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 ¶ 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 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 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 〈◊〉 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 〈◊〉 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 〈◊〉 to the sixthâ— Which was required to be proued ¶ 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 ¶ 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 certaine odde number which shall measure A by an euen number For if we should not light vpon such an odde nuÌ„ber which measureth A by an euen number we should at the length come vnto the number two and so should A be one of those numbers which are doubled from two vpward which is contrary to the supposition Wherfore A is euenly odde And it is proued that it is euenly eueÌ„ wherfore A is a number both euenly euen and euenly odde whiche was required to be demonstrated This proposition and the two former manifestly declare that which we noted vppon the tenth definition of the seuenth booke namely that Campane and Flussates and diuers other interpreters of Euclide onely Theon except did not rightly vnderstand the 8. and 9. definitions of the same booke concerning a number euenly euen and a number euenly odde For in the one definition they adde vnto Euclides wordes extant in the Greeke this word onely as we there noted and in the other this word all So that after their definitions a number can not be euenly euen vnlesse it be measured onely by euen numbers likewise a number can not be euenly odde vnlesse all the euen numbers which doo measure it doo measure it by an odde number The contrary whereof in this proposition we manifestly see For here Euclide proueth that one number may be both euenly euen and euenly odde And in the two former propositions he proued that some numbers are euenly euen onely and some euenly odde onely which word onely had bene in vaine of him added if no number euenly euen could be measured by an odde number or if all the numbers that measure a number euenly odde must needes measure it by an odde number Although Campane and Flussates to auoyde this absurdity haue wreasted the 32. proposition of this booke froÌ„ the true sence of the Greeke and as it is interpreted of Theon So also hath Flussates wreasted the 33. proposition For wheras Euclide sayth Euery nuÌ„ber produced by the doubling of two vpward is eueÌ„ly eueÌ„ only they say onely the numbers produced by the doubling of two are euenly euen Likewise whereas Euclide saith A number whose hafle part is odde is euenly odde onely Flussates sayth onely a number whose halfe part is od Is eueÌ„ly od Which their interpretatioÌ„ is not true neither can be applyed to the propositions as they are extaÌ„t in the Greeke In dede the sayd 32. and 33. propositions as they put theÌ„ are true touching those numbers which are euenly euen onely or eueÌ„ly od onely For no number is euenly euen onely but those onely which are doubled from two vpward Likewise no numbers are euenly odde onely but those onely whose halfe is an odde number But this letteth not but that a number may be euenly euen although it be not doubled from two vpward also that a number may be eueÌ„ly odde although it haue not to his halfe an odde number As in this 34. propositioÌ„ Euclide hath plainly proued Which thing could by no meanes be true if the foresayd 32. 33. propositons of this booke should haue that sence and meaning wherein they take it ¶ The 35. Theoreme The 35. Proposition If there be numbers in continuall proportion how many soeuer and if from the second and last be taken away numbers equall vnto the first as the excesse of the second is to the first so is the excesse of the last to all the nuÌ„bers going before the last SVppose that these numbers A BC D and EF be in continuall proportion beginning at A the least And from BC which is the second take away CG equall vnto the first namely to A and likewise from EF the last take away FH equall also vnto the first namely to A. Then I say that as the excesse BG is to A the first so is HE the excesse to all the numbers D BC and A which go before the last number namely EF. Forasmuch as EF is the greater for the second is supposed greater then the first put the number FL equall to the number D and likewise the number FK equall to the number BC. And forasmuch as FK is equall vnto CB of which FH is equall vnto GC therefore the residue HK is equall vnto the residue GB And for that as the whole Fâ— is to the whole FL so is the part taken away FL to the part taken away FK therefore the residue LE is to the residue KL as the whole â—E is to the whole FL by the 11. of the seuenth So likewise for that FL is to FK as FK is to FH KL shall be to HK as the whole FL is to the whole FK by the same Proposition But as FE is to FL and as FL is to FK and FK to FH so were FE to D and D to BC and BC 〈◊〉 A. Wherefore as LE is to KL and as KL is to HK so is D to BC. Wherefore alternately by the 23. of the seuenth as LE is to D so is KL to be BC and as KL is to BC so is HK to A. Wherefore also as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes Wherefore as KH is to A so are HK KL and LE to D BC and A by the 12. of the seuenth But it is proued that KH is equall vnto BG Wherefore as BG which is the excesse of the second is to A so is EH the excesse of the last vnto the numbers going before D BC and A. Wherefore as the excesse of the second is vnto the first so is the excesse of the last to all the numbers going before the last which was required to be proued ¶ The 36. Theoreme The 36. Proposition If from vnitie be taken numbers how many soeuer in double proportion continually vntill the whole added together be a prime number and if the whole multiplying the last produce any number that which is produced is a perfecte number SVppose that from vnitie be taken these numbers A B C D in double proportion continually so that all those numbers A B C D vnitie added together make a prime number and let E be the number composed of
the line AB is rationall by the definition Wherfore by the definition also of rationall figures the parallelogramme CD shall be rationall Now resteth an other caâ—e of the thirde kinde of rationall lines commensurable in length the one to the other which are to the rationall line AB first set commensurable in power onely and yet are therfore rationall lines And let the lines CE and ED be coÌ„mensurable in length the one to the other Now then let the selfe same construction remaine that was in the former so that let the lines CE and ED be rationall commensurable in power onely vnto the line AB But let them be commensurable in length the one to the other Then I say that in this case also the parallelogramme CD is rationall First it may be proued as before that the parallelogramme CD is commensurable to the square DF. Wherfore by the 12. of this booke the parallelogramme CD shall be commensurable to the square of the line ABâ— But the square of the line AB is rationall Wherefore by the definition the parallelograÌ„me CD shall be also rationall This case is well to be noted For it serueth to the demonstration and vnderstanding of the 25. Proposition of this booke ¶ The 17. Theoreme The 20. Proposition If vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth thereof shall be a rationall line and commensurable in length vnto that line wherupon the rationall parallelogramme is applied SVppose that this rationall rectangle parallelogramme AC be applied vpon the line AB which let be rationall according to any one of the foresaid wayes whether it be the first rationall line set or any other line commensurable to the rationall line first set and that in length and in power or in power onely for one of these three wayes as was declared in the Assumpt put before the 19. Proposition of this booke is a line called rationall and making in breadth the line BC. Then I say that the line BC is rationall and commensurable in length vnto the line BA Desrcribe by the 46. of the first vpon the line BA a square AD. Wherfore by the 9. definitioÌ„ of the tenth the square AD is rationall But the parallelogramme AC also is rationall by supposition Wherefore by the conuersion of the definition of rationall figures or by the 12. of this booke the square DA is commensurable vnto the parallelogramme AC But as the square DA is to the parallelogramme AC so is the line DB to the line BC by the first of the sixt Wherfore by the 10. of the tenth the line DB is commensurable vnto the line BC. But the line DB is equall vnto the line BA Wherefore the line AB is coÌ„mensurable vnto the line BC. But the line AB is rationall Wherefore the line BC also is rationall and commensurable in length vnto the line BA If therefore vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth therof shall be a rationall line commensurable in length vnto that line whereupon the rationall parallelogramme is applied which was required to be demonstrated ¶ An Assumpt A line contayning in power an irrationall superficies is irrationall Suppose that the line AB coÌ„taine in power an irrationall superficies that is let the square described vpon the line AB be equall vnto an irrationall superficies Then I say that the line AB is irrationall For if the line AB be rationall theÌ„ shall the square of the line AB be also rationall For so was it put in the definitions But by supposition it is not Wherefore the line AB is irrationall A line therefore contayning in power an irrationall superficies is irrationall ¶ The 18. Theoreme The 21. Proposition A rectangle figure comprehended vnder two rationall right lines commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall is called a mediall line SVppose that this rectangle figure AC be comprehended vnder these rationall right lines AB and BC commensurable in power onely Then I say that the superficies AC is irrationall and the line which contayneth it in power is irrationall and is called a mediall line Describe by the 46. of the first vpon the line AB a square AD. Wherefore the square AD is rationall And forasmuch as the line AB is vnto the line BC incommensurable in length for they are supposed to be commensurable in power onely and the line AB is equall vnto the line BD therefore also the lineâ— BD is vnto the line BC incommensurable in length And 〈◊〉 â—hâ— lin◠〈…〉 is to the line â— C so 〈◊〉 the square AD to the parallelogramme AC by the first of the fiuâ— Wherefore by the 10. of the tenth the square DA is vnto the parallelogramme AC incommensurable But the square DA is rationall Wherefore the parallelogramme AC is irrationall Wherefore also the line that contayneth the superficies AC in power that is whose square is equall vnto the parallelogramme AC is by the Assumpt going before irrationall And it is called a mediall line for that the square which is made of it is equall to that which is contayned vnder the lines AB and BC and therefore it is by the second part of the 17. of the sixt a meane proportionall line betwene the lines AB and BC. A rectangle figure therefore comprehended vnder rationall right lines which are commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall and is called a mediall line At this Proposition doth Euclide first entreate of the generation and production of irrationall lines And here he searcheth out the first kinde of them which he calleth a mediall line And the definition therof is fully gathered and taken out of this 21. Proposition which is this A mediall line is an irrationall line whose square is equall to a rectangled figure contayned of two rationall lines commensurable in power onely It is called a mediall line as Theon rightly sayth for two causes first for that the power or square which it producethâ— is equall to a mediall superficies or parallelogramme For as that line which produceth a rationall square is called a rationall line and that line which produceth an irrationall square or a square equall to an irrationall figure generally is called an irrationall line so iâ— thaâ— line which produceth a mediall square or a square equall to a mediall superficies called by speciall name a mediall line Secondly it is called a mediall line because it is a meane proportionall betwene the two lines coÌ„mensurable in power onely which comprehend the mediall superficies ¶ A Corollary added by Flussates A rectangle parallelogramme contayned vnder a rationall line and an â—rrationall line is irrationall For if the line AB be rationall and
the definition of a first binomiall line seâ— before the 48. proposition of this booke the line DG is a first binomiall line which was required to be proued This proposition and the fiue following are the conuerses of the sixe former propositions ¶ The 43. Theoreme The 61. Proposition The square of a first bimediall line applied to a rationall line maketh the breadth or other side a second binomiall line SVppose that the line AB be a first bimediall line and let it be supposed to be deuided into his partes in the point C of which let AC be the greater part Take also a rationall line DE and by the 44. of the first apply to the line DE the parallelograÌ„me DF equall to the square of the line AB making in breadth the line DG Then I say that the line DG is a second binomiall line Let the same constructions be in this that were in the Proposition going before And forasmuch as the line AB is a first bimediall line and is deuided into his partes in the point C therefore by the 37. of the tenth the lines AC and CB are mediall commensurable in power onely coÌ„prehending a rationall superficies Wherfore also the squares of the lines AC and CB are mediall Wherefore the parallelogramme DL is mediall by the Corollary of the 23. of the tenth and it is applied vppon the rationall line DE. Wherefore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. Againe forasmuch as that which is coÌ„tayned vnder the lines AC and CB twise is rationall therefore also the parallelogramme MF is rationall and it is applied vnto the rationall line ML Wherefore the line MG is rationall and commensurable in length to the line ML that is to the line DE by the 20. of the tenth Wherefore the line DM is incommensurable in length to the line MG and they are both rationall Wherefore the lines DM and MG are rationall commensurable in power onely Wherefore the whole line DG is a binomiall line Now resteth to proue that it is a second binomiall line Forasmuch as the squares of the lines AC and CB are greater then that which is contayned vnder the lines AC and CB twise by the Assumpt before the 60. of this booke therefore the parallelogramme DL is greater then the parallelogrrmme MF Wherefore also by the first of the sixt the line DM is greater then the line MG And forasmuch as the square of the line AC is commensurable to the square of the line CB therefore the parallelogramme DH is commensurable to the parallelogramme KL Wherefore also the line DK is commensurable in length to the line KM And that which is contayned vnder the lines DK and KM is equall to the square of the line MN that is to the fourth part of the square of the line MG Wherefore by the 17. of the tenth the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and the line MG is commensurable in length to the rationall line put namely to DE. Wherefore the line DG is a second binomiall line which was required to be proued ¶ The 44. Theoreme The 62. Proposition The square of a second bimediall line applied vnto a rationall line maketh the breadth or other side therof a third binomiall lyne SVppose that AB be a second bimediall line and let AB be supposed to be deuided into his partes in the point C so that let AC be the greater part And take a rationall line DE. And by the 44. of the first vnto the line DE apply the parallelogramme DF equall to the square of the line AB and making in breadth the line DG Then I say that the line DG is a third binomiall line Let the selfe same constructions be in this that were in the propositions next going before And forasmuch as the line AB is a second bimediall line and is deuided into his partes in the point C therfore by the 38. of the tenth the lines AC and CB are medials commensurable in power only compreheÌ„ding a mediall superficies Wherfore that which is made of the squares of the lines AC and CB added together is mediall and it is equall to the parallelogramme DL by construction Wherefore the parallelogramme DL is mediall and is applied vnto the rationall line DE wherfore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. And by the lyke reason also the line MG is rationall and incommensurable in length to the line ML that is to the line DE. Wherfore either of these lines DM and MG is rational and incommensurable in length to the line DE. And forasmuch as the line AC is incommensurable in length to the line CB but as the line AC is to the line CB so by the assumpt going before the 22. of the tenth is the square of the line AC to that which is contained vnder the lines AC and CB. Wherfore the square of the line AC is incâ—mmmensurable to that which is contayned vnder the lines AC and CB. Wherfore that that which is made of the squares of the lines AC and CB added together is incommensurable to that which is contained vnder the lines AC and CB twise that is the parallelogramme DL to the parallelogramme MF Wherfore by the first of the sixt and 10. of the tenth the line DM is incommensurable in length to the line MG And they are proued both rationall wherfore the whole line DG is a binomiall line by the definition in the 36. of the tenth Now resteth to proue that it is a third binomiall line As in the former propositions so also in this may we conclude that the line DM is greater then the line MG and that the line DK is commensurable in length to the line KM And that that which is contained vnder the lines DK and KM is equall to the square of the line MN Wherfore the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and neither of the lines DM nor MG is commensurable in length to the rational line DE. Wherfore by the definition of a third binomiâ—ll line the line DG is a third binomiall line which was required to be proued ¶ Here follow certaine annotations by M. Dee made vpon three places in the demonstration which were not very euident to yong beginners †The squares of the lines AC and Câ— are medials 〈◊〉 iâ— taught after the 21â— of this tenth and therâ—ore forasmuch as they are by supposition commeÌ„surable th' one to the other by the 15. of the teÌ„th the compound of them both is commensurable to ech part But the partes are medials therfore by the coâ—ollary of the 23. of the tenth the compound shall be
reformed by M. Dee described for it in the playne especially if ye remember the forme of the figure of the 29. proposition of this booke Only that which there ye conceaue to be the base imagine here in both the figures of this second case to be the vpper superficies opposite to the base and that which was there supposed to be the vpper superficies conceaue here to be the base Ye may describe them vpon pasted paper for your better sight taking hede ye note the letters rightly according as the construction requireth Flussas demonstrateth this proposition an otherway taking onely the bases of the solides and that after this maner Take equall bases which yet for the surer vnderstanding let be vtterly vnlike namely AEBF and ADCH and let one of the sides of eche concurre in one the same right line AED the bases being vpon one and the selfe same playne let there be supposed to be set vpon theÌ„ parallelipipedons vnder one the selfe same altitude Then I say that the solide set vpoÌ„ the base AB is equal to the solide set vpon the base AH By the poynt E draw vnto the line AC a parallel line EG which if it fall without the base AB produce the right line HC to the poynt I. Now forasmuch as AB and AH are parallelogrmaes therefore by the 24. of this booke the triangles ACI and EGL shall be equaliter the one to the other and by the 4. of the first they shal be equiangle and equall and by the first definition of the sixth and fourth Proposition of the same they shall be like Wherfore Prismes erected vppon those triangles and vnder the same altitude that the solides AB and AH aâ—e shall be equall and like by the 8. definition of this booke For they are contayned vnder like playne superficieces equall both in multitude and magnitude Adde the solide set vpon the base ACLE common to them both Wherefore the solide set vppon the base AEGC is equall to the solide set vpon the base AELI And forasmuch as the superficieces AEBF and ADHC are equall by supposition and the part taken away AG is equall to the part taken away AL therefore the residue BI shall be equall to the residue GD Wherefore as AG is to GD as AL is to BI namely equalls to equalls But as AG is to GD so iâ— the solide set vpon AG to the solide set vpon GD by the 25. of this booke for it is cut by a playne superficies set vpon the line GE which superficies is parallel to the opposite superficieces Wherefore as AL is to BI so is the solide set vpon AL to the solide set vpon BI Wherefore by the 11. of the fifth as the solide set vpon AG or vpon AL which is equall vnto it is to the solide set vpon GD so is the same solide set vpon AG or AL to the solide set vpon BI Wherefore by the 2. part of the 9. of the fifth the solides set vpon GD and BI shall be equall Vnto which solides if ye adde equall solides namely the solide set vpon AG to the solide set vpon GD and the solide set vpon AL to the solide set vpon BI the whole solides set vpon the base AH and vpon the base AB â—hall be equall Wherefore Parallelipedons consisting vpon equall bases and being vnder one and the selfe same altitude are equall the one to the other which was required to be proued ¶ The 27. Theoreme The 32. Proposition Parallelipipedons being vnder one and the selfe same altitude are in that proportion the one to the other that their bases are SVppose that these parallelipipedons AB and CD be vnder one the selfe same altitude Then I say that those parallelipipedons AB and CD are in that proportion the one to the other that their bases are that is that as the base AE is to the base CF so is the parallelipipedon AB to the parallelipipedon CD Vpon the line FG describe by the 45. of the first the parallelogramme FH equall to the parallelogramme AE and equiangle with the parallelogramme CF. And vpon the base FH describe a parallelipipedoÌ„ of the selfe same altitude that the parallelipipedoÌ„ CD is let the same be GK Now by the 31. of the eleuenth the parallelipipedon AB is equall to the parallelipipedon GK for they consist vpon equall bases namely AE and FH and are vnder one and the selfe same altitude And forasmuch as the parallelipipedon CK is cut by a plaine superficies DG being parallel to either of the opposite plaine superâ—icieces therfore by the 25. of the eleuenth as the base HF is to the base FC so is the parallelipipedon GK to parallelipipedon CD but the base HF is equal to the base AE and the parallelipipedon GK is proued equall to the parallelipipedon AB Wherfore as the base Aâ—E is to the base CF so is the parallelipedon AB to the parallelipipedon CD Wherfore parallelipipedons being vnder one and the selfe same altitude are in that proportion the one to the other that their bases are which was required to be demonstrated I neede not to put any other figure for the declaration of this demonstration for it is easie to see by the figure there described Howbeit ye may for the more full sight therof describe solides of pasted paper according to the construction there set forth which will not be hard for you to do if ye remember the descriptions of such bodies before taught A Corollary added by Flussas Equall parallelipipedons coÌ„tained vnder one and the selfe same altitude haue also their bases equal For if the bases should be vnequall the parallelipipedons also should be vnequal by this 32 propositioÌ„ And equall parallelipipedons hauing equall bases haue also one and the selfe same altitude For if they should haue a greater altitude they should exceede the equall parallelipipedons which haue the selfe same altitude But if they should haue a lesse they should want so much of those selfe same equal parallelipipedons The 28. Theoreme The 33. Proposition Like parallelipipedons are in treble proportion the one to the other of that in which their sides of like proportion are SVppose that these parallelipipedons AB and CD be like let the sides AE and CF be sides of like proportion Then I say the parallelipipedon AB is vnto the parallelipipedon CD in treble proportion of that in which the side AE is to the side CF. Extend the right lines AE GE and HE to the pointes K L M. And by the 2. of the first vnto the line CF put the line EK equal and vnto the line FN put the line EL equall and moreouer vnto the line FR put the line EM equall and make perfect the parallelogramme KL and the parallelipipedon KO Now forasmuch as these two lines EK and EL are equall to these two lines CF and FN but the angle KEL is equall to the angle CFN for the angle
a triangle and if the parallelogramme be double to the triangle those Prismes are by the 40. of the eleuenth equall the one to the other therefore the Prisme contained vnder the two triangles BKF and EHG and vnder the three parallelogrammes EBFG EBKH and KHFG is equall to the Prisme contained vnder the two triangles GFC and HKL and vnder the three parallelogrammes KFCL LCGH and HKFG And it is manifest that both these Prismes of which the base of one is the parallelogramme EBFG and the opposiâ—e vnto it the line KH and the base of the other is the triangle GFC and the opposite side vnto it the triangle KLH are greater then both these Pyramids whose bases are the triangles AGE and HKL and toppes the pointes H D. For if we drawe these right lines EF and EK the Prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is greater then the Pyramis whose base is the triangle EBF toppe the point K. But the Pyramis whose base is the triangle EBF and toppe the point K is equall to the Pyramis whose base is the triangle AEG and toppe the point H for they are contained vnder equall and like plaine superficieces Wherefore also the Prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is greater then the Pyramis whose base is the triangle AEG and toppe the point H. But the prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is equall to the prisme whose base is the triangle GFC and the opposite side vnto it the triangle HKL And the Pyramis whose base is the triangle AEG and toppe the point H is equall to the Pyramis whose base is the triangle HKL and toppe the point D. Wherefore the foresaid two prismes are greater then the foresaid two Pyramids whose bases are the triangles AEG HKL and toppes the pointes H and D. Wherefore the whole Pyramis whose base is the triangle ABC and toppe the point D is deuided into two Pyramids equall and like the one to the other and like also vnto the whole Pyramis hauing also triangles to their bases and into two equall prismes and the two prismes are greater then halfe of the whole Pyramis which was required to be demonstrated If ye will with diligence reade these fower bookes following of Euclide which concerne bodyes and clearely see the demonstrations in them conteyned it shall be requisite for you when you come to any proposition which concerneth a body or bodies whether they be regular or not first to describe of pâ—sâ—ed paper according as I taught you in the end of the definitions of the eleuenth booke such a body or bodyes as are there required and hauing your body or bodyes thus described when you haue noted it with letters according to the figure set forth vpoÌ„ a plaine in the propositioÌ„ follow the construction required in the proposition As for example in this third propositioÌ„ it is sayd that Euery pyramis hauing a triangle to â—is base may be deuided into two pyramids c. Here first describe a pyramis of pasted paper haâ—ing his base triangled it skilleth not whether it be equilater or equiangled or not only in this proposition is required that the base be a triangle Then for that the proposition supposeth the base of the pyramis to be the triangle ABC note the base of your pyramis which you haue described with the letters ABC and the toppe of your pyramis with the letter D For so is required in the proposition And thus haue you your body ordered ready to the construction Now in the construction it is required that ye deuide the lines AB BC CA. c namely the sixe lines which are the sids of the fower triangles contayning the piramis into two equall partes in the poyntet â— F G c. That is ye must deuide the line AB of your pyramis into two equall partes and note the poynt of the deuision with the letter E and so the line BC being deuided into two equall partes note the poynt of the deuision with the letter F. And so the rest and this order follow ye as touching the rest of the construction there put and when ye haue finished the construction compare your body thus described with the demonstration and it will make it very playne and easy to be vnderstaÌ„ded Whereas without such a body described of matter it is hard for young beginners vnlesse they haue a very deepe imagination fully to conceaue the demonstration by the sigâ—e as it is described in a plaine Here for the better declaration of that which I haue sayd haue I set a figure whose forme if ye describe vpon pasted paper noted with the like letters and cut the lines â—A DA DC and folde it accordingly it will make a Pyramis described according to the construction required in the proposition And this order follow ye as touching all other propositions which concerne bodyes ¶ An other demonstration after Campane of the 3. proposition Suppose that there be a Pyramis ABCD hauing to his base the triangle BCD and let his toppe be the solide angle A from which let there be drawne three subtended lines AB AC and AD to the three angles of the base and deuide all the sides of the base into two equall partes in the three poyntes E F G deuide also the three subteÌ„ded lines AB AC and AD in two equall partes in the three points H K L. And draw in the base these two lines EF and EG So shall the base of the pyramis be deuided into three superficieces whereof two are the two triangles BEF and EGD which are like both the one to the other and also to the whole base by the 2 part of the secoÌ„d of the sixth by the definitioÌ„ of like superâ—iciecâ—s they are equal the one to the other by the 8. of the first the third superficies is a quadrangled parallelogramme namely EFGC which is double to the triangle EGD by the 40. and 41. of the first Now then agayne from the poynt H draw vnto the points E and F these two subtendent lines HE and HF draw also a subtended line from the poynt K to the poynt G. And draw these lines HK KL and LH Wherefore the whole pyramis ABCD is deuided into two pyramids which are HBEF and AHKL and into two prismes of which the one is EHFGKC and is set vpon the quadrangled base CFGE the other is EGDHKL and hath to his base the triangle EGD Now as touching the two pyramids HBEF and AHKL that they are equall the one to the other and also that they are like both the one to the other and also to the whole it is manifest by the definition of equall and like bodyes and by the 10. of the eleuenth and by 2. part of the second of the sixth And that the two Prismes are equall it
it comprehendeth Wherfore the pyramis whose base is the square ABCD and altitude the self same that the cone hath is greater then the halfe of the cone Deuide by the 30. of the third euery one of the circumferences AB BC CD and DA into two equall partes in the pointes E F G and H and drawe these right lines AE EB BF FC CG GD DH and HA. Wherefore euery one of these triangles AEB BFC CGD and DHA is greater then the halfe part of the segment of the circle described about it Vppon euery one of these triangles AEB BFC CGD and DHA describe a pyramis of equall altitude with the cone and after the same maner euery one of those pyramids so described is greater then the halfe part of the segment of the cone set vpon the segment of the circle Now therefore diuiding by the 30 of the third the circumferences remaining into two equall parts drawing right lines raysing vp vpon euery one of those triangles a pyramis of equall altitude with the cone and doing this continually we shal at the length by the first of the tenth leaue certayne segmentes of the cone which shal be lesse then the excesse whereby the cone excedeth the third part of the cylinder Let those segmentes be AE EB BF FC CG GD DH and HA. Wherefore the pyramis remayning whose base is the poligonoÌ„ figure AEBFCGDH and altitude the self same with the cone is greater then the third part of the cylinder But the pyramis whose base is the poligonon figure AEBFCGDH and altitude the self same with the cone is the third part of the prisme whose base is the poligonoÌ„ figure AEBFCGDH and altitude the self same with the cylinder Whefore the prisme whose base is the poligonon figure AEBFCGDH and altitude the self same with the cylinder is greater then the cylinder whose base is the circle ABCD. But it is also lesse for it is contayned of it which is impossible Wherfore the cylinder is not in lesse proportion to the cone then in treble proportion And it is proued that it is not in greater proportion to the cone then in treble proportion wherefore the cone is the third part of the cylinder Wherfore euery cone is the third part of a cylinder hauing one the self same base and one and the selfe same altitude with it which was required to be demonstrated ¶ Added by M. Iohn Dee ¶ A Theoreme 1. The superficies of euery vpright Cylinder except his bases is equall to that circle whose semidiameter is middell proportionall betwene the side of the Cylinder and the diameter of his base ¶ A Theoreme 2. The superficies of euery vpright or Isosceles Cone except the base is equall to that circle whose semidiameter is middell proportionall betwene the side of that Cone and the semidiameter of the circle which is the base of the Cone My entent in additions is not to amend Euclideâ— Method which nedeth little adding or none at all But my desire is somwhat to furnish you toward a more general art Mathematical theÌ„ Euclides ElemeÌ„ts remayning in the termes in which they are written can sufficiently helpe you vnto And though Euclides Elementes with my Additions run not in one Methodicall race toward my marke yet in the meane space my Additions either geue light where they are annexed to Euclides matter or geue some ready ayde and shew the way to dilate your discourses Mathematicall or to inuent and practise thinges Mechanically And in deede if more leysor had happened many more straunge matters Mathematicall had according to my purpose generall bene presently published to your knowledge but want of due leasour cauâ—eth you to want that which my good will toward you most hartely doth wish you As concerning the two Theoremes here annexed their veritie is by Archimedes in his booke of the Sphere and Cylinder manifestly demonstrated and at large you may therefore boldly trust to them and vse them as suppositions in any your purposes till you haue also their demoÌ„strations But if you well remember my instructions vpon the first proposition of this booke and my other addition vpon the second with the suppositions how a Cylinder and a Cone are Mathematically produced you will not neede Archimedes demonstration nor yet be vtterly ignoraunt of the solide quantities of this Cylinder and Cone here compared the diameter of their base and heith being knowne in any measure neither can their croked superficies remayne vnmeasured Whereof vndoubtedly great pleasure and commoditie may grow to the sincere student and precise practiser ¶ The 11. Theoreme The 11. Proposition Cones and Cylinders being vnder one and the selfe same altitude are in that proportion the one other that their bases are In like sorte also may we proueâ— that as the circle EFGH is to the circle ABCD so is not the cone EN to any solide lesse then the cone AL. Now I say that as the circle ABCD is to the circle EFGH so is not the cone AL to any solide greater then the cone EN For if it be possible let it be vnto a greater namely to the solide X. Wherefore by conuersion as the circle EFGH is to the circle ABCD so is the solide X to the cone AL but as the solide X is to the cone AL so is the cone EN to some solide lesse then the cone AL as we may see by the assumpt put after thâ— second of this booke Wherefore by the 11. of the fift as the circle EFGH is to the circle ABCGâ— so is the cone EN to some solide lesse then the cone AL which we haue proued to be impossible Wherefore as the circle ABCD is to the circle EFGH so is not the cone AL to any solide greater then the cone EN And it is also proued that it is not to any lesse Wherefore as the circle ABCD is to the circle EFGH so is the cone AL to the cone EN But as the cone is to the cone so is the cylinder to the cylinder by the 15. of the fift for the one is in treble proportion to the other Wâ—erefore by the 11. of the fift as the circle ABCD is to the circle EFGH so are the cylinders which are set vpon them the one to the other the said cylinders being vnder equall altitudes with the cones Cones therefore and cylinders being vnder one the self same altitude are in that proportion the one to the other that their bases are which was required to be demonstrated ¶ The 12. Theoreme The 12. Proposition Like Cones and Cylinders are in treble proportion of that in which the diameters of their bases are Now also I say that the cone ABCDL is not to any solide greater then the cone EFGHN in treble proportion of that in which the diameter BD is to the diameter FH For if it be possible let it be to a greater namely to the solide X. Wherefore by conuersion by the
superficies or soliditie in the hole or in partâ— such certaine knowledge demonstratiue may arise and such mechanical exercise thereby be deuised that sure I am to the sincere true student great light ayde and comfortable courage farther to wade will enter into his hart and to the Mechanicall witty and industrous deuiser new maner of inuentions executions in his workes will with small trauayle for fete application come to his perceiueraunce and vnderstanding Therefore euen a manifolde speculations practises may be had with the circle his quantitie being not knowne in any kinde of smallest certayne measure So likewise of the sphere many Problemes may be executed and his precise quantitie in certaine measure not determined or knowne yet because both one of the first humane occasioÌ„s of inuenting and stablishing this Arte was measuring of the earth and therfore called Geometria that is Earthmeasuring and also the chiefe and generall end in deede is measure and measure requireth a determination of quantitie in a certayne measure by nuÌ„ber expressed It was nedefull for Mechanicall earthmeasures not to be ignorant of the measure and contents of the circle neither of the sphere his measure and quantitie as neere as sense can imagine or wish And in very deede the quantitie and measure of the circle being knowne maketh not onely the cone and cylinder but also the sphere his quantitie to be as precisely knowne and certayne Therefore seing in respect of the circles quantitie by Archimedes specified this Theoreme is noted vnto you I wil by order vpon that as a supposition inferre the conclusion of this our Theoremes Note 1. Wherfore if you deuide the one side as TQ of the cube TX into 21. equall partes and where 11. partes do end reckening from T suppose the point P and by that point P imagine a plaine passing parallel to the opposite bases to cut the cube TX and therby the cube TX to be deuided into two rectangle parallelipipedons namely TN and PX It is manifest TN to be equall to the Sphere A by construction and the 7. of the fift Note 2. Secondly the whole quantitie of the Sphere A being coÌ„tayned in the rectangle parallelipipedon TN you may easilie transforme the same quantitie into other parallelipipedons rectangles of what height and of what parallelogramme base you list by my first and second Problemes vpon the 34. of this booke And the like may you do to any assigned part of the Sphere A by the like meanes deuiding the parallelipipedon TN as the part assigned doth require As if a third fourth fifth or sixth part of the Sphere A were to be had in a parallelipipedon of any parallelograâ—â—e base assigned or of any heith assigned then deuiding TP into so many partes as into 4. if a fourth part be to be transformed or into fiue if a fifth part be to be transformed c. and then proceede â—s you did with cutting of TN from TX And that I say of parallelipipedons may in like sort by my â—â—yd two problemes added to the 34. of this booke be done in any sided columnes pyramids and prismeâ— so thâ—â— in pyramids and some prismes you vse the cautions necessary in respect of their quan 〈…〉 odyes hauing parallel equall and opposite bases whose partes 〈…〉 re in their propositions is by Euclide demonstrated And finally 〈…〉 additions you haue the wayes and orders how to geue to a Sphere or any segmeâ—â— oâ— the same Cones or Cylinders equall or in any proportion betwene two right lines geuen with many other most necessary speculations and practises about the Sphere I trust that I haue sufficiently â—raughted your imagination for your honest and profitable studie herein and also geuen you reaâ—â— â—â—tter wheâ—â— with to sâ—â—p the mouthes of the malycious ignorant and arrogant despisers of the most excellent discourses trauayles and inuentions mathematicall Sting aswel the heauenly spheres sterres their sphericall soliditie with their conueâ—e spherical superficies to the earth at all times respecting and their distances from the earth as also the whole earthly Sphere and globe it selfe and infinite other cases concerning Spheres or globes may hereby with as much ease and certainety be determined of as of the quantitie of any bowle ball or bullet which we may gripe in our handes reason and experience being our witnesses and without these aydes such thinges of importance neuer hable of vs certainely to be knowne or attayned vnto Here ende M. Iohn d ee his additions vpon the last proposition of the twelfth booke A proposition added by Flussas If a Sphere touche a playne superficiesâ— a right line drawne from the center to the touche shall be erected perpendicularly to the playne superficies Suppose that there be a Sphere BCDL whose centre let be the poynt A. And let the playne superficies GCI touch the Spere in the poynt C and extend a right line from the centre A to the poynt C. Then I say that the line AC is erected perpendicularly to tâ—e playne GIC. Let the sphere be cutte by playne superficieces passing by the right line LAC which playnes let be ABCDL and ACEL which let cut the playne GCI by the right lines GCH and KCI Now it is manifest by the assumpt put before the 17. of this booke that the two sections of the sphere shall be circles hauing to their diameter the line LAC which is also the diameter of the sphere Wherefore the right lines GCH and KCI which are drawne in the playne GCI do at the poynt C fall without the circles BCDL and ECL. Wherefore they touch the circles in the poynt C by the second definition of the third Wherefore the right line LAC maketh right angles with the lines GCH and KCI by the 16. of the third Wherefore by the 4. of the eleuenth the right line AC is erected perpendicularly to to the playne superficies GCI wherein are drawne the lines GCH and KCI If therefore a Sphere touch a playne superficies a right line drawne from the centre to the touche shall be erected perpendicularly to the playne superficies which was required to be proued The ende of the twelfth booke of Euclides Elementes ¶ The thirtenth booke of Euclides Elementes IN THIS THIRTENTH BOOKE are set forth certayne most wonderfull and excellent passions of a lyne deuided by an extreme and meane proportion a matter vndoubtedly of great and infinite vse in Geometry as ye shall both in thys booke and in the other bookes following most euidently perceaue It teacheth moreouer the composition of the fiue regular solides and how to inscribe them in a Sphere geuen and also setteth forth certayne comparisons of the sayd bodyes both the one to the other and also to the Sphere wherein they are described The 1. Theoreme The 1. Proposition If a right line be deuided by an extreme and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two
lines added together shal be quintuple to the square made of the halfe of the whole lyne SVppose that the right line AB be deuided by an extreme and meane proportioÌ„ in the point C. And let the greater segment therof be AC And vnto AC adde directly a ryght line AD and let AD be equall to the halfe of the line AB Then I say that the square of the line CD is quintuple to the square of the line DA. Describe by the 46. of the first vpon the lines AB and DC squares namely AE DF. And in the square DF describe and make complete the figure And extend the line FC to the point G. And forasmuch as the line AB is deuided by an extreme and meane proportion in the point C therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and BC is the parallelogramme CE and the square of the line AC is the square HF. Wherefore the parallelogramme CE is equall to the square HF. And forasmuch as the line BA is double to the line AD by constructâ—on 〈◊〉 the lyne BA is equall to the line KA and the line AD to the lyne AH therefore also the lyne KA is double to the line AH But as the lyne KA is to the line AH so is the parallelogramme CK to the parallelogramme CH Wherefore the parallelogramme CK is double to the parallelogramme CH. And the parallelogrammes LH and CH are double to the parallelogramme CH for supplementes of parallelogrammes are bâ— the 4â— of the first equall the one to the other Wherefore the parallelogramme CK is equall to the parallelogrammes LH CH. And it is proued that the parallelogramme CE is equall to the square FH Wherefore the whole square AE is equall to the gnâ—mon MXN And forasmuch as the line BA iâ— double to the line AD therefore the square of the line BA is by the 20. of the sixth quadruple to the square of the line DA that is the square AE to the square DH But the square AE is equall to the gnomoÌ„ MXN wherefore the gnomoÌ„ MXN is also quadruple to the square DH Wherefore the whole square DF is quintuple to the square DH But the square DF iâ— the square of the line CD and the square DH is the square of the line DA. Wherefore the square of the line CD is quintuple to the square of the line DA. If therefore a right line be deuided by an extreame and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two lines added together shal be quintuple to the square made of the halfe of the whole line Which was required to be demonstrated Thys proposition is an other way demonstrated after the fiueth proposition of this booke The 2. Theoreme The â— Proposition If a right line be in power quintuple to a segment of the same line the double of the sayd segment is deuided by an extreame and meane proportion and the greater segment thereof is the other part of the line geuen at the beginning Now that the double of the line AD that is AB is greater then the line AC may thus be proued For if not then if if it be possible let the line AC be double to the line AD wherefore the square of the line AC is quadruple to the square of the line AD. Wherefore the squares of the lines AC and AD are quintuple to the squares of the line AD. And it is supposed that the square of the line DC is quintuple to the square of the line AD wherefore the square of the line DC is equall to the square of the lines AC and AD which is impossible by the 4. of the second Wherefore the line AC is not double to the line AD. In like sorte also may we proue that the double of the line AD is not lesse then the line AC for this is much more absurd wherefore the double of the line AD is greater theÌ„ the line ACâ— which was required to be proued This proposition also is an other way demonstrated after the fiueth proposition of this booke Two Theoremes in Euclides Method necessary added by M. Dee A Theoreme 1. A right line can be deuided by an extreame and meane proportion but in one onely poynt Suppose a line diuided by extreame and meane proportion to be AB And let the greater segment be AC I say that AB can not be deuided by the sayd proportion in any other point then in the point C. If an aduersary woulde contend that it may in like sort be deuided in an other point let his other point be supposed to be D making AD the greater segment of his imagined diuision Which AD also let be lesse then our AC for the first discourse Now forasmuch as by our aduersaries opinion AD is the greater segment of his diuided lineâ— the parallelogramme conteyned vnder AB and DB is equall to the square of AD by the third definition and 17. proposition of the sixth Booke And by the same definition and proposition the parallelogramme vnder AB and CB conteyned is equall to the square of our greater segment AC Wherefore as the parallelogramme vnder AB and Dâ— is to the square of AD so i◠〈◊〉 parallelogramme vnder AB and CB to the square of AC For proportion of equality is concluded in them both But forasmuch as Dâ—â— iâ— byâ— supposition greater theÌ„ CB the parallelograÌ„me vnder AB and DB is greater then the parallelogramme vnder AC and CB by the first of the sixth for AB is their equall heith Wherefore the square of AD shal be greater then the square of AC by the 14. of the fifth But the line AD is lesse then the line AC by supposition wherefore the square of AD is lesse then the square of AC And it is concluded also to be greater then the square of AC Wherefore the square of AD is both greater then the square of ACâ— and also lesse Which is a thing impossible The square therefore of AD is not equall to the parallelogramme vnder AB and DB. And therefore by the third definition of the sixth AB is not deuided by an extreame and meane proportion in the point D as our aduersary imagined And Secondly in like sort will the inconueniency fall out if we assigne AD our aduersaries greater segment to be greater then our AC Therefore seing neither on the one side of our point C neither on the other side of the same point C any point can be had at which the line AB can be deuided by an extreame and meane proportion it followeth of necâ—ssitie that AB can be deuided by an extreame and meane proportion in the point C onely Therefore a right line can be deuided by an extreame
