfore they are not commensurable in length by the 9. of the tenth Wherefore that which is contained vnder the lines AD and DB which are rationall lines commensurable in power onely is mediall by the 22. of the tenth But that which is contained vnder the lines AD and DB is double to the triangle ABD by the 41. of the first Wherefore that which is contained vnder the lines AD and DB is equall to the whole triangle ABG which is double to the triangle ABD by the 1. of the sixt Wherefore the triangle ABG is mediall If therfore the side of an equilater triangle be rationall the superficies shall be irrationall of that kinde which is called Mediall which was required to be proued A Corollary If an Octohedron and a Tetrahedron be inscribed in a Sphere whose diameter is rationall their superficieces shall be mediall For those superficieces consiste of equilater triangles whose sides are commensurable to the diameter which is rationall by the 13. and 14. of the thirtenth and therefore they are rationall But they are commensurable in power onely to the perpendicular line and therefore they containe a mediall triangle as it was before manifest ¶ The 10. Proposition If a Tetrahedron and an Octohedron be inscribed in one and the self same Sphere the base of the Tetrahedron shall be sesquitertia to the base of the Octohedron and the supersicieces of the Octohedron shall be sesquialtera to the superficieces of the Tetrahedron FOrasmuch as the diameter of the Sphere is in power sesquialtera to the side of the Tetrahedron by the 13. of the thirtenth and the same diameter is in power duple to the side of the Octohedron by the 14. of the same booke therefore of what partes the diameter containeth in power sixe of the same the side of the Tetrahedron contayneth in power 4 and of the same the side of the Octohedron containeth in power 3. Wherefore the power of the side of the Tetrahedron is to the power of the side of the Octohedron in the same proportion that 4. is to 3 which is sesquitertia And like triangles which are the bases of the solides described of those sides shall haue the one to the other the same proportion that the squares made of those sides haue For both the triangles are the one to the other and also the squares are the one to the other in double proportion of that in which the sides are by the 20. of the sixth Wherefore of what partes one base of the Tetrahedron was 4 of the same are fower bases of the Tetrahedron 16 likewise of what partes of the same one base of the Octohedron was 3 of the same are 8. bases of the Octohedron 24. Wherfore the bases of the Octohedron are to the bases of the Tetrahedron in that proportion that 24. is to 16 which is sesquialtera If therefore a Tetrahedron and an Octohedron be inscribed in one and the selfe same Sphere the base of the Tetrahedron shall be sesquitertia to the base of the Octohedron and the superficieces of the Octohedron shall be sesquialtera to the superficieces of the Tetrahedron which was required to be proued ¶ The 11. Proposition A Tetrahedron is to an Octohedron inscribed in one and the selfe same Sphere in proportion as the rectangle parallelograÌ„me contained vnder the line which containeth in power 27. sixty fower partes of the side of the Tetrahedron vnder the line which is subsesquioctaâ—a to the same side of the Tetrahedron is to the square of the diameter of the sphere LEt vs suppose a Sphere whose diameter let be the line AB and let the centre be the point H. And in it let there be inscribed a Tetrahedron ADC and an Octohedron AEKBG And let the line NL containe in power â—7 64 of AC the side of the Tetrahedron And let the line ML be in leÌ„gth subsesquioctaua to the same side TheÌ„ I say that the Tetrahedron ACD is to the Octohedron AEB as the rectangle parallelogramme contayned vnder the lines NL and LM is to the square of the line AB Forasmuch as the line drawen froÌ„ the angle A by the centre H perpeÌ„dicularly vpon the base of the Tetrahedron falleth vpon the ceÌ„ter T of the circle which containeth that base and maketh the right line HT the sixth part of the diameter AB by the Corollary of the 13. of the thirtenth therefore the line HA which is drawen from the centre to the circuÌ„ference is triple to the line HT and therefore the whole line AT is to the line AH 〈…〉 Let the Tetrahedron ADC be cut by a plaine GHK passing by the centrâ— H and being parallel vnto the base DTC by the Corollary of the 11. of the eleuenth Now then the triangle ADC of the Tetrahedron shall be cut by the right line KG which is parallel to the line DC by the 16. of the eleuenth Wherfore as the line AT is to the line AH so is the line AC to the line AG by the 2. of the sixth Wherefore the line AC is to the line AG sesquitertia that is as 4. to 3. And forasmuch as the triangles ADC AKG and the rest which are cut by the plaine KHG are like the one to the other by the 5â— of the sixth the pyramids ADC and AKG shall be like the one to the other by the 7. definition of the â—leuenth Wherefore they are in triple proportion of that in which the sides AC and AG are by the 8. of the twelfth But the proportion of the sides AC to AG is as the proportion of 4. to 3. Now then if by the 2. of the eight ye finde out 4. of the lest numbers in continuall proportion and in that proportion that 4. is to 3 which shall be 64.48.36 and 27 it is manifest by the 15 definition of the fifth that the extremes â—4 to 27. are in triple proportion of that in which the proportion geuen 4. to 3. is Or the quantitie of the proportion of 4. to 3. which is 1. and 1 â— being twise multiplied into it selfe there shall be produced the proportion of 64. to 27. Wherefore the Pyramis
bene demonstrated FD is the diameter of the Sphere which containeth the Icosahâ—dron which diameter is in power sesquialter to AB the side of the Tetrahedron inscribed in theâ— same Sphere by the 13. of the thirtenth Wherfore the line ED the side of the Icosahedron is in power sesquialter to Gâ— the greater segment or lesse line If therefore the side of a Tetrahedron containe in power two right lines ioyned together an extreme and meane proportion the side of an Icosahedron described in the selfe same Sphere is in power sesquialter to the lesse right line ¶ The 19. Proposition The superficies of a Cube is to the superficies of an Octohedron inscribed in one and the selfe same Sphere in that proportion that the solides are SVppose that ABCDE be a Cube whose fower diameters let be the lines AC BC DC and EC produced on ech side Let also the Octohedron inscribed in the selfe same Sphere be FGHK whose three diameters let be FH GK and ON Then I say that the cube ABD is to the Octohedron FGH as the superficies of the cube is to the superficies of the Octohedron Drawe from the centre of the cube to the base ABED a perpendicular line CR. And from the centre of the Octohedron draw to the base GNH a perpendicular line â—L And forasmuch as the three diameters of the cube do passe by the 〈◊〉 C therefore by the 2. Corollary of the 15. of the thirtenth â—here shall be made of the cube sixe pyramids as thys pyramis ABDEC equall to the whole cube For there are in the cube â—ixe bases vpon which fall equall perpendiculars from the cenâ—â—â— by the Corollary of the Assumpâ— of