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Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	county_n	line_n	residue_n	120	3	16.9006	5	false

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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A00429	The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed; Elements. English	Euclid.; Dee, John, 1527-1608.; Candale, François de Foix, comte de, 1502-1594.; Billingsley, Henry, Sir, d. 1606.	1570 (1570)	STC 10560; ESTC S106699	1,020,889	884

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 F● and in the selfe same proportion And let the line which containeth in power that parallelogrāme be G. Thē I say that the line G is rational Take a rational line namely H. And vnto the line CD apply a parallelogrāme 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 cōmensurable to the names of the binomiall line that is to CE and ED and are in the selfe same proportiō But by position the lines CE and ED are cōmensurable 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 FB● 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 parallelogrāme cōtained vnder the lines CD AB is cōmensurable to the square of the line H. But the parallelogrāme 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 ot●●r 〈…〉 Flussas 〈…〉 line ●D whos● names A● and ●D let be commensurable in length vnto the names of the residuall line A● which let be AF and FB And let the li●e AE● be to the line ED● in the same proportion that the line AF is to the line F● And let the right line ● contayne in power the superficies D● Then I say tha● the li●e ● is a rationall lin● 〈…〉 l●ne which l●● b●● And vpon the line ●● describe by the 4● of the first a parallelogramme eq●all to the squar● of the line ●● and making in breadth the line DC Wherefore by the ●12 of this booke CD is a residu●ll line● whose names Which let be ●● and OD shall be co●mensurabl● in le●gth vnto the names A● and ●D and the line C o shall be vnto the line OD in the same propor●ion that the line AE is to the line ED● But as the line A● is to the line ●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 F●● Wherefore the lines CO and OD are commensurable with the lines A● and ●● by the ●● of this boke Wherfore the residue namely the line CD is to the residue namely to the line A● as the line CO is to the line AF by the 19. of the fifth But it is proued that the line CO is cōmensurable 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 D● But the parallelogramme ●● i● by construction rationall for it is equall to the square of the rationall line ● Wh●refore the parallelogramme ●D ●s also rat●●n●ll● Wher●fore the line ● which by supposition cōtayneth in power the superficies ●D● is also rationall If therfore a parallelogrāme 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 ● 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 demonstratio● 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

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 cōueniently 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 cōmensurable 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 cōmensurable 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 parallelogrāme be cōtained vnder a residuall line a binomiall lyne whose names are commensurable to the names of the residuall line and in the sel●e same proportion the lyne which contayneth in power

double to the side of the Octohedron the side is in power sequitertia to the perpēdiclar line by the 12. of this booke wherfore the diameter thereof is in power duple superbipartiens tertias to the perpendicular line Wherfore also the diameter and the perpēdicular line are rationall and commensu●able by the 6. of the tenth As touching an Icosahedron it was proued in the 16. of this booke that the side thereof is a lesse line when the diameter of the sphere is rationall And forasmuch as the angle of the inclination of the bases thereof is contayned of the perpendicular lines of the triangles and subtended of the right line which subtendeth the angle of the Pentagon which contayneth fiue sides of the Icosahedron and vnto the perpendicular lines the side is commensurable namely is in power sesquitertia vnto them by the Corollary of the 12. of this booke therefore the perpendicular lines which contayne the angles are irrationall lines namely lesse lines by the 105. of the tenth booke And forasmuch as the diameter contayneth in power both the side of the Icosahedron and the line which subtendeth the foresayd angle if from the power of the diameter which is rationall be taken away the power of the side of the Icosahedron which is irrationall it is manifest that the residue which is the power of the subtending line shal be irrationall For if it shoulde be rationall the number which measureth the whole power of the diameter and the part taken away of the subtending line should also by the 4. common sentence of the seuenth measure the residue namely the power of the side which is irrationall for that it is a lesse line which were absurd Wherefore it is manifest that the right lines which compose the angle of the inclination of the bases of the Icosahedron are Irrationall lines For the subtending line hath to the line contayninge a greater proportion then the whole