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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 |

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diameter is double to that square whose diameter it is The 34. Theoreme The 48. Proposition If the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle SVppose that ABC be a triangle and let the square which is made of one of the sides there namely of the side BC be equall to the squares which are made of the sides BA and AC Then I say that the angle BAC is a right angle Rayse vp by the 11. propositioÌ„ from the point A vnto the right line AC a perpendicular line AD. And by the thirde proposition vnto the line AB put an equall line AD. And by the first peticion draw a right line from the point D to the poinâ— C. And forasmuch as the line DA is equall to the line AB the square which is made of the line DA is equall to the square whiche is made of the line AB Put the square of the line AC common to them both VVherefore the squares of the lines DA and AC are equal to the squares of the lines BA and AC But by the proposition going before the square of the line DC is equal to the squares of the lines AD and AC For the angle DAC is a right angle and the square of BC is by supposition equall to the squares of AB and AC VVherefore the square of DC is equall to the square of BC wherefore the side DC is equall to the side BC. And forasmuch as AB is equall to AD â—nd AC is common to them both therefore these two sides DA and AC are equall to these two sides BA and AC the one to the other and the base DC is equall to the base BCâ— wherfore by the 8. proposition the angle DAC is equall to the angle BAC But the angle DAC is a right angle wherefore also the angle BAC is a right angle If therefore the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle which was required to be proued This proposition is the conuerse of the former and is of Pelitarius demonstrated by an argument leading to an impossibilitie after this maner The ende of the first booke of Euclides Elementes Â¶ The second booke of Euclides Elementes IN this second booke Euclide sheweth what is a GnomoÌ„ and a right angled parallelogramme Also in this booke are set forth the powers of lines deuided euenly and vneuenly and of lines added one to an other The power of a line is the square of the same line that is a square euery side of which is equall to the line So that here are set forth the qualities and proprieties of the squares and right lined figures which are made of lines of their parts The Arithmetician also our of this **booke** gathereth many compendious rules of reckoning and many rules also of Algebra with the equatioÌ„s therein **vsed** The groundes also of those rules are for the most part by this second **booke** demonstrated This **booke** moreouer contayneth two wonderfull **propositions** one of an obtuse angled triangle and the other of an acute which with the ayde of the 47. proposition of the first booke of Euclide which is of a rectangle triangle of how great force and profite they are in matters of astronomy they knowe which haue trauayled in that arte VVherefore if this booke had none other profite be side onely for these 2. propositions sake it were diligently to be embraced and studied The definitions 1. Euery rectangled parallelogramme is sayde to be contayned vnder two right lines comprehending a right angle A parallelogramme is a figure of fower sides whose two opposite or contrary sides are equall the one to the other There are of parallelogrammes fower kyndes a square a figure of one side longer a Rombus or diamond and a Romboides or diamond like figure as before was sayde in the 33. **definition** of the first **booke** Of these fower sortes the square and the figure of one side longer are onely right angled Parallelogrammes for that all their angles are right angles And either of them is contayned according to this definition vnder two right lynes whiâ—h concurre together and cause the right angle and containe the same Of which two lines the one is the length of the figure the other the breadth The parallelogramme is imagined to be made by the draught or motion of one of the lines into the length of the other As if two numbers shoulde be multiplied the one into the other As the figure ABCD is a parallelograme and is sayde to be contayned vnder the two right lines AB and AC which contayne the right angle BAC or vnder the two right lines AC and CD for they likewise contayne the right angle ACD of which 2. lines the one namely AB is the length and the other namely AC is the breadth And if we imagine the line AC to be drawen or moued directly according to the leÌ„gth of the line AB or contrary wise the line AB to be moued directly according