that is the right lines AG GB BD DA CA CG CB CD and IA IG IB ID are equal the one to the other Wherfore also the 8. triangles CAG CGB CBD CDA IAG IGB IBD IDA are equal and equilater And therefore AGBDCI is an Octohedron by the 23. definition of the eleueÌ„th And the sayd Octahedron is included in the dodecahedron by the first definition of this booke for that all the angles thereof doe at one time touch the sides of the dodecahedron Wherefore in the dodecahedron geuen is included an Octohedron which was required to be done ¶ The 10. Proposition The 10. Probleme In a Dodecahedron geuen to inscribe an equilater trilater Pyramis SVppose that the Dodecahedron geuen be ABCD of which Dodecahedron take thre bases meting at the poynt S namely these thre bases ALSIK DNSLE and SIBRN and of those thre bases take the three angles at the poynts A B D and draw these right lines AB BD and DA and let the diameter of the sphere containing the dodecahedron be SO and then draw thse right lines AO BO and DO Now forasmuch as by the 17. of the thirtenth the angles of the dodecahedron are set in the superficies of the sphere described about the Dodecahedronâ— therefore if vpon the diameter SO and by the poynt A be described a semicircle it shall make the angle SAO a right angle by the 31. of the third And likewise if the same semicircle be drawne by the poyntes D and B it shall also make the angles SBO and SDO right angles Wherefore the diameter SO containeth in power bothe the lines SA AO or the lines SB BO or els SD DO but the lines SA SD SB are equal the one to the other for they eche subtend one of the angles of equal pentagoÌ„s Wherfore the other lines remaining namely AO BO DO are equal the one to the other And by the same reason may be proued that the diameter HD which subtendeth the two right lines HA AD containeth in power both the said two right lines and also containeth in power bothe the right lines HB and BD which two right lines it also suhtendeth And moreouer by the same reason the diameter AC which subtendeth the right lines CB and BA containeth in power both the said right lines Câ— and BA But the right lines HA HB and CB are equal the one to the other for that eche of them also subtendeth one of the angles of equal pentagonsâ— wherfore the right lines remaining namely AD BD and BA are equal the one to the other And by the same reason may be proued that eche of those right lines AD BD and BA is equal to eche of the right lines AO BO and DO Wherefore the sixe right lines AB BD DA AO BO DO are equal the one to the other And therefore the triangles which are made of theÌ„ namely the triangles ABD AOB AOD and BOD are equal and equilater which triangles therfore do make a pyramis ABDO whose base is ABD and toppe the poynt O. Eche of the angles of which pyramis namely the angles at the pointes A B D O doe in the selfe same pointes touche the angles of the Dodecahedron Wherfore the said pyramis is inscribed in the Dodecahedron by the first diffinition of this boke Wherefore in a Dodecahedron geuen is inscribed a trilater equilater pyramis which was required to be done ¶ The 11. Proposition The 11. Probleme In an Icosahedron geuen to inscribe a cube IT was manifest by the 7. of this booke that the angles of a Dodecahedron are set in the centres of the bases of the Icosahedron And by the 8. of this boke it was proued that the angles of a cube are set in the angles of a Dodecahedron Wherefore the selfe same angles of the cube shall of necessitie be set in the centres of the bases of Icosahedron Wherfore the cube shal be inscribed in the Icosahedron by the first diffinition of this boke Wherfore in an Icosahedron geuen is included a cube which was required to be done ¶ The 12. Proposition The 12. Probleme In an Icosahedron geuen to inscribe a trilater equilater pyramis BY the former proposition it was manifest that the angles of a cube are set in the centres of the bases of the Icosâ—hedron And by the first of this booke it was playne that the foure angles of a pyramis are set in foure angles of a cube Wherefore it is euident by the first diffinition of this booke that a pyramis described of right lines ioyning together these foure centres of the bases of the Icosahedron shal be inscribed in the same Icosahedron Wherefore in an Icosadron geuen is inscribed an equilater trilater pyramis which was required to be done ¶ The 13. Probleme The 13. Proposition In a Cube geuen to inscribe a Dodecahedron TAke a Cube ADFL And diuide euery one of the sides therof into two equall partes in the pointes T H K P G L M F and pkQs And drawe these right lines TK GF pQ Hk Ps and LM which lines againe diuide into two equall partes in the pointes N V Y I Z X. And draw these right lines NY VX and IZ Now the three lines NY VX and IZ together with the diameter of the cube shall cut the one the other into two equall partes in the centre of the cube by the 3â— of the eleuenth let that centre be the point O. And not to stand long about the demonstration vnderstand all these right lines to be equall and parallels to the sides of the cube and to cut the one the other right angled wise by the 29. of the first Let their halfes namely FV GV HI and kI and the rest such like be deuided by an extreme and meane proportion by the 30. of the sixth whose greater segmeÌ„ts let be the lines FS GB HC and kE c. and drawe these right lines GI GE BC and BE. Now forasmuch as the line GI is equall to the whole line GV which is the halfe of the side of the cube and the line IE is equall to the line BV that is to the lesse segmeÌ„t therfore the squares of the lines GI and IE are triple to the square of the line GB by the 4. of the thirtenth But vnto the squares of the lines GI and IE the square of the line GE is equall by the 47. of the firstâ— for the angle GIE is a right angle Wherefore the square of the line GE is triple to the square of the line GB And forasmuch as the line FG is erected perpendicularly to the plaine AGkL by the 4. of the eleuenth for it is erected perpendicularly to the two lines AG and GI therefore the angle BGE is a right angle for the line GE is drawen in the plaine AGkL Wherefore the line BE containing in power the two lines BG and GE by the 47. of the first is in power quadruple to the
proposition after Prâ—â—lus A Corollary taken out of Flussates Demonstration leading to 〈◊〉 absurdiâ—â—e An addition oâ— Pelitarius Demonstration Three cases in this proposition The first case Construction Demonstration Three cases in this proposition The first case Euery case may happen seuen diuers wayes The like variety in ech of the other two cases Euclides construction and demostration serueth in all these cases and in their varities also Construction Demonstration How triangles are sayde to be in the selfe same parallel lines Comparison of two triangles whose sides being equal their bases and angles at the toppe are vnequall When they are lesse then two right angles Construction Demonstration Thre cases in this proposition Ech of these cases also may be diuersly Note An other addition of Pelitarius Construction Demonstration This Theoreme the conuerse of the 37. proposition An addition of Flâ—ssases An addition of Campanus Construction Demonstration leading to an absurditie This proposition is the conuerse of the 38. propositioÌ„s Demonstration Two cases in this proposition A corollary The selfe same demonstration will serue if the triangle the parallelogramme be vpon equall bases The conuerse of this proposition An other conuerse of the same propositioÌ„ Comparison of a triangle and a trapesium being vpon one the selfe same base and in the selfe same parallel lines Construction Demonstration Supplements Complementes Three cases in this Theoreme The first case This proposition called Gnomical and mistical The conuerse of this proposition Construction Demonstration Applications of spaces with excesses or wants an auncient inuention of Pithagoras How a figure is sayde to be applied to a line Three thinges geuen in this proposition The conuerse of this proposition Construction Demonstration An addition of Pelitarius To describe a square mechanically An addition of Procâ—â—â— The conuerse thereof Construction Demonstration Pithagoras the first inuenter of this proposition An addition of Pâ—lâ—tariâ—â— An other aditioÌ„ of Pelitarius An other addition of Pelitarius An other aditioÌ„ of Pelitarius A Corrollary This proposition is the conueâ—se of the former The argument of the second booke What is the power of a line Many compeÌ„dious rules of reckoning gathered one of this booke and also many rules of Algebra Two wonderfull propositions in this booke First definition What a parallelogramme is Fower kindes of parallelogrammes Second definiâ—ion A proposition added by Campane after the last proposition of the first booke Construction DemonstratioÌ„ Barlaam Barlaam Construction DemonstratioÌ„ Barlaam Construction DemonstratioÌ„ Barlaam Construction DemonstratioÌ„ A Corollary Barlaam Construction DemonstratioÌ„ Constrâ—ction Demonstration Construction Demonstration Construction DemonstratioÌ„ Many and singuler vses of this proposition This proposition can not be reduced vnto numbers Demonstration DemonstratioÌ„ A Corollary This Proposition true in all kindes of triangles Construction DemonstratioÌ„ The argument of this booke The first definition Definition of vnequall circles Second definition A contigent line Third definiâ—ion The touch of circles is 〈◊〉 in one poâ—â—â— onely Circles may touch togeâ—her two maâ—â—â— of wayes Fourth definition Fift definition Sixt definition Mixt angles Arkes Chordes Seuenth definition Difference of an angle of a Section and of an angle in a Section Eight definition Ninth definition Tenth definition Two definitions First Second Why Euclide defineth not equall Sections Constuction Demonstration leading to an impossibilitie Correlary DemonstratioÌ„ leading to an impossibilitie The first para of this Proposition Construction Demonstration The second part conuerst of the first Demonstration Demonstration leading to an impossibilitie Two cases in this Proposition Construction DemonstratioÌ„ leading to an impossibilitie Demonstratâ—on leading to an impossibilitie Two caseâ— in thys Proposition Construction The first part of this Proposition Demonstration Second part Third part This demonâ—â—rated by an argument leading to an impossibilie An other demonstration of the latter part of the Proposition leading also to an impossibilitie A Corollary Third part An other demonstration of the latter part leading also to an impossibility This Proposion is commoÌ„ly called Caâ—dâ— Panonis A Corollary Construction Demonstration An other demonstration of the same leading also to an impossibilitie Demonstration leading to an impossibilitie An other demonstration of the same leading also to an impossibilitie Construction Demonstration leading to an impossibilitie An other demonstration of the same leading also to an impossibilitie The same â—gaine demonstrated by an â—rgument leading to an absurdititie Demonstratiâ— leading to an impossibiliâ—ie An other demonstration after Pelitarius leading also to an absurditie Of circles which touch the one the other inwardly Of circles which touch the one the other outwardly An other demonstration after Pelitarius Flussates of circles which tooch the one the other outwardly Of circles which tooch the one the other inwardly The first part of this Theoreme Construction Demonstration Demonstration The second part which is the conuerse of the first An other demonstration of the first part after Campane Construction Demonstration An other demonstration after Campane The first part of this Theoreme Demonstration leading to an absurditie Second part Third part Construction Demonstration An addition of Pelitarius This Probleme commodious for the inscribing and circumscribing of figures in or abouâ— circles Demonstration leading to an impossibilitie An other deâ—onstration after Orontius Demonstration leading to an impossibilitie Two cases in thys Proposition the one when the angle set at the circumference includeth the center Demonstration The other wheÌ„ the same angle set at the circumference includeth not the center Construction Demonstration Three cases in this Proposition The first case The second case The third case Construction Demonstration Demonsâ—ration leading to an impossibilitie An adâ—ition of Campane dâ—moâ—strated by Pelitariâ—s Demonstration leading to an impossibilitie An other demonstration Construction Three cases in this Proposition The first case Demonstratioâ— The second case The third case An addition Construction Demonstration Demonstration leading to an impossibilitie Construction Demonstration The conuerse of the former Proposition Construction Demonstration Construction Demonsâ—ratioâ— Second part Thirâ— part The fift and last part An other Demonstration to proue that the angâ—e in a semicircle is a right angle A Corollary An addition of Pâ—litarius Demonstration leaâ—ing to an absurditâ—â— An addition of Campane Construction Demonstration Two cases in this Proposition Three cases in this Proposition The first case Construction Demonstratioâ— The second case Construction Demonstration The third case Construction Demonstration Consâ—â—uction Demonstration Two cases in this Proposition First case Demonstration The second câ—se Construction Demonstration Three cases in this â—roposition Construction Two cases in this Proposition The first case Demonstration The second case Construction Demonstration First Corollary Second Corollary Third Corollary This proposition is the coÌ„uerse of the former Construction Demonstration An other demonstration after Pelitarius The argument of this booke First definition Second definition The inscriptition and circumscription of rectiline â—igâ—res pertaiâ—eth only to regular figures The third definition The fourth definition The fift definition The sixt deuition Seuenth definition
to be prâ—â—â—d beâ—oâ—e 〈◊〉 â—all to the demoÌ„â—â—ration Construction Demonstration leading to an absurditie A Corollary This Probleme reduced to a Theoreme Construction Demonstration How magnitudes are sayd to be in proportion the onâ— to the other as number is to number This proâ—osition is the conuerse of formâ—r Conâ—truction Demonstration A Corollary Construction Demonstration Construction Demonstration Demonâ—tration leading to an abâ—urdiâ—ie This is the 〈…〉 demonsâ—â—ation The first part demonstratâ—d An other demonstration of the first part An othâ—r demonâ—â—raâ—ion oâ— the same first part after Montaureus Demonstration of the seconde part which is the coâ—â—erse of the former An other demonstration of the second part This Assumpâ— followeth as a Corollary of the 25 but so as it might also be here in Methode placed you shall â—inde it after the 53. of this booke absolutely demonstrated for there it serueth to the 54. his demonsâ—ration DemoÌ„stratioÌ„ of the third part DemoÌ„stratioÌ„ of the fourth part which is the coÌ„uerse of the â— Conclusion of the whole proposition A Corâ—llary Proâ—e of the first part of the Corollary Profe of the second part Profe of the third pâ—rt Proâ—e oâ— the fourth part Certayne annotations â—ut of Montauâ—â—us Rules to know whether two superficiall numbers be like or no. This assumpt is the conuerse of the 26. of the eight Demonsâ—raâ—ioÌ„ oâ— the first part Demonstration of the second partâ— A Corollary To finde out the first line incommensurable in length onely to the line geuen To finde out the second line incommensurable both in length and in power to the line geuen Construction Demonstration Tâ—is is wiâ—h Zambert an Aâ—â—â—mpt but vâ—â—eâ—ly improperly â—lâ—ssateâ— maâ—eth iâ— a Corollary but the Greeâ—e and Montaureus maâ—e it a proposition but euery way an â—nfallible truth 〈…〉 Demonstration leading to an absurditie Demonstration leading to an absurditâ—e A Corollary A Corollary Demonstration An other way to proue that the lines A E C F are proportionall Demonstration of the first part Demonstration of the second paâ—t which is the conuerse of the first A Corollary Demonstration of the first part by an argument leadindg to an absurditie Demonstration of the second paâ—t leading also to an impossibilitie And this second part is the conuerse of the first Demonstration of the second part which is the conuerse of the first How to deuide the line BC redely in such sort as iâ— required in the propositioÌ„ Demonstrâ—tion of the second part which is the conuerse of tâ—e former An other demonstrationâ—y an argumeÌ„t leading to an absurditie An Assumpt A Corollary added by Montaureuâ— Cause Cause of increasing the difficulty of this booke Note Construction Demonstration Diuers caâ—es in this proposition The second case The first kind of rationall lines commensurable in length This particle in the proposition according to any of the foresayde wayes was not in vayne put The second kinde of rationall lines coÌ„mensurable in lengâ—h The third case The third kinde of rationall lines commensurable in length The fourth case This proposition is the conuerse of the former proposition Construction Demonstration An Assumpt Constâ—uction Demonstration Diffinition of a mediall line A Corollary This assumpt is nothing els but a part of the first proposition of the sixt booke 〈◊〉 How a square is sayde to be applied vppon a line Construction Demonstration Construction Demonstration Note A Corollary Construction Demonstration leading to an absurditie Construction Demonstâ—ation Construction Demonstration * A Corollary To finde out two square nâ—mbers exceeding the one the other by a square â—umber An Assumpt Construction Demonstration Montaureus maketh this an Assumpt as the Grecke text seemeth to do likewise but without a cause Construction Demonstration This Assumpt setteth foâ—th nothing â—ls but that which the first oâ— the sâ—â—t â—etteth â—orth and therefore in sâ—me examplars it is not founde Construction Demonstration Construction Demonstration Construction Demonstration A Corollary I. Dee * The second Corollary * Therefore if you deuide the square of the side AC by the side BC the portion DC will be the product c. as in the former Corollâ—ry I. Dâ—e * The thirde Corollary * Therfore if the parallelogramme of BA and AC be deuided by BC the product will geue the pâ—â—pâ—ndicular D A. These three Corollaryes in practise Logisticall and Geometricall are profitable An other demonstration of this fourth part of the determinatioÌ„ An Assumpt Construction Deâ—onstration The first part of the dâ—termination concluded The second part coÌ„cluded The totall conclusion Construction Demonstration The first part of the determination concluded The second part coÌ„cluded The totall conclusion Construction Demonstration The first part concluded The second part coÌ„cluded The third part coÌ„cluded The totall conclusion The first Senary by composition Diffinition of a binomiall line Sixe kindes of binomiall lines Demonstration Diffinition of a first bimediall line Construction Demonstration Diffinition of a second bimediall line Demonstration Diffinition of a greater line Diffinition of a line whose power is rationall and mediall Diffinition of a liâ—e containing in power two medials An Assumpt The second Senary by composition Demonstration leading to an impossibilitie A Corollary Demonstration leading to an impossibilâ—â—e DemonstratioÌ„ leading to an impossibiliâ—ie Demonstration leading to an impâ—ssibilitie Demonstration leading to an impossibiâ—â—â—e Demonstration leading to an impossibilitie Construction Demonstration leading to an absurditie Sixe kindes of binomiall lines A binomiall line coâ—â—â—steâ—h of two paâ—tâ—s Firsâ— dâ—â—â—initiâ—n Seconâ— diffinition Third â—â—â—â—â—â—ition Fourth diffinition Fifth difâ—inition Sixth diffinition The third Senary by composition Construction Demonstratiâ—n Construction Demonstration Construction Demonstration Constâ—uction Demonstration Construction Demonstration Construction Demonstration A Corollary added by Flussates M. d ee his booke called Tyâ—â—câ—niâ—m Mathematicum This Assumpt as was before noted fâ—llâ—weth most â—riâ—fly without farther demonstration of the 25. of this booke Demonstration An Assumpt The fourth Senary by composition Construction Demonstration The first part of this demonstration concluded The secoÌ„d part of the demonstration concluded The third part coÌ„cluded The totall conclusion Demoâ—stratioâ— The first part of this demonstration concluded The third part coÌ„cluded The fourth part coÌ„cludeâ— The fift part concluded The totalâ— conclusion Demonstration Construction Demonstration Demonstration Demonstration * Looke after the Assumpt concluded at this marke for plainer opening of this place The vse of this Assumpt is in the next proposition other following The fift Senary by composition Construction Demonstration Concluded that DG is a binomiall line Construction Demonstratiâ—n Concluded that DG is a binomiall line Construction Demonstration †* ‡ DG concluded a binomiall line A Corollary added by M. Dee Construction Demonstration Construction Demonstration Construction Demonstration The sixt Senary Construction Demonstration Construction Demonstration A Corollâ—ry addâ—d by Flussetes Note Construction Demonstration An other demonstration after P. Montaureus An other demonstration after Campane Construction Demonstration An other demonstratioÌ„ afâ—â—r Campane Construction Demonstration An Assumpt An other demonstration after Campanâ— Note
VIRESCIT VULNERE VERITAS THE ELEMENTS OF GEOMETRIE of the most auncient Philosopher EVCLIDE of Megara Faithfully now first translated into the Englishe toung by H. Billingsley Citizen of London Whereunto are annexed certaine Scholics Annotations and Inuentions of the best Mathematicians both of time past and iâ— this our age With a very fruitfull Praeface made by M. I. d ee specifying the chiefe Mathematicall ScieÌ„ces what they are and wherunto commodiousâ— where also are disclosed certaine new Secrets Mathematicall and Mechâ—â—icall vntill these our daies greatly missed Imprinted at London by Iohn Daye The Translator to the Reader THere is gentle Reader nothing the word of God onely set apart which so much beautifieth and adorneth the soule and minde of maÌ„ as doth the knowledge of good artes and sciences as the knowledge of naturall and morall Philosophie The one setteth before our eyes the creatures of God both in the heauens aboue and in the earth beneath in which as in a glasse we beholde the exceding maiestie and wisedome of God in adorning and beautifying them as we see in geuing vnto them such wonderfull and manifolde proprieties and naturall workinges and that so diuersly and in such varietie farther in maintaining and conseruing them continually whereby to praise and adore him as by S. Paule we are taught The other teacheth vs rules and preceptes of vertue how in common life amongest men we ought to walke vprightly what dueties pertaine to our selues what pertaine to the gouernment or good order both of an housholde and also of a citie or common wealth The reading likewise of histories conduceth not a litle to the adorning of the soule minde of man a studie of all men coÌ„mended by it are seene and knowen the artes and doinges of infinite wise men gone before vs In histories are contained infinite examples of heroicall vertues to be of vs followed and horrible examples of vices to be of vs eschewed Many other artes also there are which beautifie the minde of man but of all other none do more garnishe beautifie it then those artes which are called Mathematicall Vnto the knowledge of which no man can attaine without the perfecte knowledge and instruction of the principles groundes and Elementes of Geometrie But perfectly to be instructed in them requireth diligent studie and reading of olde auncient authors Amongest which none for a beginner is to be preferred before the most auncient Philosopher Euclide of Megara For of all others he hath in a true methode and iuste order gathered together whatsoeuer any before him had of these Elementes written inuenting also and adding many thinges of his owne wherby he hath in due forme accomplished the arte first geuing definitions principles groundes wherof he deduceth his Propositions or conclusions in such wonderfull wise that that which goeth before is of necessitie required to the proufe of that which followeth So that without the diligent studie of Euclides Elementes it is impossible to attaine vnto the perfecte knowledge of Geometrie and consequently of any of the other Mathematicall sciences Wherefore considering the want lacke of such good authors hitherto in our Englishe tounge lamenting also the negligence and lacke of zeale to their countrey in those of our nation to whom God hath geuen both knowledge also abilitie to translate into our tounge and to publishe abroad such good authors and bookes the chiefe instrumentes of all learninges seing moreouer that many good wittes both of gentlemen and of others of all degrees much desirous and studious of these artes and seeking for them as much as they can sparing no paines and yet frustrate of their intent by no meanes attaining to that which they seeke I haue for their sakes with some charge great trauaile faithfully translated into our vulgare touÌ„ge set abroad in Print this booke of Euclide Whereunto I haue added easie and plaine declarations and examples by figures of the definitions In which booke also ye shall in due place finde manifolde additions Scholies Annotations and Inuentions which I haue gathered out of many of the most famous chiefe Mathematicies both of old time and in our age as by diligent reading it in course ye shall well perceaue The fruite and gaine which I require for these my paines and trauaile shall be nothing els but onely that thou gentle reader will gratefully accept the same and that thou mayest thereby receaue some profite and moreouer to excite and stirre vp others learned to do the like to take paines in that behalfe By meanes wherof our Englishe tounge shall no lesse be enriched with good Authors then are other straunge tounges as the Dutch French Italian and Spanishe in which are red all good authors in a manner found amongest the Grekes or Latines Which is the chiefest cause that amongest theÌ„ do florishe so many cunning and skilfull men in the inuentions of straunge and wonderfull thinges as in these our daies we see there do Which fruite and gaine if I attaine vnto it shall encourage me hereafter in such like sort to translate and set abroad some other good authors both pertaining to religion as partly I haue already done and also pertaining to the Mathematicall Artes. Thus gentle reader farewell TO THE VNFAINED LOVERS of truthe and constant Studentes of Noble Sciences IOHN d ee of London hartily wisheth grace from heauen and most prosperous successe in all their honest attemptes and exercises DIuine Plato the great Master of many worthy Philosophers and the constant auoucher and pithy perswader of Vnum Bonum and Ens in his Schole and Academie sundry times besides his ordinary Scholers was visited of a certaine kinde of men allured by the noble fame of Plato and the great commendation of hys profound and profitable doctrine But when such Hearers after long harkening to him perceaued that the drift of his discourses issued out to conclude this Vnum Bonum and Ens to be Spirituall Infinite AEternall Omnipotent c. Nothyng beyng alledged or expressed How worldly goods how worldly dignitie how health StreÌ„gth or Iustines of body nor yet the meanes how a merueâ—lous sensible and bodyly blysse and felicitie hereafter might be atteyned Straightway the fantasies of those hearers were dampt their opinion of Plato was clene chaunged yea his doctrine was by them despised and his schole no more of â—hem visited Which thing his Scholer Aristotle narrowly coÌ„sidering founde the cause therof to be For that they had no forwarnyng and information in generall whereto his doctrine tended For so might they haue had occasion either to haue forborne his schole hauntyng if they then had misliked his Scope and purpose or constantly to haue continued therin to their full satisfaction if such finall scope intent had ben to their desire Wherfore Aristotle euer after that vsed in brief to forewarne his owne Scholers and hearers both of what matter and also to what ende he tooke in hand to speake or teach While I
How often therfore these fiue sundry sortes of Operations do for the most part of their execution differre from the fiue operations of like generall property and name in our Whole numbers practisable So often for a more distinct doctrine we vulgarly account and name it an other kynde of Arithmetike And by this reason the Consideration doctrine and working in whole numbers onely where of an Vnit is no lesse part to be allowed is named as it were an Ariâ—hmetike by it selfe And so of the Arithmetike of Fractions In lyke sorte the necessary wonderfull and Secret doctrine of Proportion and proportionalytie hath purchased vnto it selfe a peculier maner of handlyng and workyng and so may seme an other forme of Arithmetike Moreouer the Astronomers for spede and more commodious calculation haue deuised a peculier maner of orderyng nuÌ„bers about theyr circular motions by Sexagenes and Sexagesmes By Signes Degrees and Minutes c. which commonly is called the Arithmetike of Astronomical or Phisicall Fractions That haue I briefly noted by the name of Arithmetike Circular By cause it is also vsed in circles not Astronomicall c. Practise hath led Numbers farder and hath framed them to take vpon them the shew of Magnitudes propertie Which is Incommensurabilitie and Irrationalitie For in puâ—e Arithmetike an Vnit is the common Measure of all Numbers And here NuÌ„bers are become as Lynes Playnes and Solides some tymes Rationall some tymes Irrationall â— And haue propre and peculier characters as √ ◠√ â— and so of other Which is to signifie Roâ—e Square Rote Cubik and so forth propre and peculier fashions in the fiue principall partes Wherfore the practiser estemeth this a diuerse Arithmetike from the other Practise bryngeth in here diuerse compoundyng of Numbers as some tyme two three foure or more Radicall nuÌ„bers diuersly knit by signes oâ— More Lesse as thus √ â— 12 + √ â— 15. Or â—hus √ â—â— 19 + √ â— 12 √ â— 2â— c. And some tyme with whole numbers or fractions of whole Number amoÌ„g them as 20 + √ â—â—4◠√ â— + 33 √ â— 10 √ â—â— 44 + 12 + √ â—9 And so infinitely may hap the varieâ—ie After this Both the one and the other hath fractions incident and so is this Arithmetike greately enlarged by diuerse exhibityng and vse of Compositions and mixtynges Consider howâ— I beyng desirous to deliuer the student from error and Cauillation do giue to this Practise the name of the Arithmetike of Radicall numbers Not of Irrationall or Surd Numbersâ— which other while are Rationall though they haue the Signe of a Rote before them which Arithmetike of whole Numbers most vsuall would say they had no such Roote and so account them Surd Numbers which generally spokeÌ„ is vntrue as Euclides tenth booke may teach you Therfore to call them generally Radicall Numbers by reason of the signe √ prefixed is a sure way and a sufficient generall distinction from all other ordryng and vsing of Numbers And yet beside all this Consider the infinite desire of knowledge and incredible power of mans Search and Capacitye how they ioyntly haue waded farder by mixtyng of speculation and practise and haue found out and atteyned to the very chief perfection almost of Numbers Practicall vse Which thing is well to be perceiued in that great Arithmeticall Arte of AEquation commonly called the Rule of Coss. or Algebra The Latines termed it Regulam Rei Census that is the Rule of the thyng and his value With an apt name comprehendyng the first and last pointes of the worke And the vulgar names both in Italian Frenche and Spanish depend in namyng it vpon the signification of the Latin word Res A thing vnleast they vse the name of Algebra And therin commonly is a dubble error The one of them which thinke it to be of Geber his inuentyng the other of such as call it Algebra For first though Geber for his great skill in Numbers Geometry Astronomy and other maruailous Artes mought haue semed hable to haue first deuised the sayd Rule and also the name carryeth with it a very nere likenes of Geber his name yet true it is that a Greke Philosopher and Mathematicien named Diophantus before Geber his tyme wrote 13. bookes therof of which six are yet extant and I had them to vse of the famous Mathematicien and my great frende Petrus Monâ—aureus And secondly the very name is Algiebar and not Algebra as by the Arabien Auicen may be proued who hath these precise wordes in Latine by Andreas Alpagus most perfect in the Arabik tung so translated Scientia faciendi Algiebar Almachabel i. Scientia inueniendi numerum ignotum per additionem Numeri diuisionem aequationem Which is to say The Science of workyng Algiebar and Almachabel that is the Science of findyng an vnknowen number by Addyng of a Number Diuision aequation Here haue you the name and also the principall partes of the Rule touched To name it The rule or Art of AEquation doth signiâ—ie the middle part and the State of the Rule This Rule hath his peculier Characters and the principal partes of Arithmetike to it appertayning do differe from the other Arithmeticall operations This Arithmetike hath NuÌ„bers Simple CoÌ„pound Mixt and Fractions accordingly This Rule and Arithmetike of Algiebar is so profound so generall and so in maner conteyneth the whole power of Numbers Application practicall that mans witt can deale with nothyngâ— more proffitable about numbers nor match with a thyng more mete for the diuine force of the Soule in humane Studies affaires or exercises to be tryed in Perchaunce you looked for long ere now to haue had some particular profe or euident testimony of the vse proffit and Commodity of Arithmetike vulgar in the Common lyfe and trade of men Therto then I will now frame my selfe But herein great care I haue least length of sundry profes might make you deme that either I did misdoute your zelous mynde to vertues schole or els mistrust your hable witts by some to gesse much more A profe then foure fiue or six such will I bryng as any reasonable man therwith may be persuaded to loue honor yea learne and exercise the excellent Science of Arithmetike And first who nerer at hand can be a better witnesse of the frute receiued by Arithmetike then all kynde of Marchants Though not all alike either nede it or vse it How could they forbeare the vse and helpe of the Rule called the Golden Rule Simple and Compoundeâ— both foâ—ward and backward How might they misse Arithmeticall helpe in the Rules of Felowshypâ— either without tyme or with tymeâ— and betwene the Marchant his â—actorâ— The Rulâ—â— of Baâ—tering in wares onelyâ— or part in wares and part in money would they gladly want Our Marchant venturers and Trauaylers ouer Sea how could they order their doynges iustly and without losse vnleast certaâ—ne and generall Rules for ExchauÌ„ge of money and Rechaunge were
Marke in your lines what numbers the water Cutteth Take the waight of the same Cube againeâ— in the same kinde of water which you had before put that also into the Pyramis or Cone where you did put the first Marke now againe in what number or place of the lines the water Cutteth them Two wayes you may conclude your purpose it is to wete either by numbers or lines By numbers as if you diuide the side of your Fundamentall Cube into so many aequall partes as it is capable of conueniently with your ease and precisenes of the diuision For as the number of your first and lesse line in your hollow Pyramis or Cone is to the second or greater both being counted from the vertex so shall the number of the side of your Fundamentall Cube be to the nuÌ„ber belonging to the Radicall side of the Cube dubble to your Fundamentall Cube Which being multiplied Cubik wise will sone shew it selfe whether it be dubble or no to the Cubik number of your Fundamentall Cube By lines thus As your lesse and first line in your hollow Pyramis or Cone is to the second or greater so let the Radical side of your FundameÌ„tall Cube be to a fourth proportionall line by the 12. proposition of the sixth boke of Euclide Which fourth line shall be the Rote Cubik or Radicall side of the Cube dubble to your Fundamentall Cube which is the thing we desired For this may I with ioy say EYPHKA EYPHKA EYPHKA thanking the holy and glorious Trinity hauing greater cause therto then Archimedes had for finding the fraude vsed in the Kinges Crowne of Gold as all men may easily Iudge by the diuersitie of the frute following of the one and the other Where I spake before of a hollow Cubik Coffen the like vse is of it and without waight Thus. Fill it with water precisely full and poure that water into your Pyramis or Cone And here note the lines cutting in your Pyramis or Cone Againe fill your coffen like as you did before Put that Water also to the firstâ— Marke the second cutting of your lines Now as you proceded before so must you here procede And if the Cube which you should Double be neuer so great you haue thus the proportion in small betwene your two litle Cubes And then the side of that great Cube to be doubled being the third will haue the fourth found to it proportionall by the 12. of the sixth of Euâ—lide Note that all this while I forget not my first Proposition Staticall here rehearsed that the Supersicies of the water is Sphaericall Wherein vse your discretion to the first line adding a small heare breadth more and to the second halfe a heare breadth more to his length For you will easily perceaue that the difference can be no greater in any Pyramis or Cone of you to be handled Which you shall thus trye For â—inding the swelling of the water aboue leuell Square the Semidiameter from the Centre of the earth to your first Waters Superficies Square then halfe the Subtendent of that watry Superficies which Subtendent must haue the equall partes of his measure all one with those of the Semidiameter of the earth to your watry Superficies Subtracte this square from the first Of the residue take the Rote Square That Roâ—e Subtracte from your first Semidiameter of the earth to your watry Superficies that which remaineth is the heith of the water in the middle aboue the leuell Which you will finde to be a thing insensible And though it were greatly sensible yet by helpe of my sixt Theoreme vpon the last Proposition of Euclides twelfth booke noted you may reduce all to a true Leuell But farther diligence of you is to be vsed against accidentall causes of the waters swelling as by hauing somwhat with a moyâ—t Sponge before made moyst your hollow Pyramis or Cone will preuent an accidentall cause of Swelling c. Experience will teach you abundantly with great ease pleasure and coÌ„moditie Thus may you Double the Cube Mechanically Treble it and so forth in any proportion Now will I Abridge your paine cost and Care herein Without all preparing of your Fundamentall Cubes you may alike worke this Conclusion For that was rather a kinde of Experimentall demoÌ„stration then the shortest way and all vpon one Mathematicall Demonstration depending Take water as much as conueniently will serue your turne as I warned before of your Fundamentall Cubes bignes Way it precisely Put that water into your Pyramis or Cone Of the same kinde of water then take againe the same waight you had before put that likewise into the Pyramis or Cone For in eche time your marking of the lines how the Water doth cut them shall geue you the proportion betwen the Radicall sides of any two Cubes wherof the one is Double to the other working as before I haue taught you sauing that for you Fundamentall Cube his Radicall side here you may take a right line at pleasure Yet farther proceding with our droppe of Naturall truth you may now geue Cubes one to the other in any proportioÌ„ geueÌ„ Rationall or Irrationall on this maner Make a hollow Parallelipipedon of Copper or Tinne with one Base waÌ„ting or open as in our Cubike Coffen FroÌ„ the bottome of that Parallelipipedon raise vp many perpendiculars in euery of his fower sides Now if any proportion be assigned you in right lines Cut one of your perpendiculars or a line equall to it or lesse then it likewise by the 10. of the sixth of Euclide And those two partes set in two sundry lines of those perpendiculars or you may set them both in one line making their beginninges to be at the base and so their lengthes to extend vpward Now set your hollow Parallelipipedon vpright perpendicularly steadie Poure in water handsomly to the heith of your shorter line Poure that water into the hollow Pyramis or Cone Marke the place of the rising Settle your hollow Parallelipipedon againe Poure water into it vnto the heith of the second line exactly Poure that water duely into the hollow Pyramis or Cone Marke now againe where the water cutteth the same line which you marked before For there as the first marked line is to the second So shall the two Radicall sides be one to the other of any two Cubes which in their Soliditie shall haue the same proportion which was at the first assigned were it Rationall or Irrationall Thus in sundry waies you may furnishe your selfe with such straunge and profitable matter which long hath bene wished for And though it be Naturally done and Mechanically yet hath it a good Demonstration Mathematicall Which is thisâ— Alwaies you haue two Like Pyramids or two Like Cones in the proportions assigned and like Pyramids or Cones are in proportion one to the other in the proportion of their Homologall sides or lines tripled Wherefore if to the first and second lines found
sydes VVherfore the angle BHA is equall to the angle EFD But the angle EFD is equall to the angle BCA VVherefore the angle BHA is equal to the angle BCA VVherefore the outward angle of the triangle AHC namely the angle BHA is equall to the inward and opposite angle namely to the angle HCA which by the 16 proposition is impossible VVherfore the syde EF is not vnequall to the syde BC wherefore it is equall And the syde AB is equall to the syde DE wherefore these two sydes AB and BC are equall to these two sydes DE and EF the one to the other and they contayne equall angles VVherfore by the 4. proposition the base AC is equall to the base DF and the triangle ABC is equall to the triangle DEF and the angle remayning namely the angle BAC is equall to the angle remayning that is to the angle EDF If therefore two triangles haue two angles of the one equall to two angles of the other ech to his correspondent angle and haue also one syde of the one equall to oâ—e syde of the other either that syde which lieth betwene the equall angles or that which is subtended vnder one of the equall angles the other sydes also of the one shal be equall to the other sydes of the other eche to his correspondent side and the other angle of the one shal be equall to the other angle of the other which was required to be proued VVhereas in this proposition it is sayde that triangles are equall which hauing two angles of the one equall to two angles of the other the one to the other haue also one side of the one equall to one side of the other either that side which lieth betwene the equall angles or that side which subtendeth one of the equall angles this is to be noted that without that caution touching the equall side the proposition shall not alwayes be true As for example The reason wherof is for that the equal side in one triangle subteÌ„deth one of the equall angles and in the other lieth betwene the equal angles So that you see that it is of necessitie that the equall side do in both triangles either subtend one of the equall angles or lie betwene the equall angles Of this proposition was Thales Milesius the inuentor as witnesseth Eudemus in his booke of Geometricall enarrations The 18. Theoreme The 27. Proposition If a right line falling vpon two right lines do make the alternate angles equall the one to the other those two right lines are parallels the one to the other SVppose that the right line EF falling vppon these two right lines AB and CD do make the alternate angles namely the angles AEF EFD equall the one to the other Then I say that AB is a parallel line to CD For if not then these lines produced shall mete together either on the side of B and D or on the syde of A C. Let them be produced therfore and let them mete if it be possible on the syde of B and D in the point G. VVherfore in the triangle GEF the outward angle AEF is equal to the inward and opposite angle EFG which by the 16. proposition is impossible VVherfore the lines AB and CD beyng produced on the side of B and D shall not meete In like sorte also may it be proued that they shall not mete on the syde of A and C. But lines whiche being produced on no syde meete together are parrallell lines by the last definition wherfore AB is a parrallel line to CD If therfore a right line falling vpon two right lines do make the alternate angles equall the one to the other those two right lines are parrallels the one to the other which was required to be demonstrated This worde alternate is of Euclide in diuers places diuersly taken somtimes for a kind of situation in place and somtime for an order in proportion in which signification he vseth it in the v. booke and in his bokes of numbers And in the first signification he vseth it here in this place and generally in all hys other bokes hâ—uing to do with lines figures And those two angles he calleth alternate which beyng both contayned within two parallel or equidistant lynes are neither angles in order nor are on the one and selfe same side but are seperated the one from the other by the line which falleth on the two lines the one angle beyng aboue and the other beneath The 19. Theoreme The 28. Proposition If a right line falling vpon two right lines make the outward angle equall to the inward and opposite angle on one and the same syde or the inwarde angles on one and the same syde equall to two right angles those two right lines shall be parallels the one to the other SVppose that the right line EF fallyng vppon these two right lines AB and CD do make the outward angle EGB equall to the inward and opposite angle GHD or do make the inward angles on one and the same side that is the angles BGH and GHD equall to two right angles Then I say that the lyne AB is a parallel line to the lyne CD For forasmuch as the angle EGB is by supposition equall to the angle GHD and the angle EGB is by the 15. proposition equall to the angle AGH therfore the angle AGH is equall to the angle GHD and they are alternate angles VVherfore by the 27. proposition AB is a parallel line to CD Agayne forasmuch as the angles BGH and GHD are by supposition equall to two right angles by the 13. proposition the angles AGH and BGH are also equall to two right angles wherefore the angles AGH and BGH are equall to the angles BGH and GHD take away the angle BGH which is common to them both VVherfore the angle remainyng namely AGH is equall to the angle remayning namely to GHD And they are alternate angles VVherfore by the former proposition AB is a parallell line to CD If therfore a right line fallyng vpon two right lines do make the outward angle equall to the inward and opposite angle on one and the same side or the inwarde angles on one and the same side equall to two right angles those two right lines shall be parallels the one to the other which was required to be proued Ptolomeus demonstrateth the second part of this proposition namely that the two inward angles on one and the same side being equall the right lines are parellels after this manner The 20. Theoreme The 29. Proposition A right line line falling vppon two parallel right lines maketh the alternate angles equall the one to the other and also the outwarde angle equall to the inwarde and opposite angle on one and the same side and moreouer the inwarde angles on one and the same side equall to two right angles And the angle AGH is by the 15. proposition equall to
AE or EC And multiply 4. into hym selfe and there is produced 16 which adde vnto 64 and there shall be produced 80 whose roote is √ â— 80 which is the line EB or the line EF by the 47. of the first And forasmuch as the line EF is √ â— 80. the lyne EA is 4. therfore the lyne AF is √ â— 80 4. And so much shall the line AH be And the line BH shall be 8 √ â— 80 4 that is 12 √ â— 80. Now theÌ„ 12 √ â— 80 multiplied into 8 shal be as much as √ â— 80 4. multiplied into it selfe For of either of them is produced 96 √ 5120. The 11. Theoreme The 12. Proposition In obtuseangle triangles the square which is made of the side subtending the obtuse angle is greater then the squares which are made of the sides which comprehend the obtuse angle by the rectangle figure which is comprehended twise vnder one of those sides which are about the obtuse angle vpon which being produced falleth a perpendicular line and that which is outwardly taken betwene the perpendicular line and the obtuse angle SVppose that ABC be an obtuseangle triangle hauing the angle BAC obtuse and from the point B by the 12. of the first draw a perpendicular line vnto CA produced and let the same be BD. Then I say that the square which is made of the side BC is greater then the squares which are made of the sides BA and AC by the rectangle figure comprehended vnder the lines CA and AD twise For forasmuch as the right line CD is by chaunce deuided in the poynt A therefore by the 4. of the second the square which is made of CD is equall to the squares which are made of CA and AD and vnto the rectangle figure contayned vnder CA and AD twise Put the square which is made of DB common vnto them both VVherefore the squares which are made of CD and DB are equall to the squares which are made of the lines CA AD and DB and vnto the rectangle figure contayned vnder the lines CA and AD twise But by the 47. of the first the square which is made of CB is equall to the squares which are made of the lines CD and DB. For the angle at the point D is a right angle And vnto the squares which are made of AD and DB by the selfe same is equall the square which is mâ—de of AB VVherfore the square which is made of CB is equall to the squares which are made of CA and AB and vnto the rectangle figure contayned vnder the lines CA and AD twise VVherfore the square which is made of CB is greater then the squares which are made of CA and AB by the rectangle figure contayned vnder the lines CA and AD twise In obtuseangle triangles therefore the square which is made of the side subtending the obtuse angle is greater then the squares which are made of the sides vvhich comprehend the obtuse angle by the rectangle figure vvhich is comprehended twise vnder one of those sides which are about the obtuse angle vpon which being produced falleth a perpendiculer lyne and that which is outwardly taken betwene the perpendiculer lyne and the obtuse angle which was required to be demonstrated Of what force thys Propositionâ— and the Proposition following touching the measuring of the obtuseangle triangle and the acuteangle triangle with the ayde of the 47. Proposition of the first booke touching the rightangle triangle he shall well perceaue which shall at any time neede the arte of triangles in which by thre thinges knowen is euer searched out three other thinges vnknowen by helpe of the table of arkes and cordes The 12. Theoreme The 13. Proposition In acuteangle triangles the square which is made of the side that subtendeth the acute angle is lesse then the squares which are made of the sides which comprehend the acute angle by the rectangle figure which is coÌ„prehended twise vnder one of those sides which are about the acuteangle vpoÌ„ which falleth a perpendiculer lyne and that which is inwardly taken betwene the perpendiculer lyne and the acute angle SVppose that ABC be an acuteangle triangle hauyng the angle at the point B acute by the 12. of the first from the point A draw vnto the lyne BC a perpendiculer lyne AD. Then I say that the square which is made of the lyne AC is lesse then the squares which are made of the lyne CB and BA by the rectangle figure conteyned vnder the lines CB and BD twise For forasmuch as the right lyne BC is by chaunce deuided in the point D therfore by the 7. of the second the squares which are made of the lines CB and BD are equall to the rectangle figure contained vnder the lines CB and DB twise and vnto the square whiche is made of line CD Put the square which is made of the line DA common vnto them both VVherfore the squares which are made of the lines CB BD and DA are equall vnto the rectangle figure contayned vnder the lines CB and BD twise and vnto the squares which are made of AD and DC But to the squares whiche are made of the lines BD and DA is equal the square which is made of the line AB for th'angle at the point D is a right angle And vnto the squares whiche are made of the lines AD and DC is equall the square whiche is made of the line AC by the 47. of the first wherfore the squares which are made of the lines CB and BA are equal to the square which is made of the line AC and to that which is contained vnder the lines CB and BD twise VVherfore the square which is made of the line AC beyng taken alone is lesse then the squares which are made of the lines CB and BA by the rectangle figure which is contained vnder the lines CB and BD twise In rectangle triangles therfore the square which is made of the side that subtendeth the acute angle is lesse then the squares which are made of the sides which comprehend the acute angle by the rectangle figure which is comprehended twise vnder one of those sides which are about the acute angle vpon which falleth a perpendicular line and that which is inwardly taken betwene the perpendicular line and the acute angle which was required to be proued ¶ A Corollary added by Orontius Hereby is easily gathered that such a perpendicular line in rectangle triangles falleth of necessitie vpon the side of the triangle that is neyther within the triangle nor without But in obtuseangle triangles it falleth without and in acuteangle triangles within For the perpendicular line in obtuseangle triangles and acuteangle triangles can not exactly agree with the side of the triangle for then an obtuse an acuteangle should be equal to a right angle contrary to the eleuenth and twelfth definitions of the first booke Likewise in obtuseangle