the 16. of the twelfth for the bases are contained in equall circlâ— of the Sphere But in the Octohedron the three diameters do make vpon the 8. bases 8. pyramids hauing their toppes in the centre by the 3. Corollary of the 14â— of the thirtenth Now the bases of the cube and of the Octohedron are contained in equall circles of the Sphere by the 13. of this booke Wherefore they shall be equally distant from the centre and the perpendicular lines CR and â— shall be equall by the Corollary of the Assumpt of the 16. of the twelfth Wherefore the pyramids of the cube shall be vnder one and the selfe same altitude with the pyramids of the Octohedron namely vnder the perpendicular line drawen from the centre to the bases Wherfore sixe pyramids of the cube are to 8. pyramids of the Octohedron being vnder one and the same altitude in that propoâ—tion that their bases are by the 6. of the twelfth that is one pyramis set vpon sixe bases of the cube and hauing to his altitude the perpendicular line which pyramis is equall to the sixe pyramids by the same 6. of the twelfth is to one pyramis set vpon the 8. bases of the Octohedron being equall to the Octohedron and also vnder onâ— and the selfe same altitude in that proportion that sixe bases of the cube which containe the whole superficies of the cube are to 8. bases of the Octohedronâ— which containe the whole superficies of the Octohedron For the solides of those pyramids are in proportion the one to the other as their bases are by the selfe same 6. of the twelfth Wherefore â—he superficies of the cube is to the superficies of the Octohedron inscribed in one and the selfe same Sphere in that proportion that the solides are which was required to be proued ¶ The 20. Proposition If a Cube and an Octohedron be contained in one the selfe same Sphere they shall be in proportion the one to the other as the side of the Cube is to the semidiameter of the Sphere SVppose that the Octohedron AECDB be inscribed in the Sphere ABCD and let the cube inscribed in the same Sphere be FGHIM whose diameter let be HI which is equall to the diameter AC by the 15. of the thirtenth let the halfe of the diameter be AE Then I say that the cube FGHIM is to the Octohedron AECDB as the side MG is to the semidiameter AE Forasmuch as the diameter AC is in power double to BK the side of the Octohedron by the 14. of the thirtenth and is in power triple to MG the side of the cube by the 15. of the same therefore the square BKDL shall be sesquialâ—er to FM the square of the cube From the line AE cut of a third part AN and froÌ„ the line MG cut of likewise a third part GO by the 9. of the sixth Now then the line EN shall be two third partes of the line AE and so also shall the line MO be of the line MG Wherefore the parallelipipedon set vpon the base BKDL and hauing his altitude the line EA is triple to the parallelipipedon set vpon the same base and hauing his altitude the line AN by the Corollary of the 31. of the eleuenth but it is also triple to the pyramis ABKDL which is set vpon the same base and is vnder the same altitude by the second Corollary of the 7. of the twelfth Wherefore the pyramis ABKDL is equall to the parallelipipedon which is set vpon the base BKDL and hath to his altitude the line AN. But vnto that parallelipipedoÌ„ is double the parallelipipedon which is set vppon the same base BKDL and hath to his altitude a line double to the line EN by the Corollary of the 31. of the first and vnto the pyramis is double the Octohedron ABKLDC by the 2. Corollary of the 14. of the thirtenth Wherefore the Octohedron ABKDLC is equall to the parallelipipedon set vpon the base BKLD hauing his altitude the line EN by the 15. of the fifth But the parallelipipedon set vpon the base BKDL which is sesquialter to the base FM and hauing to his altitude the line MO which is two third partes of the side of the cube MG is equall to the cube FG by the 2. part of the 34. of the eleuenth For it was before proued that the base BKDL is sesquialter to the base FM Now then these two parallelipipedons namely the parallelipipedon which is set vpoÌ„ the base BKDL which is sesquialter to the base of the cube and hath to his altitude the line MO which is two third partes of MG the side of the cube which parallelipipedon is proued equall to the cube and the parallelipipedon set vpon the same base BKDL and hauing his altitude the line EN which parallelipipedon is proued equall to the Octohedron these two parallelipipedons I say are the one to the other as the altitude MO is to the altitude EN by the Corollary of the 31. of the eleuenth Wherefore also as the altitude MO is to the altitude EN so is the cube FGHIM to the Octohedron ABKDLC by the 7. of the fifth But as the line MO is to the line EN so is
the whole line MG to the whole line EA by the 18. of the fifth Wherefore as MG the side of the cube is to EA the semidiameter so is the line FGHIM to the Octohedron ABKDLC inscribed in one the selfe same Sphere If therefore a cube and an Octohedron be contained in one and the selfe same Sphere they shall be in proportion the one to the other as the side of the cube is to the semidiameter of the Sphere which was required to be demonstrated A Corollary Distinctly to notefie the powers of the sides of the fiue solides by the power of the diameter of the sphere The sides of the tetrahedron and of the cube doo cut the power of the diameter of the sphere into two squares which are in proportion double the one to the other The octohedron cutteth the power of the diameter into two equall squares The Icosahedron into two squares whose proportion is duple to the proportion of a line diuided by an extreame and meane proportion whose lesse segmeÌ„t is the side of the Icosahedron And the dodecahedron into two squares whose proportion is quadruple to the proportion of a line diuided by an extreame and meane proportion whose lesse segment is the side of the dodecahedron For AD the diameter of the sphere contayneth in power AB the side of the tetrahedron and BD the side of the cube which BD is in power halfe of the side AB The diameter also of the sphere contayneth in power AC and CD two equall sides of the octohedron But the diameter contayneth in power the whole line AE and the greater segment thereof ED which is the side of the Icosahedron by the 15. of this booke Wheâ—fore their powers being in duple proportioÌ„ of that in which the sides are by the first corollary of the 20. of the sixth haue their proportion duple to the proportion of an extreame meane proportioÌ„ Farther the diameter coÌ„tayneth in power the whole line AF and his lesse segment FD which is the side of the dodecahedron by the same 15. of this booke Wherefore the whole hauing to the lesse â— double proportion of that which the extreame hath to the meane namely of the whole to the greater segment by the 10. diffinition of the fifth it followeth that the proportion of the power is double to the doubled proportion of the sides by the same first corollary of the 20. of the sixth that is is quadruple to the proportion of the extreame and of the meane by the diffinition of the sixth An aduertisment added by Flussas By this meanes therefore the diameter of a sphere being geuen there shall be geuen