to the length of the line AC you shall produce the whole rectangle parallelogramme ABCD which is sayde to be contayned of them euen as one number multiplied by an other produceth a plaine and righte angled superficiall number as ye see in the figure here set where the number of sixe or sixe vnities is multiplied by the number of fiue or by fiue vnities of which multiplication are produced 30. which number being set downe and described by his vnities representeth a playne and a right angled number VVherefore euen as equall numbers multipled by equal numbers produce numbers equall the one to the other so rectangle parallelogrames which are comprehended vnder equal lines are equal the one to the other 2. In euery parallelogramme one of those parallelogrammes which soeuer it be which are about the diameter together with the two supplementes is called a Gnomon Those perticuler parallelogrames are sayde to be about the diameter of the parallelograme which haue the same diameter which the whole parallelograme hath And supplementes are such which are without the diameter of the whole parallelograme As of the parallelograme ABCD the partial or perticuler parallelogrames GKCF and EBKH are parallelogrames about the diameter for that ech of them hath for his diameter a part of the diameter of the whole parallelogramme As CK and KB the perticuler diameters are partes of the line CB which is the diameter of the whole parallelogramme And the two parallelogrammes AEGK and KHFD are supplementes because they are wythout the diameter of the whole parallelogramme namely CB. Now any one of those partiall parallelogrammes

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 ã€ˆâ—Šã€‰ ã€ˆâ—Šã€‰ ã€ˆâ—Šã€‰ ã€ˆâ—Šã€‰ ã€ˆâ—Šã€‰ 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

first Demonstration Demonstration leading to an impossibility This proposition in discret quaÌ„titie answereth to the 23. propositioÌ„ of the fifth boke in continual quaÌ„titie This and the eleuen propositions following declare the pâ—ssions and properties ofâ— prime nuÌ„bers Demonstration leading to an impossibility This is the coÌ„uerse of the former proposition Demonstrâ—tion leading to an absurditie Demonstration leading to an absurditie Demonstration leading to an absurditie Demonstration Demonstration Deâ—onstration Demonstration of the first part leading to an absurditie Demonstration of the second part which is the conâ—câ—se of the first leanâ—ng also to an absurditiâ— Demonstrasion leading to an absurditie Demonstrasion A Corollary â—â—ded by Campaue Demonstration lâ—ading to an impossibilitie An other demonstration Demonstration Two cases in this Proposition The first case The second case Demonstration Demonstration leading to an absurditie A Corollary added by Campaâ—e Two cases in this propositioÌ„ The first case Demonstration leading to an absurditie The second caseâ— Demonstration leading to an absurditie Demonstration leading to an impossibâ—â—â—â—â— Two cases in this propositioÌ„ The first case Demonstration leaâ—iâ—g â—o an absurâ—â—â—â—e The second case Demonstration leading to an absurditie A Corollary Demonstration The coâ—â—erse of the former proposition Demonstration Construction DemonstratioÌ„ leâ—ding to an â—bsuâ—diâ—ie A Corollary adâ—ed by Campane How to â—inde out the seconde least number and the third and so â—orth â—nâ—â—nitly How to siâ—â— out the least â—â—mâ— a conâ—ayâ—â—g â—â—e paâ—â—s of parts The Arguâ—â—â—â— of the eight books Demonstration leading to an absurdâ—â—ie Construction Demonstration This proposition is the â—â—uerse of the first Demonstrationâ— Two cases in this propositioÌ„ The first case Demonstration leading to an absurditie The second case Demonstration This proposition in numbers answereth to the of the sixth touching parellelogrammes Construction Demonstration An other demonstratioÌ„ after Campane Demonstration Demonstration leading to an impossibilitie Demonstration A Corollary added by Flussates Construction Demonstration This proposition is the conuerse of the former Construction Demonstration The first part of this proposition demonstrated The second part demonstrated Construction The first part of this prâ—position deâ—â—â—strated The second part demonstrated Construction Demonstration The first part of this proposition The second part is the conuerse of the first The first part of this proposition The second part is the conuerse of the first A negatâ—ue proportion The first part of this proposition The second part is the coÌ„uerse of the first A negatiue proposition The first part of this proposition The second part is the coÌ„uerse of the first Demonstration of the fiâ—st part of this proposition Demonstration of the second part Demonstration of the first part of this