triangles it can not fall within nor in acuteangle triangles without for then the outward angle of a triangle should be lesse then the inward and opposite angle which is contrary to the 16. of the first And this is to be noted that although properly an acuteangle triangle by the definition therof geueÌ„ in the first booke be that triangle whose angles be all acute yet forasmuch as there is no triangle but that it hath an acute angle this proposition is to be vnderstanded is true generally in all kindes of triangles whatsoeuer and may be declared by them as you may easily proue The 2. Probleme The 14. Proposition Vnto a rectiline figure geuen to make a square equall SVppose that the rectiline figure geuen be A. It is required to make a square equall vnto the rectiline figure A. Make by the 45. of the first vnto the rectiline figure A an equall rectangle parallelogramme BCDE Now if the line BE be equall vnto the line ED then is the thyng done whiche was required for vnto the rectiline figure A is made an equal square BD. But if not one of these lines BE is ED the greater Let BE be the greater and let it be produced vnto the point F. And by the 3. of the first put vnto ED an equall line EF. And by the 10. of the first deuide the line BF into two equall partes in the point G. And making the centre the point G and the space GB or GF describe a semicircle BHF And by the 2. peticion extend the line DE vnto the point H. And by the 1. peticion draw a line from G to H. And forasmuch as the right line FB is deuided into two equall partes in the point G and into two vnequal partes in the point E therfore by the 5. of the second the rectangle figure comprehended vnder the lines BE and EF together with the square which is made of the line EG is equall to the square which is made of the line GF But the line GF is equall vnto the line GH VVherfore the rectangle figure comprehended vnder the lines BE and EF together with the square which is made of the line GE is equall to square which is made of the line GH But vnto the square which is made of the line GH are equall the squares whiche are made of the lines HE and GE by the 47. of the first VVherfore that which is contained vnder the lines BE and EF together with the square which is made of GE is equall to the squares which are made of HE and GE. Take away the square of the line EG common to them both VVherfore the rectangle figure contained vnder the lines BE EF is equall to the square which is made of the line EH But that whiche is contained vnder the lines BE and EF is the parallelogramme BD for the line EF is equall vnto the line ED. VVherfore the parallelogramme BD is equall to the square whiche is made of the line HE. But the parallelograme BD is equall vnto the rectiline figure A. VVherfore the rectiline figure A is equall to the square which is made of the line HE. VVherfore vnto the rectiline figure geuen A is made an equall square described of the line EH which was required to be done ¶ The ende of the second Booke of Euclides Elementes ¶ The third booke of Euclides Elementes THis third booke of Euclide entreateth of the most perfect figure which is a circle Wherefore it is much more to be estemed then the two bookes goyng before in which he did set forth the most simple proprieties of rightlined figures For sciences take their dignities of the worthynes of the matter that they entreat of But of al figures the circle is of most absolute perfection whose proprieties and passions are here set forth and most certainely demoÌ„strated Here also is entreated of right lines subtended to arkes in circles also of angles set both at the circumference and at the centre of a circle and of the varietie and differences of them Wherfore the readyng of this booke is very profitable to the attayning to the knowledge of chordes and arkes It teacheth moreouer which are circles contingeÌ„t and which are cutting the one the other and also that the angle of contingence is the least of all acute rightlined angles and that the diameter in a circle is the longest line that can be drawen in a circle Farther in it may we learne how three pointes beyng geuen how soeuer so that they be not set in a right line may be drawen a circle passing by them all three Agayne how in a solide body as in a Sphere Cube or such lyke may be found the two opposite pointes Whiche is a thyng very necessary and commodious chiefly for those that shall make instrumentes seruyng to Astronomy and other artes Definitions Equall circles are such whose diameters are equall or whose lynes drawen from the centres are equall The circles A and B are equal if theyr diameters namely EF and CD be equall or if their semidiameters whiche are lynes drawen from the center to the circumferenceâ— namely AF and BD be equall By thys also is knowen the definition of vnequall circles Circles whose diameters or semidiameters are vnequall are also vnequall And that circlâ— which hath the greater diameter or semidiameter is the greater circle and that circle which hath the lesse diameter or semidiameter is the lesse circle A right line is sayd to touch a circle which touching the circle and being produced cutteth it not As the right lyne EF drawen from the point E and passyng by a point of the circle namely by the point G to the point F only toucheth the circle GH and cutteth it not nor entreth within it For a right line entryng within a circle cutteth and deuideth the circle As the right lyne KL deuideth and cutteth the circle KLM and entreth within it and therfore toucheth it in two places But a right lyne touchyng a circle which is commonly called a coÌ„tingent lyne toucheth the circle onely in one point Circles are sayd to touch the one the other which touching the one the other cut not the one the other As the two circles AB and BC touch the one the other For theyr circumferences touch together in the poynt B. But neither of them cutteth or deuideth the other Neither doth any part of the one enter within the other And such a touch of circles is euer in one poynt onely which poynt onely is common to them both As the poynt B is in the conference of the circle AB and also 〈…〉 the â—â—â—â—â—ference of the circle BC. Circles may touch together two maner of wayes either outwardly the one wholy without the other or els the one being contayned within the other As the circles DE and DF of which the one DE contayneth the other namely DF and touch the one
the other in the poynt D and that onely poynt is common to them both neither doth the one enter into the other If any part of the one enter into any part of the other then the one cutteth and deuideth the other and toucheth the one the other not in one poynt onely as in the other before but in two pointâ—s and haue also a superficies common to them both As the circles GHK and HLK cut the one the other in two poyntes H and K and the one entreth into the other Also the superficies HK is common to them both For it is a part of the circle GHK and also it is a part of the circle HLK. Right lines in a circle are sayd to be equally distant from the centre when perpendicular lines drawen from the centre vnto those lines are equall And that line is sayd to be more distant vpon whom falleth the greater perpendicular line As in the circle ABCD whose centre is E the two lynes AB and CD haue equall distance from the centre E bycause that the lyne EF drawen from the centre E perpendicularly vpon the lyne AB and the lyne EG drawen likewise perpendilarly from the centre E vpon the lyne CD are equall the one to the other But in the circle HKLM whose centre is N the lyne HK hath greater distance from the centre N then hath the lyne LM for that the lyne ON drawen from the centre N perpendicularly vppon the lyne HK is greater then the lyne NP which is drawen froÌ„ the centre N perpendicularly vpon the lyne LM So likewise in the other figure the lynes AB and DC in the circle ABCD are equidistâ—nt from the centre G â— bycause the lynes OG and GP perpendicularly drawen from the centre G vppon the sayd lynes AB and DC are equall And the lyne AB hath greater distance from the centre G then hath the the lyne EF bycause the lyne OG perpendiculâ—rly drâ—wen from the centre G to the lyne AB is greâ—ter then the lyne HG whiche is perpendicularly drawen from the câ—â—tre G to the lyne EF. A section or segment of a circle is a figure coÌ„prehended vnder a right line and a portion of the circumference of a circle As the figure ABC is a section of a circle bycause it is comprehended vnder the right lyne AC and the circumference of a circle ABC Likewise the figure DEF is a section of a circle for that it is comprehended vnder the right lyne DF and the circuÌ„ference DEF And the figure ABC for that it coÌ„taineth within it the centre of the circle is called the greater section of a circle and the figure DEF is the lesse section of a circle bycause it is wholy without the centre of the circle as it was noted in the 16. Definition of the first booke An angle of a section or segment is that angle which is contayned vnder a right line and the circuÌ„ference of the circle As the angle ABC in the section ABC is an angle of a section bycause it is contained of the circumference BAC and the right lyne BC. Likewise the angle CBD is an angle of the section BDC bycause it is contayned vnder the circumference BDC and the right lyne BC. And these angles are commonly called mixte angles bycause they are contayned vnder a right lyne and a crooked And these portions of circumferences are commonly called arkes and the right lynes are called chordes or right lynes subtended And the greater section hath euer the greater angle and the lesse section the lesse angle An angle is sayd to be in a section wheÌ„ in the circumference is taken any poynt and from that poynt are drawen right lines to the endes of the right line which is the base of the segment the angle which is contayned vnder the right lines drawen from the poynt is I say sayd to be an angle in a section As the angle ABC is an angle is the section ABC bycause from the poynt B beyng a poynt in the circumference ABC are drawen two right lynes BC and BA to the endes of the lyne AC which is the base of the section ABC Likewise the angle ADC is an angle in the section ADC bycause from the poynt D beyng in the circuÌ„ference ADC are drawen two right lynes namely DC DA to the endes of the right line AC which is also the base to the sayd section ADC So you see it is not all one to say an angle of a section and an angle in a section An angle of a section coÌ„sisteth of the touch of a right lyne and a crooked And an angle in a section is placed on the circumference and is contayned of two right lynes Also the greater section hath in it the lesse angle and the lesse section hath in it the greater angle But when the right lines which comprehend the angle do receaue any circumference of a circle then that angle is sayd to be correspondent and to pertaine to that circumference As the right lynes BA and BC which containe the angle AB C and receaue the circumference ADC therfore the angle ABC is sayd to subtend and to pertaine to the circuÌ„ference ADC And if the right lynes whiche cause the angle concurre in the centre of a circle then the angle is sayd to be in the centre of a circle As the angle EFD is sayd to be in the centre of a circle for that it is comprehended of two right lynes FE and FD whiche concurre and touch in the centre F. And this angle likewise subtendeth the circumference EGD whiche circumference also is the measure of the greatnes of the angle EFD A Sector of a circle is an angle being set at the centre of a circle a figure contayned vnder the right lines which make that angle and the part of the circumference receaued of them As the figure ABC is a sector of a circle for that it hath an angle at the centre namely the angle BAC is coÌ„tained of the two right lynes AB and AC whiche contayne that angle and the circumference receaued by them Like segmentes or sections of a circle are those which haue equall angles or in whom are equall angles Here are set two definitions of like sections of a circle The one pertaineth to the angles whiche are set in the centre of the circle and receaue the circumfereÌ„ce of the sayd sections the other pertaineth to the angle in the section whiche as before was sayd is euer in the circumference As if the angle BAC beyng in the centre A and receaued of the circumference BLC be equall to the angle FEG beyng also in the centre E and receaued of the circumference FKG then are the two sections BCL and FGK lyke by the first definition By the same definition also are the other two sections like namely BCD and FGH for that the angle BAC is equall to the
when perpendicular lines drawen frō the centre to those lines are equall by the 4. definition of the third Wherfore the lines AB and CD are equally distant from the centre But now suppose that the right lines AB and CD be equally distant from the centre that is let the perpendicular line EF be equall to the perpendicular line EG Then I say that the line AB is equall to the line CD For the same order of construction remayning we may in like sort proue that the line AB is double to the line AF and that the line CD is double to the line CG And for asmuch as the line AE is equall to the line CE for they are drawen from the centre to the circumference therfore the square of the line AE is equall to the square of the line CE. But by the 47. of the first to the square of the line AE are equall the squares of the lines EF and FA. And by the selfe same to the square of the line CE are equall the squares of the lines EG and GC Wherfore the squares of the lines EF and FA are equall to the squares of the lines EG and GC Of which the square of the line EG is equall to the square of the line EF for the line EF is equall to the line EG Wherefore by the third common sentence the square remayning namely the square of the line AF is equall to the square of the line CG Wherefore the line AC is equall vnto the line CG But the line AB is double to the line AF and the line CD is double to the line CG Wherefore the line AB is equall to the line CD Wherefore in a circle equall right lines are equally distant from the centre And lines equally distant from the centre are equall the one to the other which was required to be proued ¶ An other demonstration for the first part after Campane Suppose that there be a circle ABDC whose centre let be the poynt E. And draw in it two equall lines AB and CD Then I say that they are equally distant from the centre Draw from the centre vnto the lines AB and CD these perpendicular lines EF and EG And by the 2. part of the 3. of this booke the line AB shall be equally deuided in the poynt F. and the line CD shall be equally deuided in the poynt G. And draw these right lines EA EB EC and ED. And for asmuch as in the triangle AEB the two sides AB and AE are equall to the two sides CD and CE of the triangle CED the base EB is equall to the base ED. therefore by the 8. of the first the angle at the point A shall be equall to the angle at the point C. And for asmuch as in the triangle AEF the two sides AE and AF are equall to the two sides CE and CG of the triangle CEG and the angle EAF is equall to the angle CEG therefore by the 4. of the first the base EF i◠equall to the base EG which for asmuch as they are perpendicular lines therefore the lines AB CD are equally distant frō the centre by the 4. definition of this booke The 14. Theoreme The 15. Proposition In a circle the greatest line is the diameter and of all other lines that line which is nigher to the centre is alwayes greater then that line which is more distant SVppose that there be a circle ABCD and let the diameter thereof be the line AD and let the centre thereof be the poynt E. And vnto the diameter AD let the line BC be nigher then the line FG. Then I say that the line AD is the greatest and the line BC is greater then the line FG. Draw by the 12. of the first from the centre E to the lines BC and FG perpendicular lines EH and EK And for asmuch as the line BC is nigher vnto the centre then the line FG therfore by the 4. definition of the third the line EK is greater then the line EH And by the third of the first put vnto the line EH an equall line EL. And by the 11. of the first from the point L raise vp vnto the line EK a perpendicular line LM and extend the line LM to the poynt N. And by the first petition draw these right lines EM EN EF and EG And for asmuch as the line EH is equall to the line EL therefore by the 14. of the third and by the 4. definition of the same the line BC is equall to the line MN Againe for asmuch as the line AE is equall to the line EM and the line ED to the line EN therefore the line AD is equall to the lines ME and EN But the lines ME and EN are by the 20. of the first greater then the line MN Wherefore the line AD is greater then the line MN And for asmuch as these two lines ME and EN are equall to these two lines FE and EG by the 15. definition of the first for they are drawen from the centre to the circumference and the angle MEN is greater then the angle FEG therefore by the 24. of the first the base MN is greater then the base FG. But it is proued that the line MN is equall to the line BC Wherefore the line BC also is greater then the line FG. Wherefore the diameter AD is the greatest and the line BC is greater then the line FG. Wherefore in a circle the greatest line is the diameter and of all the other lines that line which is nigher to the centre is alwaies greater then that line which is more distant which was required to be proued ¶ An other demonstration after Campane In the circle ABCD whose centre let be the poynt E draw these lines AB AC AD FG and HK of which let the line AD be the diameter of the circle Then I say that the line AD is the greatest of all the lines And the other lines eche of the one is so much greater then ech of the other how much nigher it is vnto the centre Ioyne together the endes of all these lines with the centre by drawing these right lines EB EC EG EK EH and EF. And by the 20. of the first the two sides EF and EG of the triangle EFG shall be greater then the third side FG. And for asmuch as the sayd sides EF EG are equall to the line AD by the definition of a circle therefore the line AD is greater then the line FG. And by the same reason it is greater then euery one of the rest of the lines if they be put to be bases of triangles for that euery two sides drawen frō the centre are equall to the line AD. Which is the first part of the Proposition Agayne for asmuch as the two sides EF and EG of the triangle EFG are equall to the
say that the lyne GFH which by the correllary of the 16. of this booke toucheth the circle is a parallel vnto the line AB For forasmuch as the right line CF fallyng vpon either of these lines AB GH maketh all the angles at the point â— right angles by the 3. of this boke and the two angles at the point Fare supposed to be right angles therfore by the 29. of the first the lines AB and GH are parallels which was required to be done And this Probleme is very commodious for the inscribing or circumscribing of figures in or about circles The 16. Theoreme The 18. Proposition If a right lyne touch a circle and from the centre to the touch be drawen a right line that right line so drawen shal be a perpendicular lyne to the touche lyne SVppose that the right line DE do touch the circle ABC in the point C. And take the centre of the circle ABC and let the same be F. And by the first petition from the poynt F to the poynt C drawe a right line FC Then I say that CF is a perpendicular line to DE. For if not draw by the 12. of the first from the poynt F to the line DE a perpendicular line FG. And for asmuch as the angle FGC is a right angle therefore the angle GCF is an acute angle Wherefore the angle FGC is greater then the angle FCG but vnto the greater angle is subtended the greater side by the 19. of the first Wherefore the line FC is greater then the line FG. But the line FC is equall to the line FB for they are drawen from the centre to the circumference Wherfore the line FB also is greater then the line FG namely the lesse then the greater which is impossible Wherefore the line FG is not a perpendicular line vnto the line DE. And in like sort may we proue that no other line is a perpendicular line vnto the line DE besides the line FC Wherfore the line FC is a perpendicular line to DE. If therefore a right line touch a circle from y centre to the touch be drawen a right line that right line so drawen shall be a perpendicular line to the touch line which was required to be proued ¶ An other demonstration after Orontius Suppose that the circle geuen be ABC which let the right lyne DE touch in the point C. And let the centre of the circle be the point F. And draw a right line from F to C. Then I say that the line FC is perpendicular vnto the line DE. For if the line FC be not a perpeÌ„diculer vnto the line DE then by the conuerse of the x. definition of the first boke the angles DCF FCE shal be vnequall therfore the one is greater then a right angle and the other is lesse then a right angle For the angles DCF and FCE are by the 13. of the first equall to two right angles Let the angle FCE if it be possible be greater then a right angle that is let it be an obtuse angle Wherfore the angle DCF â—hal be an acute angle And forasmuch as by suppositioÌ„ the right line DE toucheâ—h the circle ABC therefore it cutteth not the circle Wherefore the circumference BC falleth betwene the right lines DC CF therfore the acute and rectiline angle DCF shall be greater then the angle of the semicircle BCF which is contayned vnder the circumfereÌ„ce BC the right line CF. And so shall there be geueÌ„ a rectiline acute angle greater then the angle of a semicircle which is contrary to the 16. proposition of this booke Wherfore the angle DCF is not lesse then a right angle In like sort also may we proue that it is not greater then a right angle Wherfore it is a right angle and therfore also the angle FCE is a right angle Wherefore the right line FC is a perpendicular vnto the right line DE by the 10. definition of the firstâ— which was required to be proued The 17. Theoreme The 19. Proposition If a right lyne doo touche a circle and from the point of the touch be raysed vp vnto the touch lyne a perpendicular lyne in that lyne so raysed vp is the centre of the circle SVppose that the right line DE do touch the circle ABC in the point C. And from C raise vp by the 11. of the first vnto the line DE a perpendicular line CA. Then I say that in the line CA is the centre of the circle For if not then if it be possible let the centre be without the line CA as in the poynt F. And by the first petition draw a right line from C to F. And for asmuch as a certaine right line DE toucheth the circle ABC and from the centre to the touch is drawen a right line CF therefore by the 18. of the third FC is a perpendicular line to DE. Wherefore the angle FCE is a right angle But the angle ACE is also a right angle Wherefore the angle FCE is equall to the angle ACE namely the lesse vnto the greater which is impossibleâ— Wherefore the poynt F is not the centre of the circle ABC And in like sort may we proue that it is no other where but in the line AC If therefore a right line do touch a circle and from the point of the touch be raised vp vnto the touch line a perpendicular line in that line so raised vp is the centre of the circle which was required to be proued The 18. Theoreme The 20. Proposition In a circle an angle set at the centre is double to an angle set at the circumference so that both the angles haue to their base one and the same circumference SVppose that there be a circle ABC and at the centre thereof namely the poynt E let the angle BEC be set at the circumference let there be set the angle BAC and let them both haue one and the same base namely the circumference BC. Then I say that the angle BEC is double to the angle BAC Draw the right line AE and by the second petition extend it to the poynt F. Now for asmuch as the line AE is equall to the line EB for they are drawen from the centre vnto the circumference the angle EAB is equall to the angle EBA by the 5. of the first Wherefore the angles EAB and EBA are double to the angle EAB But by the 32. of the same the angle BEF is equall to the angles EAB and EBA Wherefore the angle BEF is double to the angle EAB And by the same reason the angle FEC is double to the angle EAC Wherefore the whole angle BEC is double to the whole angle BAC Againe suppose that there be set an other angle at the circumference and let the same be BDC And by the â—irst petition draw a line from D to E. And by the second petition extend
touching the circle ABC and let the same be DE. And by the first of the same let the point F be the centre of the circle ABC and draw these right lines FE FB and FD. Wherfore the angle FED is a right angle And for asmuch as the right line DE toucheth the circle ABC and the right line DCA cutteth the same therfore by the Proposition going before that which is contayned vnder the lines AD and DC is equall to the square of the line DE. But that which is contayned vnder the lines AD and DC is supposed to be equall to the square of the line DB. Wherefore the square of the line DE is equall to the square of the line DB. Wherefore also the line DE is equall to the line DB. And the line FE is equall to the line FB for they are drawen from the centre to the circumference Now therefore these two lines DE and EF are equall to these two lines DB and BF and FD is a common base to them both Wherefore by the 8. of the first the angle DEF is equall to the angle DBF But the angle DEF is a right angle Wherefore also the angle DBF is a right angle And the line FB being produced shall be the diameter of the circle But if from the end of the diameter of a circle be drawen a right line making right angles the right line so drawen toucheth the circle by the Correllary of the 16. of the third Wherfore the right line DB toucheth the circle ABC And the like demonstration will serue if the centre be in the line AC If therefore without a circle be taken a certaine point and from that poynt be drawen to the circle two right lines of which the one doth cut the circle and the other falleth vpon the circle and that in such sort that the rectangle parallelogramme which is contayned vnder the whole right line which cutteth the circle and that portion of the same line that lieth betwene the poynt and the vtter circumference of the circle is equall to the square made of the line that falleth vpon the circle then the line that so falleth vpon the circle shall touch the circle which was required to be proued ¶ An other demonstration after Pelitarius Suppose that there be a circle BCD whose centre let be E and take a point without it namely A And frō the poynt A drawe two right lines ABD and AC of which let ABD cut the circle in the poynt B let the other fall vpon it And let that which is contained vnder the lines AD and AB be equall to the square of the line AC Then I say that the line AC toucheth the circle For first if the line ABD do passe by the centre draw the right line CE. And by the 6. of the second that which is contayned vnder the lines AD and AB together with the square of the line EB that is with the square of the line EC for the lines EB and EC are equall is equall to the square of the line AE But that which is contained vnder the lines AD and AB is supposed to be equall to the square of the line AC Wherefore the square of the line AC together with the square of the line CE is equall to the square of the line AE Wherefore by the last of the first the angle at the point C is a right angle Wherfore by the 18. of this boke the line AC toucheth the circle But if the line ABD doo not passe by the centre drawe from the point A the line AD in which let be the centre E. And forasmuch as that which is contained vnder this whole line and his outward part is equall to that which is contained vnder the lines AD and AB by the first Corollary before put therefore the same is equall to the square of the line AC wherefore the angle ECA is a right angle as hath before bene proued in the first part of this Proposition And therfore the line AC toucheth the circle Which was required to be proued The ende of the third booke of Euclides Elementes ¶ The fourth booke of Euclides Elementes THIS FOVRTH BOOKE intreateth of the inscription circumscription of rectiline figures how one right lined figure may be inscribed within an other right lined figure and how a right lined figure may be circumscribed about an other right lined figure in such as may be inscribed and circumscribed within or about the other For all right lined figures cannot so be inscribed or circumscribed within or about the other Also it teacheth how a triangle a square and certayne other rectiline figures being regular may be inscribed within a circle Also how they may be circumscribed about a circle Likewise how a circle may be inscribed within them And how it may be circumscribed about them And because the maner of entreatie in this booke is diuers from the entreaty of the former bookes he vseth in this other wordes and termes then he vsed in them The definitions of which in order here after follow Definitions A rectiline figure is sayd to be inscribed in a rectiline figure when euery one of the angles of the inscribed figure toucheth euery one of the sides of the figure wherin it is inscribed As the triangle ABC is inscribed in the triangle DEF because that euery angle of the triangle inscribed namely the triangle ABC toucheth euery side of the triangle within which it is described namely of the triangle DEF As the angle CAB toucheth the side ED the angle ABC toucheth the side DF and the angle ACB toucheth the side EF. So likewise the square ABCD is said to be inscribed within the square EFGH for euery angle of it toucheth some one side of the other So also the Pentagon or fiue angled figure ABCDE is inscribed within the Pentagon or fiue angled figure FGHIK ◠As you see in the figure◠Likewise a rectiline figure is said to be circumscribed about a rectiline figure when euery one of the sides of the figure circumscribed toucheth euery one of the angles of the figure about which it is circumscribed As in the former descriptions the triangle DEF is said to be circumscribed about the triangle ABC for that euery side of the figure circumscribed namely of the triangle DEF toucheth euery angle of the figure wherabout it is circumscribed As the side DF of the triangle DEF circumscribed toucheth the angle ABC of the triangle ABC about which it is circumscribed and the side EF toucheth the angle BCA and the side CD toucheth the angle CAB Likewise vnderstand you of the square EFGH which is circumscribed about the square ABCD for euery side of the one toucheth some one side of the other Euē so by the same reason the Pentagon FGHIK is circumscribed about the Pentagon ABCDE as you see in the figure on the other side And thus may you
lines FB FA FE And for asmuch as the angle BCD is equall to the angle CDE the halfe of the angle BCD is the angle FCD and likewise the halfe of the angle CDE is the angle CDF Wherfore the angle FCD is equall to the angle FDC Wherefore the side FC is equall to the side FD. In like sort also may it be proued that euery one of these lines FB FA and FE is equall to euery one of these lines FC and FD. Wherefore these fiue right lines FA FB FC FD and FE are equall the one to the other Wherefore making the centre F and the space FA or FB or FC or FD or FE Describe a circle and it will passe by the pointes A B C D E and shall be described â—bout the fiue angled figure ABCDE which is equiangle and equilater Let this circle be described and let the same be ABCDE Wherefore about the Pentagon geuen beâ—â—g both equiangle and equilater is described a circle which was required to be â—one The 15. Probleme The 15. Proposition In a circle geuen to describe an hexagon or figure of sixe angles equilater and equiangle SVppose that the circle geuen be ABCDEF It is required in the circle geuen ABCDE to describe a figure of sixe angles of equall sides and of equall angles Dâ—aw the diameter of the circle ABCDEF and let the same be AD. And by the first of the third take the câ—nâ—re of the circlâ— and let the same be G. And making the centre D and the space DG describe by the third petition a circle CGEH and drawing right lines from E to G and from G to C extend them to the pointes B and F of the circumference of the circle geuen And draw these right lines AB BC CD DE EF and FA. Then I say that ABCDEF is an Hexagon figure of equall sides and of equall angles For forasmuch as the point G is the centre of the circle ABCDEF therefore by the 15. definition of the first the line GE is equall vnto the line GD Againe forasmuch as the point D is the centre of the circle CGEH therfore by the selfe same the line DE is equall vnto the line DG And it is proued that the line GE is equall vnto the lyne GD Wherfore the line GE is equall vnto the line ED by the first common sentence wherfore the triangle EGD is equilater and biâ— three angles namely EGD GDE DEG are equall the one to the other And forasmuch as by the 5. of the first in triangles of two equal sides câ—â—monâ—lâ— called Isosceles the angles at the base are equall the one to the other and the three angles of a triangle are by the 30. of the first equall â—nto two right angles therfore the angle EGD is the third part of two right angles And in lyke sorâ—e may it be proued that the angle DGC is the third parte of two right angles And forasmuch as the right line CG standing vpon the right line EB doth by the 13. of the fiâ—â—t make the two side angles â—GC and CGâ— equall to two right angles therfore the angle remayning CGB is the third part of two right angles Wherfore the angles EGD DGC and CGB are equall the one to the other Wherfore their hed angles that is BGA AGF and FGE are by the 15. of the first equall to these angles EGD DGC and CGB Wherfore these sixe angles EGD DGC CGB BGA AGF and FGE are equall the one to the other But equall angles consist vpon equall circumferences by 〈…〉 the third Therfore these sixe circumferences ABâ— BCâ— CDâ— DEâ— EFâ— and â—A are equall the one to the other But vnder equall circumferences are â—â—btâ—â—ded equall right lines by the 29. of the same Wherefore these sixe right lynes AB BC CD DE EF and FA are equall the one to the other Wherfore the Hexagon ABCDEF is equilater I say also that it is equiangle For forasmuch as the circumference AF is equall vnto the circumference ED adde the circumference ABCD common to them both Wherefore the whole circumference FABCD is equall to the whole circumference EDCBA And vppon the circumference FABCD consisteth the angle FED and vppon the circumference EDCBA consisteth the angle AFE. Wherefore the angle AFE is equall to the angle DEF In like sort also may it be proued that the rest of the angles of the Hexagon ABCDEF that is euery one of these angles FAB ABC BCD and CDE is equall to euery one of these angles AFE and FED Wherfore the Hexagon figure ABCDEF is equiangle and it is proued that it is also equilater and it is described in the circle ABCDEF Wherfore in the circle geuen ABCDEF is described a figure of sixe angles of equal sides and of equall angles Which was required to be done◠¶ An other way to do the same after Orontius Suppose that the circle geuen be ABCDEF in which first let there be described an equilater and equiangle triangle ACE by the second of thys booke Wherefore the arkes ABC CDE EFA are by the 28. of the third equall the one to the other Deuide euery one of those three arkes into two equall partes by the 30. of the same in the pointes B D F. And draw these right lines ABâ— BC CD DE EF and FA. Now then by the 2. definition of this booke there shall be described in the circle geue an Hexagon figure ABCDEF which must nedes be equilater for that euery one of the arkes which subtend the sides thereof are equall the one to the other I say also that it is equiangle For euery angle of the Hexagon figure is set vpon equall arkes namely vpon foure such partes of the circuÌ„ference wherof the whole circumference coÌ„tayneth sixe Wherfore the angles of the Hexagon figure are equall the one to the other by the 27. of the third Wherefore in the circle geuen ABCDEF is inscribed an equilater and equiangle Hexagon figure which was required to be done ¶ An other way to do the same after Pelitarius Suppose that the circle in which is to be inscribed 〈◊〉 â—â—uilâ—ter â—â—â—iangle Hexagon figure be ABCDE whose câ—ntre let be F. And from the centre draw the semidiameter FA. And from the poynt A apply by the first oâ—â—hys booke the line â—â— equall to the semidiameter Which I say is the side of an equilater and eqâ—iangle Hexagon figure to be inscribed in the circle ABCDE Draw â—â—â—ght â—ine from E to B â— And for asmuch as the line AB is equall to the line FA â— it is also equall to the line FB therfore the triangle AFB is equilater and by the 5. of the first equiangle Now then vpon the centre F describe the angle BFC equall to the angle AFâ— or to the angle FBA which is all one by drawing the right line FC And draw a line from B to C. And for asmuch as the angle AFB is the third part of two right angles by the
Poligonon figure of 24. sides Likewyse of the Hexagon AB and of the Pentagon AC shall be made a Poligonon figure of 30. sides one of whose sides shall subtend the arke BC. For the denomination of AB which is 6. excedeth the denomination of AC which is 5. onely by vnitie So also forasmuch as the denomination of AB which is 6. excedeth the denomination of AE which is 3. by 3. therefore the arke BE shall contayne 3. sides of a Poligonon figure of .18 sides And obseruing thys selfe same methode and order a man may finde out infinite sides of a Poligonon figure The end of the fourth booke of Euclides Elementes ¶ The fifth booke of Euclides Elementes THIS FIFTH BOOKE of Euclide is of very great commoditie and vse in all Geometry and much diligence ought to be bestowed therin It ought of all other to be throughly and most perfectly and readily knowne For nothyng in the bookes followyng can be vnderstand without it the knowledge of them all depende of it And not onely they and other writinges of Geometry but all other Sciences also and artes as Musike Astronomy Perspectiue Arithmetique the arte of accomptes and reckoning with other such like This booke therefore is as it were a chiefe treasure and a peculiar iuell much to be accompted of It entreateth of proportion and Analogie or proportionalitie which pertayneth not onely vnto lines figures and bodies in Geometry but also vnto soundes voyces of which Musike entreateth as witnesseth Boetius and others which write of Musike Also the whole arte of Astronomy teacheth to measure proportions of tymes and mouinges Archimides and Iordan with other writing of waightes affirme that there is proportion betwene waight and waight and also betwene place place Ye see therefore how large is the vse of this fift booke Wherfore the definitions also thereof are common although hereof Euclide they be accommodate and applied onely to Geometry The first author of this booke was as it is affirmed of many one Eudoxus who was Platos scholer but it was afterward framed and put in order by Euclide Definitions A parte is a lesse magnitude in respect of a greater magnitude when the lesse measureth the greater As in the other bookes before so in this the author first setteth orderly the definitions and declarations of such termes and wordes which are necessarily required to the entreatie of the subiect and matter therof which is proportion and comparison of proportions or proportionalitie And first he sheweth what a parte is Here is to be considered that all the definitions of this fifth booke be general to Geometry and Arithmetique and are true in both artes euen as proportion and proportionalitie are common to them both and chiefly appertayne to number neither can they aptly be applied to matter of Geometry but in respect of number and by number Yet in this booke and in these definitions here set Euclide semeth to speake of them onely Geometrically as they are applied to quantitie continuall as to lines superficieces and bodies for that he yet continueth in Geometry I wil notwithstanding for facilitie and farther helpe of the reader declare thē both by example in number and also in lynes For the clearer vnderstandyng of a parte it is to be noted that a part is taken in the Mathematicall Sciences two maner of wayes One way a part is a lesse quantitie in respect of a greater whether it measure the greater o◠no. The second way a part is onely that lesse quantitie in respect of the greater which measureth the greater A lesse quantitie is sayd to measure or number a greater quantitie when it beyng oftentymes taken maketh precisely the greater quantitie without more or lesse or beyng as oftentymes taken from the greater as it may there remayneth nothyng As suppose the line AB to contayne 3. and the lyne CD to contayne 9. thē doth the line AB measure the line CD for that if it be take certayne times namely 3. tymes it maketh precisely the lyne CD that is 9. without more or lesse Agayne if the sayd lesse lyne AB be taken from the greater CD as often as it may be namely 3. tymes there shall remayne nothing of the greater So the nūber 3. is sayde to measure 12. for that beyng taken certayne tymes namely foure tymes it maketh iust 12. the greater quantitie and also beyng taken from 12. as often as it may namely 4. tymes there shall remayne nothyng And in this meaning and signification doth Euclide vndoubtedly here in this define a part saying that it is a lesse magnitude in comparison of a greater when the lesse measureth the greater As the lyne AB before set contayning 3. is a lesse quantitie in comparison of the lyne CD which containeth 9. and also measureth it For it beyng certayne tymes taken namely 3. tymes precisely maketh it or taken from it as often as it may there remayneth nothyng Wherfore by this definition the lyne AB is a part of the lyne CD Likewise in numbers the number 5. is a part of the number 15. for it is a lesse number or quantitie compared to the greater and also it measureth the greater for beyng taken certayne tymes namely 3. tymes it maketh 15. And this kynde of part is called commonly pars metiens or mensurans that is a measuryng part some call it pars multiplicatina and of the barbarous it is called pars aliquota that is an aliquote part And this kynde of parte is commonly vsed in Arithmetique The other kinde of a part is any lesse quantitie in comparison of a greater whether it be in number or magnitude and whether it measure or no. As suppose the line AB to be 17. and let it be deuided into two partes in the poynt C namely into the line AC the line CB and let the lyne AC the greater part containe 12. and let the line BC the lesse part contayne 5. Now eyther of these lines by this definition is a part of the whole lyne AB For eyther of them is a lesse magnitude or quātity in cōparisō of the whole lyne AB but neither of thē measureth the whole line AB for the lesse lyne CB contayning 5. taken as oftē as ye list will neuer make precisely AB which contayneth 17. If ye take it 3. tymes it maketh only 15. so lacketh it 2. of 17. which is to litle If ye take it 4. times so maketh it 20. thē are there thre to much so it neuer maketh precisely 17. but either to much or to litle Likewise the other part AC measureth not the whole lyne AB for takē once it maketh but 12. which is lesse then 17. and taken twise it maketh 24. which are more then 17. by ◠So it neuer precisely maketh by takyng therof the whole AB but either more or lesse And this kynde of part they commonly call pars
of the one also to the residue of the other shal be equemultiplex as the whole is to the whole which was required to be proued The 6. Theoreme The 6. Proposition If two magnitudes be â—quemultiplices to two magnitudes any parâ—es taken away of them also be aequemultiplices to the same magnitudes the residues also of them shal vnto the same magnitudes be either equall or equemultiplices SVppose that there be two magnitudes AB and CD equemultiplices to two magnitudes E and F and let the partes takeÌ„ away of the magnitudes AB and CD namely AG and CH be equemultiplices to the same magnitudes E and F. Then I say that the residues GB and HD are vnto the selfe same magnitudes E and F either equall or els equemultiplices And in like sort may we proue that if GB be multiplex to E HD also shal be so multiplex vnto F. If therfore there be two magnitudes equemultiplices to two magnitudes and any parts taken away of them be also equemultiplices to the same magnitudes the residues also of them shall vnto the same magnitudes be either equall or equemultiplices which was required to be proued The 7. Theoreme The 7. Proposition Equall magnitudes haue to one the selfe same magnitude one and the same proportion And one and the same magnitude hath to equall magnitudes one and the selfe same proportion SVppose that A and B be equall magnitudes and take any other magnitude namely C. Then I say that either of these A and B haue vnto C one and the same proportion and that C also hath to either of these A and B one and the same proportion I say moreouer that C hath to either of these A and B one and the same proportion For the same order of constructioÌ„ remaining we may in like sort proue that D is equal vnto E there is taken an other multiplex to C namely F. Wherefore if F exceede D it also excedeth E and if it be equall it is equall and if it be lesse it is lesse But F is multiplex to C and D E are other equemultiplices to A and B. Wherfore as C is to A so is C to B. Wherfore equall magnitudes haue to one and the same magnitude one and the same proportion and one and the same magnitude hath to equall magnitudes one and the selfe same proportion which was required to be demonstrated The 8. Theoreme The 8. Proposition Vnequall magnitudes beyng taken the greater hath to one and the same magnitude a greater proportion then hath the lesse And that one and the same magnitude hath to the lesse a greater proportion then it hath to the greater SVppose that AB and C be vnequall magnitudes of which let AB be the greater and C the lesse And let there be an other magnitude whatsoeuer namely D. Then I say that AB hath vnto D a greater proportion then hath C to D and also that D hath to C a greater proportion then it hath to AB For forasmuch as AB is greater then C let there be taken a magnitude equall vnto C namely BE. Now then the lesse of these two magnitudes AE and EB being multiplied will at the length be greater then D. I say moreouer that D hath to C a greater proportion then D hath to AB For the same order of construction still remayning we may in like sort proue that N is greater then K and that it is not greater then FH And N is multiplex to D and FH and K are certayne other equemultiplices to AB and C. Wherfore D hath to C a greater proportion then D hath to AB ¶ For that Orontius seemeth to demonstrate this more plainly therefore I thought it not amisse here to set it Suppose that there be two vnequall magnitudes of which leâ— Aâ— bâ— the greâ—ter and C the lesse and let there be a certaine other magnitude namely D. Then I say first that AB hath to D a greater proportion then hath C to D. For forasmuch as by supposition AB is greater then the magnitude C therefore the magnitude AB contayâ—â—th the same magnitude C and an other magnitude besides Let Eâ— be equall vnto C and let AE be the part remayning of the same magnitude Now AE and EB are eyther vnequall or equall the one to the other First let them be vnequallâ— and leâ— AE be lesse then EB And vnto AE the lesse take any multiplex whatsoâ—uâ—r so that it be greater theÌ„ the magnitude D and let the same be FG. And how multiplex FG is to AE so multiplex let GH be to Eâ— â— and K to C. Agayne take the duple of D â— which let be L and then the triple and leâ— the same be M. And so forward alwayâ—s adding one vntill there be produced suâ—h a multiplex to D which shall be nâ—xt greater then GH that is which amongest the 〈◊〉 oâ— D â— by the conâ—inâ—all addition of one doth first beginne to exceâ—de GH and let thâ—â—â—me be N â— which let â—e quadruple to D. Now then the multiplex GH is the next multiplex lesse then N â— and theâ—fore â—s not lesse then M that is is either equall vnto it or greater then it And forasmuch â—s FG is equemultiplex to AE as GH is to Eâ— therefore how multiplex Fâ— is to AE so multiplex is FH to AB by the first of the fift But how multiplex FG is to AE so multiplex is K to C therefore how multiplex FH is to AB so multiplex is K to C. Moreouer forasmuch as GH and K are equemultiplices vnto EB and C â— and EB is by construction equall vnto C therfore by the common sentence GH is equâ—ll vnto K. But GH is not lesse then M as hath before bene shewed and FG â— was put to be greater then D. Wherefore the whole FH is greater then these two D and M. But D and M are equall vnto N. For N is quadruple to D. And M being triple to D doth together with D make quadruple vnto D. Wherefore FH is greater then N. Farther K is proued to be equall to GH Wherefore K is lesse then N. But FH and K are equemultiplices vnto AB and C vnto the first magnitude I say and the third and N is a certaine other multiplex vnto D which representeth the second the fourth magnitude And the multiplex of the first excedeth the multiplex of the second but the multiplex of the third excedeth not the multiplex of the fourth Wherefore AB the first hath vnto D the second a greater proportion then hath C the third to D the fourth by the 8. definition of thys booke But if AE be greater then EB let EB the lesse be multiplied vntill there be produced a multiplex greater then the magnitude D which let be GH And how multiplex GH is to EB so multiplex let FG be to AE and K also to C. Then take vnto D