hath to the greater segment The angle of the inclination of the bases of a dodecahedron is contayned vnder two perpendiculars of the bases of the dodecahedron and is subtended of that right line whose greater segment is the side of a Cube inscribed in the dodecahedron which right line is equall to the line which coupleth the sections into two equal parts of the opposite sides of the dodecahedron And this coupling line we say is an irrationall line for that the diameter of the sphere contayneth in power both the coupling line and the side of the dodecahedron but the side of the dodecahedron is an irrationall line namely a residuall line by the 17. of this booke Wherefore the residue namely the coupling line is an irrationall line as it is ●asy to proue by the 4. cōmon sentence of the seuēth And that the perpēdicular lines which contayne the angle of the inclination are irrationall is thus proued By the proportion of the subtending line of the foresayd angles of inclination to the lines which containe the angle is found out the obliquitie of the angle For if the subtending line be in power double to the line which contayneth the angle then is the angle a right angle by the 48. of the first But if it be in power lesse then the double it is an acute angle by the 23. of the second But if it be in power more then the double or haue a greater proportion then the whole hath to the greater segmēt● the angle shal be an obtuse angle by the 12. of the second and 4. of the thirtenth By which may be proued that the square of the whole is greater then the double of the square of the greater segment This is to be noted that that which Flussas hath here taught touching the inclinations of the bases of the ●iue regular bodies Hypsicles teacheth after the 5 proposition of the 15. booke Where he confesseth that he receiued it of one Isidorus and seking to make the mater more cleare he endeuored himselfe to declare that the angles of the inclination of the solides are geuen and that they are either acute or obtuse according to the nature of the solide although ●uclid● in all his 15. bookes hath not yet shewed what a thing geuen is Wherefore Flussas framing his demōstration vpon an other ground procedeth after an other maner which semeth more playne and more aptly hereto be placed then there Albeit the reader in that place shal not be frustrate of his also The ende of the thirtenth Booke of Euclides Elementes ¶ The fourtenth booke of Euclides Elementes IN this booke which is commonly accompted the 14. booke of Euclide is more at large intreated of our principal purpose namely of the comparison and proportion of the fiue regular bodies customably called the 5. figures or formes of Pythagoras the one to the other and also of their sides together eche to other which thinges are of most secret vse and inestimable pleasure and commoditie to such as diligently search for them and attayne vnto them Which thinges also vndoubtedly for the woorthines and hardnes thereof for thinges of most price are most hardest were first searched and found out of Philosophers not of the inferior or meane sort but of the depest and most grounded Philosophers and best exercised in Geometry And albeit this booke with the booke following namely the 15. booke hath bene hetherto of all men for the most part and is also at this day numbred and accompted amōgst Euclides bookes and supposed to be two of his namely the 14. and 15. in order as all exemplars not onely new and lately set abroade but also old monumentes written by hand doo manifestly witnes yet it is thought by the best learned in these dayes that these two bookes are none of Euclides but of some other author no lesse worthy nor of lesse estimation and authoritie notwithstanding then Euclide Apollonius a man of deepe knowledge a great Philosopher and in Geometrie maruelous whose wōderful bookes writtē of the sections of cones which exercise occupy thewittes of the wisest and best learned are yet remayning is thought and that not without iust cause to be the author of them or as some thinke Hypsicles him selfe For what can be more playnely then that which he him selfe witnesseth in the preface of this booke Basilides of Tire sayth Hypsicles and my father together scanning and peysing a writing or books of Apollonius which was of the comparison of a dodecahedron to an Icosahedron inscribed in one and the selfe same sphere and what proportion these figures had the one to the other found that Apollonius had fayled in this matter But afterward sayth he I found an other copy or booke of Apollonius wherein the demonstration of that matter was full and perfect and shewed it vnto them whereat they much reioysed By which wordes it semeth to be manifest that Apollonius was the first author of this booke which was afterward set forth by Hypsicles For so his owne wordes after in