the side of euery one of the bodies inscribed And forasmuch as three of those bodies haue their sides commensurable in power onely and not in length vnto the diameter geuen for their powers are in the proportion of a square number to a number not square wherefore they haue not the proportion of a square number to a square number by the corollary of the 25. of the eight wherefore also their sides are incommensurabe in length by the 9. of the tenth therefore it is sufficient to compare the powers and not the lengths of those sides the one to the otherâ— which powers are contained in the power of the diameter namely from the power of the diameter let there ble taken away the power of the cube and there shall remayne the power of the Tetrahedron and taking away the power of the Tetrahedron there remayneth the power of the cube and taking away from the power of the diameter halfe the power thereof there shall be left the power of the side of the octohedron But forasmuch as the sides of the dodecahedron and of the Icosahedron are proued to be irrationall for the side of the Icosahedron is a lesse line by the 16. of the thirtenth and the side of the dedocahedron is a residuall line by the 17. of the same therfore those sides are vnto the diameter which is a rationall line set incommensurable both in length and in power Wherefore their comparison can not be diffined or described by any proportion expressed by numbers by the 8. of the tenth neither can they be compared the one to the other for irrational lines of diuers kindes are incoÌ„meÌ„surable the one to the other for if they should be commensurable they should be of one and the selfe same kinde by the 103. and 105. of the tenth which is impossible Wherefore we seking to compare them to the power of the diameter thought they could not be more aptly expressed then by such proportions which cutte that rationall power of the diameter according to their sides namely diuiding the power of the diameter by lines which haue that proportioÌ„ that the greater segment hath to the lesse to put the lesse segment to be the side of the Icosahedron deuiding the sayd power of the diameter by lines hauing the proportion of the whole to the lesse segment to expresse the side of the dodecahedron by the lesse segment which thing may well be done betwene magnitudes incommensurable The ende of the fourtenth Booke of Euclides Elementes after Flussas ¶ The fiftenth booke of Euclides Elementes THis finetenth and last booke of Euclide or rather the second boke of Appollonius or Hypsicles teacheth the inscription and circumscriptioÌ„ of the fiue regular bodies one within and about an other a thing vndoutedly plesant and delectable in minde to contemplate and also profitable and necessary in act to practise For without practise in act it is very hard to se and conceiue the constructions and demonstrations of the propositions of this booke vnles a man haue a very depe sharpe fine imagination Wherfore I would wish the diligent studeÌ„t in this booke to make the study thereof more pleasant vnto him to haue presently before his eyes the bodyes formed framed of pasted paper as I taught after the diffinitions of the eleuenth booke And then to drawe and describe the lines and diuisions and superficieces according to the constructions of the propositions In which descriptions if he be wary and diligent he shall finde all things in these solide matters as clere and as manifest vnto the eye as were things before taught only in plaine or superficial figures And although I haue before in the twelfth boke admonished the reader hereof yet bicause in this boke chiefly that thing is required I thought it should not be irkesome vnto him againe to be put in minde thereof Farther this is to be noted that in the Greke exemplars are found in this 15. booke only 5. propositions which 5. are also only touched and set forthe by Hypsicies vnto which Campane addeth 8. and so maketh vp the number of 13. Campane vndoubtedly although he were very well lerned and that generally in all kinds of learning yet assuredly being brought vp in a time of rudenes when all good letters were darkned barberousnes had
the residue or of this excesse But a pyramis is to the same cube inscribed in it nonecuple by the 30. of this booke Wherefore the Dodecahedron inscribed in the pyramis and containing the same cube twise taking away the selfe same third of the lesse segment and moreouer the lesse segment of the lesse segment of halfe the residue shall containe two ninth partes of the solide of the pyramis of which ninth partes eche is equall vnto the cube taking away this selfe same excesse The solide therefore of a Dodecahedron containeth of a Pyramis circumscribed about it two ninth partes taking away a third part of one ninth part of the lesse segment of a line diuided by an extmere and meane proportion and moreouer the lesse segment of the lesse segment of halfe the residue ¶ The 36. Proposition An Octohedron exceedeth an Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side of the Icosahedron and hauing to his altitude the line which is the greater segment of halfe the semidiameter of the Octohedron SVppose that there be an Octohedron ABCFPL in which let there be inscribed an Icosahedron HKEGMXNVDSQTâ— by the â—6 of the fiuetenth And draw the diameters AZRCBROIF and the perpendicular KO â—arallel to the line AZR Then I say that the Octohedron ABCFPL is greater thâ—n the Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side HK or GE and hauing to his altitude the line KO or RZ which is the greater segment of the semidiameter AR. Forasmuch as in the same 16. it hath bene proued that the triangles KDG and KEQ are described in the bases APF and ALF of the Octohedron therefore about the solide angle there remaine vppon the base FEG three triangles KEG KFE and KFG which containe a pyramis KEFG Vnto which pyramis shall be equall and like the opposite pyramis MEFG set vpon the same base FEG by the 8. definition of the eleuenth And by the â—ame reason shall there at euery solide angle of the Octohedron remayne two pyramids equall and like namely two vpon the base AHK two vpon the base BNV two vpon the base DPS and moreouer two vpon the base QLT. Now theÌ„ there shal be made twelue pyramids set vpon a base contained of the side of the Icosahedron and vnder two leâ—â—e segmentes of the side of the Octohedron containing a right angle as for example the base GEF And forasmuch as the side GE subteÌ„ding a right angle is by the 47. of the â—irst in power duple to either of the lines EF and FG and so the â—â—deâ— KH is in power duple to either of the sides AH and AK and either of the lines AH AK or EF FG is in power duple to eyther of the lines AZ or ZK which coÌ„tayne a right angle made in the triangle or base AHK by the perpendicular AZ Wherfore it followeth that the side GE or HK is in power quadruple to the triangle EFG or AHK But the pyramis KEFG hauing his base EFG in the plaine FLBP of the Octohedron shall haue to his altitude the perpendicular KO by the 4. definition of the sixth which is the greater segment of the semidiameter of the Octohedron by the 16. of the fiuetenth Wherfore three pyramids set vnder the same altitude and vpon equall bases shall be