such a multiplex as is next greater then FG and againe let the same be N which let be quadruple to D. And in like sort as before may we proue that the whole FH is vnto AB equemultiplex as GH is to EB and also that FH K are equemultiplices vnto AB â— C and finally that GH is equall vnto K. And forasmuch as the multiplex N is next greater then FG therefore FG is not lesse then M. But GH is greater then D by construction Wherefore the whole FH is greater then D and M and so consequently is greater then N. But K excedeth not N for K is equall to GH for how multiplex K is to EB the lesse so multiplex is FG to Aâ— the greater Bâ—t those magnitudes which are equemultipliceâ— vnto vnequall magnitudâ—s are according to the same proportion vnequall Wherefore K is lesse then FG and therefore iâ— much lesse then N. Wherefore againe the multiplex of the first exceedeth the multiplex of the second but the multiplex of the third excedeth not thâ— multiplex of the fourth Wherefore by the 8. definition of the fift Aâ— the first hath to D the second a greater proportion then hath C the third to D the fourth But now if AE be equall vnto EB eyther of them shall be equall vnto C. Wherfore vnto either of thosâ— three magnitudes take equemultiplices greater then D. So that let FG be multiplex to AE and GH vnto EB and K agayne to C which by the 6. coÌ„mon sentence shall be equall the one to the other Let N also be multiplex to D and be next greater then euery one of them namely let it be qâ—adruplâ— to D. This coÌ„struction finished we may again proue that FH and K are equemultiplices to AB and C and that FH the multiplex of thâ— first magnitude exceedeth N the multiplex of the second magnitudeâ—â—nd thaâ— K tâ—â—â—ultiplex of the third excedeth not the multiplex of the fourth Wherfore we may conclude that AB hath vnto D a greater proportion then hath C to D. Now also I say that the self same magnitude D hath vnto the lesse magnitude C a greater proportion theÌ„ it hath to the greater AB And this may plainly be gathered by the foresayd discourse without chaunging the order of the magnitudes of the equemultiplices For seing that euery way it is before proued that FH excedeth N and K is exceeded of the selfe same N therefore conuersedly N excedeth K but doth not excede FH But N is multiplex to D that is to the first and third magnitude and K is multiplex to the second namely to C â— and FH is multiplex to the fourth namely to AB â— Wherefore the multiplex of the first excedeth the multiplex of the second but the multiplex of the third excedeth not the multiplex of the fourth Wherefore by the 8. definition of this fift booke D the first hath vnto C the second a greater proportion then hath D the third to AB the fourth which was required to be proued The 9. Theoreme The 9. Proposition Magnitudes which haue to one and the same magnitude one and the same proportion are equall the one to the other And those magnitudes vnto whome one and the same magnitude hath one and the same proportion are also equall SVppose that either of these two magnitudes A and B haue to C one and the same proportion Then I say that A is equall vnto B. For if it be not then either of these A and B should not haue to C one the same proportioÌ„ by the 8. of the fifth but by supposition they haue wherefore A is equall vnto B. Againe suppose that the magnitude C haue to either of these magnitudes A and B one and the same proportion Then I say that A is equall vnto B. For if it be not C should not haue to either of these A and B one and the same proportion by the former proposition but by supposition it hath wherfore A is equall vnto B. Wherfore magnitudes which haue to one and the same magnitude one and the same proportion are equall the one to the other And thosâ— magnitudes vnto whome one and the same magnitude hath one and the same proportion are also equall which was required to be proued The 10. Theoreme The 10. Proposition Of magnitudes compared to one and the same magnitude that which hath the greater proportion is the greater And that magnitude wherunto one and the same magnitude hath the greater proportion is the lesse SVppose that A haue to C a greater proportion then B hath to C. Then I say that A is greater then B. For if it be not then either A is equall vnto B or lesse then it But A cannot be equal vnto B for then either of these A and B should haue vnto C one and the same proportion by the 7 of the fifth but by supposition they haue not wherfore A is not equall vnto B. Neither also is A lesse then B for theÌ„ should A haue to C a lesse proportion then hath B to C by the 8. of the fifth but by supposition it hath not Wherfore A is not lesse then B. And it is also proued that it is not equall wherfore A is greater then B. Agayne suppose that C haue to B a greater proportion then C hath to A. Then I say that B is lesse then A. For if it be not then is it either equall vnto it or els greater but B cannot be equall vnto A for then should C haue to either of these A and B one and the same proportion by the 7. of the fifth but by supposition it hath not wherfore B is not equall vnto A. Neither also is B greater then A for then should C haue to B a lesse proportion then it hath to A by the 8. of the fifth but by supposition it hath not wherefore B is not greater then A. And it was proued that it is not equall vnto A wherfore B is lesse then A. Wherfore of magnitudes compared to one and the same magnitude that which hath y greater proportion is the greater And that magnitude wherunto one and the same magnitude hath the greater proportion is the lesse Which was required to be proued The 11. Theoreme The 11. Proposition Proportions which are one and the selfe same to any one proportion are also the selfe same the one to the other SVpppose that as A is to B so is C to D and as C is to D so is E to F. Then I say that as A is to B so is E to F. Take equemultiplices to A C and E which let be G H K. And likewise to B D and F take any other equemultiplices which let be L M and N. And because as A is to B so is C to D and to A and G are taken equemultiplices G H to B and D are take certaine other equemultiplices L
proportion then hath BC to EF. Wherefore A hath to D a greater proportion then hath BC to EF. Wherfore alternately A hath to BC a greater proportion then hath D to EF wherfore by composition ABC hath to BC a greater proportion then hath DEF to EF. Wherfore agayne alternately ABC hath to DEF a greater proportion then hath BC to EF. Wherefore by the former proposition the proportion of A to D is greater then the proportion of ABC to DEF Which was required to be proued The end of the fifth booke of Euclides Elementes ¶ The sixth booke of Euclides Elementes THIS SIXTH BOOKE is for vse and practise a most speciall booke In it are taught the proportions of one figure to an other figure of their sides the one to the other and of the sides of one to the sides of an other likewise of the angles of the one to the angles of the other Moreouer it teacheth the description of figures like to â—igures geuen and marueilous applications of figures to lines euenly or with decrease or excesse with many other Theoremes not onely of the propoâ—tions of right lined figures but also of sectors of circles with their angles On the Theoremes and Problemes of this Booke depend for the most part the compositions of all instrumentes of measuring length breadth or deâ—peâ—es and also the reason of the vse of the same instrumentes as of the Geometricall â—quarâ— the Scale of the Astrolabe the quadrant the staffe and such other The vse of which instrumentes besides all other mechanicall instrumentes of raysing vp of mouing and drawing huge thinges incredible to the ignorant and infinite other ginnes which likewise haue their groundes out of this Booke are of wonderfull and vnspeakeable profite besides the inestimable pleasure which is in them Definitions 1. Like rectiline figures are such whose angles are equall the one to the other and whose sides about the equall angles are proportionall As if ye take any two rectiline figures As for example two triangles ABC and DEF 〈…〉 of the one triangle be equall to the angles of the other namely if the angle A be equall to the angle D and the angle B equall to the angle E also the angle C equall to the angle F. And moreouer iâ— the sides which containe the equall angles be proportionall As if the side AB haue that proportion to the side BC whâ—ch the side DE hath to the side EF and also if the side BC be vnto the side CA as â—he side EF is to the side FD and morâ—ouer if the side CA be to the side AB as the side FD is to the side DE then are these two triangles sayd to be like and so iudge ye of any other kinde of figures As if in the parallelogrammes ABCD and EFGH the angle A be equall to the angle E and the angle B equall to the angle F and the angle C equall to the angle G and the angle D equall to the angle H. And farthermore if the side AC haue that proportion to the side CD which the side EG hath to the side GH and if also the side CD be to the side DB as the side GH is to the side HF and moreouer if the side DB be to the side BA as the side HF is to the side FE and finally if the side BA be to the side AC as the side FE is to the side EG then are these parallelogrammes like 2. Reciprocall figures are those when the termeâ— of proportion are both antecedentes and consequentes in either figure As if ye haue two parallelogrammes ABCD and EFGH If the side AB to the side EF an antecedent of the first figure to a consequent of the second figure haue mutually the same proportion which the side EG hath to the side AC an antecedent of the second figure to a consequent of the first figure then are these two figures Reciprocal They are called of some figures of mutuall sides and that vndoubtedly not amisse nor vnaptly And to make thys definition more plaine Campane and Pestitarius and othersâ— thus put it Reciprocall figures are when the sides of other 〈◊〉 mutually proportionall as in the example and declaration before geuen Among the barbarous they are called Mutekesia reseruing still the Arabike worde 3. A right line is sayd to be deuided by an extreme and meane proportion when the whole is to the greater part as the greater part is to the lesse As if the line AB be so deuided in the point C that the whole line AB haue the same proportion to the greater part thereof namely to AC which the same greater part AC hath to the lesse part therof namely to CB then is the line AB deuided by an extreme and meane proportion Commonly it is called a line deuided by proportion haâ—ing a meane and two extremes How to deuide a line in such sort was taught in the 11. Proposition of the second Booke but not vnder this forme of proportion 4. The alitude of a figure is a perpendicular line drawen from the toppe to the base As the altitude or hight of the triangle ABC is the line AD being drawen perpendicularly from the poynt A being the toppe or highest part of the triangle to the base therof BC. So likewise in other figures as ye see in the examples here set That which here â—ee calleth the altitude or height of a figure in the first booke in the 35. Proposition and certaine other following he taught to be contayned within two equidistant lines so that figures to haue one altitude and to be contayned within two equidistant lines is all one So in all these examples if from the highest point of the figure ye draw an equidistant line to the base therof and then froÌ„ that poynt draw a perpendicular to the same base that perpendicular is the altitude of the figure 5. A Proportion is said to be made of two proportions or more when the quantities of the proportions multiplied the one into the other produce an other quantitie An other example where the greater inequalitie and the lesse inequalitie are mixed together 6. 4. 2. 3. the denomination of the proportion of 6. to 4 is 1 â— â— of 4. to 2 is â— â— and of 2. to 3 is â— â— now if ye multiply as you ought all these denominations together ye shall produce 12. to 6 namely dupla proportion Forasmuch as so much hath hetherto bene spoken of addition of proportions it shall not be vnnecessary somewhat also to say of substraction of them Where it is to be noted that as addition of them is made by multiplicatioÌ„ of their denominations the one into the other so is the substraction of the one from the other done by diuision of the denomination of the one by the denomination of the other As if ye will from sextupla proportion subtrahe dupla proportion take
the denominations of them both The denomination of sextupla proportion is 6 the denomination of dupla proportion is 2. Now deuide 6. the denomination of the one by 2. the denomination of the other the quotient shall be 3 which is the denomination of a new proportion namely tripla so that when dupla proportion is subtrahed from sextupla there shall remayne tripla proportion And thus may ye do in all others 6. A Parallelogramme applied to a right line is sayd to want in forme by a parallelogramme like to one geuen wheÌ„ the parallelograÌ„me applied wanteth to the filling of the whole line by a parallelogramme like to one geuen and then is it sayd to exceede when it exceedeth the line by a parallelogramme like to that which was geuen As let E be a ParallelograÌ„me geuen and let AB be a right line to whom is applied the parallelogramme ACDF Now if it want of the filling of the line AB by the parallelograÌ„me DFGB being like to the parallelogramme geuen E then is the parallelogramme sayd to want in forme by a parallelogramme like vnto a parallelogramme geuen Likewise if it exceede as the parallelogramme ACGD applyed to the linâ— ABâ— if it exceede it by the parallelogramme FGBD being like to the parallelogramme F which was geuen then is the parallelogramme ABGD sayd to exceede in forme by a parallelogramme like to a parallelogramme geuen This definition is added by Flussates as it seemeth it is not in any coÌ„mon Greke booke abroad nor in any Commentary It is for many Theoremes following very necessary The 1. Theoreme The 1. Proposition Triangles parallelogrammes which are vnder one the self same altitude are in proportion as the base of the one is to the base of the other And forasmuch as the lines CB BG and GH are equall the one to the other therfore the triangles also AHG AGB and ABC are by the 38. of the first equall the one to the other Wherfore how multiplex the base HC is to the base BC so multiplex also is the triangle AHC to the triangle ABC And by the same reason also how multiplex the base LC is to the base DC so multiplex also is the triangle ALC to the triangle ADC Wherfore if the base HC be equall vnto the base CL then by the 38. of the first the triangle AHC is equall vnto the triangle ACL And if the base HC exceede the base CL then also the triangle AHC excedeth the triangle ACL and if the base be lesse the triaÌ„gle also shall be lesse Now then there are foure magnitudes namely the two basâ—s BC and CD and the two triangles ABC and ACD and to the base BC and to the triangle ABC namely to the first and the third are taken equemulâ—iplices namely the base HC and the triangle AHC and likewise to the base CD and to the triangle ADC namely to the second and the fourth are taken certaine other equemultiplices that is the base CL and the triangle ALC And it hath bene proued that if the multiplex of the first magnitude that is the base HC do exceede the multiplex of the second that is the base CL the multiplex also of the third that is the triangle AHC excedeth the multiplex of the fourthâ— that is the triangle ALC and if the said base HC be equall to the said baâ—â— CL the triangle also AHC is equall to the triangle ALC and if it be lesse it iâ— lesse Wherfore by the sixt definiâ—ion of the fifth as the first of the foresaid magnitudes is to the second so is the third to the fourth Wherfore as the base BC is to the base CD so is the triangle ABC to the triangle ACD And because by the 41. of the first the parallelogramme EC is double to the triangle ABC and by the same the parallelogramme FC is double to the triangle ACD therfore the parallelogrammes EC and FC are equemultiplices vnto the triangles ABC and ACD But the partes of equemultiplices by the 15. of the fifth haue one and the same proportion with theiâ— equemultiplices Wherfore as the triangle ABC is to the triangle ACD so is the parallelograme EC to the parallelogramme FC And forasmuch as it hath bene demonstrated that as the base BC is to the base CD so is the triangle ABC to the triangle ACD and as the triangle ABC is to the triangle ACD so is the parallelogramme EC to the parallelogramme FC Wherefore by the 11. of the fifth as the base BC is to the base CD so is the parallelogramme EC to the parallelogramme FC The parallelogrammes may also be demonstrated a part by themselues as the triangles are if we describe vpon the bases BG GH and DK KL parallelogrammes vnder the self same altitude that the parallelogrammeâ— geuen are Wherfore triangles and parallelogrammes which are vnder one and the selfe same altitude are in proportioâ— as the base of the one is to the base of the other which was required to be demonstrated Here Flussates addeth this Corollary If two right lines being geuen the one of them be deuided how so euer the rectangle figures contayned vnder the whole line vndeuided and eche of the segmentes of the line deuided are in proportion the one to the other as the segmentes are the one to the other For imaginyng the figures BA and AD in the former description to be rectangled the rectangle figures contayned vnder the whole right lyne AC and the segments of the right line BD which is cuâ— in the poynt C namely the parallelogrammes BA and AD are in proportion the one to the other as the segmeÌ„tes BC and CD are The 2. Theoreme The 2. Proposition If to any one of the sides of a triangle be drawen a parallel right line it shall cut the sides of the same triangle proportionally And if the sides of a triangle be cut proportionally a right lyne drawn from section to section is a parallel to the other side of the triangle SVppose that there be a triangle ABC vnto one of the sides whereof namely vnto BC let there be drawen a parallel line DE cuttyng the sides AC and AB in the pointes E and D. Then I say first that as BD is to DA so is CE to EA Draw a line from B to E also from C to D. Wherfore by the 37. of the first the triangle BDE is equall vnto the triangle CDE for they are set vpon one and the same base DE and are contained within the selfe same parallels DE and BC. Consider also a certaine other triangle ADE Now thinges equall by the 7. of the fifth haue to one selfe thing one and the same proportion Wherfore as the triangle BDE is to the triangle ADE so is the triangle CDE to the triangle ADE But as the triangle BDE is to the triangle ADE so is the base BD to the base DA by the first of this booke For they are vnder one
and the selfe same toppe namely E and therfore are vnder one and the same altitude And by the same reason as the triangle CDE is to the triangle ADE so is the lyne CE to the lyne EA Wherfore by the 11. of the fifth as the line BD is to the line DA so is the line CE to the line EA But now suppose that in the triangle ABC the sides AB AC be cut proportionally so that as BD is to DA so let CE be to EA draw a line from D to E. Then secondly I say that the line DE is a parallel to the lyne BC. For the same order of construction being kept for that as BD is to DA so is CE to EA but as BD is to DA so is the triangle BDE to the triangle ADE by the 1. of the sixt as CE is to EA so by the same is the triangle CDE to the triangle ADE therfore by the 11. of the fifth as the triangle BDE is to the triangle ADE so is the triangle CDE to the triangle ADE Wherfore either of these triangles BDE and CDE haue to the triangle ADE one and the same proportion Wherefore by the 9. of the fifth the triangle BDE is equall vnto the triangle CDE and they are vpon one and the selfe base namely DE. But triangles equall and set vpon one base are also contained within the same parallel lines by the 39. of the first Wherfore the line DE is vnto the line BC a parallel If therfore to any one of the sides of a triangle be drawn a parallel line it cutteth the other sides of the same triangle proportionally And if the sides of a triangle be cut proportionally a right lyne drawen from section to section is parallel to the other side of the triangle which thing was required to be demonstrated ¶ Here also Flussates addeth a Corollary If a line parallel to one of the sides of a triangle do cut the triangle it shall cut of from the whole triangle a triangle like to the whole triangle For as it hath bene proued it deuideth the sides proportionally So that as EC is to EA so is BD to DA wherfore by the 18. of the fifth as AC is to AE so is AB to AD. Wherfore alternately by the 16. of the fifth as AC is to AB so is AE to AD wherefore in the two triangles EAD and CAB the sides about the common angle A are proportional The sayd triangles also are equiangle For forasmuch as the right lynes AEC and ADB do fall vpon the parallel lynes ED and CB therefore by the 29. of the firs◠they make the angles AED and ADE in the triangle ADE equall to the angles ACB and ABC in the triangle ACB Wherefore by the first definition of this booke the whole triangle ABC is like vnto the triangle cut of ADE The 3. Theoreme The 3. Proposition If an angle of a triangle be deuided into two equall partes and if the right line which deuideth the angle deuide also the base the segmentes of the base shall be in the same proportion the one to the other that the other sides of the triangle are And if the segmētes of the base be in the same proportion that the other sides of the sayd triangle are a right drawen from the toppe of the triangle vnto the section shall deuide the angle of the triangle into two equall partes SVppose that there be a triangle ABC and by the 9. of the first let the angle BAC be deuided into two equall partes by the right lyne AD which let cut also the base BC in the point D. Then I say that as the segment BD is to the segment DC so is the side BA to the side AC For by the point C by the 31. of the first draw vnto the line DA a parallel line CE and extende the line BA till it concurre with the line CE in the point E and do make the triangle BEC But the lyne BA shall concurre with the line CE by the 5. peticion for that the angles EBC and BCE are lesse then two right angles For the angle ECB is equall to the outwarde and opposite angle ADB by the 29. of the first And the two angles ADB and DBA of the triangle BAD are lesse then two right angles by the 17. of the first Now forasmuch as vpon the parallels AD and EC falleth the right line AC therefore by the 29. of the first the angle ACE is equall vnto the angle CAD But vnto the angle CAD is the angle BAD supposed to be equall Wherfore the angle BAD is also equall vnto the angle ACE Againe because vpon the parallels AD and EC falleth the right line BAE the outward angle BAD by the 28. of the first is equall vnto the inward angle AEC But before it was prouell that the angle ACE is equall vnto the angle BAD wherfore the angle ACE is equall vnto the angle AEC Wherefore by the 6. of the first the side AE is equall vnto the side AC And because to one of the sides of the triangle BCE namely to EC is drawen a parallel line AD therfore by the 2. of the sixt as BD is to DC so is BA to AE But AE is equall vnto AC therfore as BD is to DC so is BA to AC But now suppose that as the segment BD is to the segment DC so is the side BA to the side AC draw a line from A to D. Then I say that the angle BAC is by the right line AD deuided into two equall partes For the same order of construction remayning for that as BD is to DC so is BA to AC but as BD is to DC so is BA to AE by the 2. of the sixt for vnto one of the sides of the triangle BCE namely vnto the side EC is drawn a parallel line AD. Wherefore also as BA is to AC so is BA to AE by the 11. of the fifth Wherfore by the 9. of the fifth AC is equall vnto AE Wherfore also by the 5. of the first the angle AEC is equall vnto the angle ACE but the angle AEC by the 29. of the first is equall vnto the outward angle BAD and the angle ACE is equall vnto the angle CAD which is alternate vnto him wherefore the angle BAD is equall vnto the angle CAD Wherfore the angle BAC is by the right line AD deuided into two equall partes Wherefore if an angle of a triangle be deuided into two equall partes and if the right line which deuideth the angle cut also the base the segmentes of the base shall be in the same proportion the one to the other that the other sides of the said triangle are And if the segmentes of the base be in the same proportion that the other sides of the sayd triangle are a right line drawen from the toppe of the
to BG and draw a lyne from A to G. Now forasmuch as AB is to BC as DE is to EF therfore alternately by the 16. of the fifth as AB is to DE so is BC to EF. But as BC is to EF so is EF to BG wherfore also by the 11. of the fifth as AB is to DE so is EF to BG Wherfore the sides of the triangles ABG DEF which include the equal angles are reciprokally proportionall But if in triangles hauing one angle of the one equall to one angle of the other the sides which include the equall angles be reciprokal the triangles also by the 15. the sixth shal be equall Wherfore the triangle ABG is equall vnto the triangle DEF And for that as y line BC is to the line EF so is the line EF to the line BG but if there be three lines in proportion the first shall haue to the third double proportion that it hath to the second by the 10. definition of the fifth therfore the line BC hath vnto the line BG double proportion that it hath to the line EF. But as BC is to BG so by the 1. of the sixth is the triangle ABC to the triangle ABG Wherfore the tiangle ABC is vnto the triangle ABG in double proportion that the side BC is to the side EF. But the triangle ABG is equall to the triangle DEF Wherfore also the triangle ABC is vnto the triangle DEF in double proportion that the side BC is to the side EF. Wherfore lyke triangles are one to the other in double proportion that the sides of like proportion are which was required to be proued Corollary Hereby it is manifest that if there be three right lines in proportion as the first is to the third so is the triangle described vpon the first vnto the triangle described vpon the second so that the sayd triangles be like and in lyke â—ort described for it hath bene proued that as the lyne CB is to the line BG so is the triangle ABC to the triangle DEF which was required to be demonstrated The 14. Theoreme The 20. Proposition Like Poligonon figures are deuided into like triangles and equall in number and of like proportion to the whole And the one Poligonon figâ—re is to the other Poligonon figure in double proportion that one of the sides of like proportion is to one of the sides of like proportion SVppose that the like Poligonon figures be ABCDE FGHKL hauing the angle at the point F equall to the angle at the point A and the angle at the point â— equall to the angle at the point B and the angle at the point H equall to y angle at the point C and so of the rest And moreouer as the side AB is to the side BC so let the side FG be to the side GH and as the side BC is to the side CD so let the side GH be to the side HK and so forth And let the sides AB FG be sides of like proportion Then I say first that these Poligonon figures ABCDE FGHKL are deuided into like triangles and equall in number For draw these right lines AC AD FH FKâ— And forasmuch as by supposition that is by reason the figure ABCDE is like vnto the figure FGHKL the angle B is equall vnto the angle G and as the side AB is to the side BC so is the side FG to the side GH it followeth that the two triangles ABC and FGH haue one angle of the one equall to one angle of the other and haue also the sides about the equall angles proportionall Wherefore by the 6. of the sixt the triangle ABC is equiangle vnto the triangle FGH And those angles in theÌ„ are equall vnder which are subtended sides of like proportion namely the angle BAC is equall to the angle GFH and the angle BCA to the angle GHF Wherefore by the 4. of the sixt the sides which are about the equall angles are proportionall and the sides which are subtended vnder the equall angles are of like proportion Wherfore as AC is to BC so is FH to GH But by supposition as BC is to CD so is GH to HK Wherefore of equalitie by the 22. of the fift as AC is to CD so is FH to HK And forasmuch as by supposition the whole angle BCD is equall to the whole angle GHK and it is proued that the angle BCA is eâ—â—all to the angle GHF therefore the angle remayning ACD is equall to the angle remayning FHK by the 3. common sentence Wherefore the â—riaâ—glâ—s ACD and FHK haue againe one angle of the one equall to one angle of the other and the sides which are about the equall sides are proportionall Wherefore by the â—ame sixt of this booke the triangles ACD FHK are equiangle And by the 4. of this booke the sides which are about the equall angles 〈◊〉 proportionall And by the same reason may we proue that the triangle ADâ— is equiangle vnto the triangle FKL And that the sides which are about the equall angles are proportionall Wherefore the triangle ABC is like to the triangle FGH and the triangle ACD to the triangle FHK and also the triangle ADE to the triangle FKL by the first definition of this sixt booke Wherfore the Poligonon figures geuen ABCDE and FGHKL are deuided into triangles like and equall in number I say moreouer that the triangles are the one to the other and to the whole Poligonon figures proportionall that is as the triangle ABC is to the triangle FGH so is the triangle ACD to the triangle FHK and the triangle ADE to the triangle FKL and as the triangle ABC is to the triangle FGH so is the Poligonon figure ABCDE to the Poligonon figure FGHKL For forasmuch as the triangle ABC is like to the triangle FGH and AC and FH are sides of like proportion therfore the proportion of the triangle ABC to the triangle FGH is double to the proportion of the side AC to the 〈◊〉 FH by the former Proposition And therefore also the proportion of the triangle ACD to the triangle FKH is double to the proportion that the same side AC hath to the side FH Wherefore by the 11. of the fift as the triangle ABC is to the triangle FGH so is the triangle ACD to the triangle FHK Againe forasmuch as the triangle ACD is like to the triangle FHK and the sides AD FK are of like proportion therefore the proportion of the triangle ACD to the triangle FHK is double to the proportion of the side AD to the side FK by the foresayd 19. of the sixt And by the same reason the proportion of the triangle ADE to the triangle FKL is double to the proportion of the same side AD to the side FK Wherfore by the 11. of the fift as the triangle ACD is to the triangle FHK so is the triangle ADE to the triangle FKL But
EG the sides which include the equall angles are proportionall Wherefore the parallelogramme ABCD is by the first definition of the sixth like vnto the parallelogramme EG And by the same reason also the parallelogramme ABCD is like to the parallelogramme KH wherefore either of these parallelogrammes EG and KH is like vnto the parallelogramme ABCD. But rectiline figures which are like to one and the same rectiline figure are also by the 21. of the sixth like the one to the other Wherefore the parallelogramme EG is like to the parallelogramme HK Wherfore in euery parallelogramme the parallelogrammes about the dimecient are like vnto the whole and also like the one to the other Which was required to be proued ¶ An other more briefe demonstration after Flussates Suppose that there be a parallelograÌ„me ABCD whose dimeâ—ient let bâ— Aâ— about which let consist these parallelogrammes EK and TI hauing the angles at the pointes â— and 〈…〉 with the whole parallelogramme ABCD. Then I say that those parallelogrammes EK and TI are like to the whole parallelogramme DB and also alâ— like the one to the other For forasmuch as BD EK and TI are parallelogrammes therefore the right line AZG falling vpon these parallell lines AEB KZT and DI G or vpon these parallell lines AKD EZI and BTG maketh these angles equall the one to the other namely the angle EAZ to the angle KZA the angle EZA to the angle KAZ and the angle TZG to the angle ZGI and the angle TGZ to the angle IZG and the angle BAG to the angle AGD and finally the angle BGA to the angle DAG Wherefore by the first Corollary of the 32. of the first and by the 34. of the first the angles remayning are equall the one to the other namely the angle B to the angle D and the angle E to the angle K and the angle T to the angle I. Wherefore these triangles are equiangle and therefore like the one to the other namely the triangle ABG to the triangle GDA and the triangle AEZ to the triangle ZKA the triangle ZTG to the triangle GIZ. Wherefore as the side AB is to the side BG so is the side AE to the side EZ and the side ZT to the side TG Wherefore the parallelogrammes contayned vnder those right lines namely the parallelogrammes ABGD EK TI are like the one to the other by the first definition of this booke Wherefore in euery parallelogramme the parallelogrammes c. as before which was required to be demonstrated ¶ A Probleme added by Pelitarius Two equiangle Parallelogrammes being geuen so that they be not like to cut of from one of them a parallelogramme like vnto the other Suppose that the two equiangle parallelogrammes be ABCD and CEFG which let not be like the one to the other It is required from the Parallelogramme ABCD to cut of a parallelogramme like vnto the parallelogramme CEFG Let the angle C of the one be equall to the angle C of the other And let the two parallelogrammes be so 〈◊〉 that the lines BC CG may make both one right line namely BG Wherefore also the right lines DC and CE shall both make one right line namely DE. And drawe a line from the poynt F to the poynt C and produce the line FC till it coÌ„curre with the line AD in the poynt H. And draw the line HK parallell to the line CD by the 31. of the first Then I say that from the parallelogramme AC is cut of the parallelograÌ„me CDHK like vnto the parallelograÌ„me EG Which thing is manifest by thys 24. Proposition For that both the sayd parallelogrammes are described about one the selfe same dimetient And to the end it might the more plainly be seene I haue made complete the Parallelogramme ABGL ¶ An other Probleme added by Pelitarius Betwene two rectiline Superficieces to finde out a meane superficies proportionall Suppose that the two superficieces be A and B betwene which it is required to place a meane superficies proportionall Reduce the sayd two rectiline figures A and B vnto two like parallelograÌ„mes by the 18. of this booke or if you thinke good reduce eyther of them to a square by the last of the second And let the said two parallelogrammes like the one to the other and equall to the superficieces A and B be CDEF and FGHK And let the angles F in either of them be equall which two angles let be placed in such sort that the two parallelogrammes ED and HG may be about one and the selfe same dimetient CK which is done by putting the right lines EF and FG in such sort that they both make one right line namely EG And make coÌ„plete the parallelograÌ„me CLK M. Then I say that either of the supplements FL FM is a meane proportionall betwene the superficieces CF FK that is betwene the superficieces A and B namely as the superficies HG is to the superficies FL so is the same superficies FL to the superficies ED. For by this 24. Proposition the line HF is to the line FD as the line GF is to the line FE But by the first of this booke as the line HF is to the line FD so is the superficies HG to the superficies FL and as the line GF is to the line FE so also by the same is the superficies FL to the superficies ED. Wherfore by the 11. of the fift as the superficies HG is to the superficies FL so is the same superficies FL to the superficies ED which was required to be done The 7. Probleme The 25. Proposition Vnto a rectiline figure geuen to describe an other figure lyke which shal also be equall vnto an other rectiline figure geuen SVppose that the rectiline figure geueÌ„ wherunto is required an other to be made like be ABC and let the other rectiline figure whereunto the same is required to be made equal be D. Now it is required to describe a rectiline figure like vnto the figure ABC and equall vnto the figure D. Vppon the line BC describe by the 44. of the first a parallelogramme BE equall vnto the triangle ABC and by the same vpon the line CE describe the parallelogramme C M equall vnto the rectiline figure D and in the said parallelogramme let the angle FCE be equall vnto the angle CBL And forasmuch as the angle FCE is by construction equall to the angle CBL adde the angle BCE common to them both Wherefore the angles LBC and BCE are equall vnto the angles BCE and ECF but the angles LBC and BCE are equall to two right angles by the 29. of the first wherfore also the angles BCE and ECF are equall to two right angles Wherfore the lines BC and CF by the 14. of the first make both one right line namely BF and in like sort do the lines LE and EM make both one right line namely LM Then by the
figure comprehended vnder the lines GB and GE is equall to the square made of the line AG by the 36. of the third it followeth that the touch line GA is a meane proportionall betwene the extremes GB and GE by the second part of the 17. of the sixt for that by that Proposition the lines GB GA and GE are proportionall And by the same reason may it be proued that the line GA is a meane proportionall betwene the lines GD and GC and so of all others If therefore from a poynt geuenâ— câ— which was required to be demonstrated The end of the sixth booke of Euclides Elementes ¶ The seuenth booke of Euclides Elementes HETHERTO IN THE SIXE bookes before hath Euclide passed through and entreated of the Elementes of Geometrie without the ayde and succor of nomber But the matters which remayne to be taught and to be spoken of in these his Geometricall bookes which follow as in the tenth eleuenth and so forth he could by no meanes fully and clearely make plaine demonstrate without the helpe and ayd of nombers In the tenth is entreated of lines irrationall and vncertaine and that of many sondry kindes and in the eleuenth the other following he teacheth the natures of bodyes and compareth theyr sides and lines together All which for the most part are also irrationall And as rationall quantites and the comparisons and proportions of them cannot be knowen nor exactly tried but by the meane of nomber in which they are first sene and perceiued euen so likewise cannot irrationall quantities be knowen and found out without nomber As straightnes is the triall of crokednes and inequalitie is tried by equalitie so are quantities irrationall perceiued and knowen by quantities rationall which are â—irst and chiefely found among nombers Wherefore in these three bookes following being as it were in the middest of his Elementes he is compelled of necessitie to entreate of nombers although not so fully as the nature of nombers requireth yet so much as shall seme to be fit and sufficiently to serue for his purpose Wherby is sene the necessitie that the one Arte namely Geometrie hath of the other namely of Arithmeticke And also of what excelleÌ„cy and worthines Arithmeticke is aboue Geometrie in that Geometrie boroweth of it principles ayd and succor and is as it were maymed with out it Whereas Arithmeticke is of it selfe sufficient and neadeth not at all any ayde of Geometrie but is absolute and perfit in it selfe and may well be taught and attayned vnto without it Agayne the matter or subiect where about Geometrie is occupied which are lines figures and bodyes are such as offer them selues to the sences as triangles squares circles cubes and other are sene iudged to be such as they are by the sight but nomber which is the subiect and matter of Arithmeticke falleth vnder no sence nor is represented by any shape forme or figure and therefore cannot be iudged by any sence but only by consideration of the minde and vnderstanding Now thinges sensible are farre vnder in degree then are thinges intellectuall and are of nature much more grosse then they Wherefore nomber as being only intellectuall is more pure more immateriall and more subtile farre then is magnitude and exteÌ„deth it selfe farther For Arithmeticke not onely aydeth Geometrie but ministreth principles and groundes to many other nay rather to all other sciences and artes As to musicke Astronomy naturall philosophy perspectiue with others What other thing is in musicke entreated of then nomber contracted to sound and voyce In Astronomie who without the knowledge of nomber can doo any thing either in searching out of the motions of the heauens and their courses either in iudging and foreshewing the effectes of them In natural philosophie it is of no small force The wisest and best learned philosophers that haue bene as Pithagoras Timeus Plato and their followers found out taught most pithely and purely the secret and hidden knowledge of the nature and condicion of all thinges by nombers and by the proprieties and passions of them Of what force nomber is in perspectiue let him declare and iudge who hath any thing traueled therein Yea to be short what can be worthely and with prayse practised in common life of any man of any condition without the knowledge of nomber Yea it hath bene taught of the chiefest amongst philosophers that all naturall thinges are framed and haue their constitucion of nomber Boetius sayth Hoc fuit principalâ— in animâ— câ—â—ditoris exemplar Nomber sayth he was the principall example and patron in the minde of the creator of the world Doth not that great philopher Timaus in his booke also Plato in his Timâ—â— following him shew how the soule is composed of harmonicall nombers and consonantes of musicke Nomber compaseth all thinges and is after these men the being and very essence of all thinges And ministreth ayde and helpe as to all other knowledges so also no small to Geometrie Which thing causeth Euclidâ— in the midest of his booke of Geometrie to inserte and place these three bookes of Arithmeticke as without the ayde of which he could not well passe any father In this seuenth booke he â—irst placeth the generall principles and first groundes of Arithmetickâ— and setteth the diffinitions of nomber or multitude and the kinds therof as in the first boke he did of magnitude and the kindes and partes thereof After that he entreateth of nombers generally and of theyr partes and searcheth and demonstrateth in generall the most common passions and proprieties of the same and chieâ—ely of nombers prime or incommensurable and of nombers composed or commensurable and of their proprieties and partely also of the comparison oâ— proportion of one nomber to an other ¶ Definitions 1 Vnitie is that whereby euery thing that is is sayd to be on As a point in magnitude is the least thing in magnitude and no magnitude at all yet the ground and beginning of all magnitudes euen so is vnitie in multitude or nomber the least thing in nomber and no nomber at all and yet the ground and beginning of all nombers And therefore it is here in this place of Euclide first defined as in the first booke for the like reason and cause was a point first defined Vnitie sayth Euclide is that whereby euery thing is sayd to be one that is vnitie is that whereby euery thing is deuided and seperated from an other and remayneth on in it selfe pure and distinct froÌ„ all others Otherwise were not this vnitie whereby all thinges are seioyned the on from the other all things should suffer mixtion and be in confusion And where confusion is there is no order nor any thing can be exactly knowen either what it is or what is the nature and what are the properties thereof Vnitie therefoâ—e is that which maketh euery thing to be that which it is Boetius sayth very aptly
Vnum quodque idea est quia vnum numero est that is euery thing therfore is that is therefore hath his being in nature and is that it is for that it is on in nomber According whereunto Iordane in that most excellent and absolute worke of Aâ—ithmeticke which he wrote defineth vnitie after this maner Vnitas est res per se discretio that is vnitie is properly and of it selfe the difference of any thing That is vnitie is that whereby euery thing doth properly and essentially differ and is an other thing from all others Certainely a very apt deâ—inition and it maketh playne the definition here set of Euclide 2 Number is a multitude composed of vnities As the number of three is a multitude composed and made of three vnities Likewise the number of fiue is nothing ells but the composition putting together of fiue vnities Although as was before sayde betwene a poynt in magnitude and vnitie in multitude there is great agreement and many thiâ—â—â—â— are comâ—on to them both for as a poynt is the beginning of magnitude so is vnitie the beginning of nomber And as a poynt in magnitude is indiuisible so is also vnitie in number indiuisible yet in this they differ and disagree There is no line or magnitude made of pointes as of his partes So that although a point be the beginning of a lyne yet is it no part therof But vnitie as it is the beginning of number so is it also a part therof which is somewhat more manifestly set of Boetius in an other dâ—ffinition of number which he geueth in his Arithmetike which is thus Numerus est quantitatâ— acernus ex vnitatibus profusus that is Number is a masse or heape of quantities produced of vnities which diffinition in substance is all one with the first wherin is said most plainly that the heape or masse that is the whole substance of the quantitie of number is produced made of vnities So that vnitie is as it were the very matter of number As foure vnities added together are the matter wherof the number 4. is made eche of these vnities is a part of the number foure namely a fourth part or a quarter Vnto this diffinition agreeth also the definition geuen of Iordane which is thus Number is a quantitie which gathereth together thinges seuered a sonder As fiue men beyng in themselues seuered and distincte are by the number fiue brought together as it were into one masse and so of others And although vnitie be no number yet it contayneth in it the vertue and power of all numbers and is set and taken for them In this place for the Farther elucidation of thinges partly before set and chiefly hereafter to be set because Euclide here doth make mention of diuers kyndes of numbers and also defineth the same is to be noted that number may be considered three maner of wayes First number may be considered absolutely without comparyng it to any other number or without applieng it to any other thing onely vewing â—nd paysing what it is in it selfe and in his owne nature onely and what partes it hath and what proprieties and passions As this number sixe may be considered absolutely in his owne nature that it is an euen number and that it is a perfect number and hath many mo conditions and proprieties And so conceiue ye of all other numbers whatsoeuer of 9. 12. and so forth An other way number may be coÌ„sidered by way of coÌ„parison and in respect of some other number either as equall to it selfe or as greater theÌ„ it selfe or as lesse theÌ„ it selfe As 12. may be coÌ„sidered as coÌ„pared to 12. which is equall vnto it or as to 24. which is greater then it for 12 is the halfe thereof of as to 6. which is lesse then it as beyng the double therof And of this consideration of numbers ariseth and springeth all kyndes and varieties of proportioÌ„ as hath before bene declared in the explanation of the principles of the fift booke so that of that matter it is needelesse any more to be sayd in this place Thus much of this for the declaration of the thinges following 3 A part is a lesse number in comparison to the greater when the lesse measureth the greater As the number 3 compared to the number 12. is a part For 3 is a lesse number then is 12. and moreouer it measureth 12 the greater number For 3 taken or added to it selfe certayne times namely 4 tymes maketh 12. For 3 foure tymes is 12. Likewise is â— a part of 8 2 is lesse then 8 and taken 4 tymes it maketh 8. For the better vnderstandyng of this diffinition and how this worde Parte is diuersly taken in Arithmetique and in Geometry read the declaration of the first diffinition of the 5. booke 4 Partes are a lesse number in respect of the greater when the lesse measureth not the greater As the number 3 compared to 5 is partes of 5 and not a part For the number 3 is lesse then the nuÌ„ber 5 and doth not measure 5. For taken once it maketh but 3. once 3 is 3 which is lesse then 5. and 3 taken twise maketh 6 which is more then 5. Wherfore it is no part of 5 but partes namely three fifth partes of 5. For in the number 3 there are 3 vnities and euery vnitie is the fifth part of 5. Wherfore 3 is three fifth partes of 5 and so of others 5 Multiplex is a greater number in comparison of the lesse when the lesse measureth the greater As 9 compared to 3 is multiplex the number 9 is greater then the number 3. And moreouer 3 the lesse number measureth 9 the greater number For 3 taken certaine tymes namely 3 tymes maketh 9. three tymes three is 9. For the more ample and full knowledge of this definition read what is sayd in the explanation of the second definition of the 5 booke where multiplex is sufficiently entreated of with all his kyndes 6 An euen number is that which may be deuided into two equal partes As the number 6 may be deuided into 3 and 3 which are his partes and they are equall the one not exceding the other This definition of Euclide is to be vnderstand of two such equall partes which ioyned together make the whole number as 3 and 3 the equall partes of 6 ioyned together make 6 for otherwise many numbers both euen and odde may be deuided into many equall partes as into 4. 5. 6â— or mo and therfore into 2. As 9 may be deuided into 3 and 3 which are his partes and are also equall for the one of them excedeth not the other yet is not therfore this number â— an euen number for that 3 and 3 these equall partes of 9 added together make not 9 but onely 6. Likewise taking the definition so generally euery number whatsoeuer should be an euen numberâ— for
number euenly euen is that which may be diuided into two euen partes and that part agayne into two euen partes and so continually deuiding without stay 〈◊〉 come to vnitie As by exampleâ— 64. may be deuided into 31 and. 32. And either of these partes may be deuided into two euen partes for 32 may be deuided into 16 and 16. Againe 16 may be deuided into 8 and 8 which are euen partes and 8 into 4 and 4. Againe 4 into â— and â— and last of all may â— be deuided into one and one 9 A number euenly odde called in latine pariter impar is that which an euen number measureth by an odde number As the number 6 which 2 an euen number measureth by 3 an odde number thre times 2 is 6. Likewise 10. which 2. an euen number measureth by 5 an odde number In this diffinition also is found by all the expositors of Euclide the same want that was found in the diffinition next before And for that it extendeth it selfe to large for there are infinite numbers which euen numbers do measure by odde numbers which yet after their mindes are not eueÌ„ly odde nuÌ„bers as for example 12. For 4 an eueÌ„ nuÌ„ber measureth 12 by â— an odde numberâ— three times 4 is 12. yet is not 12 as they thinke an euenly odde number Wherfore Campane amendeth it after his thinking by adding of this worde all as he did in the first and defineth it after this maner A number euenly odde is when all the euen numbers which measure it do measure it by vneuen tymes that is by an odde number As 10. is a number euenly odde for no euen number but onely 2 measureth 10. and that is by 5 an odde number But not all the euen numbers which measure 12. do measure it by odde numbers For 6 an euen number measureth 12 by 2 which is also euen Wherfore 12 is not by this definition a number euenly odde Flussates also offended with the ouer large generalitie of this definition to make the definition agree with the thing defined putteth it after this maner A number euenly odde is that which an odde number doth measure onely by an euen number As 14. which 7. an odde number doth measure onely by 2. which is an euen number There is also an other definition of this kinde of number commonly geuen of more plaines which is this A number euenly odde in that which may be deuided into two equall partes but that part cannot agayne be deuided into two equall partes as 6. may be deuided into two equall partes into 3. and 3. but neither of them can be deuided into two equall partes for that 3. is an odde number and suffereth no such diuision 10 A number oddly euen called im lattin in pariter par is that which an odde number measureth by an euen number As the number 12 for 3. an odde number measureth 12. by 4. which is an euen number three times 4. is 12. This definition is not founde in the greeke neither was it doubtles euer in this maner written by Euclide which thing the slendernes and the imperfection thereof and the absurdities following also of the same declare most manifestly The definition next before geuen is in substance all one with this For what number soeuer an euenâ— number doth measure by an odde the selfe same number doth an odde number measure by an euen As 2. an eueÌ„ number measureth 6. by 3. an odde number Wherfore 3. an odde number doth also measure the same number 6. by 2. an eueÌ„ nuÌ„ber Now if these two definitions be definitions of two distinct kindes of numbers then is this number 6. both euenly euen and also euenly odde and so is contayned vnder two diuers kindes of numbers Which is directly agaynst the authoritie of Euclide who playnely pâ—ouoâ—h here after in the 9. booke that euery nomber whose halfe is an odde number is a number euenly odde onely Flussates hath here very well noted that these two euenly odde and oddely euen were taken of Euclide for on and the selfe same kinde of nomber But the number which here ought to haue bene placed is called of the best interpreters of Euclide numerus pariter par nupar that is a number eueÌ„ly eueÌ„ and eueÌ„ly odde Yeâ— and it is so called of Euclide him selfe in the 34. proposition of his 9. booke which kinde of number Campanus and Flussates in steade of the insufficient and vâ—apt definition before geuen assigne this definition A number euenly euen and euenly â—dde is that which an euen number doth measure sometime by an euen number and sometime by an odde As the number 12 for 2. an euen number measureth 12. by 6. an euen number two times 6. iâ— 12. Also 4. an euen number measureth the same number 12. by 3. an odde number Add therefore is 12. a number euenly euen and euenly odde and so of such others There is also an other definition geuen of this kinde of number by Boetius and others commonly which is thus A number eueâ—ly euen and euenly odde is that which may be deuided into two equall partes and eche of them may aâ—ayne be deuided into two equall partes and so forth But this deuision is at lenghth stayd and continueth not till it come to vnitie As for example 48 which may be deuided into two equall partes namely into 24. and 24. Agayne 24. which is on of the partes may be deuided into two equall partes 12. and 12. Agayne 12. into 6. and 6. And agayne 6 may be deuided into two equall partes into 3. and 3 but 3. cannot be deuided into two equall partes Wherefore the deuision there stayeth and continueth not till it come to vnitie as it did in these numbers which are euenly euen only 11 A number odly odde is that which an odde number doth measure by an odde number As 25 which 5. an odde number measureth by an odde number namely by 5. Fiue times fiue is 25 Likewise 21. whom 7. an odde number doth measure by 3 which is likewise an odde number Three times 7. is 21. Flussatus geueth this definition following of this kinde of number which is all one in substance with the former definition A number odly odde it that which onely an odde number doth measure As 15. for no number measureth 15. but onely 5. and 3 also 25 none measureth it but onely 5. which is an odde number and so of others 12 A prime or first number is that which onely vnitie doth measure As 5.7.11.13 For no number measureth 5 but onely vnitie For v. vnities make the number 5. So no number measureth 7 but onely vnitie .2 taken 3. times maketh 6. which is lesse then 7 and 2. taken 4. times is 8 which is more then 7. And so of 11.13 and such others So that all prime numbers which also are called first numbers and numbers vncomposed haue