proposition The second part This proposition is the conuerse of the 18. proposition Construction Demonstration This proposition is the conuerse of the 19. proposition Construction Demonstration Demonstration Demonstration Demonstration Demonstration A Corollary added by Flussates Construction Construction Demonstration A Corollary added by Flussates Another Corollary added by Flussates The ArgumeÌ„t of the niâ—th booke Demonstration This proposition is the conuâ—rse oâ— tâ—e formâ—â— Demonstration A Corollary aâ—ded by Campane Demonstration Demonstration Demonstration A Corollary added by Campane Demonstration Demonstration Demonstration of the first part The second part demonstrated DemostratioÌ„ of the third part Demostration of the first part of this proposition The second pâ—rt demonstrated Demonstration of the first part leauing to an absuâ—â—itie Demonstration of the â—â—cond pâ—â—â— leading alâ—o to an absurditie Demonstration Demonstration leading to an absurditie An other demonstratioÌ„ aâ—ter Flussates Demonstration leading to an absurditie An other demonstratioÌ„ after Campane Demoâ—stration leading to an absurditie A propositioâ— added by Campane Construcâ—ion Demonstration Demonstration to proue that the numbers A and C are prime to B. Demonstratiou This proposition is the coÌ„uerse of the former Demonstration This answereth to the 2. of the second Demonstration This answereth to the 3. of the thirds Demonstration This answerâ—th to thâ— 4. of the second Demonstration This answereth to the 5. of the second Demonstration This answereth to the 6. of the second Demonstration This answereth to the 7. of the second Demonstration This answereth to the 8. of the second Demonstratition This answereth to thâ— 9. of the second Demonstration This answereth to the 10. oâ— the second Demonstration A negatiue propositiâ—n Demonstration leaâ—ing to an impossibilitie Demonstration leading to an absurditie Demonstration leading to an abjurditie Three cases in this proposition The first case The second case The third case Diuert cases â—n this proposition The first case Two cases in this Proposition The first case The second case Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration A proposition added by Campaâ—e An other added by him Demonstration leading to an absurditie Demonstration Demonstration Demonstration leading to an absurditie An other demonstration Demonstration Demonstration This proposition teachâ—th how to finde out a perfect number Construction Demonstration Demonstration leading to an absurditie The ArgumeÌ„t of the tenth booke Difference betwene number and magnitude A line is not made of points as number is made of vnities This booke the hardest to vnderstand of all the bookes of Euclide In this booke is entreated of a straunger maner of matter then in the former Many euen of the well learned haue thought that this booke can not well be vnderstanded without Algebra The nine former bookes the principles of this â—ooke well vnderstoode this booke will not be hard to vnderstand The fâ—rst definition The second definition Contraryes made manifest by the comparing of the one to the other The thirde definition What the power of a line is The fourth **definition** Vnto the supposed line first set may be compared infinite lines Why some mislike that the line first set should be called a rational line Flussates calleth this line a line certaine This rational line the grouÌ„d in a maner of all the **propositions** in this tenth **booke** Note The line Rationall of purpose The sixth **deâ—inition** Campâ—nus â—ath caused much oâ—scuritie in this tenth **booke** The seuenth **definition** Flussates in steede of this word irrationall **vseth** this word vncertayne Why they are called irrationall lines The cause of the obscurity and confusednes in this **booke** The eighth **definition** The ninth **definitâ—on** The tenth **deâ—inition** The eleuenth **deâ—inition** Construction Demonstration A Corollary Construction Demonstration This **proposition** teacheth that incontinuall quantitie which the first of the seuenth taught in discrete quantity Construction Demonstration leading to an abâ—urditie Two cases in this propositioÌ„ The first case This proposition teacheth that in continual quantity which the 2. of the sâ—â—ith taught in numbers The second case Demonstration leading to an absurditie A Corollary This Probleme reduced to a Theoreme This proposition teacheth that in continual quantity which the 3. of the second taught in numbers Construction Two cases in this Proposition The first case Demonstration leading to an absurditie The second case A Leâ—ma necesâ—ary