equall to one prisme set vpon the same base and vnder the same altitude by the 1. Corollary of the 7. of the twelfth Wherefore 4. prismes set vpon the base GEF quadrupled which is equall to the square of the side GE and vnder the altitude KO or RZ the greater segment which is equall to KO shall containe a solide equall to the twelue pyramids which twelue pyramids make the excesse of the Octohedron aboue the Icosahedron inscribed in it An Octohedron therefore excedeth an Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side of the Icosahedron and hauing to his altitude the line which is the greater segment of halfe the semidiameter of the Octohedron ¶ A Corollary A Pyramis exceedeth the double of an Icosahedron inscribed in it by a solide set vpon the square of the side of the Icosahedron inscribed in it and hauing to his altitude that whole line of which the side of the Icosahedron is the greater segmeÌ„t For it is manifest by the 19. of the fiueteÌ„th that an octohedroÌ„ an IcosahedroÌ„ inscribed in it are inscribed in one the self same pyramis It hath moreouer bene proued in the 26. of this boke that a pyramis is double to an octohedroÌ„ inscribed in it Wherfore the two excesses of the two octohedrons vnto which the pyramis is equal aboue the two Icosahedrons inscribed in the said two octohedrons being brought into an solide the said solide shal be set vpon the selfe same square of the side of the Icosahedron and shall haue to his altitude the perpendicular KO doubled whose double coupling the opposite sides HK and XM maketh the greater segment the same side of the Icosahedron by the first and second corollary of the 14. of the fiuâ—â—enâ—h The 37. Proposition If in a triangle hauing to his base a rational line set the sides be commensurable in power to the base and from the toppe be drawn to the base a perpendicular line cutting the base The sections of the base shall be commensurable in length to the whole base and the perpendicular shall be commensurable in power to the said whole base And now that the perpendicular AP is commensurable in power to the base BG iâ— thus proued Forasmuch as the square of AB is by supposition commensurable to the square of BG and vnto the rational square of AB is commensurable the rational square of BP by the 12. of the eleuenth Wherfore the residue namely the square of PA is commensurable to the same square of BP by the 2. part of the 15. of the eleuenth Wherefore by the 12. of the tenth the square of PA is commensurable to the whole square of BG Wherefore the perpendicular AP is commensurable in power to the base BG by the 3. diffinition of the tenth which was required to be proued In demonstrating of this we made no mention at all of the length of the sides AB and AG but only of the length of the base BG for that the line BG is the rational line first set and the other lines AB and AG are supposed to be commensurable in power only to the line BG Wherefore if that be plainely demonstrated when the sides are commensurable in power only to the base much more easily wil it follow if the same sides be supposed to be commensurable both in length and in power to the base that is if their lengthes be expressed by the rootes of square nombers ¶ A Corollary 1. By the former things demonstrated it is manifest that if from the powers of the base and of one of the sides be taken away the
from a cube and an octohedrō cōposed into one solide there shal be left an exocthedrō Moreouer the solide angles taken away from two pyramids composed into one solide there shal be left an Octohedron Flussas after this setteth forth certaine passions and properties of the fiue simple regular bodies which although he demonstrateth not yet are they not hard to be demonstrated if we wel pease and conceiue that which in the former bookes hath bene taught touching those solides Of the nature of a trilater and equilater Pyramis A trilater equilater Pyramis is deuided into two equal partes by three equal squares which in the centre of the pyramis cutte the one the other into two equal partes and perpendicularly and whose angles are set in the midle sections of the sides of the pyramis From a pyramis are taken away 4. pyramids like vnto the whole which vtterly take away the sides of the pyramis and that which is left is an octohedrō inscribed in the pyramys in which all the solides inscribed in the pyramis are contained A perpendicular drawne from the angle of the pyramis to the base is double to the diameter of the cube inscribed in it And a right line coupling the midle sections of the opposite sides of the pyramis is triple to the side of the selfe same cube The side also of the pyramis is triple to the diameter of the base of the cube Wherefore the same side of the pyramis is in power duple to the right line which coupleth the midle sections of the opposite sides And it is in power sesquialter to the perpendicular which is drawne from the angle to the base Wherefore the perpendicular is in power sesquitertia to the line which coupleth the midle sections of the opposite sides A pyramis and an Octohedron inscribed in it also an Icosahedron inscribed in the same Octohedron doo containe one and the selfe same sphere Of the nature of an Octohedron Foure perpendiculars of an Octohedron drawne in 4. bases therof from two opposite angles of the said Octohedron and coupled together by those 4. bases describe a Rhombus or diamond figure one of whose diameters is in power duple to the other diameter For it hath the same proportiō that the diameter of the Octohedron hath to the side of the Octohedron An Octohedron an Icosahedron inscribed in it do containe one and the selfe same sphere The diameter of the solide of the Octohedron is in power sesquialter to the diameter of the circle which containeth the base and is in power triple to the right line which coupleth the cētres of the opposite bases and is in power duple superbipartiens tercias to the perpēdicular or side of the foresaid Rhombus and moreouer is in lēgth triple to the line which coupleth the centres of the next bases The angle of the inclination of the bases of the Octohedron doth with the angle of the inclination of the bases of the pyramis make angles equal to two right angles Of the nature of a Cube The diameter of a cube is in power sesquialter to the diameter of his base and is in power triple to his side and vnto the line which coupleth the centres of the next bases it is in power sextuple Moreouer the side of the cube is to the side of the Icosahedron inscribed in it as the whole is to the greater segment vnto the side of the Dodecahedron it is as the whole is to the lesse segment vnto the side of the Octohedron it is in power duple and vnto the side of the pyramis it is in power subduple Moreouer the cube is triple to the pyramis but to the cube the Dodecahedron is in a maner duple Wherfore the same Dodecahedron is in a maner sextuple to the sayd pyramis Of the nature of an Icosahedron Fiue triangles of an Icosahedron do make a solide angle the bases of which triangles make a pentagon If therfore from the opposite bases of the Icosahedron be taken the other pentagon by them described these pentagons shall in such sort cut the diameter of the Icosahedron which coupleth