no part to measure theÌ„ but onely vnitie 13 Numbers prime the one to the other are they which onely vnitie doth measure being a common measure to them As 15. and 22. be numbers prime the one to the other .15 of it selfe is no prime number for not onely vnitie doth measure it but also the numbeâ—â— 5. and 3 for â— times 5. is lx Likewise 22. is of it selfe no prime number for it is measured by 2. and 11 besides vnitie For 11. twise or 2. eleuen times make 22. So that although neither of these two numbers 15. and 22. be a prime or incomposed number but eyther haue partes of his owne whereby it may be measured beside vnitie yet compared together they are prime the one to the other for no one number doth as a common measure measure both of them but onely vnitie which is a common measure to all numbers The numbers 5. and 3. which measure 15. will not measure 22 againe the numbers 2. and 11. which measure 22 do not measure 15. 14 A number composed is that which some one number measureth A number composed is not measured onely by vnitie as was a prime number but hath some number which measureth it As 15 the number 3. measureth 15 namely taken 5. times Also the number 5. measureth 15 namely taken 3. times 5. times â— and 3. times 5 is 15. Likewise 18. is a composed number it is measured by these numbers 6.3.9.2 and so of others These numbers are also called commonly second numbers as contrary to prime or first numbers 15 Numbers composed the one to the other are they which some one number being a common measure to them both doth measure As 12. and 8. are two composed numbers the one to the other For the number 4 is a common measure to them both 4. taken three times maketh 12 and the same 4. taken two tymes maketh 8. So are 9. and 15 3. measureth them both Also 10. and 25 for 5. measureth both of them and so infinitely of others In thys do numbers composed the one to the other or second numbers differre from numbers prime the one to the other for that two numbers being composed the one to the other ech of them seuerally is of necessitie a composed number As in the examples before 8. and 12. are composed numbers likewise 9. and 15 also 10 and 25 but if they be two numbers prime the one to the other â—it is not of necessitie that ech of them seuerally be a prime number As 9. and 22. are two numbers prime the one to the other no one number measureth both of them and yet neither of them in it selfe and in his owne nature is a prime number but ech of them is a composed number For 3. measureth 9 and 11. and 2. measure 22. 16 A number is sayd to multiply a number when the number multiplyed is so oftentimes added to it selfe as there are in the number multiplying vnities and an other number is produced In multiplication are euer required two numbers the one is whereby ye multiply commonly called the multiplier or multiplicant the other is that which is multiplied The number by which an other is multiplied namely the multiplyer is sayd to multiply As if ye will multiply 4. by 3 then is three sayd to multiply 4 therefore according to this definition because in 3. there are three vnities adde 4,3 times to it selfe saying 3. times 4 so shall ye bring forth an other number namely 12 which is the summe produced of that multiplication and so of all other multiplications 17 When two numbers multiplying them selues the one the other produce an other the number produced is called a plaine or superficiall number And the numbers which muliply them selues the one by the other are the sides of that number As let these two numbers 3. and 6 multiply the one the other saying 3. times 6 or sixe tymes 3 they shall produce 18. Thys number 18. thus produced is called a plaine number or a superficiall number And the two multiplying numbers which produced 12 namely 3. and 6 are the sides of the same superficiall or plaine number that is the length and breadth thereof Likewise if 9. multiply 11 or eleuen nine there shal be produced 99. a plaine number whose sides are the two numbers 9. and 11 â— as the length and breadth of the same They are called plaine and superficiall numbers because being described by their vnities on a plaine superficies they represent some superficiall forme or figure Geometricall hauing length and breadth As ye see of this example and so of others And all such plaine or superficiall numbers do euer represent right angled figures as appeareth in the example 18 When three numbers multiplyed together the one into the other produce any number the number produced is called a solide number and the numbers multiplying them selues the one into the other are the sides therof As taking these three nuÌ„bers 3.4 5. multiply the one into the other First 4. into 5. saying foure times 5. is 20 then multiply that number produced nameâ—y 20. into 3 which is the third number so shall ye produce 60. which is a solide number and the three numbers which produced the number namely 3.4 and 5. are the sides of the same And they are called solide numbers because being described by their vnities they represent solide and bodylicke figures of Geometry which haue length breadth and thicknes As ye see this number 60. expressed here by hys vnities Whose length is hys side 5 his breadth is 3 and thicknes 4. And thus may ye do of all other three nuÌ„bers multiplying the one the other 19 A square number is that which is equally equall or that which is contayned vnder two equall numbers As multiply two equall numbers the one into the other As 9. by 9. ye shall produce 81 which is a square number Euclide calleth it a number equally equall because it is produced of the multiplication of two equall numbers the one into the other Which numbers are also sayd in the second definition to contayne a square number As in the definitions of the second booke two lines are sayd to containe a square or a parallelogramme figure It is called a square number because being described by his vnities it representeth the figure of a square in Geometry As ye here see doth the number 81. whose sides that is to say whose length and breadth are 9. and 9 equall numbers which also are sayd to contayne the square number 81 and so of others 20 A cube number is that which is equally equall equally or that which is contayned vnder three equall numbers As multiply three equall numbers the one into the other as 9 9 and 9 first 9. by 9 so shall ye haue 81 which agayne multiply by 9 so shall ye produce 729. which is a cube number And Euclide calleth
it a number equally equall equally because it is produced of the multiplication of three equall numbers the one into the other which three numbers are sayd in the second definition wherein he speaketh more applying to Geometry to contayne the cube number It is therefore called a cube number because being described by hys vnities it representeth the forme of a cube in Geometry whose sides that is to say whose length breadth and thicknes are the three equall numbers 9 9 and 9 of which he was produced which three sides also are sayd to containe the cube number 729 beholde here the description therof 21 Numbers proportionall are when the first is to the second equemultiplex as the third is to the fourth or the selfe same part or the selfe same partes Here he defineth which numbers are called proportionall that is what numbers haue one and the selfe same proportion For example 6. to 3 and 4. to 2 are numbers proportionall and haue one and thâ— self same proportion for 6. the first is to 3. the secoÌ„d equemultiplex as 4. the third is to 2. the fourth â— is double to 3 and so is 4. double to 2. Likewise these foure numbers are in like proportion 3 9.4.1â— for what part 3. is of 9 such part is 4. of 12 â—3 of 9. is a third part so is also 4. of 12. a third part So are these foure numbers also in proportion â— 5 4. 10 what partes 2. are of 5 such partes are 4. of 10 2 of 5 are two â—ift partes likewise 4. of 10 are two fift partes Moreouer these numbers 8.6 1â— 9 be in proportion for what and how many parts 8. are of 6 such so many parts are 12. of 9 8. of 6 is foure third partes for one third part of 6. is 2 which taken foure times maketh 8 so 12. of 9 is also foure thyrd partes for one third part of 9. is 3 which taken foure times make 1â— And so conceaue ye of all other proportiâ—â—â—ll numbers In the 〈◊〉 definitioâ— of the v. booke Euclide gaue a â—arre other definition of magnitudes proportionall and much vnlike to this which he here geueth of numbers proportionall the reason is as there also was partly noted for that there he gaue a definition common to all quantities discrete and continuall rationall and irrationall and therefore was constrayned to geue the definition by the excesse equalitie or want of their equemultiplices and that generally onely for that irrationall quantities haue no certayne part or common measure to be measured by or knowen neyther can they be expressed by any certayne numbers But here in this place because in numbers there are no irrationall quantities but all are certaynly knowen so that both they and the proportions betwene them may be expressed by numbers certayne and knowen by reason of their partes certayne and for that they haue some common measure to measure them at the lest vnitie which is a common measure to all numbers he geueth here this definition of proportionall numbers by that the one is like equemultiplex to the other or the same part or the same partes which definition is much easier then was the other and is not so large as is the other neither extendeth it self generally to all kinde of quantities rationall and irrationall but contayneth it selfe within the limites and bondes of rationall quantitie and numbers 22 Like plaine numbers and like solide numbers are such which haue their sides proportionall Before he shewed that a plaine number hath two sides and a solide number three sides Now he sheweth by this definition which plaine numbers and which solide numbers are like the one to the other The likenes of which numbers dependeth altogether of the proportions of the sides of these numbers So that if the two sides of one plaine number haue the same proportion the one to the other that the two sides of the other plaine number haue the one to the other then are such two plaine numbers like For an example 6 and 24 be two plaine numbers the sides of 6 be 2 and 3 two tymes 3 make 6 the sides of 24 be 4 and 6 foure times 6 makes 24. Againe the same proportion that is betwene 3 and 2 the sides of 6. is also betwene 6 and 4 the sides of 24. Wherfore 24 and 6 be two like plaine and superficiall numbers And so of other plaine numbers After the same manner is it in solide numbers If three sides of the one be in like proportion together as are the three sides of the other then is the one solide number like to the other As 24 and 192 be solide numbers the sides of 24. are 2. 3. and 4 two tymes there taken 4 times are 24. the sides of 192 are 4.6 and 8 for foure tymes 6. 8 times make 192. Againe the proportion of 4 to 3 is sesquitercia the proportion of 3 to 2 is sesquialtera which are the proportions of the sides of the one solide number namely of 24 the proportion betwene 8 and 6 is sesquâ—â—ercia the proportion betwene 6 and 4 is sesquialtera which are the proportions of the sides of the other solide number namely of 191. And they are one and the same with the proportione of the 〈◊〉 of the other wherfore thâ—se two solide numbers 24 and 192 be like and so of other solide nuÌ„bers 23 A perfect number is that which is equall to all his partes As the partes of 6 are 1. 2. 3. three is the halfe of 6 two the third part and 1. the sixth part and mo pâ—rtes 6 hath not which thrâ—â— paâ—tes 1. â— 3 added together make 6 the whole number whose partes they are Wherfore 6 is a perfect number So likewise is 28 a perfect number the partes whereof are these numbers 14. 7. 2. and 1 14 is the halfe therof 7 is the quarter 4 is the seuenth part 2 is a fourtenth part and 1 an 28 part and these are all the partes of 28. all which namely 1 2 3 4 7 and 14 added together make iustly without more or lâ—sse 28. Wherfore 〈…〉 a perfect number and so of others the like This kinde of numbers is very rare and seldome found From 1 to 10 there is but one perfect number namely 6. From 10 to an 100 there is also but one that is 28. Also from 100 to 1000 there is but one which is 496. From 1000 to 10000 likewise but one So that betwene euery sâ—ay in numbring which is euer in the tenth place there is found but one perfect number And for their rarenes and great perfection they are of maruelous vse in magike and in the secret part of philosophy This kinde of number is called perfect in respect 〈…〉 numbers which are iâ—perfect For as the nature of a perfect number standeth in this that all partes added together are equall to the whole and make the whole so in an imperfect nuÌ„ber all
other number DE is of an other nuÌ„ber F and let AB be lesse then DE. Then I say that alternately also what part or partes AB is of DE the selfe same partes or part is C of F. Forasmuch as what partes AB is of C the selfe same partes is DE of F therefore how many partes of C there are in AB so many partes of F also are there in DE. Deuide AB into the partes of C that is into AG and GB And likewise DE into the partes of F that is DH and HE. Now then the multitude of these AG and GB is equall vnto the multitude of these DH and HE. And forasmuch as what part AG is of C the selfe same part is DH of F therefore alternately also by the former what part or partes AG is of DH the selfe same part or partes is C of F. And by the same reason also what part or partes GB is of HE the same part or partes is C of F. Wherefore what part or partes AG is of DH the selfe same part or partes is AB of DE by the 6. of the seuenth But what part or partes AG is of DH the selfe same part or partes is it proued that C is of F. Wherefore what partes or part AB is of D E the selfe same partes or part is C of F which was required to be proued ¶ The 9. Theoreme The 11. Proposition If the whole be to the whole as a part taken away is to a part taken away then shall the residue be vnto the residue as the whole is to the whole SVppose that the whole number AB be vnto the whole number CD as the part takeÌ„ away AE is to the part takeÌ„ away CF. TheÌ„ I say that the residue EB is to the residue FD as the whole AB is to the whole CD For forasmuch as AB is to CD as AE is to CF therfore what part or partes AB is of CD the selfe same part or partes is AE of CF. Wherfore also the residue EB is of the residue FD by the 8. of the seuenth the selfe same parte oâ— partes that AB is of CD Wherefore also by the 21. definition of this booke as EB is to FD so is AB to CD which was required to be proued ¶ The 10. Theoreme The 12. Proposition If there be a multitude of numbers how many soeuer proportionall as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes SVppose that there be a multitude of nuÌ„bers how many soeuer proportional namely A B C D so that as A is to B so let C be to D. Then I say that as one of the antecedentes namely A is to one of the consequentes namely to B or as C is to D so are all the antecedentes namely A and C to all the consequentes namely to B and D. For forasmuch as by supposition as A is to B so is C to D therfore what parte or partes A is of B the selfe same part or partes is C of D by the 21. definition of this booke wherefore alternately what part or partes A is of C the selfe same parte or partes is B of D by the ninth and tenth of the seuenth wherefore both these numbers added together A and C are of both these numbers B and D added together the selfe same part or partes that A is of B by the 5. and 6. of the seuenth wherfore by the 21. definition of the seuenth as one of the antecedents namely A is to one of the consequentes namely to B so are all the antecedentes A and C to all the consequentes B D. Which was required to be proued ¶ The 11. Theoreme The 13. Proposition If there be foure numbers proportionall then alternately also they shall be proportionall SVppose that there be foure numbers proportional A B C D so that as A is to B so let C be to D. Then I say that alternately also they shal be proportional that is as A is to C so is B to D. For forasmuch as by supposition as A is to B so is C to D therfore by the 21. definition of this booke what part or partes A is of B the selfe same part or partes is C of D. Therfore alternately what part or partes A is of C the selfe same part or partes is B of D by the 9. of the seuenth also by the 10. of the same wherfore as A is to C so is B to D by the 21. definition of this booke which was required to be proued Here is to be noted that although in the foresayd example and demonstration the number A be supposed to be lesse then the number B and so the number C is lesse then the number D yet will the same serue also though A be supposed to be greater then B wherby also C shall be greater then D as in thâ—s example here put For for that by supposition as A is to B so is C to D and A is supposed to be greater then B and C greater then D therefore by the 21. definition of this Booke how multiplex A is to B so multiplex is C to D and therefore what part or partes B is of A the selfe same part or partes is D of C. Wherefore alternately what part or partes B is of D the selfe same part or partes is A of C and therefore by the same definition B is to D as A is to C. And so must you vnderstand of the former Proposition next going before ¶ The 12. Theoreme The 14. Proposition If there be a multitude of numbers how many soeuer and also other numbers equall vnto them in multitude which being compared two and two are in one and the same proportion they shall also of equalitie be in one and the same proportion SVppose that there be a multitude of numbers how many soeuer namely A B C and let the other numbers equall vnto them in multitude be D E F which being compared two and two let be in one and the same proportion that is as A to B so let D be to E and as B is to C so let E be to F. Then I say that of equalitie as A is to C so is D to F. For forasmuch as by supposition as A is to B so is D to E therefore alternately also by the 13 of the seuenth as A is to D so is B to E. Againe for that as B is to C so is E to F therfore alternately also by the self same as B is to E so is C to F. But as B is to E so is A to D. VVherfore by the seuenth common sentence of the seuenth as A is to D so is C to F. Wherâ—ore alternately by the 13. of the seuenth as A is to C so is D to F which
was required to be demonstrated After this Proposition Campane demonstrateth in numbers these foure kindes of proportionalitie namely proportion conuerse composed deuided and euerse which were in continual quantitie demonstrated in the 4. 17. 18. and 19. propositions of the fift booke And first he demonstrateth conuerse proportion in this maner But if A be greater then B C also is greater then D and what part or partes B is of A the selfe same part or partes is D of C. Wherefore by the same definition as B is to A so iâ— D to C which was required to be proued Proportionalitie deuided is thus demonstrated Suppose that the number AB be to the number B as the number CD is to the number D. Then I say that deuided also as A is to B so is C to D. For for that as AB is to B so is CD to D therâ—fore alternately by the 14. of this booke as AB is to CD so is B to D. Wherefore by the 11. of this booke as AB is to CD so is A to C. Wherefore as B is to D so is A to C and for that as A is to C so is B to D theâ—efore alternately as A is to B so is C to D. Proportionalitie composed is thus demonstrated If A be vnto B as C is to D then shall AB be to B as CD is to D. For alternately A is to C as B is to D. Wherefore by the 13. of this booke as AB namely all the antecedentes are to CD namely to all the consequentes so is B to D namely one of the antecedentes to one of the consequentes Wherfore alteâ—nately as AB is to B so is CD to D. Euerse proportionalitie is thus proued Supposâ— that AB be to B as CD is to D then shall AB be to A as CD is to C. For alternately AB is to CD aâ— B is to D. Wherefâ—râ— by the 13. of this booâ—â— Aâ— is â— CD as A is to C. Wherefore alternately AB iâ— to A aâ— CD iâ— to C whiâ—h was required to be proued ¶ A proportion here added by Campane If the proportion of the first to the second be as the proportion of the third to the foâ—rth and if the proportiâ—n of â—he fift to the second be as the propâ—rtion of the sixt to the fourth then the proportion of the first and the fifth taken together shall be to the second as the proportion â—f the third and the sixt taken together to the fourth And after the same maner may you proue the conuerse of this Proposition If B be to A as D is to Câ— and if also B be vnto E as D is to F Then shall B be to AE as D is to CF. For by conuerse proportionalitie A is to B as C is to D. Wherefore of equalitie A is to E as C is to F. Wherefore by composition A and E are to E as C and F are to F. Wherefore conuersedly E is to A and E as F is to C and F. But by supposition B is to E as D is to F. Wherefore agayne by Proportion of equalitie B is to A and E as D is to C and F which was required to be proued A Corollary By this also it is manifest that if the proportion of numbers how many soeuer vnto the first be as the proportion of as many other numbers vnto the second then shall the proportion of the numbers composed of all the numbers that were antecedentes to the first be to the first as the number composed of all the numbers that were antecedentes to the second is to the second And also conuersedly if the proportion of the first to nuÌ„bers how many soeuer be as the proportion of the second to as many other numbers then shall the proportion of the first to the number composed of all the numbers that were consequentes to it selfe be as the proportion of the second to the number composed of all the numbers that were consequenâ—es to it selfe ¶ The 13. Theoreme The 15. Proposition If vnitie measure any number and an other number do so many times measure an other number vnitie also shall alternately so many times measure the third number as the second doth the fourth SVppose that vnitie A do measure the number BC and let an other nuÌ„ber D so many times measure some other nuÌ„ber namely EF. Then I say that alternately vnitie A shall so many times measure the number D as the number BC doth measure the number EF. For forasmuch as vnitie A doth so many times measure BC as D doth EF therefore how many vnities there are in BC so many numbers are there in EF equall vnto D. Deuide I say BC into the vnities which are in it that is into BG GH and HC And deuide likewise EF into the numbers equall vnto D that is into EK KL and LF Now then the multitude of these BG GH and HC is equall vnto the multitude of these EK KL LF And forasmuch as these vnities BG GH and HC are equall the one to the other and these numbers EK KL LF are also equall the one to the other and the multitude of the vnities BG GH and HC are equall vnto the multitude of the numbers EK KL LF therefore as vnitie BG is to the number EK so is vnitie GH to the number KL and also vnitie HC to the number LF Wherfore by the 12â— of the seueâ—th as one of the antecedeâ—tâ—â—â—s to one of the consequentes so are all the antecedenâ—es to all the consequentes Wherfore as vnitie BG is to the number EK so is the number BC to the number EF. But vnitie BG is equall vnto vnitie A and the number EK to the number D. VVherefore by the 7. common sentence as vnitie A is to the number D so is the number BC to the number EF. VVherefore vnitie A measureth the nuÌ„ber D so many times as BC measureth EF by the 21 definition of this booke which was required to be proued ¶ The 14. Theoreme The 16. Proposition If two numbers multiplying them selues the one into the other produce any numbers the numbers produced shall be equall the one into the other SVppose that there be two numbers A and B and let A multiplying B produce C and let B multiplying A produce D. Thâ—n I say that the number Câ— equall vnto the nâ—mber D. Take any vnitie namelyâ— E. And forasmuch as A multiplying B produced C therefore B measureth C by the vnities which are in A. And vnitie E measureth the number A by those vnities which are in the number A. VVhereâ—ore vnitie E so many times measureth A as B measureth C. VVherefore alternately by the 15. of the seuenth vnitie E measureth the number B so many times as A measureth C. Againe for that B multiplying A produced D therefore A measureth D by thâ— vnities which are in B. And vnitie E
measureth B by the vnities which are in B. VVherefore vnitie E so many times measureth the number B as A measureth D. But vnitie E so many times measureth the number B as A measureth C. VVherefore A measureth either of these numbers C and D a like VVherefore by the 3. common sâ—ntence of this booke C is equall vnto D which was required to be demonstrated The 15. Theoreme The 17. Proposition If one number multiply two numbers and produce other numbers the numbers produced of them shall be in the selfe same proportion that the numbers multiplied are SVppose that the number A multiplieng two numbâ—rs B and C do produce the numbers D and E. Then I say that as B is to C so is D to E. Take vnitie namely F. And â—orasmuch as A multiplieng B produced D therfore B measureth D by those vnities that are in A. And vnitie F measureth A by thâ—sâ— vâ—itiâ—â— whih are in A. Wherfore vnitie F so many times measureth the number A as B measureth D. VVherfore as vnitie I is to the number A. so is the number B to the number D by the 21 definition of this booke And by the same reason as vnitie F is to the number A so is the number C to the number E wherefore also by the 7. common sentence of this booke as B is to D so is C to E. VVherfore alternately by the 15. of the seuenth as B is to C so is D to E. If therfore one number multiply two numbersâ— and produce other numbers the numbers produced of them shall be in the selfe same proportion that the numbers multiplied are which was required to be proued Here Fluâ—â—tes addeâ—h thiâ— Coâ—ollary If two numberâ— hauing one and the samâ— proportiâ—â— with two other numbers do multiply thâ— oâ—e the other alternately and produce any numbers the numbers produced of them shall be equall the one to the other Suppose that there be two numberâ— â— and B and also two other numbers C and D hauing thâ— same proportion that the numbers A and B haue and let the numbers A and B multiply the numberâ— C D alternately that is let A multiplieng D produce F and let B multiplieng C produce E. Then I say that the numbers produced namely E F are equall Let A and B multiply the one the other in such sort that let A multiplieng B produce G and let B multiplieng A produce H Now then the numbers G and H are equal by the 16. of this bookeâ— And forasmuch as A multipliâ—ng the two numbers B and D produced the numbers G and F therfore G is to â— as B is to D by this proposition So likewise B multiplieng the two numbers A and C produced the two numbers H and E. Wherfore by the same H is to E as A is to C. But alternately by the 13. of this booke A is to C as B is to D but as A is to C so is H to E and as â— is to D so is G to â— Wherfore by the seuenth common sentence as H is to E â— so is G to F. Wherfore aâ—ternately by the 13. of this booke H is to G as E is to F. But it is proued that G H are equall Wherfore E and F which haue the same proportion that A and B haue are equall If therefore there be two numbers c. Which was required to be proued ¶ The 16. Theoreme The 18. Proposition If two numbers multiply any number produce other numbers the numbers of them produced shall be in the same proportion that the numbers multiplying are SVppose that two numbers A and B multiplieng the number C doo produce the numbers D and E. Then I say that as A is to B so is D to E. For forasmuch as A multiplieng C produced D therfore C multiplieng A produceth also D by the 16. of this booke And by the same reason C multiplieng B produceth E. Now then one number C multiplieng two numbers A and B produceth the numbers D and E. VVherfore by the 17. of the seuenth as A is to B so is D to E which was required to be demonstrated This Proposition and the former touching two numbers may be extended to numbers how many soeuer So that if one number multiply numbers how many soeuer and produce any numbers the proportion of the numbers produced and of the numbers multiplied shall be one and the selfe same Likewise if numbers how many soeuer multiply one number and produce any numbers the proportion of the numbers producedâ— and of the numbers multiplieng shall be one and the selfe same which thing by this and the former proposition repeted as often as is needefull is not hard to proue ¶ The 17. Theoreme The 19. Proposition If there be foure numbers in proportion the number produced of the first and the fourth is equall to that number which is produced of the second and the third And if the number which is produced of the first and the fourth be equall to that which is produced of the second the third those foure numbers shall be in proportion But now againe suppose that E be equall vnto F. Then I say that as A is to B so is C to D. For the same order of construction remayning still forasmuch as A multiplieng C D produced G and E therfore by the 17. of the seuenth as C is to D so is G to E but E is equall vnto F But if two numbers be equall one number shall haue vnto them onâ— and the same proportion wherfore as G is to E so is G to F. But as G is to E so is C to D. Wherefore as C is to D so is G to F but as G is to F so is A to B by the 18. of the seuenth wherfore as A is to B so is C to D which was required to be proued Here Campane addeth that it is needeles to demonstrate that if one number haue to two numbers one and the same proportion the said two numbers shall be equall or that if they be equal one number hath to them one and the same proportion For saith he if G haue vnto E and F one and the same proportion theÌ„ either what part or partes G is to E the same part or parts is G also of F or how multiplex G is to E so multiplex is G to F by the 21. definition And therfore by the 2 and 3 common sentence the said numbers shall be equall And so conuersedly if the two numbers E and F be equal then shall the numbers E and F be either the selfe same parte or partes of the number G or they shall be equemultiplices vnto it And therfore by the same definition the number G shall haue to the numbers E and F one and the same proportion ¶ The 18. Theoreme The 20. Proposition If there be three numbers in proportion the number
all those numbers A B C D vnitie added together and let E multiplying D which is the last number produce the number FG. Then I say that FG is a perfect number How many in multitude A B C D are so many in continuall double proportion take beginning at E which let be the numbers E HK L and M. VVherefore of equalitie by the 13. of the seuenth as A is to D so is E to M. VVherefore that which is produced of E into D is equall to that which is produced of A into M. But that which is produced of E into D is the number FG. VVherefore that which is produced of A into M is equall vnto FG. VVherefore A multiplying M produceth FG. VVherefore M measureth FG by those vnities which are in A. But A is the number two VVherefore FG is double to M. And the numbers M L HK and E are also in continuall double proportion VVherefore all the numbers E HK L M and FG are continually proportionall in double proportion Take from the second number KH and from the last FG a number equall vnto the first namely to E and let those numbers taken be HN FX VVherefore by the Proposition going before as the excesse of the second number is to the first number so is the excesse of the last to all the numbers going before it VVherefore as NK is to E so is XG to these numbers M L KH and E. But NK is equall vnto E for it is the halfe of HK which is supposed to be double to E VVherefore XG is equall vnto these numbers M L HK and E. But XF is equall vnto E and E is equall vnto these numbers A B C D and vnto vnitie Wherfore the whole number FG is equall vnto these numbers E HK L M and also vnto these numbers A B C D and vnto vnitie Moreouer I say that vnitie and all the numbers A B C D E HK L and M do measure the number FG. That vnitie measureth it it needeth no proufe And forasmuch as FG is produced of D into E therefore D and E do measure it And forasmuch as the double from vnitie namely the nūbers A B C do measure the number D by the 13. of this booke therefore they shall also measure the number FG whom D measureth by the ◠common sentence By the same reason forasmuch as the nūbers E HK L and M are vnto FG as vnitie and the numbers A ◠C are vnto D namely in subduple proportion and vnitie and the numbers A B C do 〈◊〉 D therefore also the numbers E HK L and M shall measure the number FG No◠I say also that no other number measureth FG besides these numbers A B C D E HK L M and vnitie For if it be possible let O measure FG. And let O not be any of these numbers A B C D E HK L and M. And how often O measureth FG 〈◊〉 vnities let there be in P. Wherefore O multiplying P produceth FG. But E also multiplying D produced FG. Wherefore by the 19. of the seuenth as E is to O so is P to D. Wherefore alternately by the 9. of the seuenth as E is to P so is O to D. And forasmuch as from vnitie are these numbers in continuall proportion A B C D and the number A which is next after vnitie is a prime number therfore by the 13. of the ninth no other number measureth D besides the numbers A B C. And it is supposed that O is not one and the same with any of these nūbers A B C. Wherefore O measureth not D. But as O is to D so is E to P. Wherefore neither doth E measure P. And E is a prime number But by the 31. of the seuenth euery prime number is to euery number that it measureth not a prime number Wherefore E and P are prime the one to the other yea they are prime and the least But by the 21. of the seuenth the least measure the numbers that haue one and the same proportion with them equally the antecedent the antecedent and the consequent the consequent And as E is to P so is O to D. Wherefore how many times E measureth O so many times P measureth D. But no other number measureth D besides the numbers A B C by the 13. of this booke Wherefore P is one and the same with one of these numbers A B C. Suppose that P be one and the same with B how many B C D are in multitude so many take from E vpward namely E HK and L. But E HK and L are in the same proportion that B C D are VVherefore of equalitie as B is to D so is E to L. VVherefore that which is produced of B into L is equall to that which is produced of D into E. But that which is produced of D into E is equall to that which is produced of P into O. VVherefore that which is produced of P into O is equall to that which is produced of B into L. VVherefore as P is to B so is L to O and P is one the same with B wherefore L also is one and the same with O which is impossible For O is supposed not to be one and the same with any of the numbers geuen VVherefore no number measureth FG besides these numbers A B C D E HK L M and vnitie And it is proued that FG is equall vnto these numbers A B C D E HK L M and vnitie which are the partes therof by the 39. of the seuenth But a perfect nūber by the definition is that which is equall vnto all his partes VVherfore FG is a perfect number◠which was required to be proued The end of the ninth booke of Euclides Elementes ¶ The tenth booke of Euclides Elementes IN THIS TENTH BOOKE doth Euclide entreat of lines and other magnitudes rationall irrationall but chiefly of irrationall magnitudes commensurable and incommensurable of which hitherto in al his former 9. bokes he hath made no mention at all And herein differeth number from magnitude or Arithmetike from Geometry for that although in Arithmetike certayne numbers be called prime numbers in consideration of themselues or in respect of an other and so are called incommensurable for that no one number measureth them but onely vnitie Yet in dede and to speake absolutely and truely there are no two numbers incommensurable but haue one common measure which measureth thē both if none other yet haue they vnitie which is a common part and measure to all numbers and all numbers are made of vnities as of their partes As hath before bene shewed in the declaration of the definitiōs of the seuenth booke But in magnitude it is farre otherwise for although many lines plaine figures and bodies are commensurable and may haue one measure to measure them yet all haue not so nor can haue For that a line is not made of pointes as
number is made of vnities and therfore cannot a point be a common part of all lines and measure them as vnitie is a common part of all numbers and measureth them Vnitie taken certayne tymes maketh any number For there are not in any number infinite vnities but a point taken certayne tymes yea as often as ye list neuer maketh any line for that in euery line there are infinite pointes Wherfore lines figures and bodies in Geometry are oftentymes incommensurable and irrationall Now which are rationall and which irrationall which commensurable and which incommensurable how many and how sundry sortes and kindes there are of them what are their natures passions and properties doth Euclide most manifestly shew in this booke and demonstrate them most exactly This tenth booke hath euer hitherto of all men and is yet thought accompted to be the hardest booke to vnderstand of all the bookes of Euclide Which coÌ„mon receiued opinion hath caused many to shrinke and hath as it were deterred them from the handeling and treatie thereof There haue bene in deede in times past and are presently in these our dayes many which haue delt and haue taken great and good diligence in commenting amending and restoryng of the sixe first bookes of Euclide and there haue stayed themselues and gone no farther beyng deterred and made afrayde as it seemeth by the opinion of the hardnes of this booke to passe forth to the bookes following Truth it is that this booke hath in it somewhat an other strauÌ„ger maner of matter entreated of theÌ„ the other bokes before had and the demonstrations also thereof the order seeme likewise at the first somewhat straunge and vnaccustomed which thinges may seeme also to cause the obscuritie therof and to feare away many from the reading and diligent study of the same so much that many of the well learned haue much complayned of the darkenes and difficultie thereof and haue thought it a very hard thing and in maner impossible to attayne to the right and full vnderstanding of this booke without the ayde and helpe of some other knowledge and learnyng and chiefly without the knowledge of that more secret and subtill part of Arithmetike commonly called Algebra which vndoubtedly first well had and knowne would geue great light therunto yet certainly may this booke very well be entred into and fully vnderstand without any straunge helpe or succour onely by diligent obseruation of the order and course of Euclides writinges So that he which diligently hath perused and fully vnderstandeth the 9. bookes goyng before and marketh also earnestly the principles and definitions of this â—enth booke he shal well perceiue that Euclide is of himselfe a sufficient teacher and instructer and needeth not the helpe of any other and shall soone see that this tenth booke is not of such hardnes and obscuritie as it hath bene hetherto thought Yea I doubt not but that by the trauell and industry taken in this translation and by addicions and emendations gotten of others there shall appeare in it no hardnes at all but shall be as easie as the rest of his bookes are Definitions 1 Magnitudes commensurable are suchwhich one and the selfe same measure doth measure First he sheweth what magnitudes are commensurable one to an other To the better and more cleare vnderstanding of this definition note that that measure whereby any magnitude is measured is lesse then the magnitude which it measureth or at least equall vnto it For the greater can by no meanes measure the lesse Farther it behoueth that that measure if it be equall to that which is measured taken once make the magnitude which is measured if it be lesse then oftentimes taken and repeted it must precisely render and make the magnitude which it measureth Which thing in numbers is easely sene for that as was before said all numbers are commensurable one to an other And although Euclide in this definition comprehendeth purposedly onely magnitudes which are continuall quantities as are lines superficieces and bodies yet vndoubtedly the explication of this and such like places is aptly to be sought of numbers as well rationall as irrationall For that all quantities commensurable haue that proportion the one to the other which number hath to numbers In numbers therfore 9 and 12 are commensurable because there is one common measure which measureth them both namely the number 3. First it measureth 12 for it is lesse then 12. and being taken certaine times namely 4 times it maketh exactly 12 3 times 4 is 12 it also measureth 9 for it is lesse then 9. and also taken certaine times namely 3 times it maketh precisely 9 3 times 3 is 9. Likewise is it in magnitudes if one magnitude measure two other magnitudes those two magnitudes so measured are said to be commensurable As for example if the line C being doubled do make the line B and the same lyne C tripled do make the line A then are the two lines A and B lines or magnitudes commensurable For that one measure namely the line C measureth theÌ„ both First the line C is lesse theÌ„ the line A and alsolesse theÌ„ the line B also the line C taken or repeted certaine times namely 3 times maketh precisely the line A and the same line C taken also certain times namely two times maketh precisely the line B. So that the line C is a common measure to them both and doth measure them both And therfore are the two lines A and B lines commensurable And so imagine ye of magnitudes of other kyndes as of superficiall figures and also of bodies 2 Incommensurable magnitudes are such which no one common measure doth measure This diffinition neadeth no explanation at all it is easely vnderstanded by the diffinition going before of lines commensurable For contraries are made manifest by comparing of the one to the other as if the line C or any other line oftentimes iterated doo not render precisely the line A nor the line B theÌ„ are the lines A and B incommensurable Also if the line C or any other line certayne times repeted doo exactly render the line A and doo not measure the line B or if it measure the line B and measureth not also the line A the lines A and B are yet lines incoÌ„mensurable so of other magnitudes as of superficieces and bodyes 3 Right lines commensurable in power are such whose squares one and the selfe same superficies area or plat doth measure To the declaration of this diffinition we must first call to minde what is vnderstanded ment by the power of a line which as we haue before in the former bookes noted is nothing ells but the square thereof or any other plaine figure equall to the square therof And so great power habilitie â—s a line said to haue as is the quantitie of the square which it is able to describe or a figure superficial equal to the square
therof 〈◊〉 This iâ— also to be noted that of lines some are commensurable in length the one to the other and some are commensurable the one to the other in power Of lines commensurable in length the one to the other was geuen an example in the declaration of the first diffinitioÌ„ namely the lines A and B which were commensurable in length one and the selfe measure namely the line C measured the length of either of them Of the other kinde is geuen this diffinition here set for the opening of which take this example Let there be a certaine line namely the line BC and let the square of that line be the square BCDE Suppose also an other line namely the line FH let the square thereof be the square FHIK and let a certayne superficies namely the superficies A measure the square BCDE taken 16. times which is the number of the litle areas squares plats or superficieces coÌ„tained and described within the sayd squares ech of which is equall to the superficie A. Agayne let the same superficies A measure the square FHIK 9. times taken according to the number of the fieldâ—s or superficieces contayned and described in the same Ye see theÌ„ that one and the selfe same superficies namely the superficies A is a common measure to both these squares and by certayne repeticions thereof measureth them both Wherefore the two lines BC and FH which are the sides or lines producing these squares and whose powers these squares are are by this diffinition lines commensurable in power 4 Lines incommensurable are such whose squares no one plat or superficies doth measure This diffinition is easy to be vnderstanded by that which was sayd in the diffinition last set before this and neadeth no farther declaration And thereof take this example If neither the superficies A nor any other superficies doo measure the two squares B CDE and FHIK or if it measure the one â—â—rely BCDE and not the other FHIK or if it measure the square FHIK and not the square BCDE the two lines BC and FH are in power incommensurable and therfore also incommeÌ„surable in length For whatsoeuer lines are incommeÌ„surable in power the same are also incommensurable in length as shall afterward in the 9. proposition of this booke be proued And therfore such lines are here defined to be absolutely incommensurable These thinges thus standing it may easely appeare that if a line be assigned and layd before vs there may be innumerable other lines commensurable vnto it and other incommensurable vnto it of commensurable lines some are commensurable in length and power and some in power onely 5 And that right line so set forth is called a rationall line Thus may ye see how to the supposed line first set may be compared infinite lines some commensurable both in length power and some commensurable in power onely and incommensurable in length and some incommensurable both in power in length And this first line so set whereunto and to whose squares the other lines and their squares are compared is called a rationall line commonly of the most part of writers But some there are which mislike that it should be called a rationall line that not without iust cause In the Greeke copy it is called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 rete which signifieth a thing that may be spokeÌ„ expressed by word a thing certayne graunted and appoynted Wherefore Flussates a man which bestowed great trauell and diligence in restoring of these elementes of Euclide leauing this word rationall calleth this line supposed and first set a line certaine because the partes thereof into which it is deuided are certaine and known and may be expressed by voyce and also be coumpted by number other lines being to this line incommensurable whose parts are not distinctly known but are vncertayne nor can be expressed by name nor assignd by number which are of other men called irrationall he calleth vncertaine and surd lines Petrus Montaureus although he doth not very wel like of the name yet he altereth it not but vseth it in al his booke Likewise wil we doo here for that the word hath bene and is so vniuersally receiued And therefore will we vse the same name and call it a rationall line For it is not so great a matter what names we geue to thinges so that we fully vnderstand the thinges which the names signifie This rationall line thus here defined is the ground and foundation of all the propositions almost of this whole tenth booke And chiefly from the tenth proposition forwardes So that vnlesse ye first place this rationall line and haue a speciall and continuall regard vnto it before ye begin any demonstration ye shall not easely vnderstand it For it is as it were the touch and triall of all other lines by which it is known whether any of them be rationall or not And this may be called the first rationall line the line rationall of purpose or a rationall line set in the first place and so made distinct and seuered from other rationall lines of which shall be spoken afterwarde And this must ye well commit to memory 6 Lines which are commensurable to this line whether in length and power or in power onely are also called rationall This definition needeth no declaration at all but is easily perceiued if the first definition be remembred which â—heweth what magnitudes are commensurable and the third which â—heweth what lines are commensurable in power Here notâ— how aptly naturally Euclide in this place vseth these wordes commensurable either in length and power or in power onely Because that all lines which are commensurable in length are also commensurable in powerâ— when he speaketh of lines commensurable in leÌ„gth he euer addeth and in power but when he speaketh of lines commensurable in power he addeth this worde Onely and addeth not this worde in length as he in the other added this worde in power For not all lines which are commensurable in power are straight way commensurable also in length Of this definition take this example Let the first line rationall of purpose which is supposed and laide forth whose partes are certaine known and may be expressed named and nuÌ„bred be AB the quadrate wherof let be ABCD then suppose againe an other lyne namely the line EF which let be commensurable both in length and in power to the first rationall line that is as before was taught let one line measure the length of eche line and also lâ—t one superâ—icies measure the two squares of the said two lines as here in the example is supposed and also appeareth to the eie then is the line E F also a rationall line Moreouer if the lyne EF be commensurable in power onely to the rationall line AB first set and supposed so that no one line do measure the two lines AB and EF As in example yâ— see to be for