about the diameter together with the two supplementes make a gnomon As the parallelograme EBKH with the two supplementes AEGK and KHFD make the gnomon FGEH Likewise the parallelogramme GKCF with the same two supplementes make the gnomon EHFG And this **diffinition** of a gnomon extendeth it selfe and is generall to all kyndes of parallelogrammes whether they be squares or figures of one side longer or Rhombus or Romboides To be shorte if you take away from the whole parallelogramme one of the partiall parallelogrammes which are about the diameter whether ye will the rest of the figure is a gnomon Campaâ—e after the last **proposition** of the first **booke** addeth this **propositioÌ„** Two squares being geuen to adioyne to one of them a Gnomon equall to the other square which for that as then it was not taught what a Gnomon is I there omitted thinking that it might more aptly be placed here The doing and demonstration whereof is thus Suppose that there be two squares AB and CD vnto one of which namely vnto AB it is required to adde a Gnomon equall to the other square namely to CD Produce the side BF of the square AB directly to the point E. and put the line FE equall to the side of the square CD And draw a line from E to A. Now then forasmuch as EFA is a rectangle triangle therefore by the 47. of the first the square of the line EA is equall to the squares of the lines EF FA. But the square of the line EF is equall to the square CD the square of the side FA is the square AB Wherefore the square of the line AE is equall to the two squares CD and AB But the sides EF and FA are by the 21. of the first longer then the side AE and the side FA is equall to the side FB Wherfore the sides EF and FB are longer theÌ„ the side AE Wherefore the whole line BE is longer then the line AE From the line BE cut of a line equall to the line AE which let be BC. And by the 46. proposition vpon the line BC describe a square which let be BCGH which shal be equal to the square of the line AE but the square of the line AE is equal to the two squares AB and DC Wherefore the square BCGH is equal to the same squares Wherfore forasmuch as the square BCGH is composed of the square AB and of the gnomon FGAH the sayde gnomon shal be equall vnto the square CD which was required to be done An other more redy way after Pelitarius Suppose that there be two squares whose sides let be AB and BC. It is required vnto the square of the line AB to adde a gnomon equall to the square of the line BC. Set the lines AB and BC in such sort that they make a right angle ABC And draw a line froÌ„ A to C. And vpoÌ„ the line AB describe a square which let be ABDE And produce the line BA to the point F and put the line BF equall to the line AC And vpon the line BF describe a square which let be BFGH which shal be equal to the square of the line AC wheÌ„ as the lines BF and AC are equal and therefore it is equal to the squares of the two lines AB and BC. Now forasmuch as the square BFGH is made complete by the square ABDE and by the gnomon FEGD the gnomon FEGD shal be equal to the square of the line BC which was required to be done The 1. Theoreme The 1. Proposition If there be two right lines and if the one of them be deuided into partes howe many soeuer the rectangle figure comprehended vnder the two right lines is equall to the rectangle figures whiche are comprehended vnder the line vndeuided and vnder euery one of the partes of the other line SVppose that there be two right lynes A and BC and let one of them namely BC be deuided at all aduentures in the pointes D and E. Then I say that the rectangle figure comprehended vnder the lines A and BC is equall vnto the rectangle figure comprehended vnder the lines A and BD vnto the rectangle figure which is coÌ„prehended vnder the lines A and DE and also vnto the rectangle figure which is comprehended vnder the lines A and EC For from the pointe Brayse vp by the 11. of the first vnto the right line BC a perpendiculer line BF vnto the line A by the third of the first put the line BG equall and by the point G by the 31. of the first draw a parallel line vnto the right line BC and let the same be GM and by the selfe same by the points D E and C draw vnto the line BG these parallel lines DK EL and CH. Now then the parallelograme BH is equall to these parallelogrammes BK DL and EH But the parallelograme BH is equall vnto that which is contayned vnder the lines A and BC. For it is compreheÌ„ded vnder the lines GB BC and the line GB is equall vnto the line A And the parallelograme BK is equall to that which is contayned vnder the lines A and BD for it is comprehended vnder the line GB and BD and BG is equall vnto A And the parallelograme DL is equall to that which is contayned vnder the lines A and DE for the line DK that is BG is equal vnto A And moreouer likewise the parallelograme EH is equall