Demonstration * The circles so made or so considered in the sphere are called the greatest circles All other not hauing the center of the sphere to be their center also● are called lesse circles Note these descriptions * An other Corollary * An other Corollary Construction * This is also proued in the As●umpt before added out o● Flussas Note what a greater or greatest circle in a Spere is First part of the Construction Note● * You know full well that in the superficies of the sphere ●●●ly the circumferences of the circles are but by th●se circumferences the limitatiō and assigning of circles is vsed and so the circumference of a circle vsually called a circle which in this place can not offend This figure is restored by M. Dee his diligence For in the greeke and Latine Euclides the line GL the line AG and the line KZ in which three lynes the chiefe pinch of both the demonstrations doth stand are vntruely drawen as by comparing the studious may perceaue Note You must imagine 〈◊〉 right line AX to be perpēdicular vpon the diameters BD and CE though here AC the semidiater seme to be part of AX. And so in other pointes in this figure and many other strengthen your imagination according to the tenor of constructions though in the delineatiō in plaine sense be not satisfied Note BO equall to BK in respect of M. Dee his demonstration following † Note ●his point Z that you may the better vnderstand M. Dee his demōstration Second part of the construction Second part of the demonstration ✚ Which of necessity shall fall vpon Z as M. Dee proueth it and his profe is set after at this marke ✚ following I. Dee * But AZ is greater thē AG as in the former propositiō KM was euident to be greater then KG so may it also be made manifest that KZ doth neyther touch nor cut the circle FG●H An other proue that the line AY is greater thē the line AG. * This as an assumpt is presently proued Two cases in this proposition The first case Demonstration leading to an impossibilitie Second case * As it is ●asi● to gather by the ●●●umpt put after the seco●● of this boo●● Note a generall rule The second part of the Probleme two wayes executed An vpright Cone The second part of the Probleme The second ●a●● o● the ●robleme ☜ * This may easely be demonstrated as in th● 17. proposition the section of a sphere was proued to be a circle * For taking away all doubt this a● a Lemma afterward is dem●strated A Lemma as it were presently demonstrated Construction Demonstration The second part of the Probleme * Construction Demonstration An other way of executing this probleme The conuerse of the assūpt A great error commonly maintained Betwene straight and croked all maner of proportiō may be geuen Construction Demonstration The diffini●iō of a circ●e ●●ap●●d in a sp●er● Construction Demonstration This is manifest if you consider the two triangles rectangles HOM and HON and then with all vse the 47. of the first of Euclide Construction Demonstratiō Construction Demonstratiō This in maner of a Lemm● is presently proued Note here of Axe base soliditie more then I nede to bring any farther proofe for Note * I say halfe a circ●lar reuolution for that su●●iseth in the whole diameter ST to describe a circle by i● it be moued ●●out his center Q c. Lib 2 prop 2. de Sphe●a Cylindr● Note * A rectangle parallelipipedon geu●n equall to a Sphere geuen To a Sphere or to any part of a Sphere assigned as a third fourth fifth c to geue a parallelipipedon equall Sided Columes Pyramids and prismes to be geuen equall to a Sphere or to any certayne part thereof To a Sphere or any segment or sector of the same to geue a cone or cylinder equall or in any proportion assigned Farther vse of Sphericall Geometrie The argument of the thirtenth booke Construction Demonstration * The Assūpt proued * Because AC is supposed greater then AD therefore his residue is lesse then the residue of AD by the common sentence Wherefore by the supposition DB is greater then ●C The chie●e line in all Euclides Geometrie What is ment here by A section in one onely poi●t Construction Demonstration * Note how CE and the gnonom XOP are proued equall for it serueth in the conuerse demonstrated by M. Dee here next after This proposition ●the conuerse of the former * As we ha●e noted the place of the peculier pro●e there ●in the demōstration of the 3. * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated Construction * Though I say perpēdicular yes you may perceue how infinite other p●s●●iōs will serue so that DI and AD make an angle for a triangle to haue his sides proportionally cut c. Demonstration Demonstration I. Dee This is most euident of my second Theoreme added to the third propositiō For to adde to a whole line a line equall to the greater segmēt to adde to the greater segment a line equall to the whole line is all one thing in the line produced By the whole line I meane the line diuided by extreme and meane proportion This is before demonstrated most euidently and briefly by M. Dee after the 3. proposition Note Note 4. Proportional lines Note two middle proportionals Note 4. wayes of progres●ion in the proportion of a line deuided by extreme and middle proportion What resolution and composition is hath before bene taught in the beginning of the first booke * Proclus in the Greeke in the 58. page Construction Demonstration Two cases in this proposition Construction Th● first case Demonstration The second case Construction Demonstration Construction Demonstration This Corollary is the 3. proposition of the ●4 booke after Campane Demonstration of the first part Demonstration of the second part Construction Dem●nstration Construstion Demonstration Constr●yction Demonstration This Corollary is the 11. prop●sition of the 14. booke after Campane This Corollary is the 3. Corollary after the 17. proposition of the 14 booke after Campane * By the name o● a Pyramis both here i● this booke following vnderstand a Tetrahedron An other construction and demonstration of the second part after F●ussas Third part of the demonstration This Corollary is the 15. proposition of the 14. booke after Campane This Corollary Campane putteth as a Corollary after