quadrates but all other kindes of rectiline figures playne plats superficieses What so euer so that if any such figure be coÌ„mensurable vnto that rationall squareâ— it is also rationall As suppose that the square of the rationall line which is also rationall be ABCD suppose 〈◊〉 so some other square as the square EFGH to be commensurable to the same theÌ„ is the square EFGH also rational So also if the rectiline figure KLMN which is a figure on the one side longer be commensurable vnto the sayd square as is supposed in this exampleâ— it is also a rational superficies and so of all other superficieses 10 Such which are incommensurable vnto it are irrationall Where it is sayd in this diffinition such which are incommensurable it is generally to be taken as was this word coÌ„mensurable in the diffinitioÌ„ before For al superficieses whether they be squares or figures on the one side longer or otherwise what maner of right lined figure so euer it be if they be incommensurable vnto the rationall square supposed theÌ„ are they irrationall As let thâ— square ABCD be the square of the supposed rationall line which square therefore is also rationall suppose also also an other square namely the square E suppose also any other figure as for example sake a figure of one side longer which let be F Now if the square E and the figure F be both incommensurable to the rationall square ABCD then is 〈◊〉 of these figures E F irrationall And so of other 11 And these lines whose poweres they are are irrationall If they be squares then are their sides irrationall If they be not squares but some other rectiline figures then shall the lines whose squares are equall to these rectiline figures be irrationall Suppose that the rationall square be ABCD. Suppose also an other square namely the square E which let be incoÌ„meÌ„surable to the rationall square therefore is it irrationall and let the side or line which produceth this square be the line FG then shall the line FG by this diffinition be an irrationall line because it is the side of an irrationall square Let also the figure H being a figure on the one side longer which may be any other rectiline figure rectangled or not rectangled triangle pentagone trapezite or what so euer ells be incommensurable to the rationall square ABCD then because the figure H is not a square it hath no side or roote to produce it yet may there be a square made equall vnto it for that all such figures may be reduced into triangles and so into squares by the 14. of the second Suppose that the square Q be equall to the irrationall figure H. The side of which figure Q let be the line KL then shall the line KL be also an irrational line because the power or square thereof is equal to the irrationall figure H and thus conceiue of others the like These irrationall lines and figures are the chiefest matter and subiect which is entreated of in all this tenth booke the knowledge of which is deepe and secret and pertaineth to the highest and most worthy part of Geometrie wherein standeth the pith and mary of the hole science the knowlede hereof bringeth light to all the bookes following with out which they are hard and cannot be at all vnderstoode And for the more plainenes ye shall note that of irrationall lines there be diâ—ers sortes and kindes But they whose names are set in a table here following and are in number 13. are the chiefe and in this teÌ„th boke sufficiently for Euclides principall purpose discoursed on A mediall line A binomiall line A first bimediall line A second bimediall line A greater line A line containing in power a rationall superficies and a mediall superficies A line containing in power two mediall superficieces A residuall line A first mediall residuall line A second mediall residuall line A lesse line A line making with a rationall superficies the whole superficies mediall A line making with a mediall superficies the whole superficies mediall Of all which kindes the diffinitions together with there declarations shal be set here after in their due places ¶ The 1. Theoreme The 1. Proposition Two vnequall magnitudes being geuen if from the greater be taken away more then the halfe and from the residue be againe taken away more then the halfe and so be done still continually there shall at length be left a certaine magnitude lesser then the lesse of the magnitudes first geuen SVppose that there be two vnequall magnitudes AB and C of which let AB be the greater Then I say that if from AB be taken away more then the halfe and from the residue be taken againe more then the halfe and so still continually there shall at the length be left a certaine magnitude lesser then the lesse magnitude geueÌ„ namely then C. For forasmuch as C is the lesse magnitude therefore C may be so multiplyed that at the length it will be greater then the magnitude AB by the 5. definition of the fift booke Let it be so multiplyed and let the multiplex of C greater then AB be DE. And deuide DE into the partes equall vnto C which let be DF FG and GE. And from the magnitudes AB take away more then the halfe which let be BH and againe from AH take away more then the halfe which let be HK And so do continually vntill the diuisions which are in the magnitude AB be equall in multitude vnto the diuisions which are in the magnitude DE. So that let the diuisions AK KH and HB be equall in multitude vnto the diuisions DF FG and GE. And forasmuch as the magnitude DE is greater then the magnitude AB and from DE is taken away lesse then the halfe that is EG which detraction or taking away is vnderstand to be done by the former diuision of the magnitude DE into the partes equall vnto C for as a magnitude is by multiplication increased so is it by diuision diminished and from AB is taken away more then the halfe that is BH therefore the residue GD is greater then the residue HA which thing is most true and most easie to conceaue if we remeÌ„ber this principle that the residue of a greater magnitude after the taking away of the halfe or lesse then the halfe is euer greater then the residue of a lesse magnitude after the taking away of more then the halfe And forasmuch as the magnitude GD is greater then the magnitude HA and from GD is taken away the halfe that is GF and from AH is taken away more then the halfe that is HK therefore the residue DF is greater then the residue AK by the foresayd principle But the magnitude DF is equall vnto the magnitude C by supposition Wherefore also the magnitude C is greater then the magnitude AK Wherefore the magnitude AK is lesse then the magnitude C. Wherefore of the magnitude
An Assumpt Forasmuch as in the eight booke in the 26. proposition it was proued that like playne numbers haue that proportion the one to the other that a square number hath to a square number and likewise in the 24. of the same booke it was proued that if two numbers haue that proportion the one to the other that a square number hath to a square number those numbers are like plaine numbers Hereby it is manifest that vnlike plaine numbers that is whose sides are not proportionall haue not that proportion the one to the other that a square number hath to a square number For if they haue then should they be like plaine numbers which is contrary to the supposition Wherfore vnlike plaine numbers haue not that proportion the one to the other that a square number hath to a square nuÌ„ber And therfore squares which haue that proportion the one to the other that vnlike plaine numbers haue shall haue their sides incommensurable in length by the last part of the former proposition for that those squares haue not that proportion the one to the other that a square number hath to a square number ¶ The 8. Theoreme The 10. Proposition If foure magnitudes be proportionall and if the first be commensurable vnto the second the third also shal be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth SVppose that these foure magnitudes A B C D be proportionall As A is to B so let C be to D and let A be commensurable vnto B. Then I say that C is also commensurable vnto D. For forasmuch as A is commensurable vnto B it hath by the fift of the tenth that proportion that number hath to number But as A is to B so is C to D. Wherfore C also hath vnto D that proportion that number hath to number Wherfore C is commensurable vnto D by the 6. of the tenth But now suppose that the magnitude A be incommensurable vnto the magnitude B. Then I say that the magnitude C also is incommensurable vnto the magnitude D. For forasmuch as A is incommensurable vnto B therfore by the 7. of this booke A hath not vnto B such proportion as number hath to number But as A is to B so is C to D. Wherefore C hath not vnto D such proportion as number hath to number Wherfore by the 8. of the tenth C is incommensurable vnto D. If therefore there be foure magnitudes proportionall and if the first be commensurable vnto the second the third also shall be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth which was required to be proued ¶ A Corollary added by Montaureus If there be foure lines proportionall and if the two first or the two last be commensurable in power onely the other two also shall be commensurable in power onely This is proued by the 22. of the sixt and by this tenth proposition And this Corollary Euclide vseth in the 27. and 28. propositions of this booke and in other propositions also ¶ The 3. Probleme The 11. Proposition Vnto a right line first set and geuen which is called a rationall line to finde out two right lines incommensurable the one in length onely and the other in length and also in power SVppose that the right line first set and geuen which is called a rationall line of purpose be A. It is required vnto the said line A to finde out two right lines incommensurable the one in length onely the other both in length and in power Take by that which was added after the 9. proposition of this booke two numbers B and C not hauing that proportion the one to the other that a square number hath to a square number that is let them not be like plaine numbers for like plaine numbers by the 26. of the eight haue that proportion the one to the other that a square number hath to a square number And as the number B is to the number C so let the square of the line A be vnto the square of an other line namely of D how to do this was taught in the assumpt put before the 6. proposition of this booke Wherfore the square of the line A is vnto the square of the line D commensurable by the sixt of the tenth And forasmuch as the number B hath not vnto the number C that proportion that a square number hath to a square nuÌ„ber therfore the square of the line A hath not vnto the square of the line D that proportioÌ„ that a square number hath to a nuÌ„ber Wherfore by the 9. of the tenth the line A is vnto the line D incommensurable in length onely And so is found out the first line namely D incommensurable in length onely to the line geuen A. Agayne take by the 13. of the sixt the meane proportionall betwene the lines A and D and let the same be E. Wherfore as the line A is to the line D so is the square of the line A to the square of the line E by the Corollary of the 20. of the sixt But the line A is vnto the line D incommensurable in length Wherfore also the square of the line A is vnto the square of the line E incommensurable by the second part of the former proposition Now forasmuch as the square of the line A is incoÌ„meÌ„surable to the square of the line E it followeth by the definition of incommensurable lynes that the line A is incommensurable in power to the line E. Wherfore vnto the right line geuen and first set A which is a rationall line and which is supposed to haue such diuisions and so many partes as ye list to conceyue in minde as in this example 11 whereunto as was declared in the 5. definition of this booke may be compared infinite other lines either commensurable or incommensurable is found out the line D incommensurable in length onely Wherfore the line D is rationall by the sixt definitioÌ„ of this booke for that it is incommensurable in length onely to the line A which is the first line set and is by suppositioÌ„ rational There is also found out the line E which is vnto the same line A incommensurable not onely in length but also in power which line E compared to the rationall line A is by the definition irrationall For Euclide alwayes calleth those lines irrationall which are incommensurable both in length and in power to the line first set and by supposition rationall ¶ The 9. Theoreme The 12. Proposition Magnitudes commensurable to one and the selfe same magnitude are also commensurable the one to the other SVppose that either of these magnitudes A and B be commensurable vnto the magnitude C Then I say that the magnitude A is commensurable vnto the magnitude B. For â—orasmuch as the
in like sort be deuided as the line AB is by that which hath bene demonstrated in the 66. Proposition of this bookeâ— let it be so deuided in the poynt E. Wherefore it can not be so deuided in any other poynt by the 42â— of this booke And for that the line AB â—â— to the line DZ as the line AG is to the line DE but the lines AG DE namely the greater names are commensurable in length the one to the other by the 10. of this booke for that they are commensurable in length to 〈◊〉 and the selfe same rationall line by the first definition of binomiall lines Wherefore the lines AB and DZ are commensurable in length by the 13. of this booke But by supposition they are commensurable in power onely which is impossible The selfe same demonstration also will serue if we suppose the line AB to be a second binomial line for the lesse names GB and EZ being commensurable in length to one and the selfe same rationall line shall also be commensurable in length the one to the other And therefore the lines AB and DZ which are in the selfe same proportion with them shall also be commensurable in length the one to the other which is contrary to the supposition Farther if the squares of the lines AB and DZ be applyed vnto the rationall line CF namely the parallelogrammes CT and HL they shall make the breadthes CH and HK first binomiall lines of what order soeuer the lines AB DZ whose squares were applyed vnto the rational line are by the 60. of this booke Wherefore it is manifest that vnder a rationall line and a first binomiall line are confusedly contayned all the powers of binomiall lines by the 54. of this booke Wherfore the onely commensuration of the powers doth not of necessitie bryng forth one and the selfe same order of binomiall lines The selfe same thyng also may be proued if the lines AB and DZ be supposed to be a fourth or fifth binomiall line whose powers onely are conmmensurable namely that they shall as the first bring forth binomiall lines of diuers orders Now forasmuch as the powers of the lines AG and GB and DE and EZ are commensurable proportionall it is manifest that if the line AG be in power more then the line GB by the square of a line commensurable in length vnto AG the line DE also shall be in power more then the line EZ by the square of a line commensurable in length vnto the line DE by the 16. of this booke And so shall the two lines AB and DZ be eche of the three first binomiall lines But if the line AG be in power more then the line GB by the square of a line incommensurable in length vnto the line AG the line DE shall also be in powâ—r 〈◊〉 then the line EZ by the square of a line incomensurable in length vnto the line DE by the selfâ— same Pâ—oposition And so shall eche of the lines AB and DZ be of the three last binomiall lines But why it is not so in the third and sixt binomiall lines the reason is For that in them neither of the nameâ— is commensurable in length to the rationall line put FC ¶ The 50. Theoreme The 68. Proposition A line commensurable to a greater line is also a greater line SVppose that the line AB be a greater line And vnto the line AB let the line CD be commensurable Then I say that the line CD also is a greater line Deuide the line AB into his partes in the point E. Wherfore by the 39. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ„ mediall And let the rest of the construction be in this as it was in the former And for that as the line AB is to the line CD so is the line AE to the line CF thâ— line EB to the line FD but the line AB is commensurable to the line CD by suppositioÌ„ Wherfore the line AE is commensurable to the line CF and the line EB to the line FD. And for that as the line AE is to the line CF so is the line EB to the line FD. Therfore alternately by the 16. of the fift as the line AE is to the line EB so is the line CF to the line FD. Wherfore by composition also by the 18. of the fift as the line AB is to the line EB so is the line CD to the line FD. Wherefore by the 22. of the sixt as the square of the line AB is to the square of the line EB so is the square of the line CD to the square of the line FD. And in like sort may we proue that as the square of the line AB is to the square of the line AE so is the square of the line CD to the square of the line CF. Wherfore by the 11. of the fift as the square of the lyne AB is to the squares of the lines AE and EB so is the square of the line CD to the squares of the lines CF and FD. Wherfore alternately by the 16. of the fift as the square of the line AB is to the square of the line CD so are the squares of the lines AE and EB to the squares of the lines CF and FD. But the square of the line AB is commensurable to the square of the line CD for the line AB is commensurable to the line CD by suppositioÌ„ Wherfore also the squares of the lines AE and EB are commensurable to the squares of the lines CF and FD. But the squares of the lines AE and EB are incommensurable and being added together are rationall Wherfore the squares of the lines CF and FD are incommensurable being added together are also rationall And in like sort may we proue that that which is contained vnder the lines AE and EB twise is commensurable to that which is contained vnder the lines CF and FD twise But that which is contained vnder the lines AE and EB twise is mediall wherfore also that which is contained vnder the lines CF and FD twise is medial Wherfore the lines CF and FD are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ„ mediall Wherfore by the 39. of the tenth the whole line CD is irrationall is called a greater line A line therfore commensurable to a greater line is also a greater line An other demonstration of Peter Montaureus to proue the same Suppose that the line AB be a greater line and vnto it let the line CD be commensurable any way that is either both in length and in power or els in power onely Then I say that the line CD also is a greater
two other propositions going next before it so farre misplaced that where they are word for word before duâ—ly placed being the 105. and 106. yet here after the booke ended they are repeated with the numbers of 116. and 117. proposition Zambert therein was more faythfull to follow as he found in his greke example than he was skilfull or carefull to doe what was necessary Yea and some greke written auncient copyes haue them not so Though in deede they be well demonstrated yet truth disorded is halfe disgracedâ— especially where the patterne of good order by profession is auouched to be But through ignoraunce arrogancy and â—emerltie of vnskilfull Methode Masters many thinges remayne yet in these Geometricall Elementes vnduely tumbled in though true yet with disgrace which by helpe of so many wittes and habilitie of such as now may haue good cause to be skilfull herein will I hope ere long be taken away and thinges of importance wanting supplied The end of the tenth booke of Euclides Elementes ¶ The eleuenth booke of Euclides Elementes HITHERTO HATH â—VCLIDâ— IN THâ—Sâ— former bookes with a wonderfull Methode and order entreated of such kindes of figures superficial which are or may be described in a superficies or plaine And hath taught and set forth their properties natures generations and productions euen from the first roote ground and beginning of them namely from a point which although it be indiuisible yet is it the beginning of all quantitie and of it and of the motion and slowing therof is produced a line and consequently all quantitie coÌ„tinuall as all figures playne and solide what so euer Euclide therefore in his first booke began with it and from thence went he to a line as to a thing most simple next vnto a point then to a superficies and to angles and so through the whole first booke he intreated of these most simple and plaine groundes In the second booke he entreated further and went vnto more harder matter and taught of diuisions of lines and of the multiplication of lines and of their partes and of their passions and properties And for that rightlined â—igures are far distant in nature and propertie from round and circular figures in the third booke he instructeth the reader of the nature and conditioÌ„ of circles In the fourth booke he compareth figures of right lines and circles together and teacheth how to describe a figure of right lines with in or about a circle and contraâ—iwiâ—e a circle with in or about a rectiline figure In the fifth booke he searcheth out the nature of proportion a matter of wonderfull vse and deepe consideration for that otherwise he could not compare â—igure with figure or the sides of figures together For whatsoeuer is compared to any other thing is compared vnto it vndoubtedly vnder some kinde of proportion Wherefore in the sixth booke he compareth figures together one to an other likewise their sides And for that the nature of proportion can not be fully and clearely sene without the knowledge of number wherein it is first and chiefely found in the seuenth eight and ninth bookes he entreatâ—th of number of the kindes and properties thereof And because that the sides of solide bodyes for the most part are of such sort that compared together they haue such proportion the one to the other which can not be expresâ—ed by any number certayne and therefore are called irrational lines he in the teÌ„th boke hath writteÌ„ taught which lineâ— are coÌ„meÌ„surable or incoÌ„meÌ„surable the one to the other and of the diuersitie of kindes of irrationall lines with all the conditions proprieties of them And thus hath Euclide in these ten foresayd bokes fully most pleÌ„teously in a meruelous order taught whatsoeuer semed necessary and requisite to the knowledge of all superficiall figures of what sort forme so euer they be Now in these bookes following he entreateth of figures of an other kinde namely of bodely figures as of Cubes Piramids Cones Columnes Cilinders Parallelipipedons Spheres and such othersâ— and sheweth the diuersitie of theÌ„ the generation and production of them and demonstrateth with great and wonderfull art their proprieties and passions with all their natures and conditions He also compareth one oâ— them to an other whereby to know the reason and proportion of the one to the other chiefely of the fiue bodyes which are called regular bodyes And these are the thinges of all other entreated of in Geometrie most worthy and of greatest dignitie and as it were the end and finall entent of the whole are of Geometrie and for whose cause hath bene written and spoken whatsoeuer hath hitherto in the former bookes bene sayd or written As the first booke was a ground and a necessary entrye to all the râ—st â—ollowing so is this eleuenth booke a necessary entrie and ground to the rest which follow And as that contayned the declaration of wordes and definitions of thingeâ— requisite to the knowledge of superficiall figures and entreated of lines and of their diuisions and sections which are the termes and limites of superficiall figures so in this booke is set forth the declaration of wordes and definitions of thinges pertayning to solide and corporall figures and also of superficieces which are the termes limites of solides moreouer of the diuision and intersection of them and diuers other thinges without which the knowledge of bodely and solide formes can not be attayned vnto And first is set the definitions as followâ—th Definitions A solide or body is that which hath length breadth and thicknes and the terme or limite of a solide is a superficies There are three kindes of continuall quantitie a line a superficies and a solide or body the beginning of all which as before hath bene sayd is a poynt which is indiuisible Two of these quantities namely a line and a superficies were defined of Euclide before in his first booke But the third kinde namely a solide or body he there defined not as a thing which pertayned not then to his purpose but here in this place he setteth the definitioÌ„ therof as that which chiefely now pertayneth to his purpose and without which nothing in these thinges can profitably be taught A solide sayth he is that which hath leÌ„gth breadth and thicknes or depth There are as before hath bene taught three reasons or meanes of measuring which are called coÌ„monly dimensions namely lâ—ngth breadth and thicknes These dimensions are ascribed vnto quantities onely By these are all kindes of quantitie deâ—ined â—â— are counted perfect or imperfect according as they are pertaker of fewer or more of them As Euclide defined a line ascribing vnto it onely one of these dimensions namely length Wherefore a line is the imperfectest kinde of quantitie In defining of a superficies he ascribed vnto it two dimensions namely length and breadth whereby a superficies is a quantitie of
if the angle LMK be an acute angle then is that angle the inclination of the superficies ABCD vnto the superficies EFGH by this definition because it is contained of perpendicular lines drawen in either of the superficieces to one and the self same point being the common section of them both 5 Plaine superficieces are in like sort inclined the on◠〈…〉 her when the sayd angles of inclination are equall the one to the o 〈…〉 This definition needeth no declaration at all but is most manifest by the definition last going before For in considering the inclinations of diuers superficieces to others if the acute angles contayned vnder the perpendicular lines drawen in them from one point assigned in ech of their common sections be equall as if to the angle LMK in the former example be geuen an other angle in the inclination of two other superficieces equall then is the inclination of these superficieces like and are by this definition sayd in like sort to incline the one to the other Now also let there be an other ground plaine superficies namely the superficies MNOP vnto whom also let leane and incline the superficies Qâ—â—T and let the common section or segment of them be the line QR And draw in the superficies MNOP to some one point of the coÌ„mon section as to the point X the line VX making with the common section right angles namely the angle VXR or the angle VXQ also in the superficies STQR draw the right line YX to the same point X in the common section making therwith right angles as the angle YXâ— or the angle YXQ. Now as sayth the definition if the angles contayned vnder the right lines drawen in these superficieces making right angles with the common section be in the pointes that is in the pointes of their meting in the common section equall then is the inclination of the superficieces equall As in this example if the angle LGH contayned vnder the line LG being in the inclining superficies â—KEF and vnder the line HG being in the ground superficies ABCD beÌ„ equall to the angle YXV contayned vnder the line VX being in the ground superficies MNOP and vnder the line YX being in the inclining superficies STQR then is the inclination of the superâ—icies IKEF vnto the superficies ABCD like vnto the inclination of the superficies STQR vnto the superficies MNOP And so by this definition these two superficieces are sayd to be in like sort inclined 6 Parallell plaine superficieces are those which being produced or extended any way neuer touch or concurre together Neither needeth this definition any declaration but is very easie to be vnderstanded by the definition of parallell lines â—or as they being drawen on any part neuer touch or come together so parallel plaine superâ—icieces are such which admitte no touch that is being produced any way infinitely neuer meete or come together 7 Like solide or bodily figures are such which are contained vnder like plaine superficieces and equall in multitude What plaine superâ—icieces are called like hath in the beginning of the sixth booke bene sufficiently declared Now when solide figures or bodies be contained vnder such like plaine superficieces as there are defined and equall in number that is that the one solide haue as many in number as the other in their sides and limites they are called like solide figures or like bodies 8 Equall and like solide or bodely figures are those which are contained vnder like superficieces and equall both in multitude and in magnitude In like solide figures it is suâ—ficient that the superficieces which containe them be like and equall in number onely but in like solide figures and equall it is necessary that the like superficieces contaynyng them be also equal in magnitude So that besides the likenes betwene them they be eche being compared to his correspondent superâ—icies oâ— one greatnes and that their areas or fieldes be equal When such superâ—icieces contayne bodies or solides then are such bodies equall and like solides or bodies 9 A solide or bodily angle is an inclination of moe then two lines to all the lines which touch themselues mutually and are not in one and the selfe same superâ—icies Or els thus A solide or bodily angle is that which is contayned vnder mo then two playne angles not being in one and the selfe same plaine superficies but consisting all at one point Of a solide angle doth Euclide here geue two seueâ—all definitioÌ„s The first is geuen by the concurse and touch of many lines The second by the touch concurse of many superficiall angles And both these definitions tende to one and are not much different for that lynes are the limittes and termes of superficieces But the second geuen by superâ—iciall angles is the more naturall definition because that supeâ—ficieces aâ—e the next and immediate limites of bodies and so are not lines An example of a solide angle cannot wel and at â—ully be geueÌ„ or described in a plaâ—â—e superficies But touchyng this first definitioÌ„ lay before you a cube or a die and coÌ„sider any of the corners or angles therof so shal ye see that at eueâ—y angle there concurre thre lines for two lines coÌ„curring cannot make a solide angle namely the line or edge of his breadth of his leÌ„gth and of his thicknes which their so inclining coÌ„curring touether make a solide angle and so of others And now coÌ„câ—rning the second definitioÌ„ what superâ—icial or plaine angles be hath bene taught before in the first bokâ— namely that it is the touch of two right lines And as a superâ—iciall or playne angle is caused coÌ„tained of right lines so si a solide angle caused coÌ„tayned of plaine superficiall angles Two right lines touching together make a plaine angle but two plaine angles ioyned together can not make a solide angle but according to the definitioÌ„ they must be moe theÌ„ two as three â—oure â—iue or moâ— which also must not be in one the selfe same superficiâ—s but must be in diuers superficieces â—eeting at one point This definition is not hard but may easily be coÌ„ceiued in a cube or a die where ye see three angles of any three superficieces or sides of the die concurre and meete together in one point which three playne angles so ioyned together make a solide angle Likewise in a Pyramiâ— or a spiâ—e of a steple or any other such thing all the sides therof teÌ„ding vpward narower and narower at length ende their angles at the heigâ—â— or toppe therof in one point So all their angles there ioyned together make a solide angle And for the better â—igâ—t thereof I haue set here a figure wherby ye shall more easily conceiue â—â— the base of the figure is a triangle namely ABC if on euery side of the triangle ABC ye rayse vp a triangle as vpon the side AB ye raise vp the triangle AFB and vpon the side AC the
triangle AFC and vpon the side BC the triangle BFC and so bowing the triangles raised vp that their toppes namely the pointes F meete and ioyne together in one point ye shal easily and plainly see how these three superficiall angles AFBBFC CFA ioyne and close together touching the one the other in the point F and so make a solide angle 10 A Pyramis is a solide figure contained vnder many playne superficieces set vpon one playne superficies and gathered together to one point Two superficieces raysed vpon any ground can not make a Pyramis for that two superficiall angles ioyned together in the toppe cannot as before is sayd make a solide angle Wherfore wheÌ„ thre foure fiue or moe how many soeuer superficieces are raised vp froÌ„ one superficies being the ground or base and euer asceÌ„ding diminish their breadth till at the leÌ„gth all their angles coÌ„curre in one point making there a solide angle the solide inclosed bounded and terminated by these superficieces is called a Pyramis as ye see in a taper of foure sides and in a spire of a towre which containeth many sides either of which is a Pyramis And because that all the superficieces of euery Pyramis ascend from one playne superficies as from the base and tende to one poynt it must of necessitie come to passe that all the superficieces of a Pyramis are trianguler except the base which may be of any forme or figure except a circle For if the base be a circle then it ascendeth not with sides or diuers superficieces but with one round superficies and hath not the name of a Pyramis but is called as hereafter shall appeare a Cone 11 A prisme is a solide or a bodily figure contained vnder many plaine superficieces of which the two superficieces which are opposite are equall and like and parallells all the other superficieces are parallelograÌ„mes Flussas here noteth that Theon and Campane disagree in defining a Prisme and he preferreth the definition geuen of Campane before the deâ—inition geuen of Euclide which because he may seme with out lesse offence to reiect he calleth it Theons definition and following Campane he geueth an other definition which is this A Prisme is a solide figure which is contayned vnder fiue playne superficieces of which two are triangles like equall and parallels and the rest are parallelogrammes The example before set agreeth likewise with this definition and manifestly declareth the same For in it were â—iue superficieces the base the two erected superficieces and the two endes of which the two endes are triangles like equall and parallels and all the other are parallelogrammes as this definition requireth The cause why he preferreth the difinion of Campane before the difinition of Theon as he calleth it but in very deede it is Euclides definition as certainely as are all those which are geuen of him in the former bookes neither is there any cause at all why it should be doubted in this one definition more then in any of the other as he him selfe alledgeth is for that it is as he sayth to large and comprehendeth many mo kindes of solide figures besides Prismes as Columnes hauing sides and all Parallelipipedons which a definition should not doo but should be conuertible with the thing defined and declare the nature of it onely and stretch no farther Me â—hinketh Flussas ought not to haue made so much a doo in this matter nor to haue bene so sharpe in sight and so quicke as to see and espy out such faultes which can of no man that will see rightly withouâ— affection be espyed for such great faultes For it may well be aunswered that these faultes which he noteth if yet they be faultes are not to be found in this definion It may be sayd that it extendeth it selfe not â—arther then it should but declareth onely the thing defined namely a Prisme Neither doth it agree as â—lussas cauilleth with all Parallelipipedons and Columnes hauing sides All Parallelipipedons what so euer right angled or not right angled which are described of equidistant sides or superficieces haue their sides opposit So that in any of them there is no one side but it hath a side opposit vnto it So likewise is it of eueÌ„ sided Columnes eche hath his opposite side directly agaynst it which agreeth not with this definition of Euclide Here it is euidently sayd that of all the superficieces the two which are opposite are equall like and parallels meaning vndoubtedly onely two no moe Which is manifest by that which followeth The other sayth he are parallelogrammes signifiing most euidently that none of the rest besides the two aforesayd which are equall like and parallels are opposite but two of necessitie are raysed vp and concurre in one common line and the other is the base So that it contayneth not vnder it the figures aforesayd that is sided Columnes al Parallelipipedons as Flussas hath not so aduisedly noted Agayne where Flussas setteth in his definition as an essentiall part thereof that of the fiue superficieces of which a Prisme is contayned two of them must be triangles that vndoubtedly is not of necessitie they may be of some other figure Suppose that in the figure before geuen that in the place of the two opposite figures which there were two trianglâ—s were placed two pentagoÌ„s yet should the figure remayne a Prisme still and agree with the definition of Euclide and â—alleth not vnder the definition of Flussas So that his definitioÌ„ semeth to be to narrow and stretcheth not so farre as it ought to do nor declareth the whole nature of the thing defined Wherefore it is not to be preferrd before Euclides definition as he woulde haue it This figure of Euclide called a Prisme is called of Campane and certayne others Figura Serrâ—tilis for that it represâ—teth in some maner the forme of a Sawe And of some others it is called Cuneus that is a Wedge because it beareth the figure of a wedge Moreouer although it were so that the definitioÌ„ of a Prisme should be so large that it should coÌ„taine all these figures noted of Flussas as sided Columnes all Parallelipipedons yet should not Flussas haue so great a cause to finde so notably a fault so vtterly to reiect it It is no rare thing in all learninges chiefely in the Mathematicalls to haue one thing more generall then an other Is it not true that euery Isosceles is a triangle but not euery triangle is an Isosceles And why may not likewise a Prisme be more generall then a Parallelepipedon or a Columne hauing sides and contayne them vnder it as a triangle coÌ„tayneth vnder it an Isosceles and other kinds of triangles So that euery Prallelipipedon or euery sided Columne be a Prisme but not euery Prisme a ParallelipipedoÌ„ or a sided Columne This ought not to be so much offensiue And indeede it semeth manifestly of many yea of the learned so to be takeÌ„ as clearely appeareth by
the wordes of Psellus in his Epitome of Geometrie where he entreateth of the production and constitution of these bodyes His wordes are these All râ—ctiliâ—e figures being erected vpon their playnes or bases by right angles make Prismes Who perceaueth not but that a Pentagon erected vpoÌ„ his base of â—iue sides maketh by his motion a sided Columne of fiue sides Likewise an Hexagon erected at right angles produceth a Columne hauing sixe sides and so of all other rectillne figures All which solides or bodyes so produced whether they be sided Columnes or Parallelipipedons be here in most plaine words of this excelleÌ„t and auncient Greke author Psellus called Prismes Wherfore if the definitioÌ„ of a Prisme geueÌ„ of Euclide should extend it selfe so largely as Flussas imagineth and should enclude such figures or bodyes as he noted he ought not yet for all that so much to be offended and so narowly to haue sought faultes For Euclide in so defining mought haue that meaning sense of a Prisme which Psellus had So ye see that Euclide may be defended either of these two wayes either by that that the definition extendeth not to these figures and so not to be ouer generall nor stretch farther then it ought or ells by that that if it should stretch so far it is not so haynous For that as ye se many haue takâ— it in that sense In deede coÌ„monly a Prisme is taken in that significatioÌ„ and meaning in which Campaâ—â—â— Flussas and others take it In which sense it semeth also that in diuers propositions in these bookes following it ought of necessitie to be taken 12 A Sphere is a figure which is made when the diameter of a semicircle abiding fixed the semicircle is turned round about vntill it returne vnto the selfe same place from whence it began to be moued To the end we may fully and perfectly vnderstand this definition how a Sphere is produced of the motion of a semicircle it shall be expedient to coÌ„sider how quantities Mathematically are by imagination conceaued to be produced by flowing and motion as was somewhat touched in the beginning of the first booke Euer the lesse quantitie by his motion bringeth forâ—h the quaÌ„titie next aboue it As a point mouing flowing or gliding bringeth forth a line which is the first quantitie and next to a point A line mouing produceth a superficies which is the second quantitie and next vnto a line And last of all a superficies mouing bringeth forth a solide or body which is the third last quantitie These thinges well marked it shall not be very hard to attaine to the right vnderstanding of this definition Vpon the line AB being the diameter describe a semicircle ACB whose centre let be D the diameter AB being sixed on his endes orâ— pointes imagine the whole superficies of the semicircle to moue round from some one point assigned till it returne to the same point againe So shall it produce a perfect Sphere or Globe the forme whereof you see in a ball or bowle And it is fully round and solide for that it is described of a semicircle which is perfectly round as our countrey man Iohannes de Sacro Busco in his booke of the Sphere of this definition which he taketh out of Euclide doth well collecte But it is to be noted and taken heede of that none be deceaued by the definition of a Sphere geuen by Iohannes de Sacro Busco A Sphere sayth he is the passage or mouing of the circumference of a semicircle till it returne vnto the place where it beganne which agreeth not with Euclide Euclide plainly sayth that a Sphere is the passage or motion of a semicircle and not the passage or motion of the circumference of a semicircle neither can it be true that the circumference of a semicircle which is a line should describe a body It was before noted that euery quantitie moued describeth and produceth the quantitie next vnto it Wherefore a line moued can not bring forth a body but a superficies onely As if ye imagine a right line fastened at one of his endes to moue about from some one point till it returne to the same againe it shall describe a plaine superficies namely a circle So also if ye likewise conceaue of a crooked line such as is the circumference of a semicircle that his diameter fastened on both the endes it should moue from a point assigned till it returne to the same againe it should describe produce a â—ound superficies onely which is the superficies and limite of the Sphere and should not produce the body and soliditie of the Sphere But the whole semicircle which is a superficies by his motion as is before said produceth a body that is a perfect Sphere So see you the errour of this definition of the author of the Sphere which whether it happened by the author him selfe which I thinke not or that that particle was thrust in by some one after him which is more likely it it not certaine But it is certaine that it is vnaptly put in and maketh an vntrue definition which thing is not here spoken any thing to derogate the author of the booke which assuredly was a man of excellent knowledgeâ— neither to the hindrance or diminishing of the worthines of the booke which vndoubtedly is a very necessary booke then which I know none more meere to be taught and red in scholes touching the groundes and principles of Astronomie and Geographie but onely to admonishe the young and vnskilâ—ull reader of not falling into errour Theodosius in his booke De Sphericis a booke very necessary for all those which will see the groundes and principles of Geometrie and Astronomie which also I haue translated into our vulgare tounge ready to the presse defineth a Sphere after thys maner A Sphere is a solide or body contained vnder one superficies in the midle wherof there is a point froÌ„ which all lines drawen to the circumference are equall This definition of Theodosius is more essentiall and naturall then is the other geuen by Euclide The other did not so much declare the inward nature and substance of a Sphere as it shewed the industry and knowledge of the producing of a Sphere and therfore is a causall definition geuen by the cause efficient or rather a description then a definition But this definition is very esâ—entiall declaring the natuâ—e and substance of a Sphere As if a circle should be thus defined as it well may A circle is the passage or mouing of a line from a point till it returne to the same point againeâ— it is a causall definition shewing the efficient cause wherof a circle is produced namely of the motion of a line And it is a very good description fully shewing what a circle is Such like description is the deâ—inition of a Sphere geuen oâ— Euclide â— by the motion of a semicircle But when a circle is defined to be a plaine superficies
in the middest wherof is a point from which all lines drawen to the circumference therof are equall this definition is essentiall and formall and declareth the very nature of a circle And vnto this definition of a circle is correspondent the deâ—inition of a Sphere geueÌ„ by Theodosius saying that it is a solide oâ— body in the middest whereof there is a point from which all the lines drawen to the circumference are equall So see you the affinitie betwene a circle and a Sphere For what a circle is in a plaine that is a Sphere in a Solide The fulnes and content of a circle is described by the motion of a line moued about but the circumference therof which is the limite and border thereof is described of the end and point of the same line moued about So the fulnes content and body of a Sphere or Globe is described of a semicircle moued about But the Sphericall superficies which is the limite and border of a Sphere is described of the circumference of the same semicircle moued about And this is the superficies ment in the definition when it is sayd that it is contained vnder one superficies which superficies is called of Iohannes de â—acro Busco others the circumference of the Sphere Galene in his booke de diffinitionibus mediciâ— â— geueth yet an other definitioÌ„ of a Sphere by his propertie or coÌ„mon accideÌ„ce of mouing which is thus A Sphere is a figure most apt to all motion as hauing no base whereon th stay This is a very plaine and witty deâ—inition declaring the dignitie thereof aboue all figures generally All other bodyes or solides as Cubes Pyramids and others haue sides bases and angles all which are stayes to rest vpon or impedimentes and lets to motion But the Sphere hauing no side or base to stay one nor angle to let the course thereof but onely in a poynt touching the playne wherein 〈◊〉 standeth moueth freely and fully with out let And for the dignity and worthines thereof this circular and Sphericall motion is attributed to the heauens which are the most worthy bodyes Wherefore there is ascribed vnto them this chiefe kinde of motion This solide or bodely figure is also commonly called a Globe 13 The axe of a Sphere is that right line which abideth fixed about which the semicircle was moued As in the example before geuen in the definition of a Sphere the line AB about which his endes being fixed the semicircle was moued which line also yet remayneth after the motion ended is the axe of the Sphere described of that semicircle Theodosius defineth the axe of a Sphere after this maner The axe of a Sphere is a certayne right line drawen by the centre ending on either side in the superficies of the Sphere about which being fixed the Sphere is turned As the line AB in the former example There nedeth to this definition no other declaration but onely to consider that the whole Sphere turneth vpon that line AB which passeth by the centre D and is extended one either side to the superficies of the Sphere wherefore by this definition of Theodosius it is the axe of the Sphere 14 The centre of a Sphere is that poynt which is also the centre of the semicircle This definition of the centre of a Sphere is geuen as was the other definition of the axe namely hauing a relation to the definition of a Sphere here geuen of Euclide where it was sayd that a Sphere is made by the reuolution of a semicircle whose diameter abideth fixed The diameter of a circle and of a semicrcle is all one And in the diameter either of a circle or of a semicircle is contayned the center of either of them for that they diameter of eche euer passeth by the centre Now sayth Euclide the poynt which is the center of the semicircle by whose motion the Sphere was described is also the centre of the Sphere As in the example there geuen the poynt D is the centre both of the semicircle also of the Sphere Theodosius geueth as other definition of the centre of a Sphere which is thus The centre of a Sphere is a poynt with in the Sphere from which all lines drawen to the superficies of the Sphere are equall As in a circle being a playne figure there is a poynt in the middest from which all lines drawen to the circumfrence are equall which is the centre of the circle so in like maner with in a Sphere which is a solide and bodely figure there must be conceaued a poynt in the middest thereof from which all lines drawen to the superficies thereof are equall And this poynt is the centre of the Sphere by this definition of Theodosius Flussas in defining the centre of a Sphere comprehendeth both those definitions in one after this sort The centre of a Sphere is a poynt assigned in a Sphere from which all the lines drawen to the superficies are equall and it is the same which was also the centre of the semicircle which described the Sphere This definition is superfluous and contayneth more theÌ„ nedeth For either part thereof is a full and sufficient diffinition as before hath bene shewed Or ells had Euclide bene insufficient for leauing out the one part or Theodosius for leauing out the other Paraduenture Flussas did it for the more explication of either that the one part might open the other 15 The diameter of a Sphere is a certayne right line drawen by the ceÌ„tre and one eche side ending at the superficies of the same Sphere This definitioÌ„ also is not hard but may easely be couceaued by the definitioÌ„ of the diameter of a circle For as the diameter of a circle is a right line drawne froÌ„ one side of the circuÌ„frence of a circle to the other passing by the centre of the circle so imagine you a right line to be drawen from one side of the superficies of a Sphere to the other passing by the center of the Sphere and that line is the diameter of the Sphere So it is not all one to say the axe of a Sphere and the diameter of a Sphere Any line in a Sphere drawen from side to side by the centre is a diameter But not euery line so drawen by the centre is the axe of the Sphere but onely one right line about which the Sphere is imagined to be mouedâ— So that the name of a diameter of a Sphere is more general then is the name of an axe For euery axe in a Sphere is a diameter of the same but not euery diameter of a Sphere is an axe of the same And therefore Flussas setteth a diameter in the definition of an axe as a more generall word â—n this maner The axe of a Sphere is that fixed diameter aboue which the Sphere is moued A Sphere as also a circle may haue infinite diameters but it can haue but