to that which is contained vnder the lines A EC VVherfore that which is compreheÌ„ded vnder the lines A BC is equall to that which is comprehended vnder the lines A BD vnto that which is compreheÌ„ded vnder the lines A and DE and moreouer vnto that which is comprehended vnder the lines A and EC If therfore there be two right lines and if the one of them be deuided into partes how many soeuer the rectangle figure comprehended vnder the two right lines is equall to the rectangle figures which are comprehended vnder the line vndeuided and vnder euery one of the partes of the other line which was required to be demonstrated Because that all the Propositions of this second booke for the most part are true both in lines and in numbers and may be declared by both therefore haue I haue added to euery Proposition conuenient numbers for the manifestation of the same And to the end the studious and diligent reader may the more fully perceaue and vnderstand the agrement of this art of Geometry with the science of Arithmetique and how nere deare sisters they are together so that the one cannot without great blemish be without the other I haue here also ioyned a little booke of Arithmetique written by one Barlaam a Greeke authour a man of greate knowledge In whiche booke are by the authour demonstrated many of the selfe same proprieties and passions in number which Euclide in this his second boke hath demonstrated in magnitude

M. If therfore G exceede L then also H excedeth M and if it be equall it is equall and if it be lesse it is lesse by the conuerse of the 6â— definition of the fifth Agayne because that as C is to D so is E to F and to C and E are taken â—â—â—emâ—ltiplices H â—â—d K and likewise to D F are takeÌ„ certaine other equemultiplices M N. If therfore H exceede M then also K excedeth N and if it be equall it is equall and if it be lesse it is lesse by the same conuerse But if K exceede M then also G excedeth L and if it be equal it is equall and if it be lesse it is lesse by the same conuerse Wherfore if G excede L then K also excedeth N and if it be equal it is equall and if it be lesse it is lesse But G K are equemultiplices of A E. And L N are certaine other equemultiplices of B F. Wherfore by the 6. definition as A is to B so is E to F. Proportions therfore which are one and the selfe same to any one proportion are also the selfe same one to the other which was required to be proued The 12. Theoreme The 12. Proposition If there be a number of magnitudes how many soeâ—â—r proportionall as one of the antecedentes is to one of the coÌ„sequentes so are all the antecedentes to all the consequentes SVppose that there be a number of magnitudes how many soeuer namely A B C D E F in proportion so that as A is to B so let C be to D and E to F. Then I say that as A is to B so ã€ˆâ—Šã€‰ A C E to B D F. Take equemultiplices to A C and E. And let the same be G H K. And likewise to B D and F â—ake any other equemultiplices which to be L M N. And because that ã€ˆâ—Šã€‰ A is to B so iâ— C to D and E to F. And to A C E are taken â—quemultiplices G H K and likewise to â— D F are taken certaine other equemâ—â—tipliâ—â—s L M N. If therefore G exceede L H also exceedeth M and KN and if it be equall it is equall and if it be lesse it is lesse â—y the conuerse of the sixâ— definition of the fift Wherfore if G exceede L then G H K also exceede L M N and if they be equall they are equall and if they be lesse they are lesse by the same But G and G H K are equemultiplices to the magnitude A and to the magnitudes A C E. For by the first of the fift if there be a number of magnitudes equemultiplices to a like number of magnitudes ech to ech how multiplex one magnitude is to one so multiplices are all the magnitudes to all And by the same reason also L and L M N are equemultiplices to the magnitude B and to the magnitudes B D F Wherefore as A is to B so is A C E to B D F by the sixt definition of the fift If therefore there be a number of magnitudes how many soeuer proportionall as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes which was required to be proued The 13. Theoreme The 13. Proposition If the first haue vnto the second the self same proportion that the third hath to the fourth and if the third haue vnto the fourth a greater proportioÌ„ theÌ„ the fifth hath to the sixth theÌ„ shall the first also haue vnto the second a greater proportion then hath the fifth to the sixth SVppose that there be sixe magnitudes of which let A be the first B the second C the third D the fourth E the fifth and F the sixth Suppose that A the first haue vnto B the second the self same proportion that C the third hath to D the fourth And let C the third haue vnto D the fourth a greater proportion then hath E the