E produced D wherfore A measureth D but it also measureth it not which is impossible Wherfore it is impossible to finde out a fourth number proportionall with these numbers A B C whensoeuer A measureth not D. ¶ The 20. Theoreme The 20. Proposition Prime numbers being geuen how many soeuer there may be geuen more prime numbers SVppose that the prime numbers geuen be A B C. Then I say that there are yet more prime numbers besides A B C. Take by the 38. of the seuenth the lest number whom these numbers A B C do measure and let the same be DE. And vnto DE adde vnitie DF. Now EF is either a prime number or not First let it be a prime number then are there found these prime numbers A B C and EF more in multitude then the prime numbers ●irst geuen A B C. But now suppose that EF be not prime Wherefore some prime number measureth it by the 24. of the seuenth Let a prime number measure it namely G. Then I say that G is none of these numbers A B C. For if G be one and the same with any of these A B C. But A B C measure the nūber DE wherfore G also measureth DE and it also measureth the whole EF. Wherefore G being a number shall measure the residue DF being vnitie● which is impossible Wherefore G is not one and the same with any of these prime numbers A B C and it is also supposed to be a prime number Wherefore there are ●ound these prime numbers A B C G being more in multitude then the prime numbers geuen A B C which was required to be demonstrated A Corollary By thys Proposition it is manifest that the multitude of prime numbers is infinite ¶ The 21. Theoreme The 21. Proposition If euen nūbers how many soeuer be added together the whole shall be euē SVppose that these euen numbers AB BC CD and DE be added together Then I say that the whole number namely AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an euen number therefore euery one of them hath an halfe Wherefore the whole AE also hath an halfe But an euen number by the definition is that which may be deuided into two equall partes Wherefore AE is an euen number which was required to be proued ¶ The 22. Theoreme The 22. Proposition If odde numbers how many soeuer be added together if their multitude be euen the whole also shall be euen SVppose that these odde numbers AB BC CD and DE being euen in multitude be added together Then I say that the whole AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an odde number is ye take away vnitie from euery one of them that which remayneth o● euery one of thē is an euen number Wherefore they all added together are by the 21. of the ninth an euen number and the multitude of the vnities taken away is euen Wherefore the whole AE is an euen number which was required to be proued ¶ The 23. Theoreme The 23. Proposition If odde numbers how many soeuer be added together and if the multitude of them be odde the whole also shall be odde SVppose that these odde numbers AB BC and CD being odde in multitude be added together Then I say that the whole AD is an odde number Take away from CD vnitie DE wherefore that which remaineth CE is an euen number But AC also by the 22. of the ninth is an euen number Wherfore the whole AE is an euen number But DE which is vnitie being added to the euen number AE maketh the whole AD a● odde number which was required to be proued● ¶ The 24. Theoreme The 24. Proposition If from an euen number be takē away an euen number that which remaineth shall be an euen number SVppose that AB be an euen number and from it take away an euen number CB. Then I say that that which remayneth namely AC is an euen number For forasmuch as AB is an euen euen number it hath an halfe and by the same reason also BC hath an halfe Wherfore the residue CA hath an halfe Wherfore AC is an euen number which was required to be demonstrated ¶ The 25. Theoreme The 25. Proposition If from an euen number be taken away an odde number that which remaineth shall be an odde number SVppose that AB be an euen number and take away from it BC an odde number Then I say that the residue CA is an odde number Take away from BC vnitie CD Wherfore DB is an euen number And AB also is an euen number wherefore the residue AD is an euen number by the ●ormer proposition But CD which is vnitie being taken away from the euen nūber AD maketh the residue AC an odde number which was required to be proued ¶ The 26. Theoreme The 26. Proposition If from an odde number be taken away an odde number that which remayneth shall be an euen number SVppose that AB be an odde number and from it take away an odde number BC. Thē I say that the residue CA is an euen number For forasmuch as AB is an odde number take away from it vnitie BD. Wherfore the residue AD is euen And by the same reason CD is an euen number wherfore the residue CA is an euen number by the 24. of this booke ● which was required to be proued ¶ The 27. Theoreme