the forsaid opposite angles that that part which is contained betweene the plaines of those two pentagons shal be the greater segment and the residue which is drawne from the plaine to the angle shall be the lesse segment If the opposite angles of two bases ioyned together be coupled by a right line the greater segment of that right line is the side of the Icosahedron A line drawne from the centre of the Icosahedron to the angles is in power quintuple to halfe that line which is takē betwene the pentagōs or of the halfe of that line which is drawne from the centre of the circle which cōtayneth the foresaid pentagon which two lines are therefore equall The side of the Icosahedron contayneth in power either of them and also the lesse segment namely the line which falleth from the solide angle to the pentagon The diameter of the Icosahedron contayneth in power the whole line which coupleth the opposite angles of the bases ioyned together and the greater segment thereof namely the side of the Icosahedron The diameter also is in power quintuple to the line which was taken betwene the pentagons or to the line which is drawne from the centre to the circumference of the circle which containeth the pentagon cōposed of the sides of the Icosahedron The dimetient contayneth in power the right line which coupleth the centres of the opposite bases of the Icosahedron and the diameter of the circle which contayneth the base Moreouer the sayd dimetient contayneth in power the diameter of the circle which contayneth the pentagon and also the line which is drawne from the centre of the same circle to the circumference That is it is quintuple to the line drawne from the centre to the circumference The line which coupleth the centres of the opposite bases contayneth in power the line which coupleth the centres of the next bases and also the rest of that line of which the side of the cube inscribed in the Icosahedron is the greater segment The line which coupleth the middle sections of the opposite sides is triple to the side of the dodecahedron inscribed in it Wherefore if the side of the Icosahedron and the greater segment thereof be made one line the third part of the whole is the side of the dodecahedron inscribed in the Icosahedron Of the nature of a Dodecahedron The diameter of a dodecahedron contayneth in power the side of the dodecahedron and also that right line vnto which the side of the dodecahedron is the lesse segment and the side of the cube inscribed in it is the greater segmēt which line is that which subtendeth the angle of the inclination of the bases contayned vnder two perpendiculars of the bases of the dodecahedron If there be taken two bases of the dodecahedron distant the one from the other by the length of one of the sides
to be prâ—â—â—d beâ—oâ—e 〈◊〉 â—all to the demoÌ„â—â—ration Construction Demonstration leading to an absurditie A Corollary This Probleme reduced to a Theoreme Construction Demonstration How magnitudes are sayd to be in proportion the onâ— to the other as number is to number This proâ—osition is the conuerse of formâ—r Conâ—truction Demonstration A Corollary Construction Demonstration Construction Demonstration Demonâ—tration leading to an abâ—urdiâ—ie This is the 〈…〉 demonsâ—â—ation The first part demonstratâ—d An other demonstration of the first part An othâ—r demonâ—â—raâ—ion oâ— the same first part after Montaureus Demonstration of the seconde part which is the coâ—â—erse of the former An other demonstration of the second part This Assumpâ— followeth as a Corollary of the 25 but so as it might also be here in Methode placed you shall â—inde it after the 53. of this booke absolutely demonstrated for there it serueth to the 54. his demonsâ—ration DemoÌ„stratioÌ„ of the third part DemoÌ„stratioÌ„ of the fourth part which is the coÌ„uerse of the â— Conclusion of the whole proposition A Corâ—llary Proâ—e of the first part of the Corollary Profe of the second part Profe of the third pâ—rt Proâ—e oâ— the fourth part Certayne annotations â—ut of Montauâ—â—us Rules to know whether two superficiall numbers be like or no. This assumpt is the conuerse of the 26. of the eight Demonsâ—raâ—ioÌ„ oâ— the first part Demonstration of the second partâ— A Corollary To finde out the first line incommensurable in length onely to the line geuen To finde out the second line incommensurable both in length and in power to the line geuen Construction Demonstration Tâ—is is wiâ—h Zambert an Aâ—â—â—mpt but vâ—â—eâ—ly improperly â—lâ—ssateâ— maâ—eth iâ— a Corollary but the Greeâ—e and Montaureus maâ—e it a proposition but euery way an â—nfallible truth 〈…〉 Demonstration leading to an absurditie Demonstration leading to an absurditâ—e A Corollary A Corollary Demonstration An other way to proue that the lines A E C F are proportionall Demonstration of the first part Demonstration of the second paâ—t which is the conuerse of the first A Corollary Demonstration of the first part by an argument leadindg to an absurditie Demonstration of the second paâ—t leading also to an impossibilitie And this second part is the conuerse of the first Demonstration of the second part which is the conuerse of the first How to deuide the line BC redely in such sort as iâ— required in the propositioÌ„ Demonstrâ—tion of the second part which is the conuerse of tâ—e former An other demonstrationâ—y an argumeÌ„t leading to an absurditie An Assumpt A Corollary added by Montaureuâ— Cause Cause of increasing the difficulty of this booke Note Construction Demonstration Diuers caâ—es in this proposition The second case The first kind of rationall lines commensurable in length This particle in the proposition according to any of the foresayde wayes was not in vayne put The second kinde of rationall lines coÌ„mensurable in lengâ—h The third case The third kinde of rationall lines commensurable in length The fourth case This proposition is the conuerse of the former proposition Construction Demonstration An Assumpt Constâ—uction Demonstration Diffinition of a mediall line A Corollary This assumpt is nothing els but a part of the first proposition of the sixt booke 〈◊〉 How a square is sayde to be applied vppon a line Construction Demonstration Construction Demonstration Note A Corollary Construction Demonstration leading to an absurditie Construction Demonstâ—ation Construction Demonstration * A Corollary To finde out two square nâ—mbers exceeding the one the other by a square â—umber An Assumpt Construction Demonstration Montaureus maketh this an Assumpt as the Grecke text seemeth to do likewise but without a cause Construction Demonstration This Assumpt setteth foâ—th nothing â—ls but that which the first oâ— the sâ—â—t â—etteth â—orth and therefore in sâ—me examplars it is not founde Construction Demonstration Construction Demonstration Construction Demonstration A Corollary I. Dee * The second Corollary * Therefore if you deuide the square of the side AC by the side BC the portion DC will be the product c. as in the former Corollâ—ry I. Dâ—e * The thirde Corollary * Therfore if the parallelogramme of BA and AC be deuided by BC the product will geue the pâ—â—pâ—ndicular D A. These three Corollaryes in practise Logisticall and Geometricall are profitable An other