fifth to F the sixth Then I say that A the first hath to B the second a greater proportion then hath E the fifth to F the sixt For forasmuch as C hath to D a greater proportion then hath E to F therfore there are certaine equemultiplices to C and E and likewise any other equemultiplices whatsoeuer to D and F which being compared together the multiplex to C shall exceede the multiplex to D but the multiplex to E shall not exceede the multiplex to F by the conuerse of the eight **definition** of this **booke** Let those multiplices be taken and suppose that the equemultiplices to C and E be G and H and likewise to D and F take any other equemultiplices whatsoeuer and let the same be K and L so that let G exceede K but let not H exceede L. And how multiplex G is to C so multiplex let M be to A. And how multiplex K is to D so multiplex also let N be to B. And because that as A is to B so is C to D and to A and C are taken equemultiplices M and G. And likewise to B and D are taken certayne other equemultiplices N K if therfore M exceede N G also excedeth K and if it be equall it is equall and if it be lesse it is lesse by the conuersion of the sixt definition of the fifth But by construction G excedetâ— K wherfore M also excedeth N but H excedeth not L. But M H are equemultiplices to A E and N L are certaine other equemultiplicesâ— whatsoeuer to B and F. Wherfore A hath vnto B a greater proportion then E hath to F by the 8. definition If therefore the first haue vnto the second the selfe same proportion that the third hath to the fourth and if the third haue vnto the fourth a greater proportion then the fifth hath to the sixth then shall the firsâ— also haue vnto the second a greater proportion then hath the ã€ˆâ—Šã€‰ to the sixthâ— Which was required to be proued Â¶ An addition of Campane If there be foure quantities and if the first haue vnto the second a greater proportion theÌ„ hath the third to the fourth then shall there be some equemultiplices of the first and the third which beyng compared to some equemultiplices of the second and the fourth the multiplex of the first shall be greater then the multiplex of the second but the multiplex of the third shall not be greater then the multiplex of the fourth Although this **proposition** here put by Campane nedeth no demonstration for that it is but the conuerse of the 8. **definition** of this **booke** yet thought I it not worthy to be omitted for that it reacheth the way to finde out such equemultiplices that the multiplex of the first shall excede the multiplex of the second but the multiplex of the third shall not exceede the multiplex of the fourth The 14. Theoreme The 14.

and the same proportion wherfore by the 9. of the fifth the figure NH is equal vnto the figure SR And it is vnto it like and in like sort situate But in like and equall rectiline figures beyng in like sort situate the sides of like proportion on which they are described are equall Wherfore the line GH is equall vnto the line QR And because as the lyne AB is to the line CD so is the line EF to the line QR but the line QR is equall vnto the line GH therfore as the line AB is to he line CD so is the line EF to the line GH If therefore there be foure right lines proportionall the rectiline figures also described vpon them beyng like and in lyke sort situate shall be proportionall And if the rectiline figures vpon them described beyng like and in like sort situate be proportionall those right lines also shall be proportional which was required to be proued An Assumpt And now that in like and equall figures being in like sort situate the sides of like proportion are also equall which thing was before in this proposition taken as graunted may thus be proued Suppose that the rectiline figures NH and SR be equall and like and as HG is to GN so let RQ be to QS and let GH and QR be sides of like proportion Then I say that the side RQ is equall vnto the side GH For if they be vnequall the one of them is greater then the other let the side RQ be greater then the side HG And for that as the line RQ is to the line QS so is the line HG to the line GN and alternately also by the 16. of the fifth as the line RQ is to the line HG so is the line QS to the lyne GN but the line RQ is greater then the line HG Wherfore also the line QS is greater then the line GN Wherefore also the figure RS is greater then the figure HN but by supposition it is equall vnto it which is impossible Wherfore the line QR is not greater then the line GH In like sorte also may we proue that it is not lesse then it wherfore it is equall vnto it which was required to be proued Flussates demonstrateth this second part more briefly by the first corollary of the â—0 of this boke thus Forasmuch as the rectiline