The 27. Proposition If from an odde number be taken a way an euen number the residue shall be an odde number SVppose that AB be an odde number and from it take away an euen number BC. Then I say that the residue CA is an odde number Take away frō AB vnitie AD. Wherfore the residue DB is an euē number BC is by supposition euen Wherfore the residue CD is an euen number Wherefore DA which is vnitie beyng added vnto CD which is an euen number maketh the whole AC an ●dde number which was required to be proued ¶ The 28. Theoreme The 28. Proposition If an odde number multiplieng an euen number produce any number the number produced shall be an euen number SVppose that A being an odde number multiplieng B being an euen number do produce the number C. Then I say that C is an euen number For forasmuch as A multiplieng B produced C therfore C is composed of so many numbers equall vnto B as there be in vnities in A. But B is an euen nūber wherfore C is composed of so many euen numbers as there are vnities in A. But if euē numbers how many soeu●r be added together the whole by the 21. of the ninth is an euen number wherfore C is an euen number which was required to be demonstrated ¶ The 29. Theoreme The 29. Proposition I● an odde number multiplying an

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 th● 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 A● 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 six● 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 cōtained 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 frō 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 nūber For it was proued by the first of the ninth that if two like plaine numbers multiplieng the one the other produce any nūber 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 demōstrated 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 numb●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 A● 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 C● into two equall par●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 num●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 thē 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 nūber 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 nūber

of these triangles EKF FLG GMH and HNE is greater then the halfe of the segment of the circle which is described about it Now then deuiding the circumferences remaining into two equall partes and drawing right lines from the pointes where those diuisions are made so continually doing this we shall at the length by the 1. of the tenth leaue certaine segmentes of the circle which shall be lesse then the excesse wherby the circle EFGH excedeth the superficies S. For it hath bene proued in the first Proposition of the tenth booke that two vnequall magnitudes being geuen if from the greater be taken away more then the halfe and likewise againe from the residue more then the hal●e and so continually there shall at the length be left a certaine magnitude which shall be lesse then the lesse magnitude geuen Let there be such segmentes left let the segmentes of the circle EFGH namely which are made by the lines EK KF FL LG GM MH HN and NE be lesse then the excesse whereby the circle EFGH excedeth the superficies S. Wherefore the residue namely the Poligonon figure EKFLGMHN is greater then the superficies S. Inscribe in the circle ABCD a Poligonon figure like to the Poligonon figure EKFLGMHN and let the same be AXBOCPDR Wherefore by the Proposition next going before as the square of the line BD is to the square of the line FH so is the Poligonon figure AXBOCPDR to the Poligonon figure EKFLGMHN But as the square of the line BD is to the square of the line FG so is the circle ABCD supposed to be to the superficies S. Wherefore by the 11. of the fift as the circle ABCD is to the superficies S so is the Poligonon figure AXBOCPDR to the Poligonon figure EKFLGMHN Wherefore alternately by the 16. of the fift as the circle ABCD is to the Poligonon figure described in it so is the superficies S to the Poligonon figure EKFLGMHN But the circle ABCD is greater then the Poligonon figure described in it Wherefore also the superficies S is greater then the Poligonon figure EKFLGHMN but it is also lesse which is impossible Wherefore as the square of the line BD is to the square of the line FH so is not the circle ABCD to any superficies lesse then the circles EFGH In like sort also may wproue that as the square of the line FH is to the square of the line BD so is not the circle EFGH to any superficies lesse then the circle ABCD. I say namely that as the square of the line BD is to the square of the line FH so is not the circle ABCD to any superficies greater thē then the circle EFGH For if it be possible let it be to a greater namely to the superficies S. Wherfore by conuersion as the square of the line FH is to the square of the line BD so is the superficies S to the circle ABCD. But as the s●perficies S is to the circle ABCD so is the circle EFGH to some supe●ficies l●sse thē the circle ABCD. Wherefore by the 11. of the fift as the square of the line FH is to the square of the line BD so is the circle EFGH to some superficies lesse then the circle ABCD which is in the first case proued to be impossible Wherefore as the square