demonstration of this fourth part of the determinatioÌ„ An Assumpt Construction Deâ—onstration The first part of the dâ—termination concluded The second part coÌ„cluded The totall conclusion Construction Demonstration The first part of the determination concluded The second part coÌ„cluded The totall conclusion Construction Demonstration The first part concluded The second part coÌ„cluded The third part coÌ„cluded The totall conclusion The first Senary by composition Diffinition of a binomiall line Sixe kindes of binomiall lines Demonstration Diffinition of a first bimediall line Construction Demonstration Diffinition of a second bimediall line Demonstration Diffinition of a greater line Diffinition of a line whose power is rationall and mediall Diffinition of a liâ—e containing in power two medials An Assumpt The second Senary by composition Demonstration leading to an impossibilitie A Corollary Demonstration leading to an impossibilâ—â—e DemonstratioÌ„ leading to an impossibiliâ—ie Demonstration leading to an impâ—ssibilitie Demonstration leading to an impossibiâ—â—â—e Demonstration leading to an impossibilitie Construction Demonstration leading to an absurditie Sixe kindes of binomiall lines A binomiall line coâ—â—â—steâ—h of two paâ—tâ—s Firsâ— dâ—â—â—initiâ—n Seconâ— diffinition Third â—â—â—â—â—â—ition Fourth diffinition Fifth difâ—inition Sixth diffinition The third Senary by composition Construction Demonstratiâ—n Construction Demonstration Construction Demonstration Constâ—uction Demonstration Construction Demonstration Construction Demonstration A Corollary added by Flussates M. d ee his booke called Tyâ—â—câ—niâ—m Mathematicum This Assumpt as was before noted fâ—llâ—weth most â—riâ—fly without farther demonstration of the 25. of this booke Demonstration An Assumpt The fourth Senary by composition Construction Demonstration The first part of this demonstration concluded The secoÌ„d part of the demonstration concluded The third part coÌ„cluded The totall conclusion Demoâ—stratioâ— The first part of this demonstration concluded The third part coÌ„cluded The fourth part coÌ„cludeâ— The fift part concluded The totalâ— conclusion Demonstration Construction Demonstration Demonstration Demonstration * Looke after the Assumpt concluded at this marke for plainer opening of this place The vse of this Assumpt is in the next proposition other following The fift Senary by composition Construction Demonstration Concluded that DG is a binomiall line Construction Demonstratiâ—n Concluded that DG is a binomiall line Construction Demonstration †* ‡ DG concluded a binomiall line A Corollary added by M. Dee Construction Demonstration Construction Demonstration Construction Demonstration The sixt Senary Construction Demonstration Construction Demonstration A Corollâ—ry addâ—d by Flussetes Note Construction Demonstration An other demonstration after P. Montaureus An other demonstration after Campane Construction Demonstration An other demonstratioÌ„ afâ—â—r Campane Construction Demonstration An Assumpt An other demonstration after Campanâ— Note
Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both SubstaÌ„tiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Mixt Which with aide of Geometrie principall demonstrateth some Arithmeticall Conclusion or Purpose The vse whereof is either In thinges Supernaturall â—ternall Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both SubstaÌ„tiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Geometrie Simple Which dealeth with Magnitudes onely and demonstratâ—th all their properties passions and appertenances whose Point is Indiuisible The vse whereof is either In thinges Supernaturall â—ternall Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both SubstaÌ„tiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Mixt Which with aide of Arithmetike principall demonstrateth some Geometricall purpose as EVCLIDES ELEMENTES The vse whereof is either In thinges Supernaturall â—ternall Diuine By Application Ascending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Mathematicall without farther Application The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue In thinges Naturall both SubstaÌ„tiall Accidentall Visible Inuisible c. By Application Descending The like Vses and Applications are though in a degree lower in the Artes Mathematicall Deriuatiue Deriuatiue froÌ„ the Principalls oâ— which some haue The names of the Principalls as Arithmetike vulgar which considereth Arithmetike of most vsuall whole Numbers And of Fractions to them appertaining Arithmetike of Proportions Arithmetike Circular Arithmetike of Radicall NuÌ„bers Simple Compound Mixt And of their Fractions Arithmetike of Cossike NuÌ„bers with their Fractions And the great Arte of Algiebar Geometrie vulgar which teacheth Measuring At hand All Lengthes Mecometrie All Plaines As Land Borde Glasse c. Embadometrie All Solids As Timber Stone Vessels c. Stereometrie With distaÌ„ce from the thing Measured as How farre from the Measurer any thing is of him sene on Land or Water called Apomecometrie Of which are growen the Feates Artes of Geodesie more cunningly to Measure and Suruey Landes Woods Waters c. Geographie Chorographie Hydrographie Stratarithmetrie How high or deepe from the leuell of the Measurers standing any thing is Seene of hym on Land or Water called Hypsometrie Of which are growen the Feates Artes of Geodesie more cunningly to Measure and Suruey Landes Woods Waters c. Geographie Chorographie Hydrographie Stratarithmetrie How broad a thing is which is in the Measurers vew so it be situated on Land or Water called Platometrie Of which are growen the Feates Artes of Geodesie more cunningly to Measure and Suruey Landes Woods Waters c. Geographie Chorographie Hydrographie Stratarithmetrie Propre names as Perspectiue Which demonstrateth the maners and properties of all Radiations Directe Broken and Reflected Astronomie Which demonstrateth the Distances Magnitudes and all Naturall motions Apparences and Passions proper to the Planets and fixed Starres fâ—r any time past prâ—sent and to come in respecte of a certaine Horizon or without respecte of any Horizon Musike Which demonstrateth by reason and teacheth by sense perfectly to iudge and order the diuersitie of Soundes hiâ— or lâ—w Cosmographie Which wholy and perfectly maketh description of the Heauenly and also Elementall part of the World and of these partes maketh hâ—mâ—lâ—gall application and mutuall collation necessary Astrologie Which reasonably demonstrateth the operations and effectes of the naturall beaâ—es of light and 〈◊〉 Inâ—luence of the Planets and fixed Starres 〈◊〉 euery Element and Elementall body at all times in any Horizâ—n assigned Statike Which demonstrateth the causes of heauines and lightnes of all thinges and of the motions and properties to heauines and lightnes belonging Anthropographie Which describeth the NuÌ„ber Measure Waight Figure Situation and colour of euery diuers thing contained in the perfectâ— body of â—â— AN and geueth certaine knowledge of the Figure Symmetriâ— Waight Characterization due Locall motion of any pâ—rcell of the sayd body assigned and of numbers to the said pâ—rcell appertaining Trochilike Which demonstrateth the properties