figures are by supposition in one and the same proportion and the same proportion is double to the proportion of the sides AB to CD and EF to GH by the foresaid corollary the proportion also of the sides shall be one and the selfe same by the 7. common sentence namely the line AB shall be vnto the line CD as the line EF is to the line GH The 17. Theoreme The 23. **Proposition** Equiangle Parallelogrammes haue the one to the other that proportion which is composed of the sides Flussates demonstrateth this Theoreme without taking of these three lines K L M after this maner Forasmuch as sayth he it hath bene declared vpon the 10. **definition** of the fift **booke** and â—ift **definition** of this **booke** that the proportions of the extremes consist of the proportions of the meanes let vs suppose two equiangle parallelograÌ„mes ABGD and GEZI and let the angles at the poynt G in eyther be equall And let the lines BG and GI be set directly that they both make one right line namely BGI. Wherefore EGD also shall be one right line by the conuerse of the 15. of the first Make complete the parallelogramme GT Then I say that the proportion of the parallelogrammes AG GZ is composed of the proportions of the sides BG to GI and DG to GE. For forasmuch as that there are three magnitudes AG GT and GZ and GT is the meane of the sayd magnitudes and the proportion of the extremes AG to GZ consisteth of the meane proportions by the 5. **definition** of this **booke** namely of the proportion of AG to GT and of the proportion GT to GZ But the proportion of AG to GT is one and the selfe same with the proportion of the sides BG to GI by the first of this booke And the proportion also of GT to GZ is one and the selfe same with the proportion of the other sides namely DG to GE by the same Proposition Wherefore the proportion of the parallelogrammes AG to GZ consisteth of the proportions of the sides BG to GI and DC to GE. Wherefore equiangle parallelogrammes are the one to the other in that proportion which is composed of theyr sides which was required to be proued The 18. Theoreme The 24. Proposition In euery parallelogramme the parallelogrammes about the dimecient are lyke vnto the whole and also lyke the one to the other SVppose that there be a parallelogramme ABCD and let the dimecient therof be AC and let the parallelogrammes about the dimecient AC be EG and HK Then I say that either of these parallelogrames EG and HK is like vnto the whole parallelogramme ABCD and also are lyke the one to the other For forasmuch as to one of the sides of the triangle ABC namely to BC is drawen a parallel lyne EF therfore as BE is to EA so by the 2. of the sixt is CF to FA. Agayne forasmuch as to one of the sides of the triangle ADC namely to CD is drawen a parallel lyne Fâ— therefore by the same as CF is to FA so is DG to GA. But as CF is to FA so is it proâ—ued that BE is to EA Wheâ—fore as BE is to EA so by the 11. of the fifth â— is DG to GA. Wherfore by composition by the 18. of the fifth as BA is to AEâ— so is DA to AG. And alternately by the 16. of the fifth as BA is to AD so is EA to AG. Wherfore in the parallelogrammesâ— ABCD and EG the sides which are about the common angle BAD are proportionall And because the line GF is a parallel vnto the lyne DCâ— therfore the angle AGF by the 29â— of theâ— first is equall vnto the angle ADC â— the angle GFA equall vnto the angle DCA and the angle DAC is common to the two triangles ADC and AFG Wherfore the triangle DAC is equiangle vnto the triangle AGF And by the same reason the triangle ABC is equiangle vnto the triangle AEF Wherfore the whole parallelogramme ABCD is equiangle vnto the parallelograÌ„me EG Wherfore as AD is in proportion to DC so by the 4. of the sixth is AG to GF and as DC is to CA so is GF to FA. And as AC is to CB so is AF to FE And moreouer as CB is to BA so is FE to EA And forasmuch as it is proued that as Dâ— is to CA so is GF to FA but as AC is to Câ— so is AF to FE Wherfore of equalitie by the 22. of the fifth as DC is to CB so is GF to FE Wherefore in the parallelogrammes ABCD and

which was required to be proued Â¶ The 34. Theoreme The 34. Proposition If a number be neither doubled from two nor hath to his half part an odde number it shall be a number both euenly euen and euenly odde SVppose that the nuÌ„ber A be a nuÌ„ber neither doubled froÌ„ the nuÌ„ber two neither also let it haue to his halfe part an odde nuÌ„ber Then I say that A is a nuÌ„ber both euenly euen and euenly odde That A is euenly euen it is manifest for the halfe therof is not odde and is measured by the number 2. which is an euen number Now I say that it is euenly odde also For if we deuide A into two equall partes and so continuing still we shall at the length light vpon a