of the line BD is to the square of the line FH so is not the circle ABCD to any superficies greater then the circle EFGH And it is also proued that it is not to any lesse Wherefore as the square of the l●ne BD is to the square of the line FH so is the circle ABCD to the circle EFGH Wherefore circles are in that proportion the one to the other that the squares of their diameters are which was required to be proued ¶ An Assumpt I say now that the superficies S being greater then the circle EFGH as the superficies S is to the circle ABCD so is the circle EFGH to some superficies lesse then the circle ABCD. For as the superficies S is to the circle ABCD so let the circle EFGH be to the superficies T. Now I say that the superficies T is lesse then the circle ABCD. For for that as the superficies S is to the circle ABCD so is the circle EFGH to the superficies T therefore alternately by the 16. of the fift as the superficies S is to the circle EFGH so is the circle ABCD to the superficies T. But the superficies S is greater then the circle EFGH by supposition Wherefore also the circle ABCD is greater then the superficies T by the 14. of the fift Wherefore as the superficies S is to the circle ABCD so is the circle EFGH to some superficies lesse then the circle ABCD which was required to be demonstrated ¶ A Corollary added by Flussas Circles haue the one to the other that proportion that like Poligonon figures and in like sort described in them haue For it was by the first Proposition proued that the Poligonon figures haue that proportiō the one to the other that the squares of the diameters haue which proportion likewise by this Proposition● the circles haue ¶ Very needefull Problemes and Corollaryes by Master Ihon Dee inuented whose wonderfull vse also be partely declareth A Probleme 1. Two circles being geuē to finde two right lines which haue the same proportion one to the other that the geuen circles haue o●e to the other● Suppose A and B to be the diameters of two circles geuen I say that two right lines are to be foūde hauing that proportiō that the circle of A hath to the circle of B. Let to A B by the 11 of the sixth a third proportionall line be found which suppose to be C. I say now that A hath to C that proportion which the circle of A hath to the circle of B. For forasmuch as A B and C are by construction three proportionall lines the square of A is to the square of B as A is to C by the Corollary of the 20. of the sixth ● but as the square of the line A is to the square of the line B so is the circle whose diameter is the line A to the circle whose diameter is the line B by this second of the eleuēth Wherfore the circles of the line● A and B are in the proportion of the right lines A and C. Therefore two circles be●ng geuen we haue found two right lines hauing the same proportion betwene thē that the circles geuen haue one to the other which ought to be done A Probleme 2. Two circles being geuen and a right line to finde an other right line to which the line geuē shall haue that proportion which the one circle hath to the other Suppose two circles geuē which let be A B a right line geuē which let be C I say that an other right line is to be ●ounde to which the line C shall haue that proportion that

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 HKEGMXNVDSQT● by the ●6 of the fiuetenth And draw the diameters AZRCBROIF and the perpendicular KO ●arallel to the line AZR Then I say that the Octohedron ABCFPL is greater th●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 ●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 thē there shal be made twelue pyramids set vpon a base contained of the side of the Icosahedron and vnder two le●●e segmentes of the side of the Octohedron containing a right angle as for example the base GEF And forasmuch as the side GE subtēding a right angle is by the 47. of the ●irst in power duple to either of the lines EF and FG and so the ●●de● 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 cōtayne 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 segmēt For it is manifest by the 19. of the fiuetēth that an octohedrō an Icosahedrō 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 octohedrō 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 fiu●●en●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 i● 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

in the poynt E. And vnto the line CG put the line CL equall Now forasmuch as the lines AG and GC are the greater seg●●●tes of halfe the line AB for ●che of them is the halfe of the greater segment of the whole line AB the lines EB and EC shall be the lesse segmentes of halfe the line AB Wherefore the whole line C● is the greater segment and the line CE is the lesse segment But as the line CL is to the line CE so is the line CE to the residue EL. Wherfore the line EL is the greater segment of the line CE or of the line EB which is equall vnto it Wherfore the residue LB is the lesse segment of the same EB which is the les●e segment of half● the side of the cube But the lines AG GC and CL are three greater segmentes of the halfe of the whole line AB which thre greater segmentes make the altitude of the foresayd solide wherefore the altitude of the sayd solide wanteth of AB the side of the