of all Circular motions Simple and Compound Helicosophie Which demonstrateth the designing of all Spirall lines in Plaine on Cylinder Coâ—â— Sphâ—re Câ—nâ—id and Spharoâ—d and their properties Pneumatithmie Which demonstrateth by close hollow Geometricall figures Regular and Irregular the straunge properties in motion or stay of the Water Ayre Smoke and Fire in their Continuitiâ— and as they are ioyned to the Elementes next them Menadrie Which demonstrateth how about Natures Vertue and power simple Vertue and force may be multiplied and so to directe to lifâ— to pull to aâ—d to put or cast fro any multiplied or simple determined Vertue Waight or Force naturally not so directible or moueable Hypogeiodie Which demonstrateth how vnder the Spharicall Superficieâ— of the Eâ—rth at â—ny depth to any perpendicular line assigned whose distance from the perpendicular of the entrance and the Aziâ—uth likewise 〈◊〉 respeâ—â—e of the sayd entrance is knowen certaine way may be prescribed and gâ—ne c. Hydragogie Which demonstrâ—teth the possible leading of water by Natures lâ—â— and by artificiall helpe frâ—â— any head being Spring standing or running water to any other place assigned Horometrie Which demonstrateth how at all times appointed the precise vsuall denomination of time â—â—y â—e knowâ—n for any place assigned Zographie Which demonstrateth and teacheth how the Intersection of all vsuall 〈…〉 assigned the Center distancâ— and lightes bâ—ing determined may be by lines and proper colâ—urs repreâ—â—â—â— Architecture Which is a Sciâ—â—â—â— garâ—ished with many doctrines and 〈…〉 are iudged Nauigation Thaumaturgike Archemastrie ¶ The first booke of Euclides Elementes IN THIS FIRST BOOKE is intreated of the most simple easie and first matters and groundes of Geometry as namely of Lynes Angles Triangles Parallels Squares and Parallelogrammes First of theyr definitions showyng what they are After that it teachâ—th how to draw Parallel lynes and how to forme diuersly figures of three sides foure sides according to the varietie of their sides and Angles
28. Proposition To finde out mediall right lynes commensurable in power onely contayning a mediall parallelogramme LEt there be put three rationall right lines commensurable in power only namely A B and C and by the 13. of the sixt take the meane proportional betwene the lines A and B let thâ— same be D. And as the line B is to the line C so by the 12. of the sixt let the line D be to the line E. And forasmuch as the lines A and B are rationall commensurable in power onely therefore by the 21. of the tenth that which is contained vnder the lines A and B that is the square of the line D is mediall Wherfore D is a mediall line And forasmuch as the lines B and C are commensurable in power onely and as the line B is to the line C so is the line D to the line E wherfore the lines D and E are commensurable in power onely by the corollary of the tenth of this booke but D is a mediall line Wherefore E also is a mediall line by the 23. of this booke Wherfore D E are mediall lines commensurable in power onely I say also that they containe a mediall parallelograme For for that as the line B is to the line C so is the line D to the line E therfore alternately by the 16 of the fift as the line B is to the line D so is the line C to the line E. But as the lyne B is to the line D so is the line D to the line Aâ— by conuerse proportion which is proued by the corollary of the fourth of the fifth Wherfore as the line D is to the line A so is the line C to the line E. Wherfore that which is contained vnder the lines A C is by the 16. of the sixâ— equall to that which is contayned vnder the lines D E. But that which is contained vnder the lines A and C is medial by the 21. of the tenth Wherfore that which is coÌ„tained vnder the lines D and E is mediall Wherfore there are found out mediall lines commensurable in power onely containing a mediall superficies which was required to be done An Assumpt To finde out two square numbers which added together make a square number Let there be put two like superficiall numbers AB and BC which how to finde out hath bene taught after the 9. proposition of this booke And let them both be either euen numbers or odde And let the greater number be AB And forasmuch as if from any euen number be taken away an euen number or froÌ„ an odde number be taken away an odde number the residue shall be euen by the 24. and 26 of the ninth If therfore from AB being an euen number be taken away BC an euen number or from AB being an odde number be taken away BC being also odde the residue AC shall be euen Deuide the number AC into two equall partes in D wherefore the number which is produced of AB into BC together with the square number of CD is by the sixt of the second as Barlaam demonstrateth it in numbers equall to the square number of BD. But that which is produced of AB into BC is a square nuÌ„ber For it was proued by the first of the ninth that if two like plaine numbers multiplieng the one the other produce any nuÌ„ber the number produced shal be a square number Wherfore there are found out two square numbers the one being the square number which is produced of AB into BC and the other the square number produced of CD which added together make a square number namely the square number produced of BD multiplied into himselfe forasmuch as they were demoÌ„strated equall to it A Corollary And hereby it is manifest that there are found out two square numbers namely the 〈◊〉 the square number of BD and the other the square number of CD so that that numbâ—r wherin th one excedeth the other the number I say which is produced of AB into BC is also a square number namely when Aâ— BC are like playne numbers But when they are not like playne numbers then are there found out two square numbers the square number of BD and the square number of DC whose excesse that is the number wherby the greater excedeth the lesse namely that which is produced of AB into BC is not a square number ¶ An Assumpt To finde out two square numbers which added together make not a square number Let AB and BC be like playne numbers so that by the first of the ninth that which is produced of AB into BC is a square number and let AC be an euen number And deuide Câ— into two equall parâ—es in D. Now by that which hath before bene sayd in the former assumpt it is manifest that the square number produced of AB into BC together with the square number of CD is equall to the square number of BD. Take away from CD vnitie DE. Wherfore that which is produced of AB into BC together with the square of CE is lesse then the square number of BD. Now then I say that the square numâ—er produced of AB into BC added to the square number of CE make not a square number For if they do make a square number then that square number which they make is either greater theÌ„ the square number of BE or equall vnto it or lesse then it First greater it cannot be for it is already proued that the square number produced of AB into BC together with the square number of CE is lesse then the square number of BD. But betwene the square number of BD and the square number of BE there is no meane square number For the number BD excedeth the number BE onely by vnitie which vnitie can by no meanes be deuided into numbers Or if the number produced of AB into BC together with the square of the nuÌ„ber CE should be greater then the square of the number BE then should the selfe same number produced of AB into BC together with the square of the number CE be equall to the square of the number BD the contrary wherof is already proued Wherfore if it be possible let that which is produced of AB into BC together with the square number of the number CE be equall to the square number of BE. And let GA be double to vnitie DE that is let it be the number two Now forasmuch as the whole number AC is by supposition double to the whole number CD of which the number AG is double to vnitie DE therfore by the 7. of the seuenth the residue namely the number GC is double to the residue namely to the number EC Wherfore the number GC is deuided into two equall partes in E. Wherefore that which is produced of GB into BC together with the square number of CE is equall to the square nuÌ„ber