â—ignifieth Last of all a Dodecahedron for that it is made of Pâ—ntagoâ— whose angles are more ample and large then the angles of the other bodies and by that â—eaâ—â—â— draw more â—â— rounâ—nes 〈◊〉 to the forme and nature of a sphere they assigned to a sphere namely 〈…〉 Who so will 〈…〉 in his Tineus shall â—ead of these figures and of their mutuall proportionâ—â—â—raunge maâ—terâ— which hâ—re are not to be entreated of this which is sayd shall be sufficient for the 〈◊〉 of them and for thâ— declaration of their diffinitions After all these diffinitions here set of Euclide Flussas hath added an other diffinition which 〈◊〉 of a Parallelipipedon which bicause it hath not hitherto of Euclide in any place bene defined and because it is very good and necessary to be had I thought good not to omitte it thus it is A parallelipipedon is a solide figure comprehended vnder foure playne quadrangle figures of which those which are opposite are parallels Because these fiue regular bodies here defined are not by these figures here set so fully and liuely expressed that the studious beholder can throughly according to their definitions conceyue them I haue here geuen of them other descriptions drawn in a playne by which ye may easily attayne to the knowledge of them For if ye draw the like formes in matter that wil bow and geue place as most aptly ye may do in fine pasted paper such as pastwiues make womeÌ„s pastes of theÌ„ with a knife cut euery line finely not through but halfe way only if theÌ„ ye bow and bende them accordingly ye shall most plainly and manifestly see the formes and shapes of these bodies euen as their definitions shew And it shall be very necessary for you to hadâ—â—tore of that pasted paper by you for so shal yoâ— vpon it 〈…〉 the formes of other bodies as Prismes and Parallelipopedons 〈…〉 set forth in these fiue bookes following and see the very 〈◊〉 of thâ—se bodies there meÌ„cioned which will make these bokes concerning bodies as easy vnto you as were the other bookes whose figures you might plainly see vpon a playne superficies Describe thiâ— figurâ— which consistâ—th of twâ—luâ—â—quilâ—â—â—â— and â—quianglâ— Pâ—ntâ—â—â—â—â— vpoâ— the foresaid mattâ—r and finely cut as before was â—â—ught tâ—â—â—lâ—uâ—n lines containâ—d within thâ— figurâ— and bow and folde the Penâ—â—gonâ— accordingly And they will so close toâ—ethâ—â— thaâ— thâ—y will â—â—kâ— thâ— very forme of a Dodecahedron If ye describe this figure which consisteth of twentie equilater and equiangle triangles vpon the foresaid matter and finely cut as before was shewed the ninâ—tâ—ne lines which are contayned within the figure and then bowe and folde them accordingly they will in such sort close together that therâ— will be made a perfecte forme of an Icosahedron Because in these fiue bookes there are sometimes required other bodies besides the foresaid fiue regular bodies as Pyramises of diuers formes Prismes and others I haue here set forth three figures of three sundry Pyramises one hauing to his base a triangle an other a quadrangle figure the other â— Pentagonâ— which if ye describe vpon the foresaid matter finely cut as it was before taught the lines contained within ech figure namely in the first three lines in the second fower lines and in the third fiue lines and so bend and folde them accordingly they will so close together at the toppes that they will â—ake Pyramids of that forme that their bases are of And if ye conceaue well the describing of these ye may most easily describe the body of a Pyramis of what forme so euer ye will. Because these fiue bookes following are somewhat hard for young beginners by reason they must in the figures described in a plaine imagine lines and superficieces to be eleuated and erected the one to the other and also conceaue solides or bodies which for that they haue not hitherto bene acquainted with will at the first sight be somwhat sâ—raunge vnto theÌ„ I haue for their more â—ase in this eleuenth booke at the end of the demonstration of euery Proposition either set new figures if they concerne the eleuating or erecting of lines or superficieces or els if they concerne bodies I haue shewed how they shall describe bodies to be compared with the constructions and demonstrations of the Propositions to them belonging And if they diligently weigh the maner obserued in this eleuenth booke touching the description of new figures agreing with the figures described in the plaine it shall not be hard for them of them selues to do the like in the other bookes following when they come to a Proposition which concerneth either the eleuating or erecting of lines and superficieces or any kindes of bodies to be imagined ¶ The 1. Theoreme The 1. Proposition That part of a right line should be in a ground playne superficies part eleuated vpward is impossible FOr if it be possible let part of the right line ABC namely the part AB be in a ground playne superficies and the other part therof namely BC be eleuated vpwarde And produce directly vpoÌ„ the ground playne superficies the right line AB beyond the point B vnto the point D. Wherfore vnto two right lines geuen ABC and ABD the line AB is a common section or part which is impossible For a right line can not touche a right line in 〈◊〉 pointes then one vâ—lesse those right be exactly agreing and laid the one vpon the other Wherfore that part of a right line should be in a ground plaine superficies and part eleuated vpward is impossible which was required to be proued This figure more plainly setteth forth the foresaid demonstratioÌ„ if ye eleuate the superficies wheriâ— the line BC. An other demonstration after Flâ—sâ—â—s If it be possible let there be a right line ABG whose part AB let be in the ground playne superficies AED and let the rest therof BG be eleuated on high that is without the playne AED Then I say that ABG is not one right line For forasmuch as AED is a plaine superficies produce directly equally vpon the sayd playne AED the right lyne AB towardes D which by the 4. definition of the first shall be a right line And from some one point of the right line ABD namely from C draâ— vnto the point G a right lyne CG Wherefore in the triangle 〈…〉 the outward angâ—â— ABâ— is eqâ—â—ll to the two inward and opposite angles by the 32. of the first and therfore it is lesse then two right angles by the 17. of the same Wherfore the lyne ABG forasmuch as it maketh an angle is not â— right line Whâ—refore that part of a right line should be in a ground playne superficies and part eleuated vpward is impossible If ye marke well the figure before added for the playâ—er declaration of Euclides demonstration iâ— will not be hard for you to coâ—â—â—â—e this figure which â—lussâ—s putteth for his demonstâ—â—tion â— wherein
same superficies Wherefore these right lines AB BD and DC are in one and the selfe same superficies and either of these angles ABD and BDC is a right angle by supposition Wherefore by the 28. of the first the line AB is a parallel to the line CD If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be proued Here for the better vnderstanding of this 6. proposition I haue described an other figure as touching which if ye erect the superficies ABD perpendicularly to the superficies BDE and imagine only a line to be drawne from the poynt A to the poynt E if ye will ye may extend a thred from the saide poynt A to the poynt E and so compare it with the demonstration it will make both the proposition and also the demonstration most cleare vnto you ¶ An other demonstration of the sixth proposition by M. Dee Suppose that the two right lines AB CD be perpendicularly erected to one the same playne superficies namely the playne superficies OP Then I say that â—â— and CD are parallels Let the end points of the right lines AB and CD which touch the plaine supâ—â—â—â—cies Oâ— be the poyntes â— and D froÌ„â— to D let a straight line be drawne by the first petition and by the second petition let the straight line â—D be exteÌ„ded as to the poynts M N. Now forasmuch as the right line AB from the poynt â— produced doth cutte the line MN by construction Therefore by the second proposition of this eleuenth booke the right lines AB MN are in one plainâ— superficies Which let be QR cutting the superficies OP in the right line MN By the same meanes may we conclude the right line CD to be in one playne superficies with the right line MN But the right line MN by supposition is in the plaine superficies QR wherefore CD is in the plaine superficies QR And Aâ— the right line was proued to be in the same plaine superficies QR Therfore AB and CD are in one playne superficieâ— namely QR And forasmuch as the lines Aâ— and CD by supposition are perpendicular vpon the playne superficies OP therefore by the second definition of this booke with all the right lines drawne in the superficies OP and touching AB and CD the same perpeÌ„diculars Aâ— and CD do make right angles But by construction MN being drawne in the plaine superficies OP toucheth the perpendiculars AB and CD at the poyntes â— and D. Therefore the perpendiculars Aâ— and CD make with the right line MN two right angles namely ABN and CDM and MN the right line is proued to be in the one and the same playne superficies with the right lines AB CD namely in the playne superficies QR Whâ—refore by the second part of the 28. proposition of the first booke the right lineâ— AB and CD are parallelâ— If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be demonstrated A Corollary added by M. Dee Hereby it is euident that any two right lines perpendicularly erected to one and the selfe same playne superficies are also them selues in one and the same playne superficies which is likewisâ— perpendicularly erected to the same playne superficies vnto which the two right lines are perpendicular The first part hereof is proued by the former construction and demonstration that the right lines AB and CD are in one and the same playne superficies Qâ— The second part is also manifest that is that the playne superficies QR is perpendicularly erected vpon the playne superficies OP for that Aâ— and CD being in the playne superficies QR are by supposition perpendicular to the playne superficies OP wherefore by the third definition of this booke QR is perpendicularly erected to or vpon OP which was required to be proued Io. d ee his aduise vpon the Assumpt of the 6. As concerning the making of the line DE equall to the right line AB verely the second of the first without some farther consideration is not properly enough alledged And no wonder it is for that in the former bookeâ— whatsoeâ—â—â—â—aâ—h of lines bene spoken the same hath alwayâ—s bene imagined to be in one onely playne superficies considered or executed But here the perpendicular line AB is not in the same playnâ— superficies that the right line DB is Therfore some other helpe must be put into the handes of young beginners how to bring this probleme to execution which is this most playne and briefe Vnderstand that BD the right line is the common section of the playne superficies wherein the perpendiculars AB and CD are of the other playne superficies to which they are perpendiculars The first of these in my former demonstration of the 6 â— I noted by the playne superficies QR and the other I noted by the plaine superficies OP Wherfore BD being a right line common to both the playne supâ—rficieces QR OP therby the ponits B and D are coÌ„mon to the playnes QR and OP Now from BD sufficiently extended cutte a right line equall to AB which suppose to be BF by the third of the first and orderly to BF make DE equall by the 3. oâ— the first if DE be greater then BF Which alwayes you may cause so to be by producing of DE sufficiently Now forasmuch as BF by construction is cutte equall to AB and DE also by construction put equâ—ll to BF therefore by the 1. common sentence DE is put equall to AB which was required to be done In like sort if DE were a line geuen to whome AB were to be cutte and made equall first out of the line DB suâ—â—iciently produced cutting of DG equall to DE by the third of the first and by the same 3. cutting from BA sufficiently produced BA equall to DG then is it euideÌ„t that to the right line DE the perpeÌ„dicular line AB is put equall And though this be easy to conceaue yet I haue designed the figure accordingly wherby you may instruct your imagination Many such helpes are in this booke requisite as well to enforme the young studentes therewith as also to master the froward gaynesayer of our conclusion or interrupter of our demonstrations course ¶ The 7. Theoreme The 7. Proposition If there be two parallel right lines and in either of them be taken a point at all aduentures a right line drawen by the said pointes is in the self same superficies with the parallel right lines SVppose that these two right lines AB and CD be parallels and in either of theÌ„ take a point at all aduentures namely E and F. Then I say that a right line drawen from the point E to the point F is in the selfe same plaine superficies that the
an other from the point X to the point F. And forasmuch as these two parallel superficieces KL and MN are cut by the superâ—icies EBDX therâ—ore their common sections which are the lines EX and BD are by the 16. of the eleuenth parallels the one to the other And by the same reason also â—orasmuch as the two parallel superficies GH and KL be cut by the superâ—icies AXFC their common sections AC and XF are by the 16. of the eleuenth parallels And â—orasmuch as to one of the sides of the triangle ABDâ— namely to the side BD is drawne a parallel line EX therfore by the 2. of the sixt proportionally as the line AE is to the line EB so is the line AX to the line XD Againe forasmuch as to one of the sides of the triangle ADC namely to the side AC is drawen a parallel line XF therfore by the 2. of the sixt proportionally as the line AX is to the line XD so is the line CF to the line FD. And it was proued that as the line AX is to the line XD so is the line AE to the line EB therefore also by the 11. of the fift as the line AE is to the line EB so is the line CF to the line FD. If therfore two right lines â—e deuided by plaine superâ—icieces being parallels the parts of the lines deuided shal be proportionall which was required to be demonstrated In this figure it is more easy to see the former demonstration if ye erect perpendicularly vnto the ground superficies ACBD the thre superficieces GH KL and MN or if ye so â—râ—ct them that thâ—y be equedistant one to the other ¶ The 16. Theoreme The 18. Proposition If a right line be erected perpeÌ„dicularly to a plaine superficies all the superficieces extended by that right line are erected perpendicularly to the selfe same plaine superficies SVppose that a right line AB be erected perpendicularly to a ground superficies TheÌ„ I say that all the superficieces passing by the line AB are erected perpendicularly to the ground superficies Extend a superficies by the line AB and let the same be ED let the coÌ„mon section of the plaine superficies and of the ground superficies be the right line CE. And take in the line CE a point at all aduentures and let the same be F and by the 11. of the first from the point F drawe vnto the line CE a perpendicular line in the superficies DE and let the same be FG. And forasmuch as the line AB is erected perpendicularly to the ground superficies therefore by the 2. definition of the eleuenth the line AB is erected perpendicularly to all the right lines that are in the ground plaine superficies and which touch it Wherfore it is erected perpendicularly to the line CE. Wherefore the angle ABE is a right angle And the angle GFB is also a right angle by construction Wherefore by the â—8 of the first the line AB is a parallel to the line FG. But the line AB is erected perpendicularly to the ground superficies wherefore by the 8. of the eleuenth the line FG is also erected perpendicularly to the ground superficies And forasmuch as by the 3. definition of the eleuenth a plaine superficies is then erected perpendicularly to a plaine superficies when all the right lines drawen in one of the plaine superficieces vnto the common section of those two plaine superficieces making therwith right angles do also make right angles with the other plaine superficies and it is proued that the line FG drawen in one of the plaine superficieces namely in DE perpendicularly to the common section of the plaine superficieces namely to the line CE is erected perpendicularly to the ground superficies wherefore the plaine superficies DE is erected perpendicularly to the ground superficies In like sort also may we proue that all the plaine superficieces which passe by the line AB are erected perpendicularly to the ground superficies If therefore a right line be erected perpendicularly to a plaine superficies all the superficieces passing by the right line are erected perpendicularly to the selfe same plaine superficies which was required to be demonstrated In this figure here set ye may erect perpeÌ„dicularly at your pleasure the superficies wherin are drawen the lines DC GF AB and HE to the ground superficies wherin is drawen the line CFBE and so plainly compare it with the demonstration before put ¶ The 17. Theoreme The 19. Proposition If two plaine superficieces cutting the one the other be erected perpendicularly to any plaine superficies their common section is also erected perpendicularly to the selfe same plaine superficies SVppose that these two plaine superâ—icieces AB BC cutting the one the other be erected pâ—rpâ—ndicularly to a ground superficies and let their common section be the line BD. Then I say that the line BD is erected perpendicularly to the ground superâ—icies For if not then by the 11. of the first from the point D draw in the superficies AB vnto the right line DA a perpendicular line DE. And in the superficies CB draw vnto the line DC a perpendicular line DF. And forasmuch as the superficies AB is erected perpendicularly to the ground superficies and in the plaine superficies AB vnto the common section of the plaine superficies and of the ground superficies namely to the line DA is erected a perpendicular line DE therefore by the conuerse of the 3. deâ—inition of this booke the line DE is erected perpendicularly to the ground superâ—icies And in like sort may we proue that the line DF is erected perpendicularly to the ground superficies Wherefore from one and the selfe same point namely from D are erected perpendicularly to the ground superficies two right lines both on one and the self same side which is by the 15. of the eleuenth impossible Wherfore from the point D can not be erected perpendicularly to the ground superficies any other right lines besides BD which is the common section of the two superficieces AB and BC. If therefore two plaine superâ—icieces cutting the one the other be erected perpendicularly to any plaine superâ—icies their common section is also erected perpendicularly to the selfe same plaine superâ—icies which was required to be proued Here haue I set an other figure which will more plainly shewe vnto you the former demonstration if ye erecte perpendicularly to the ground superficies AC the two superficieces AB and BC which cut the one the other in the line BD. The 18. Theoreme The 20. â—roposition If a solide angle be contayned vnder three playne superficiall angles euery two of those three angles which two so euer be taken are greater then the third SVppose that the solide angle A be contayned vnder three playne superficiall angles that is vnder BAC CAD and DAB Then I say that two of these superficiall angles how so euer they be taken are greater then the third If the
the line DC Wherfore the superficies AC is a parallelogramme In like sort also may we proue that euery one of these superficices CE GF BG FB and AE are parallelogrammes Draw a right line from the point A to the point H and an other from the point D to the point F. Aud forasmuch as the line AB is proued a parallel to the line CD and the lyne BH to the line CF therfore these two right lines AB and BH touching the one the other are parallels to these two right lines DC and CF touching also the one the other and not being in one and the selfe same plaine superficies Wherfore by the 10. of the eleuenth they comprehend equall angles Wherfore the angle ABH is equall to the angle DCF And forasmuch as these two lines AB and BH are equall to these two lines DC and CF and the angle ABH is proued equall to the angle DCFâ— therfore by the 4. of the first the base AH is equall to the base DF and the triangle ABH is equall to the triangle DCF And forasmuch as by the 41. of the first the parallelogramme BG is double to the triangle ABH and the parallelogramme CE is also double to the triangle DCF therfore the parallelogramme BG is equall to the parallelogramme CE. In like sort also may we proue that the parallelogramme AC is equall to the parallelogramme GF and the parallelograme AE to the parallelogramme FB If therfore a solide or body be contained vnder sixe parallel plaine superficieces the opposite plaine superficieces of the same body are equal parallelogrammes which was required to be demonstrated I haue for the better helpe of young beginners described here an other figure whose forme if it be described vpon pasted paper with the letters placed in the same order that it is here and then if ye cut finely these lines AG DE and CF not through the paper and folde it accordingly compare it with the demonstration and it will shake of all hardenes from it The 22. Theoreme The 25. Proposition If a ParallelipipedoÌ„ be cutte of a playne superficies beyng a parallel to the two opposite playne superficieces of the same body then as the base is to the base so is the one solide to the other solide I haue for the better sight of the coÌ„structioÌ„ demoÌ„stration of the former 25. propositioÌ„ here set another figure whose forme if ye describe vppon pasted paper and finely cut the three lines XI BS and TY not through the paper but halfe way and then fold it accordingly and compare it with the construction and demoÌ„stration you shall see that it will geue great light therunto Here Flussas addeth three Corollaries First Corollary If a Prisme be cutte of a playne superficies parallel to the opposite superficieces the seâ—â—ions of the Prisme shall be the one to the other in that proportion that the sections of the base are the one to the other For the sections of the bases which bases by the 11. definitioÌ„ of this booke are parallelogrammes shall likewise be parallelogrammes by the 16. of this booke when as the superficies which cutteth is parallelel to the opposite superâ—icieces and shal also be equiangle Wherfore if vnto the bases by producing the right lines be added like and equall bases as was before shewed in a parallelipipedon of those sections shal be made as many like Prismes as ye will. And so by the same reason namely by the commoÌ„ excesse equalitie or want of the multiplices of the bases of the sections by the 5. definitioÌ„ of the fifth may be proued that euery section of the Prisme multiplyed by any multiplycation whatsoeuer shall haue to any other section that proportion that the sections of the bases haue Second Corollary Solides whos 's two opposite superficieâ—es are poligonon figures like equall and parallels the other superficies which of necessitie are parallelogrammes being cutte of a playne superficies parallel to the two opposite superficies the sections of the base are the one to the other as the sections of the solide are thâ— one to the other Which thing is manifest for such solides are deuided into Prismes which haue one coÌ„mon side namely the axe or right line which is drawne by the centers of the opposite bases Wherefore how many paâ—allelograÌ„mes or bases are set vpon the opposite poligonon figures of so many Prismes shal the whole solide be coÌ„posed For those poligonon figures are deuided into so many like triangles by the 20. of the sixth which describe Prismes Which Prismes being cut by a superficies parallel to the opposite superficieces the sectioÌ„s of the bases shal by the former Corollary be proportional with the sectioÌ„s of the Prismes Wherefore by the â—â— of the fifth as the sections of the one are the one to the other so are the sections of the whole the one to the other Of these solides there are infinite kindes according to the varietie of the opposite and parallel poligonon figures which poligonon figures doo alter the angles of the parallelogrammes set vpon them according to the diuersitie oâ— their situation But this is no let at all to this corollary for that which we haue proued will alwayes follow When as the superficieces which are set vpoÌ„ the opposite like equal poligonon and parallel superficieces are alwayes parallelogrammes Third Corollary Tâ—e foresayd solides â—omposed of Prismes being cutte by a playne superficies parallel to the oppositâ— superficieces are the one to the other as the heades or higher parts cutte are For it is proued that they are the one to the other as the bases are Which bases forasmuch as they are parâ—llelogrammes are the one to the other as the right lines are vpon which they are set by the first of the sixth which right lines are the heddes or higher parts of the Prismes The 4. Probleme The 26. Proposition Vpon a right lyne geuen and at a point in it geuen to make a solide angle equall to a solide angle geuen In thesâ— two 〈…〉 here put you may in 〈◊〉 clearely concerne the â—â—â—â—mer construction and dâ—â—monstratioÌ„ if ye erect peâ—â—pendicularly vnto the ground superficies the triangles ALB and DCE eleuate the triangles ALH and DCF that the lynes LA and CD of them may exactly agree with the lineâ— LA and CD of the â—riangles erecâ—edâ— For so ordering them if ye compare the former construction and demonstration with them they will be playnâ— vnto you Although Euclides demoÌ„stration be touching solide angles which are contained only vnder three superficiall angles that is whose bases are triangles yet by it may ye performe the Probleme touching solide angles contained vnder superficiall angles how many soeuer that is hauing to their bases any kinde of Poligonon figures For euery Poligonon figure may by the 20. of the sixt be resolued into like tringles And so also shall the solide angle be deuided into so many solide angles as there be
at all aduentures namely D V G S and a right line is drawen from the point D to the point G and an other from the point V to the point S. Wherefore by the 7. of the eleuenth the lines DG and VS are in one and the selfe same plaine superficies And forasmuch as the line DE is a parallel to the line BG therefore by the 24. of the first the angle EDT is equall to the angle BGT for they are alternate angles and likewise the angle DTV is equall to the angle GTS Now then there are two triangles that is DTV and GTS hauing two angles of the one equall to two angles of the other and one side of the one equall to one side of the other namely the side which subtendeth the equall angles that is the side DV to the side GS for they are the halfes of the lines DE and BG Wherefore the sides remayning are equall to the sides remayning Wherfore the line DT is equall to the line TG and the line VT to the line T S If therefore the opposite sides of a Parallelipipedon be deuided into two equall partes and by their sections be extended plaine superficieces the common section of those plaine superficieces and the diameter of the Parallelipipedon do deuide the one the other into two equall partes which was required to be demonstrated A Corollary added by Flussas Euery playne superficies extended by the center of a parallelipipedon diuideth that solide into two equall partes and so doth not any other playne superficies not extended by the center For euery playne extended by the center cutteth the diameter of the parallelipipedon in the center into two equall partes For it is proued that playne superficieces which cutte the solide into two equall partes do cut the dimetient into two equall partes in the center Wherefore all the lines drawen by the center in that playne superficies shall make angles with the dimetient And forasmuch as the diameter falleth vpon the parallel right lines of the solide which describe the opposite sides of the sayde solide or vpon the parallel playne superficieces of the solide which make angels at the endes of the diameter the triangles contayned vnder the diameter and the right line extended in that playne by the center and the right line which being drawen in the opposite superficieces of the solide ioyneth together the endes of the foresayde right lines namely the ende of the diameter and the ende of the line drawen by the center in the superficies extended by the center shall alwayes be equall and equiangle by the 26. of the first For the opposite right lines drawen by the opposite playne superficieces of the solide do make equall angles with the diameter forasmuch as they are parallel lines by the 16. of this booke But the angles at the ceÌ„ter are equall by the 15. of the first for they are head angles one side is equall to one side namely halfe the dimetient Wherefore the triangles contayned vnder euery right line drawen by the center of the parallelipipedon in the superficies which is extended also by the sayd center and the diameter thereof whose endes are the angles of the solide are equall equilater equiangle by the 26. of the first Wherfore it followeth that the playne superficies which cutteth the parallelipipedon doth make the partes of the bases on the opposite side equall and equiangle and therefore like and equall both in multitude and in magnitude wherefore the two solide sections of that solide shal be equall and like by the 8. diffinition of this booke And now that no other playne superficies besides that which is extended by the center deuideth the parallelipipedon into two equall partes it is manifest if vnto the playne superficies which is not extended by the center we extend by the center a parallel playne superficies by the Corollary of the 15. of this booke For forasmuch as that superficies which is extended by the center doth deuide the parallelipipedoÌ„ into two equall parâ—â— it is manifest that the other playne superficies which is parallel to the superficies which deuideth the solide into two equall partes is in one of the equall partes of the solide wherefore seing that the whole is euer greater then his partes it must nedes be that one of these sections is lesse then the halfe of the solide and therefore the other is greater For the better vnderstanding of this former proposition also of this Corollary added by Flussas it shal be very nedefull for you to describe of pasted paper or such like matter a parallelipipedoÌ„ or a Cube and to deuide all the parallelograÌ„mes therof into two equall parts by drawing by the câ—Ì„ters of the sayd parallelogrammes which centers are the poynts made by the cutting of diagonall lines drawen froÌ„ thâ— opposite angles of the sayd parallelograÌ„mes lines parallels to the sides of the parallelograÌ„mes as in the former figure described in a plaine ye may see are the sixe parallelograÌ„mes DE EH HA AD DH and CG whom these parallel lines drawen by the ceÌ„ters of the sayd parallelograÌ„mes namely XO OR PR and PX do deuide into two equall parts by which fower lines ye must imagine a playne superficies to be extended also these parallel lynes KL LN NM and Mâ— by which fower lines likewise yâ— must imagine a playne superficies to be extended ye may if ye will put within your body made thus of pasted paper two superficieces made also of the sayd paper hauing to their limites lines equall to the foresayde parallel lines which superficieces must also be deuided into two equall partes by parallel lines drawen by their centers and must cut the one the other by these parallel lines And for the diameter of this body exteÌ„d a thred from one angle in the base of the solide to his opposite angle which shall passe by the center of the parallelipipedon as doth the line DG in the figure before described in the playne And draw in the base and the opposite superficies vnto it Diagonall lines from the angles from which is extended the diameter of the solide as in the former description are the lines BG and DE. And when you haue thus described this body compare it with the former demonstration and it will make it very playne vnto you so your letters agree with the letters of the figure described in the booke And this description will playnely set forth vnto you the corollary following that proposition For where as to the vnderstanding of the demonstration of the proposition the superficieces put within the body were extended by parallel lynes drawen by the ceÌ„ters of the bases of the parallelipipedon to the vnderstanding of the sayd Corollary ye may extende a superficies by any other lines drawen in the sayd bases so that yet it passe through the middest of the thred which is supposed to be the center of the parallelipipedon ¶ The 35. Theoreme The 40. Proposition If there be
and so of such like Who can not readily fall into Archimedes reckoning and account by his method To finde the proportion of the circumference of any circle to his diameter to be almost triple and one seuenth of the diameter but to be more then triple and ten one seuentithes that is to be lesse then 3 1 â— and more then 3 10 71. And where Archimedes vsed a Poligonon figure of 96. sides he that for exercise sake or for earnest desire of a more nerenes will vse Polygonon figures of 384. sides or more may well trauaile therein till either wearines cause him stay or els he finde his labour fruitles In deede Archimedes concluded proportion of the circumference to the diameter hath hitherto serued the vulgare andâ— Mechanicall worâ— men wherewith who so is not concented let his owne Methodicall trauaile satisfie his desire or let him procure other therto For narrower termes of greater and lesse found and appointed to the circumference will also winne to the Area of the circle a nearer quantitie seing it is well demâ—â—â—rated of Archimedâ—s that a triangle rectangle of whose two sides contayning the right angle one is equall to the semidiameter of the circle and the other to the circumference of the same is equall to the Area of that circle Vpon which two Theoremes it followeth that the square made of the diameter is in that proportion to the circle very neare in which 14 is to 11â— Wherefore euery circle is eleuen fowertenthes well neare of the square about him described The one side then of that square deuide into 14. equall partes and from that point which endeth the eleuenth part drawe to the opposite side a line parallel to the other sidâ—s and so make peâ—fecte the parallelogramme Then by the last Proposition of the second booke vnto that parallelogramme whose one side hath those 11. equall partes make a square equall Then is it euident that square to be equall to the circle about which the first square is described As ye may here beholde in these figures Gentle frend the great desire which I haue that both with pleasure and also profite thou mayest spend thy time in these excellent studies doth cause me here to furnishe thee somewhat extraordinarily about the circle not onely by pointing vnto thee the welspring of Archimedes his so much wondreâ— at and iustly commended trauaile in the former 3. Theoremes here repeated but also to make thee more apt to vnderstand and practise this and other bookes following where vse of the circle may be had in any consideration as in Cones Cylinders and Spheres c. ¶ A Corollary 1. By Archimedes second Theoreme as I haue here alleaged them it is manifest that a parallelogramme contained either vnder the semidiameter and halfe the circumference or vnder the halfe semidiameter and whole circumference of any circle is equall to the circle by the 41. of the first and first of the six◠¶ A Corollary 2. Likewise it is euident that the parallelogramme contayned vnder the semidiameter and halfe of any portion of the circumference of a circle geuen is equall to that sector of the same circle to which the whole portion of the circumference geuen doth belong Or you may vse the halfe semidiameter and the whole portion of the circumference as sides of the said parallelogramme The farther winning and inferring I commiâ—â—â— to your skill care and â—â—udyâ— But in an other sort will I geue you newe ayde and instruction here ¶ A Theoreme Of all circles the circumferences to their owne diameters haue one and the same proportion in what one circle soeuer they are assigned That is as Archimedes hath demonstrated almost as 22. to 7 or nearer if nearer be fouâ—d vntill the very precise proportion be demonstrated Which what soeuer it be in all circumferences to their proper diameters will be demonstrated one and the same A Corollary 1. Wherefore if two circles be propounded which suppose to be A and B as the circumference of A is to the circumference of B so is the diameter of A to the diameter of B. For by the former Theoreme as the circumference of A is to his own diameter so is the circumference of B to his own diameter Wherfore alternately as the circumference of A is to the circumference of B so is the diameter of A to the diameter of B Which was required to be demonstrated A Corollary 2. It is now then euident that we can geue two circles whose circumferences one to the other shall haue any proportion geuen in two right lines The great Mechanicall vse besides Mathematicall considerations which these two Corollaryes may haue in Wheeles of Milles Clockes Cranes and other engines for water workes and for warres and many other purposes the earnest and wittie Mechanicien will soone boult out gladly practise â— Iohn Dee ¶ The 2. Theoreme The 2. Proposition Circles are in that proportion the one to the other that the squares of their diameters are SVppose that there be two circles ABCD and EFGH and let their diameters be BD and FH Then I say that as the square of the line DB is to the square of the line FH so is the circle ABCD to the circle EFGH For if the circle ABCD be not vnto the circle EFGH as the square of the line BD is to the square of the line FH then the square of the line BD shall be to the square of the line FH as the circle ABCD is to a superficies either lesse then the circle EFGH or greater First let the square of the line BD be to the square of the line FH as the circle ABCD is to a superficies lesse then the circle EFGH namely to the superficies S. Describe by the 6. of the fourth in the circle EFGH a square EFGH Now this square thus described is greater then the halfe of the circle EFGH For if by the pointes E F G H we drawe right lines touching the circle the square EFGH is the halfe of the square described about the circle but the square described about the circle is greater then the circle Wherefore the square EFGH which is inscribed in the circle is greater then the halfe of the circle EFGH Deuide the circumferences EF FG GH and HE into two equall partes in the pointes K L M N. And drawe these right lines EK KF FL LG GM MH HN and NE. Wherefore euery one of these triangles EKF FLG GMH and HNE is greater then the halfe of the segmeÌ„t of the circle which is described about it For if by the pointes K L M N be drawen lines touching the circle and then be made perfecte the parallelograÌ„mes made of the right lines EF FG GH HE euery one of the triangles EKF FLG GMH HNE is the halfe of the parallelogragraÌ„me which is described about it by the 41. of the first but the segmeÌ„t described about it is lesse then the parallelogramme Wherefore euery one
and meane proportion but in one onely point which was requisite to be demonstrated A Theoreme 2. What right line so euer being deuided into two partes hath those his two partes proportionall to the two segmentes of a line deuided by extreame and meane proportion is also it selfe deuided by an extreame and meane proportion and those his two partes are his two segments of the sayd proportion Suppose AB to be a line deuided by an extreame and meane proportion in the point C and AC to be the greater segment Suppose also the right line DE to be deuided into two partes in the point F and that the part DF is to FE as the segment AC is to CB or DF to be to AC as FE is to CB. For so these paâ—tes are proportionall to the sayd segmentes I say now that DE is also deuided by an extreame and meane proportion in the point F. And that DF FE are his segmentes of the sayd proportion For seing as AC is to CB so is DF to FE by supposition Therfore as AC and CB which is AB are to CB so is DF and FE which is DE to FE by the 18. of the fifth Wherefore alternately as AB is to DE so is CB to FE And therefore the residue AC is to the residue DF as AB is to DE by the fifth of the fift And then alternately AC is to AB as DE is to DF. Now therefore backward AB is to AC as DE is to DF. But as AB is to AC so is AC to CB by the third definition of the sixth booke Wherefore DE is to DF as AC is to CB by the 11. of the fifth And by supposition as AC is to CB so is DF to FE wherefore by the 11. of the fifth as DE is to DF so is DF to FE Wherefore by the 3. definition of the sixth DE is deuided by an extreame and meane proportion in the point F. Wherefore DF and FE are the segmentes of the sayd proportion Therefore what right line so euer being deuided into two partes hath those his two partes proportionall to the two segmentes of a line deuided by extreame and meane proportionâ— is also it selfe deuided by an extreme and meane proportion and those his two partes are his two segmentes of the sayd proportionâ— which was requisite to be demonstrated Note Many wayes these two Theoremes may be demonstrated which I leaue to the exercise of young studentes But vtterly to want these two Theoremes and their demonstrations in so principall a line or rather the chiefe piller of Euclides Geometricall pallace was hetherto and so would remayne a great disgrace Also I thinke it good to note vnto you what we meant by one onely poynt We mâ—â—â—â— that the quantities of the two segmentes can not be altered the whole line being once geuen And though from either end of the whole line the greater segment may begin And so as it were the point of section may seeme to be altered yet with vs that is no alteration forasmuch as the quantities of the segmentes remayne all one I meane the quantitie of the greater segment is all one at which end so euer it be taken And therefore likewise the quantitie of the lesse segment is all one c. The like confideration may be had in Euclides tenth booke in the Binomiall lines c. Ioâ—n Dee 1569. Decemb. 18. The 3. Theoreme The 3. Proposition If a right line be deuided by an extreme and meane proportion and to the lesse segment be added the halfe of the gerater segment the square made of those two lines added together is quintuple to the square made of the half line of the greater segment SVppose that the right line AB be deuided by an extreme and meane proportion in the point C. And let the greater segment thereof be AC And deuide AC into two equall partes in the point D. Then I say that the square of the line BD it quintuple to the square of the line DC Describe by the 46. of the first vpon the line AB a square AE And describe and make perfect the figure that is diuide the lyne AT like vnto the diuision of the line AB by the 10. of the sixth in the pointes R H by which pointes drawe by the 31. of the first vnto the line AB parallel lines RM and HN. So likewise draw by the pointes D C vnto the line BE these parallel lines DL and CS draw the diameter BT And forasmuch as the line AC is double to the line DC therefore the square of AC is quadruple to the square of DC by the 20. of the sixth that is the square RS to the square FG And forasmuch as that which is contayned vnder the lines AB and BC is equall to the square of the line AC and that which is contayned vnder the lines AB and BC is equall to the parallelogramme CE the square of the line AC is the square RS wherefore the parallelogramme CE is equall to the square RS. But the square RS is quadruple to the square FG wherefore the parallelogramme CE also is quadruple to the square FG. Agayne forasmuch as the line AD is equall to the line DC therfore the line HK is equall the line KF wherefore also the square GF is equall to the square HL wherefore the line GK is equall to the line KL that is the line MN to the line NE wherefore the parallelogramme MF is equall to the parallelogramme FE But the parallelogramme MF is equall to the parallelogramme CG wherfore the parallelogramme CG is also equal to the parallelogramme FE Put the parallelogramme CN common to theÌ„ both Wherefore the gnomon XOP is equall to the parallelogramme CK But the parallelogramme CE is proued to be quadruple to Gâ— the square wherefore the gnomon XOP is quadruple to the square GF Wherefore the square DN is quintuple to the square FG And DN is the square of the line DB and GF the square of the line DC Wherefore the square of the line DB is quintuple to the square of the line DC If therefore a right line be deuided by an extreme and meane proportion and to the lesse segment be added the halfe of the greater segment the square made of those two lines added together is quintuple to the square made of the halfe line of the greater segmeÌ„t Which was required to be demonstrated Ye shall finde this proposition an other way demonstrated after the fiueth proposition of this booke Here foloweth M. Dee his additions ¶ A Theoreme 1. If a right line geuen be quintuple in power to the powre of a segment of him self the double of that segment and the other part remayning of the first geuen line make a line diuided by extreme and meane proportion and that double of the segment is the greater part thereof Forasmuch as this is the conuerse of Euclides
third proposition we will vse the same suppositions and constructions there specified so farre as they shall serue our purpose Beginning therefore at the conclusion we must infer the part of the proposition before graunted It was concluded that the square of the line DB is quintuple to the square of the line DC his owne segment Therefore DN the square of DB is quintuple to GF the square of DC But the squaâ—e of AC the double of DC which is RS is quadruple to GF by the second Corollary of the 20. of the sixth and therefore RS with GF are quintuple to GF and so it is euident that the square DN is equall to the square RS together with the square GF Wherefore from those two equalles taking the square GF common to them both remayneth the square RS equall to the Gnomon XOP But to the Gnomon XOP the parallelogramme CE is equall Wherefore the square of the line AC which is RS is equâ—ll to the parallelograÌ„me Câ— Which parallelogamme is coÌ„tained vnder BE equall to AB and CB the part remayning of the first line gâ—uen which was DB. And the line AB is made of the double of the segment DC and of CBâ— the other part of the line DB first gouen Wherefore the double of the segment DC with CB the part remayning which altogether is the whole line AB is to AC the double of the segment DC as that same AC is to CB by the second part of the 16. of the sixth Therfore by the 3. definitioÌ„ of the sixth booke the whole line AB is deuided by an extreme and meane proportion AC the double of the segmeÌ„t DC being middell proportionall is the greater part therof Wheâ—efore if a right line be quintuple in power c. as in the proposition which was to be demonstrated Or thus it may be demonstrated Forasmuch as the square DN is quinâ—uple to the square GF I meane the square of DB the line geueÌ„ to the square oâ— DC the segmeÌ„t And the same square DN is equall to the parallelograÌ„me vnder AB CB with the square made of the line DC by the sixth of the second for vnto the line AC equally deuided the line CB is as it were adioyned Wherefore the parallelogramme vnder AB CB together with the square of DC which is GF is quintuple to the square GF made oâ— thâ— line DC Taking then that square GF â—rom the parallelogramme vnder AB CB that parallelogramme vnder AB CB remayning alone is but quadruple to the sayd square of the line DC But by the 4. of the second or the second Corollary of the 20. of the sixth RS â—he square of the line AC is quadrupla to the same square GFâ— Wherfore by the 7. of the fifth the square of the line AC is equall to the parallelogramme vnder AB CB and so by the second part of the 16. of the sixth AB AC and CB are three lines in continuall proportion And seing AB is greater theÌ„ AC the same AC the double of the line DC shall be greater then the part BC remayning Wherfore by the 3. definition of the sixth AB composed or made of the double of DC and the other part of DB remaining is deuided by an extreme and middel proportion and also his greater segment is AC the double of the segment DC Wherfore If a right line be quintuple in power c. as in the propositionâ— which was to be demonstrated A Theoreme 2. If a right line deuided by an extreme and meane proportion be geuen and to the great segment â—herof he directly adioyned a line equal to the whole line geuen that adioyned line and the said greater segment do make a line diuided by extreme and meane proportion whose greater segment is the line â—dioyned Suppose the line geuen deuided by extreame and meane proportion to be AB deuided in the point C and his greater segment let be AC vnto AC directly adioyne a line equall to AB let that be AD I say that AD together with AC that is DC is a deuided by extreme and middel proportion whose greater segment is AD the line adioyned Deuide AD equally in the point E. Now forasmuch as AE is the halfe of AD by construction it is also the halfe of AB equall to AD by construction Wherfore by the 1. of the thirtenth the square of the line composed of AC and AE which â—ne is EC is quintuple to the square of the line AE Wherefore the double of AE and the line AC composed as in one right line is a line deuided by extreme and meane proportion by the conuerse of this third by me demonstrated and the double of AE is the greater segment But DC is the line composed of the double of AE the line AC and with all AD is the double of AE Wherfore DC is a line deuided by extreme and meane proportion and AD iâ— hiâ— greater segment If a right line therefore deuided by extreme and meane proportion be geuen and to the greater segment thereof be directly adioyned a line equall to the whole line geuen that adioyned line and the sayd greater segment do make a line diuided by extreame and meane proportion whose greater segment is the line adioyned Which was required to be demonstrated Two other briefe demonstrations of the same Forasmuch as AD is to AC as AB is to AC because AD is equall to AB by construction but as AB is to AC so is AC to CB by supposition Therefore by the 11. of the fifth as AC is to CB so is AD to AC Wherefore as AC and CB which is AB is to CB so is AD and AC which is DC to AC Therefore euersedly as AB is to AC so is DC to AD. And it is proued AD to be to AC as AC is to CB. Wherefore as AB is to AC and AC to CB so is DC to AD and AD to AC But AB AC and CB are in continuall proportion by supposition Wherfore DC AD and AC are in continuall proportion Wherefore by the 3. definition of the sixth booke DC is deuided by extreme and middell proportion and his greatest segment is AD. Which was to be demonstrated Note from the marke how this hath two demonstrations One I haue set in the margent by ¶ A Corollary 1. Vpon Euclides third proposition demonstrated it is made euident that of a line deuided by extreame and meane proportion if you produce the lesse segment equally to the length of the greater the line therby adioyned together with the sayd lesse segment make a new line deuided by extreame and middle proportion Whose lesse segment is the line adioyned For if AB be deuided by extreme and middell proportion in the point C AC being the greater segment and CB be produced from the poynt B making a line with CB equall to AC which let be CQ and the