cube by the line LB which is the lesse s●gment of the line BE. Which line BE agayne is the lesse segment of halfe the side AB of the cube Wherefore the foresayd solide consisting of the sixe solides whereby the dodecahedron exceedeth the cube inscribed in it is set vpon a base which wanteth of the base of the cube by a third part of the lesse segment and is vnder an altitude wanting of the side of the cube by the lesse segment of the lesse segment of halfe the side of the cube The solide therefore of a dodecahedron exceedeth the solide of a cube inscribed in it by a parallelipipedon whose base wanteth of the base of the cube by a third part of the lesse segment and whose altitude wanteth of the altitude of the cube by the lesse segment of the lesse segment of halfe the side of the cube ¶ A Corollary A Dodecahedron is double to a Cube inscribed in it taking away the third part of the lesse segment of the cube and moreouer the lesse segment of the lesse segment of halfe of that excesse For if there be geuen a cube from which is cut of a solide set vpon a third part of the lesse segment of the base and vnder one and the same altitude with the cube that solide taken away hath to the whole solide the proportion of the section of the base to the base by the 32. of the eleuenth Wherefo●e from the cube is taken away a third ●art of the lesse segment Farther forasmuch as the residue wanteth of the altitude of the cube by the lesse segment of the lesse segment of halfe the altitude or side and that residue is a parallelipipedon if it be cut by a plaine superficies parallel to the opposite plaine superficieces cutting the altitude of the cube by a point it shall take away from that parallelipipedon a solide hauing to the whole the proportion of the section to the altitude by the 3. Corollary of the 25. of the eleuenth Wherefore the excesse wanteth of the same cube by the thi●d part of the lesse segment and moreouer by the lesse segment of the lesse segment of halfe of that excesse ¶ The 34. Proposition The proportion of the solide of a Dodecahedron to the solide of an Icosahedron inscribed in it consisteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron and of the proportion of the side of the Cube to the side of the Icosahedron inscribed in one and the selfe same Sphere SVppose that AHBCK be a Dodecahedron● whose diamet●r let be AB and let the line which coupleth the cētres of the opposite bases be KH● and let the Icosahedron inscribed in the Dodecahedron ABC be d ee whose diameter let be DE. Now forasmuch a● o●e and the selfe same circle cōtaineth the pentagon of a Dodecahedron the triangle of an Icosahedro● described in one and the selfe same Sphere by the 14. of the fourtenth Let that circle be IGO. Wherfore IO is the side of the cube and IG the side of the Icosahedron by the same Thē I say that the proportion of the Dodecahedron AHBCK to the Icosahedron DEF inscribed in it cōsisteth of the proportiō tripled of the line AB to the line KH and of the proportion of the line IO to the line IG For ●o●asmuch as the Icosahedron DEF is inscribed in the Dodecahedrō ABC by suppositiō the diameter DE shal be equal to the line KH by the 7. of the fiuetenth Wherefore the Dodecahedron set vpō the diameter KH shall be inscribed in the same Sphere wherein the Icosahedron DEF is inscribed but the Dodecahedron AHBCK is to the Dodecahedron vpon the diameter KH in triple proportion of that in which the diameter AB is to the diameter KH by the Corollary of the 17. of the twelfth and the same Dodecahedron which is set vpon the diameter KH hath to the Icosahedron DEF which is set vpon the same diameter or vpon a diameter equall vnto it namely DE that proportion which IO the side of the cube hath to● IG the side of the Icosahedron inscribed in one the selfe same Sphere by the 8 of the fouretenth Wherefore the proportion of the Dodecahedron AHBCK to the Icosahedron DEF inscribed in it consisteth of the proportion tripled of the diameter AB to the line KH which coupleth the centres of the opposite bases of the Dodecahedron which proportion is that which the Dodecahedron AHBCK hath to the Dodecahedron set vpon the diameter KH and of the proportion of IO the side of the cube to IG the side of the Icosahedron which is the proportion of the Dodecahedron set vpon the diameter KH to the Icosahedron DEF described in one and the selfe same Sphere by the 5. definition of the sixth The proportion therefore of the solide of a Dodecahedron to the solide of an Icosahedron inscribed in it con●isteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron and of the prop●●tion of the side of the cube to the side of the Icosahedron inscribed in one and the selfe same Sphere The 35. Proposition The solide 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 extreme and meane proportion and moreouer the lesse segment of the lesse segment of halfe the residue IT hath bene proued that the Dodecahedron together with the cube inscribed in it is contained in one and the selfe same pyramis by the Corollary of the first of this booke And by the Corollary of the 33. of this booke it is manifest that the Dodecahedron is double to the same cube taking away the third part of the lesse segment and moreouer the lesse segment of the lesse segment of halfe