contayned vnder a rationall line and a fourth residual line is irrationall and the line which contayneth in power the same parallelogramme is also irrationall and is called a lesse line by the 94. of the tenth But the line AB contayneth in power the parallelogramme contayned vnder the lines HB and BM for if we drawe a right line from the point A to the point H the triangle ABH shall be like to the triangle ABM by the 8. of the sixth For from the right angle BAH is drawen to the base BH a perpendicular line And therefore as the line BH is to the line BA so is the line AB to the line BM this followeth also of the coâ—ollary of the sayd 8. of the sixth Wherefore the line AB which is the side of the pentagon figure is an irrationall line of that kinde which is called a lesse line If therefore in a circle hauing a rationall line to his diameter be inscribed an equilater pentagon the side of the pentagon is an irratioâ—all line and is of that kinde which is called a lesse line which was required to be demonstrated ¶ The 12. Theoreme The 12. Proposition If in a circle be described an equilater triangle the square made of the side of the triangle is treble to the square made of the line which is drawen from the centre of the circle to the circumference SVppose that ABC be a circle and in it describe an equilater triangle ABC Then I say that the square made of the side of the triangle ABC is treble to the square made of the line drawen from the center of the circle ABC to the circumference Take by the 1. of the third the centre of the circle and let the same be D. Aâ—d draw a right line from the point A to the poynt D and extend it to the point E. And draw a right line from the point B to the poynt E. Now forasmuch as the triangle ABC is equilater tâ—â—refore eche of these three circumferences AB AC BEC is the third part of the whole circumference of the circle ABC wherefore the circumference BE is the sixth part of the circuÌ„ference of the circle for the circumfereÌ„ce of the semicircle ABE is equall to the circumference of the semicircle ACE from which taking away equal circumferences AB and AC the circumference remayning BE shal be equal to the circumference remayning EC wherefore the right line BE is the side of an equilater hexagon figure described in the circle Wherefore it is equall to the line drawen from the centre of the circle to the circumference that is vnto the line DE by the corollary of the 15. of the sixth And forasmuch as the line AE is double to the line DE therefore the square of the line AE is quadruple to the square of the line DE by the 4. of the second that is to the square of the line BE. But the square of the line AE is equall to the squares of the lines AB and BE by the 47. of the first for the angle ABE is by the 31. of the third a right angle Wherfore the squares of the line AB BE are quadruple to the square of the line BE. Wherefore taking away the square of the line BE the squaâ—e of the line AB shâ—lbe treble to the square of BE but the line BE is equall to the line DE. Whâ—râ—fore the square of the line AB is treble to the square of the line DE. Wherefore the square made of the side of the triangle is treble to the square made of the line drawen froÌ„ the centre of the circle to the circumference which was required to be proued A Corollary added by Campane Hereby it is manifest that the line BC which is the side of the equilater triangle diuideth the semidiameter DE into two equall parts For let the poynt of the diuision be F. And suppose a line to be drawen from the poynt D to the B and an other from the poynt D to the poynt C. Now it is manifest by the 4. of the first that the line BF is equall to the line FC and therefore by the 3. of the third all the angles at the poynt F are â—ight angles Wherefore by the 47. of the first the square of the line BD is equall to the squares oâ— the lineâ—â—F and FD and by the same the square of the line BE is equall to the squares of the lines BF and FE bâ—t the line BD is equall to the line BE as hath before bene proued Wherefore by the common sentence the twoo squares of the two lines BF and FD are equall to the two squares of the lineâ— BF and FE Wherefore taking away the square of the line BF which is coÌ„mon to them both the residue namely the square of the line DF shal be equall to the residue namely to the square of the line FE Wherfore also the line FD is equal to the line FE Wherfore hereby it is manifest that a â—erpendicular line drawen from the centre of a circle to the side of an equilater triangle inscribed in it is equall to halfe of the line drawen from the centre of the same circle to the circumference thereof A Corollary added by Flussas The side of an equilater triangle is in power sesquitertia to the perpendicular line which is draweÌ„ from one of the angles to the opposite side For of what parts the line AB contayneth in power 12. of such parts the line BF which is the halfe of AB contayned in power 3. Wherefore the residue namely the perpendicular line AF contayneth in power of such parts 9. for the squares of the lines AF and BF are by the 47. of the first equall to the square of the line AB Now 1â— to 9. is sesquitertiaâ— wherfore the power of the line AB is to the power of the line AF in sesquitertia proportion Moreouer the side of the triangle is the meane proportionall betwene the diameter and the perpendicular line For by the Corollary of the 8. of the sixth the line AE is to the line AB as the line AB is to the line AF. Farther the perpendicular line drawen from the angle diuideth the base into two equal parts and passeth by the center For if there should be drawen any other right line froÌ„ the point A to the poynt F theÌ„ that which is drawen by the point D two right lines should include a superficies which is impossible Wherefore the contrary followeth namely that the line which being drawen from the angle passeth by the center is a perpendicular line to the base by the 3. of the third The 1. Probleme The 13. Proposition To make a Pyramis and to comprehend it in a sphere geuen and to proue that the diameter of the sphere is in power sesquialtera to the side of the Pyramis In the semicircle ADB of the former figure drawe the line DN And diuide the