the line AB is diuided by an extreme and meane proportion in the poynt C and the greater segment thereof is the line ACâ— therfore that which is contayned vnder the lines AB and BC is equall to the square of the line AC Wherfore that which is coÌ„tayned vnder the lines AB and BC twise is double to the square of AC Wherfore that which is contayned vnder the lines AB and BC twise together with the square of the line AC is treble to the square of the line AC But that which is contayned vnder the lines AB and BC twise together with the square of the line AC is the squares of the lines AB and BC by the 7. of the second Wherefore the squares of the lines AB and BC are treble to the square of the line AC which was required to be demonstrated Resolution of the 5. Theoreme Suppose that a certaine right line AB be diuided by an extreme and meane proportion in the point C. And let the greater segment therof be the line AC And vnto the line AB adde a line equall to the line AC and let the same be AD. Theâ— I say that the line DB is diuided by an extreme and meane proportion in the point A. And the greater segment therof is the line AB For forasmuch as the line DB is diuided by an extreme meane proportion in the point A and the greater segment thereof is the line AB therfore as the line DB is to the line BA so is the line BA to the line AD but the line AD is equall to the line AC wherefore as the line DB is to the line BA so is the line BA to the line AC Wherfore by conuersioÌ„ as the line BD is to the line DA so is the line AB to the line BC by the corollary of the 19. of the fifth wherfore by diuision by the 17. of the fifth as the line BA is to the line AD â—o is the line AC to the line CB. But the line AD is equall to the line AC Wherfore as the line BA is to the line AC so is the line AC to the line CB. And so it is indeede for the line AB is by supposition diuided by an extreme and meane proportion in the point C. Composition of the 5. Theoreme Now forasmuch as the line AB is diuided by an extreme and meane proportion in the point C therefore as the line BA is to the line AC so is the line AC to the line CB but the line AC is equall to the line AD. Wherefore as the line BA is to the line AD so is the line AC to the line CB. Wherfore by composition by the 18. of the fifth as the line BD is to the line DA so is the line AB to the line BC. Wherefore by conuersion by the corollary of the 19. of the fiueth as the line DB is to the line BA so is the line BA to the line AC but the line AC is equall to the line AD. Wherefore as the line DB is to the line BA so is the line BA to the line AC Wherfore the line DB is deuided by an extreme and meane proportion in the point A and his greater segment is the line AB which was required to be demonstrated An Aduise by Iohn Dee added SEing it is doubteles that this parcel of Resolution and Composition is not of Euclides doyng it can not â—ustly be imputed to Euclide that he hath therby eyther superfluitie or any part disproportioned in his whole Composition Elementall And though for one thing one good demonstration well suffiseth for stablishing of the veritie yet oâ— one thing diuersly demonstrated to the diligent examiner of the diuerse meanes by which that varietie ariseth doth grow good occasions of inuenting demonstrations where matter is more straunge harde and barren Also though resolution were not in all Euclide before vsed yet thankes are to be geuen to the Greke Scholic writter who did leaue both the definition and also so short and easy examples of a Method so auncient and so profitable The antiquity of it is aboue 2000. yeares it is to weâ—e euer since Plato his time and the profite therof so great that thus I finde in the Greeke recorded 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Proclus hauing spoken of some by nature excellent in inuenting demonstrations pithy and breif sayeth Yet are there Methods geuen for that purpose And in dede that the best which by Resolution reduceth the thing inquired of to an vndoubted principle Which Method Plato taught Leodamas as iâ— reported And he is registred thereby to haue bene the inuenter of many things in Geometry And verely in Problemes it is the chief ayde for winning and ordring a demonstration first by Supposition of the thing inquired of to be done by due and orderly Resolution to bring it to a stay at an vndoubted veritie In which point of Art great abundance of examples are to be seen in that excellent and mighty Mathematiciâ—n Archimedes in his expositor Eutocius in Menaechmus likewise and in Diocles booke de Pytiâ—s and in many other And now for as much as our Euclide in the last six Propositions of this thirtenth booke propoundeth and concludeth those Problemes which were the ende Scope and principall purpose to which all the premisses of the 12. bookes and the rest of this thirtenth are directed and ordered It shall be artificially done and to a great commodity by Resolution backward from these 6. Problemes to returne to the first definition of the first booke I meane to the definition of a point Which is nothing hard to do And I do counsaile all such as desire to atteinâ— to the profound knowledge of Geometrie Arithmeticke or any braunche of the sciences Mathematicall so by Resolution discreatly and aduisedly to resolue vnlose vnioynt and disseauer euery part of any worke Mathematicall that therbyâ— aswell the due placing of euery verity and his proofe as also what is either superfluous or wanting may euidently appeare For so to inuent there with to order their writings was the custome of them who in the old time were most excellent And I for my part in writing any Mathematicall conclusion which requireth great discourse at length haue found by experience the commoditie of it such that to do other wayes were to me a confusion and an vnmethodicall heaping of matter together besides the difficulty of inuenting the matter to be disposed and ordred I haue occasion thus to geue you friendely aduise for your beâ—ofeâ— because some of late haue inueyed against Euclide or Theon in this place otherwise than I would wish they had The 6. Theoreme The 6. Propositionâ— If a rationall right line be diuided by an extreme and meane proportion eyther of the segments is an irrationall line of that kinde which is called a residuall line SVppose that AB beyng a rationall line be deuidedly 〈◊〉 extreme
namely the part greater then the whole which is impossible Wherefore the circumference shall not cut the line CD Now I say that it shall not passe aboue the line CD and not touch it in the point D. For if it be possible let it passe aboue it and extend the line CD to the circuÌ„ference and let it cut it in the point â— And draw the lines FB and FA and it shall followe as before that the line CD is greater then the line CF which is impossible Wherefore that is manifest which was required to be proued ¶ Second Assumpt If there be a right angle vnto which a base is subtended and if vpon the same be described a semicircle the circumference thereof shall passe by the point of the right angle The conuerse of this was added after the demonstration of the 31. of the third out of Pelitarius And these two Assumptes of Campane are necessary for the better vnderstanding of the demonstration of the secoâ—d part of this 13. Proposition wherein is proued that the pyramis is contained in the Sphere geuen ¶ Certaine Corollaryes added by Flussas First Corollary The diameter of the Sphere is in power quadruple sesquialtera to the line which is drawen from the centre to the circumference of the circle which containeth the base of the pyramis For forasmuch as it hath bene proued that the diameter KL is in power sesquialter to the side EF and it is proued also by the 12. of this booke that the side EF is in power triple to the line EH which is drawen from the centre of the circle contayning the triangle EFG But the proportion of the extremes namely of the diameter to the line EH consisteth of the proportions of the meanes namely of the proportion of the diameter to the line EF and of the proportion of the line EF to the line EH by the 5. definition of the sixt which proportions namely triple and sesquialter added together make quadruple sesquialter as it is easie to proue by that which was taught in the declaration of the 5. definition of the sixt booke Wherefore the Corollary is manifest ¶ Second Corollary Onely the line which is drawen from the angle of the pyramis to the base opposite vnto it passing by the center of the Sphere is perpendicular to the base and falleth vpon the centre of the circle which containeth the base For if any other line then the line KMH which is drawen by the centre of the Sphere to the centre of the circle should fall perpendicularly vpon the plaine of the base then from one and the selfe same point should be drawen to one and the selfe same plaine two perpendicular lines contrary to the 13. of the eleuenth which is impossible Farther if from the toppe K should be drawen to the center of the base namely to the point H any other right line not passing by the centre M two right liues shoulde include a superficies contrary to the last common sentence which were absurde Wherefore onely the line which is drawen by the center of the Sphere to the centre of the base is perpendicular to the sayd base And the line which is drawen from the angle perpendicularly to the base shall passe by the centre of the Sphere Third Corollary The perpendicular line which is drawen from the centre of the Sphere to the base of the pyramis is equall to the sixt part of the diameter of the Sphere For it is before proued that the line MH which is drawen from the centre of the Sphere to the centre of the base is equall to the line NC which line NC is the sixt part of the diameter AB and therfore the line MH is the sixt part of the diameter of the Sphere For the diameter AB is equall to the diameter of the Sphere as hath also before bene proued ¶ The 2. Probleme The 14. Proposition To make an octohedron and to coÌ„prehend it in the sphere geuen namely that wherein the pyramis was comprehended and to proue that the diameter of the sphere is in power double to the side of the octohedron TAke the diameter of the former sphere geuen which let be the line AB and diuide it by the 10. of the first into two equall partes in the point C. And describe vpon the line AB a semicircle ADB And by the 11. of the first froÌ„ the point C rayse vp vnto the line AB a perpendicular line CD And draw a right line from the point D to the point B. And describe a square EFGH hauing euery one of his sides equall to the line BD. and draw the diagonal lines FH EG cutting the one the other in the point K. And by the 12. of the eleuenth from the point K namely the point where the lines FH and EG cut the one the other rayse vp to the playne superficies wherein the square EFGH is a perpendicular line KL and extend the line KL on the other side of the playne superficies to the point M. And let eche of the lines KL and KM be put equall to one of these lines KE KF KH or KG And draw these right lines LE LF LG LH ME MF MG and MH Now forasmuch as the line KE is by the corollary of the 34. of the first equall to the line KH and the angle EKH is a right angle therefore the square of HE is double to the square of EK by the 47. of the first Agayne forasmuch as the line LK is equall to the line KE by position and the angle LKE is by the second diffinition of the eleuenth a right angle therefore the square of the line EL is double to the square of the line EK and it is proued that the square of the line HE is double to the square of the line EK Wherefore the square of the line LE is equall to the square of the line EH Wherefore also the line LE is equall to the line EH And by the same reason the line LH is also equall to the line HE. Wherefore the triangle LHE is equilater In like sort may we proue that euery one of the rest of the triangles whose bases are the sides of the square EFGH and toppes the pointes L and M are equilater The sayd eight triangles also are equall the one to the other for euery side of eche is equall to the side of the square EFGH Wherfore there is made an octohedron coÌ„tained vnder eight triangles whose sides are equall Now it is required to comprehend it in the sphere geuen and to proue that the diameter of the sphere is in power double to the side of the octohedron Forasmuch as these three lines LK KM and KE are equall the one to the other therfore a semicircle described vpon the line LM shall passe also by the point E. And by the same reason if the semicircle be turned round about vntill it returne vnto the selfe same
place from whence first it began to be moued it shal passe by the pointes F G H and the octohedron shall be comprehended in a sphere I say also that it is comprehended in the sphere geuen For forasmuch as the line LK is equal to the line KM by position and the line KE is common to them both and they contayne right angles by the 3. diffinition of the eleuenth therefore by the 4. of the first the base LE is equall to the base EM And forasmuch as the angle LEM is a right angle by the 31. of the third for it is in a semicircle as hath bene proued therefore the square of the line LM is double to the square of the line LE by the 47. of the first Againe forasmuch as the line AC is equall to the line BC therefore the line AB is double to the line BC by the diâ—finition of a circle But as the line AB is to the line BC so is the square of the line AB to the square of the line BD by the corollaries of the 8. and â—0 of the sixt Wherefore the square of the line AB is double to the square of the line BD. And it is proued that the square of the line LM is double to the square of the line LE. Wherefore the square of the line BD is equall to the square of the line LE. For the line EH which is equall to the line LF is put to be equall to the line DB. Wherefore the square of the line AB is equall to the square of the line LM Wherefore the line AB is equall to the line LM And the line AB is the diameter of the sphere geueÌ„ wherefore the line LM is equall to the diameter of the sphere geuen Wherefore the octoedron is contayned in the sphere geuen and it is also proued that the diameter of the sphere is in power double to the side of the octohedron Wherefore there is made an octohedron and it is comprehended in the sphere geuen wherein was comprehended the Pyramis and it is proued that the diameter of the sphere is in power double to the side of the octohedrn which was required to be doone and to be proued Certayne Corollaries added by Flussas First Corollary The side of a Pyramis is in power sesquitertia to the side of an octâ—hedron inscribed in the same Sphere For forasmch as the diameter is in power double to the side of the octohedron therefore of what partes the diameter contayneth in power 6. of the same the side of the octohedron coÌ„tayneth in power 3. but of what partes the diameter contayneth 6. of the same the side of the pyramis contayneth 4. by the 13. of this booke Wherefore of what partes the side of the pyramis contayneth 4. of the same the side of the octohedron contayneth 3. Second Corollary An octohedron is deuided into two equall and like Pyramids The common bases of these Pyramids are set vpon euery square contayned of the sides of the octohedron vpon which square are set the â—â— triangles of the octohedron which pyramids are by the â— diffinition of the eleuenth equall and like And the foresayd square common to those Pyramids is the halfe of the square of the diameter of the sphere for it is the square of the side of the octohedron Third Corollary The three diameters of the octohedron do cutte the one the other perpendicularly into two equall parts in the center of the sphere which contayneth the sayd octohedron As it is manifest by the three diameters EG FH and LM which cutte the one the other in the center K equally and perpendicularly ¶ The 3. Probleme The 15. Proposition To make a solide called a cube and to comprehend it in the sphere geuen namely that Sphere wherein the former two solides were comprehendâ—dâ— and to proue that the diameter of the sphere is in power treble to the side of the cube TAke the diameter of the sphere geuen namely AB and diuide it in the point Câ— So that let the line AC be double to the line BC by the 9. of the sixt And vpon the line AB describe a semicircle ADB And by the 11. of the first from the pâ—ynt C râ—yse vp vnto the line AB a perpeÌ„dicular line CD And draw a right linâ— DB. And describe a squarâ— EFGH hauing euery one of his sides equall to the line DB And from the pointes E F G H rayse vp by the 12. of the eleuenth vnto the playne superficies of the square EFGH perpendicular lines EK FL GM and HN and let euery one of the lines EK FL GM and HN be put equall to one of the lines EF FG GH or HE which are the sides of the square and draw these right lines KL LM MN and NK Wherfore there is made a cube namely FN which is contayned vnder six equall squares Now it is required to comprehend the same cube in the sphere geuen and to proue that the diameter of the sphere is in power ble to the side of the cube Draw these right lines KG and EG And forasmuch as the angle KEG is a right angle for that the line KE is erected perpendicularly to the playne superficies Eâ— and therefore also to the right line EG by the 2. diffinitioÌ„ of the eleuenth wherefore a semicircle described vpon the line KG shall passe by the poynt E. Agayne forasmuch as the line FG is erected perpendicularly to either of these lines FL and FE by the diffinition of a square by the 2. diffinition of the eleuenth therefore the line FG is erected perpendicularly to the playne superficies FK by the 4. of the eleuenth Wherefore if we draw a right line from the point F to the point K the line GF shall be erected perpendicularly to the line KF by the 2. diffinition of the eleuenth And by the same reason agayne a semicircle described vpon the line GK shall passe also by the point F. And likewise shall it passe by the rest of the pointes of the angles of that cube If now the diameter KG abiding fixed the semicircle be turned round about vntill it returne into the selfe same place from whence it began first to be moued the cube shal be compreheÌ„ded in a sphere I say also that it is comprehended in the sphere geuen For forasmuch as the line GF is equall to the linâ—â—E and the angle F is a right angle therefore the square of the line EG is by the 47. of the first double to the square of the line â—F But the line EF is equall to the line EK Wherefore the square of the line EG is double to the square of the line EK Wherfore the squares of EG and EK that is the square of the line GK by the 47. of the first are treble to the square of the line EK And forasmuch as the line AB is treble to the line BC but
F. And draw these right lines FA FB FC FD FE Wherefore those lines do diuide the angles of the pentagon into two equall partes in the poyntes A B C D E by the 4. of the first And â—orasmuch as the fiue angles that are at the poynt F aâ—e equall to fower right angles by the corollary of the 15. of the first and they are equall the one to the other by the 8. of the first therfore one of those angles as â—or example sake the angle AFB is a fiâ—th part lesse then a right angle Wherfore the angles remayning namely FAB ABF are one right angle and a fifth part ouer But the angle FAB is equall to the angle FBC Wherefore the whole angle ABC being one of the angles of the pentagon is a right angle and a fifth part more then a right angle which was required to be proued ¶ A Corollary added by Flussas Now let vs teach how those fiue solides haue eche like inclinations of theyr bases â—iâ—st let vs take a Pyramis and diuide one of the sides thereof into two equall parts and from the two angles opposite vnto that side dâ—aw perpeÌ„diculars which shall fall vpon the section by the corollary of the 12. of the thirtenth and at the sayd poynt of diuision as may easily be proued Wherfore they shal containe the angâ—e of the inclination of the plaines by the 4. diffinition of the eleuenth which angle is subtended of the opposite side of the pyramis Now forasmuch as the rest of the angles of the inclination of the playnes of the Pyramis are contayned vnder two perpeÌ„dicular lines of the triangles and are subtended of the side of the Pyramis it foloweth by the 8. of the firâ—t that those angles are equall Wherâ—foâ—e by the 5. diffinition of the eleueÌ„th the superficieces are in like sort inclined the one to the other One of the sides of a Cube being diuided into two equall parts if from the sayd section be drawen in two of the bases thereof two perpendicular lines they shal be parallels and equall to the sides of the square which coÌ„tayne a right angle And forasmuch as all the angles of the bases of the Cube are right angles therefore those perpendiculars falling vpon the section of the side common to the two bases shall contâ—yne a right angle by the 10. of the eleuenth which selfe angle is the angle of inclination by the 4. diffinition of the eleuenth and is subtended of the diameter of the base of the Cube And by the same reason may we proue that the rest of the angles of the inclination of the bases of the cube are right angles Wherefore the inclinations of the superficieces of the cube the one to the other are equal by the 5. diffinition of the eleuenth In an Octohedron take the diameter which coupleth the two opposite angles And from those opposite angles draw to one and the selâ—e same side of the Octohedron in two bases thereof two perpendicular lines which shall diuide that side into two equall parts and perpendicularly by the Corollary of the 12. of the thirtenth Wherefore those perpendiculars shall contayne the angle of the inclination of the bases by the 4. diffinition of the eleueÌ„th and the same angle is subtended of the diameter of the OctohedroÌ„ Wherfore the rest of the angles after the same maner described in the rest of the bases being comprehended and subtended of equall sides shall by the 8. of the first be equall the one to the other And therefore the inclinations of the playnes in the Octohedron shal by the 5. diffinition of the eleuenth be equall In an Icosahedron let there be drawen from the angles of two of the bases to one side common to both the sayd bases perpendiculars which shall contayne the angle of the inclination of the bases by the 4. diffinition of the eleuenth which angle is subtended of the right line which subtendeth the angle of the pentagon which contayneth fiue sides of the Icosahedron by the 16. of this booke for it coupleth the twoo opposite angles of the triangles which are ioyned together Wherefore the rest of the angles of the inclination of the bases being after the same maner found out they shal be contayned vnder equall sides and subtended of equall bases and therefore by the 8. of the fiâ—st those angles shal be equall Wherfore also al the inclinations of the bases of the Icosahedron the one to the other shalbâ— equall by the 5. diffinition of the eleuenth In a Dodecahedron from the two opposite angles of two next pentagons draw to theyr common side perpendicular lines passing by the centres of the sayd pentagons which shal where they fal diuide the side into two equall parts by the 3. of the third For the bases of a Dodecahedron are contayned in a circle And the angle contaynâ—d vnder those perpendicular lines is the inclination of those bases by the 4. diffinition of the eleuenth And the foresayd opposite angles are coupled by a right line equal to the right line which coupleth the opposite sections into two equall parts of the sides of the dodecahedroÌ„ by the 33. of the first For they couple together the halfe sids of the dodecahedroÌ„ which halfes are parallels and equall by the 3. corollary of the 17. of this booke which coupling lines also are equall by the same corollary Wherefore the angles being contayned of equal perpendicular lines and subtended of equall coupling lines shall by the 8. of the first be equal And they are the angles of the inclinations Wherefore the bases of the dodecahedron are in like sort inclined the one to the other by the 5. diffinition of the eleuenth Flussas after this teacheth how to know the rationality or irrationality of the sides of the triangles which contayne the angles of the inclinations of the superficieces of the foresayd bodies In a Pyramis the angle of the inclinatioÌ„ is contayned vnder two perpâ—dicular lines of the triangles and is subtended of the side of the Pyramis Now the side of the pyramis is in power sesquitertia to the perpendicular line by the corollary of the 12. of this booke and therfore the triangle coÌ„tained of those perpeÌ„dicular lines and the side of pyramis hath his sides rational commensurable in power the one to the other Forasmuch as the twoo sides of a Cube or right lines equall to them subtended vnder the diameter of one of the bases doo make the angle of the inclination and the diameter of the cube is in power sesquialter to the diameter of the base which diameter of the base is in power double to the side by the 47. of the first therefore those lines are rationall and commensurable in power In an Octohedron whose two perpendiculars of the bases contayne the angle of the inclination of the Octohedron which angle also is subtended of the diameter of the Octohedron the diameter is in power
the whole line MG to the whole line EA by the 18. of the fifth Wherefore as MG the side of the cube is to EA the semidiameter so is the line FGHIM to the Octohedron ABKDLC inscribed in one the selfe same Sphere If therefore a cube and an Octohedron be contained in one and the selfe same Sphere they shall be in proportion the one to the other as the side of the cube is to the semidiameter of the Sphere which was required to be demonstrated A Corollary Distinctly to notefie the powers of the sides of the fiue solides by the power of the diameter of the sphere The sides of the tetrahedron and of the cube doo cut the power of the diameter of the sphere into two squares which are in proportion double the one to the other The octohedron cutteth the power of the diameter into two equall squares The Icosahedron into two squares whose proportion is duple to the proportion of a line diuided by an extreame and meane proportion whose lesse segmeÌ„t is the side of the Icosahedron And the dodecahedron into two squares whose proportion is quadruple to the proportion of a line diuided by an extreame and meane proportion whose lesse segment is the side of the dodecahedron For AD the diameter of the sphere contayneth in power AB the side of the tetrahedron and BD the side of the cube which BD is in power halfe of the side AB The diameter also of the sphere contayneth in power AC and CD two equall sides of the octohedron But the diameter contayneth in power the whole line AE and the greater segment thereof ED which is the side of the Icosahedron by the 15. of this booke Wheâ—fore their powers being in duple proportioÌ„ of that in which the sides are by the first corollary of the 20. of the sixth haue their proportion duple to the proportion of an extreame meane proportioÌ„ Farther the diameter coÌ„tayneth in power the whole line AF and his lesse segment FD which is the side of the dodecahedron by the same 15. of this booke Wherefore the whole hauing to the lesse â— double proportion of that which the extreame hath to the meane namely of the whole to the greater segment by the 10. diffinition of the fifth it followeth that the proportion of the power is double to the doubled proportion of the sides by the same first corollary of the 20. of the sixth that is is quadruple to the proportion of the extreame and of the meane by the diffinition of the sixth An aduertisment added by Flussas By this meanes therefore the diameter of a sphere being geuen there shall be geuen the side of euery one of the bodies inscribed And forasmuch as three of those bodies haue their sides commensurable in power onely and not in length vnto the diameter geuen for their powers are in the proportion of a square number to a number not square wherefore they haue not the proportion of a square number to a square number by the corollary of the 25. of the eight wherefore also their sides are incommensurabe in length by the 9. of the tenth therefore it is sufficient to compare the powers and not the lengths of those sides the one to the otherâ— which powers are contained in the power of the diameter namely from the power of the diameter let there ble taken away the power of the cube and there shall remayne the power of the Tetrahedron and taking away the power of the Tetrahedron there remayneth the power of the cube and taking away from the power of the diameter halfe the power thereof there shall be left the power of the side of the octohedron But forasmuch as the sides of the dodecahedron and of the Icosahedron are proued to be irrationall for the side of the Icosahedron is a lesse line by the 16. of the thirtenth and the side of the dedocahedron is a residuall line by the 17. of the same therfore those sides are vnto the diameter which is a rationall line set incommensurable both in length and in power Wherefore their comparison can not be diffined or described by any proportion expressed by numbers by the 8. of the tenth neither can they be compared the one to the other for irrational lines of diuers kindes are incoÌ„meÌ„surable the one to the other for if they should be commensurable they should be of one and the selfe same kinde by the 103. and 105. of the tenth which is impossible Wherefore we seking to compare them to the power of the diameter thought they could not be more aptly expressed then by such proportions which cutte that rationall power of the diameter according to their sides namely diuiding the power of the diameter by lines which haue that proportioÌ„ that the greater segment hath to the lesse to put the lesse segment to be the side of the Icosahedron deuiding the sayd power of the diameter by lines hauing the proportion of the whole to the lesse segment to expresse the side of the dodecahedron by the lesse segment which thing may well be done betwene magnitudes incommensurable The ende of the fourtenth Booke of Euclides Elementes after Flussas ¶ The fiftenth booke of Euclides Elementes THis finetenth and last booke of Euclide or rather the second boke of Appollonius or Hypsicles teacheth the inscription and circumscriptioÌ„ of the fiue regular bodies one within and about an other a thing vndoutedly plesant and delectable in minde to contemplate and also profitable and necessary in act to practise For without practise in act it is very hard to se and conceiue the constructions and demonstrations of the propositions of this booke vnles a man haue a very depe sharpe fine imagination Wherfore I would wish the diligent studeÌ„t in this booke to make the study thereof more pleasant vnto him to haue presently before his eyes the bodyes formed framed of pasted paper as I taught after the diffinitions of the eleuenth booke And then to drawe and describe the lines and diuisions and superficieces according to the constructions of the propositions In which descriptions if he be wary and diligent he shall finde all things in these solide matters as clere and as manifest vnto the eye as were things before taught only in plaine or superficial figures And although I haue before in the twelfth boke admonished the reader hereof yet bicause in this boke chiefly that thing is required I thought it should not be irkesome vnto him againe to be put in minde thereof Farther this is to be noted that in the Greke exemplars are found in this 15. booke only 5. propositions which 5. are also only touched and set forthe by Hypsicies vnto which Campane addeth 8. and so maketh vp the number of 13. Campane vndoubtedly although he were very well lerned and that generally in all kinds of learning yet assuredly being brought vp in a time of rudenes when all good letters were darkned barberousnes had
ouerthrowne and ouerwhelmed the whole world he was vtterly rude and ignorant in the Greke tongue so that certenly he neuer redde Euclide in the Greke nor of like translated out of the Greke but had it translated out of the Arabike tonge The Arabians were men of great study and industry and commonly great Philosophers notable Phisitions and in mathematicall Artes most expert so that all kinds of good learning flourished and raigned amongst them in a manner only These men turned whatsoeuer good author was in the Greke tonge of what Art and knowledge so euer it were into the Arabike tonge And froÌ„ thence were many of theÌ„ turned into the Latine and by that meanes many Greeke authors came to the handes of the Latines and not from the first fountaine the Greke tonge wherin they were first written As appeareth by many words of the Arabike tonge yet remaining in such bokes as are Zenith nadir helmuayn helmuariphe and infinite suche other Which Arabians also in translating such Greke workes were accustomed to adde as they thought good for the fuller vnderstanding of the author many things as is to be sene in diuers authors as namely in Theodosius de Sphera where you see in the olde translation which was vndoubteldy out of the Arabike many propositions almost euery third or fourth leafe Some such copye of Euclide most likely did Campanus follow wherein he founde those propositioÌ„s which he hath more aboue those which are found in the Greke set out by Hypsicles and that not only in this 15. boke but also in the 14. boke wherin also ye finde many propositions more theÌ„ are founde in the Greeke set out also by Hypsicles Likewise in the bookes before ye shall finde many propositions added and manye inuerted and set out of order farre otherwise then they are placed in the Greeke examplars Flussas also a diligent restorer of Euclide a man also which hath well deserued of the whole Art of Geometrie hath added moreouer in this booke as also in the former 14. boke he added 8. proâ—ositioÌ„s 9. propositioÌ„s of his owne touching the inscription and circumscriptâ—on 〈…〉 bodies very siâ—gular â—ndoubtedly and wittye All which for that nothing should want to the desirous louer of knowledge I haue faithfully with no small paines turned And whereas Flâ—ssâ—â— in the beginning of the eleuenth booke namely in the end of the diffinitions there â—eâ— putteth two diffinitions of the inscription and circumscription of solides or corporall figures within or about the one the other which certainely are not to be reiected yet for that vntill this present 15. boke there is no mention made of the inscription or circumscription of these bodyes I thought it not so conuenient thâ—râ— to place them but to referre theÌ„ to the beginning of this 15. booke where they are in maner of necessitie required to the elucidation of the Proposiâ—ions and dâ—monstrationâ— of the same The diffinitions are these Diffinition 1. A solide figure is then â—aid to be inscribed in a solide figure when the angles of the figure inscribed touche together at one time either the angles of the figure circumscribed or the superficieces or the sides Diffinition 2. A solide figure is then said to be circumscribed about a solide figure when together at one time either the angles or the superficieces or the sides of the figure circumscribed â—ouch the angles of the figure inscribed IN the fourâ—â— booke in the diffinitions of the inscription or circumscription of playne rectiline figures one with in or about an other was requâ—red that all the angles of the figuâ—â— inscribed should at one time touch all the sides of the figure circumscribed but in the fiue regular solides â—o whome chefely these two diffinitions pertaine for that the nomber of their angles superficieces sides are not equal one compared to an other it is not of necessitie that all the angles of the solide inscribed should together at one time touch either all the angles or all the superficieces or all the sides of the solide circumscribed but it is sufficient that those angles of the inscribed solide which touch doe at one time together eche touch some one angle of the figure circumscribed or some one base or some one side so that if the angles of the inscribed figure do at one time touche the angles of the figure circumscribed none of them may at the same time touche either the bases or the sides of the same circumscribed figure and so if they touch the bases they may touche neither angles nor sides and likewise if they touche the sides they may touch neither angles nor bases And although sometimes all the angles of the figure inscribed can not touch either the angles or the bases or the sides of the figure circumscribed by reason the nomber of the angles bases or sides of the said figure circumscribed wanteth of the nomber of the angles of the â—igure inscribed yet shall those angles of the inscribed figure which touch so touch that the void places left betwene the inscribed and circumscribed figures shal on euery side be equal and like As ye may afterwarde in this fiftenth booke most plainely perceiue ¶ The 1. Proposition The 1. Probleme In a Cube geuen to describe a trilater equilater Pyramis SVppose that the cube geuen be ABCDEFGH In the same cube it is required to inscribe a Tetrahedron Drawe these right lines AC CE AE AH EH HC Now it is manifest that the triangles AEC AHE AHC and CHE are equilater for their sides are the diameters of equall squares Wherfore AECH is a trilater equilater pyramis or Tetrahedron it is inscribed in the cube geueÌ„ by the first definition of this booke which was required to be done ¶ The 2. Proposition The 2. Probleme In a trilater equilater Pyramis geuen to describe an Octohedron SVppose that the trilater equilater pyramis geueÌ„ be ABCD whose sides let be diuided into two equall partes in the pointes E Z I K L T. And draw these 12. right lines EZ ZI IE KL LT TK EK KZ ZL LI IT and TE Which 12. right lines are by the 4. of the first equall For they subtend equall plaine angles of the bases of the pyramis and those equall angles are contained vnder equall sides namely vnder the halfes of the sides of the pyramis Wherefore the triangles TKL TLI TIE TEK ZKL ZLI ZIB ZEK are equilater and they limitate and containe the solide TKLEZI Wherefore the solide TKLEZI is an Octohedron by the 23. definition of the eleuenth And the angles of the same Octohedron do touch the sides of the pyramis ABCD in the pointes E Z I T K L. Wherefore the Octohedron is inscribed in the pyramis by the 1. definition of this booke Wherefore in the trilater equilater pyramis geuen is inscribed an Octohedron which was required to be done A Corollary added by Flussas Hereby it is manifest that a pyramis is cut into two
equall partes by euery one of the three equall squares which diuide the Octohedron into two equall partes and perpendicularly For the three diameters of those squares do in the centre cut the one the other into two equall partes and perpendicularly by the third Corollary of the 1â— of the thirtenth which squares as for example the square EKLI do diuide in sunder the pyramids and the prismes namely the pyramis KLTD and the prisme KLTEIA from the pyramis EKZB and the prisme EKZILG which pyramids are equall the one to the other and so also are the prismes equall the one to the other by the 3. of the twelfth And in like sort do the rest of the squares namely KZIT and ZLTE which squares by the second Corollary of the 14. of the thirtenth do diuide the Octohedron into two equall partes ¶ The 3. Proposition The 3. Probleme In a cube geuen to describe an Octohedron TAke a Cube namely ABCDEFGH And diuide euery one of the sides thereof into two equall partes And drawe right lines coupling together the sections as for example these right lines PQ and RS which shall be equall vnto the side of the cube by the 33. of the first and shall diuide the one the other into two equall parts in the middest of the diameter AG in the point I by the Corollary of the 34. of the first Wherefore the point I is the centre of the base of the cube And by the same reason may be found out the centres of the rest of the bases which let be the pointes K L O N M. And drawe these right lines LI IM MO OL KI KL KM KO NI NL NM NO And now forasmuch as the angle IPL is a right angle by the 10. of the eleuenth for the lines IP and PL are parallels to the lines RA and AB And the right line IL subtendeth the right angle IPL namely it subtendeth the halfe sides of the cube which containe the right angle IPL and likewise the right line IM subtendeth the angle IQM which is equall to the same angle IPL and is contained vnder right lines equall to the right lines which containe the angle IPL Wherefore the right line IM is equall to the right line IL by the 4. of the first And by the same reason may we proue that euery one of the right lines MO OL KI KL KM KO NI NL NM and NO which subtend angles equall to the selfe same angle IPL and are coÌ„tained vnder sides equall to the sides which containe the angle IPL are equall to the right line IL. Wherefore the triangles KLI KLO KMI KMO and NLI NLO NMI NMO are equilater and equall and they containe the solide IKLONM Wherefore IKLONM is an Octohedron by the 23. definition of the eleuenth And forasmuch as the angles thereof do altogether in the pointes I K L O N M touch the bases of the cube which containeth it it followeth that the Octohedron is inscribed in the cube by the first definition of this booke Wherefore in the cube geuen is described an Octohedron which was required to be done ¶ A Coâ—ollary aâ—ded by â—lussâ—â— Hereby it is manifest that right lines ioyning together the â—entres of the opposite bases of the cube do cut the one the other into two equall parts and perpendicularly in the centre of the cube or in the centre of the Sphere which containeth the cube For forasmâ—ch as the right lines LM and IO which knâ—â— together the centres of the opposite bases of the cube do also knit together the opposite anglâ—s of the Octâ—hedron inscribed in the cube it followeth by the 3. Corollary of the 14. of the thirtenth that those lines LM and IO do cut the one the other into two equall partes in a point But the diameters of the cube do also cut the one the other into two equall partes by the 39. of the eleuenth Wherfore that point shall be the centre of the sphere which containeth the câ—â—â— For making that point the centre and the space some one of the semidiameters describe a sphere and it shall passe by the angles of the cube and likewise making the same point the centre and the space halfe of the line LM describe a sphere and it shall also passe by the angles of the Octohedron ¶ The 4. Proposition The 4. Probleme In an Octohedron geuen to describe a Cube SVppose that the Octohedron geuen be ABGDEZ And let the two pyramids thereof be ABGDE and ZBGDE And take the centres of the triangles of the pyramis ABGDE that is take the centres of the circles which containe those triangles and let those centres be the pointâ—s T I K L. And by these centres let there be drawen parallâ—l lines â—o the sides of the square BGDE which parallel â—igâ—â— linâ—â— let be MTN NLX XKO OIM. And forasmuch as thâ—se parallel right lines do by the 2. of the sixth cut the equall right lines AB AG AD and AE proportionally therfore they concurre in the pointes M N X O. Wherefore the right lines MN NX XO and OM which subtend equall angles set at the point A contained vndâ—r â—quall right lines are equall by the 4. of the first And moreouer seing that they are parallels vnto the lines BG GD DE Eâ— which make a square therefore MNXO is also a square by the 10. of the eleuenth Wherefore also by the 15. of the â—ame the square MNXO is parallel to the squarâ— BGDE For all tâ—e right lines touch the one the other in the pointes of their sections From the centres T I K L drawe these right lines TI IK KL LTâ— And drawe the right line AIC And forasmuch as I is the centre of the equilater triangle ABE therefore the right line AI being extended cutteth the right line BE into two equall partes by the Corollary of the 12. of the thirtenth And forasmuch as MO is a parallel to BE therefore the triangle AIO is like to the whole triangle ACE by the Corollary of the 2. of the sixth And the right line MO is diuided into two equall partes in the point I by the 4. of the sixth And by the same reason may we proue that the right lines MN NX XO are diuided into two equall partes in the pointes T L K. Wherefore also againe the bases TI IK KL LT which subtend the angles set at the pointes M O X N which angles are right angles and are contained vnder equall sides those bases I say are equall And forasmuch as TIM is an Isosceles triangle therefore the angles set at the base namely the angles MTI and MIT are equal by the â—â— of the first But the angle M is a right angle wherefore eche of the angles MIT and MTI is the halfe of a right angle And by the same reason the angles OIK OKI are equall Wherefore the angle remayning namely TIK is a right angle
by the 13. of the first For the right lines TI and IK are set vpon the line MO. And by the same reason may the rest of the angles namely IKL KLT LTI be proued right angles and they are in one and the self same plaine superficies namely MNXO by the 7. of the eleueÌ„th Wherefore the right lines which ioyne together the centres of the plaine superficiall triangles which make the solide angle A do make the square ITKL And by the same reason may be proued that the plaine superficiall triangles of the rest of the fiue solide angles of the Octohedron set at the pointes B G Z D E do in the centres of their bases receaue squares So that there are in number sixe squares for euery Octohedron hath sixe solide angles and those squares are equall for their sides do containe equall angles of inclinations contained vnder equall sides namely vnder those sides which are drawen from the centre to the side of the equall triangles by the 2. Corollary of the 18. of the thirtenth Wherefore ITKLRPVS is a cube by the 21. definition of the eleuenth and hath his angles in the centres of the bases of the Octohedron and therefore is inscribed in it by the first definition of this booke Wherefore in an Octohedron geuen is described a cube which was required to be done The 5. Proposition The 5. Probleme In an Icosahedron geuen to describe a Dodecahedron TAke an Icosahedron one of whose solide angles let be Z. Now forasmuch as by those thinges which haue bene proued in the 16. of the thirtenth the bases of the triangles which contayne the angle of the Icosahedron doo make a pentagon inscribed in a circle let that pentagon be ABGDE which is made of the fiue bases of the triangles whose playne superficiall angles remayning make the solide angle geuen namely Z. And take the centres of the circles which contayne the foresaid triangles which centers let be the poyntes I T K M L and draw these right lines IT TK KM ML LI. Now then a perpendicular line drawne from the poynt Z to the playne superficies of the pentagon ABGDE shall fall vpon the centre of the circle which contayneth the pentagon ABGDE by those thinges which haue bene proued in the selfe same 16. of the thirtenth Moreouer perpendicular lines drawne from the centre to the sides of the pentagon ABGDE shall in the poyntes C N O where they fall cut the right lines AB BG GD into two equall partes by the 3. of the third Draw these right lines CN and NO And forasmuch as the angles CBN and NGO are equall and are contained vnder equall sides therefore the base CN is equall to the base NO by the 4. of the first Moreouer perpendicular lines drâ—â—â—e from the poynt Z to the bâ—sâ—â— of the pentagon ABGDE shall likewise cutte the bases into two equall partes by thâ—â— of the third For the perpendiculars passe by the centre by the corollary of the 12. of the thirteÌ„th Wherfore thâ—se perpendicular lines shall fal vpoÌ„ the points C N O. And now forasmuch as the right lines ZI IG are equall to the right lines ZT TN also to the right lines ZK KO by reason of the likenes of the equall triangles therefore the line IT is a parallell to the line CN and so also is the line TK to the line NO by the 2. of the sixt Wherfore the angles ITK and CNO are equal by the 11. of the eleuenth Agayne forasmuch as the triangles CBN and NGO are Isoscels triangles therefore the angles BCN and BNC are equall by the 5. of the first And by the same reason the angles GNO and GON are equall And moreouer the angles BCN and BNC are equall to the angles GNO and GON for that the triangles CBN and NGO are equall and like Bâ—â— the three angles BNC CNO ONG are equall to two right angles by the 13. of the first for that vpon the right line Bâ— are set the right lines CN ON And the three angles of the triangle CBN namely the angles BNC BCN or GNO for the angle GNO is equall to the angle BCN as it hath bene proued and NBC are also equall to two right angles by the 32. of the first Wherefore taking away the angles BNC GNO the angle remayning namely CNO is equall to the angle remayning namely to CBN Wherfore also the angle ITK which is proued to be equall to the angle CNO is equall to the angle CBN Wherefore ITK is the angle of a pentagon And by the same reason may be proâ—ed that the rest of the angles namelyâ— the angles TIL ILM LMK MKT are equall to the rest of the angles namely to BAE AED EDG DGB Wherefore ITKML is an equilater and equiangle pentagon by the 4. of the first For the equall bases of the pentagon ITKML doo subtend equall angles set at the point Z and comprehended vnder equall sides Moreouer it is manifest that the pentagon I TKML is in one and the selfe same playne superficies For forasâ—uch as the angles ONC and NCP are in one and the selfe sâ—me playne superficies namely in the superficies ABGDE But vnto the same playne superficies the playne superficieces of the angles KTI and TIL are parallels by the 15. of the eleuenth And the triangles KTI and TIL concurre wherefore they are in one and the selfe same playne superficies by the corollary of the 16. of the eleueÌ„th And by the same reasoÌ„ so may we proue that the triangles ILM LMK MKT are in the selfe same playne superficies wherein are the triangles KTI and TIL Wherefore the pentagon ITKML is in one and the selfe same playne superficies Wherefore the solide angle of the Icosahedron namely the solide angle at the poynt Z subtendeth an equilater and equiangle pentagon plaine superficies which pentagon hath his plaine superficiall angles in the centres of the triangles which make the solide angle Z. And in like sort may we proue that the other eleuen solide angles of the Icosahedron eche of which eleuen solide angles are equall and like to the solide angle Z by the 16. of the thirtenth are subtended vnto pentagons equall and like and in like sort set to the pentagon ITKML And forasmuch as in those peÌ„tagons the right lines which ioyne together the centers of the bases are common sides it followeth that those 12. pentagons include a solide which solide is therefore a dâ—decâ—hedron by the 24. diffinition of the eleuenth and is by the first diffinition of this booke described in the Icosahedron fiue sides whâ—reof 〈◊〉 set vpon the pentagon ABGDE Wherefore in an Icosahedron geuen iâ— inscribed a dodecahedron which was required to be done An annotation of Hypsiâ—les This is to be noted that if a man should demaund 〈◊〉 many sides an Icosahedron hath we may thus answere It is manifest that an Icosahâ—râ—n is contayned vnder 20. triangles and that euery triangle consisteth of three
right linâ—s Now then multiply the 20. triangles into the sides of one of the triangles and so shall there be produced 6â— â—he halfe of which is 30. And so many sides hath an Icosahedron And in like sort in a dodecahedron forasmuch as 12. pentagons make a dodecahedron and euery pentagon contayneth â— right linesâ— multiply â—â— into 12. and there shall be produced 60. the halfe of which is 30. And so many are the sides of a dodecahedron And the reason why we take the halfe iâ— for that euery side whether it be of a triangle or of a pentagon or of a square as in a cube â—s taken twise And by the same reason may you finde out how many sides are in a cube and in a pyramis and in an octohedron But now agayne if ye will finde out the number of the angles of euery one of the solide figures when ye haue done the same multiplication that ye did before diâ—idâ— the same sides by the number of the plaine superficieces which comprehend one of the angles of the solides As for example forasmuch as 5. triangles contayne the solide angle of an Icosahedron diuide 60. by 5. and there will come forth 12. and so many solide angles hath an Icosahedâ—on In a dodecahedron forasmuch as three pentagons comprehend an angle diuide 60. by 3. and there will come forth 20 and so many are the angles of a dodecahedron And by the same reason may you finde out how many angles are in eche of the rest of the solide figures If it be required to be knowne how one of the plaines of any of the fiue solides being geuen there may be found out the inclination of the sayd plaines the one to the other which contayne eche of the solides This as sayth Isidorus our greate master is foâ—â—d out after this maner It is manifest that in a cube the plaines which contayne iâ— do◠〈◊〉 the one the other by a right angle But in a Tetrahedron one of the triangles being geuen let the endes of one of the sides of the sayd triangle be the centers and let the space be the perpendicular line drawne from the toppe of the triangle to the base and describe circumferâ—nces of a circle which shall cutte the one the other and from the intersection to the centers draw right lines which shall containe the inclination of the plaines coÌ„tayning the Tetrahedron In an Octoâ—edron take one of the sides of the triangle therâ—of and vpon it describe a square and draw the diagonall line and making the centres the endes of the diagonall line and the space likewise the perpendicular line drawne from the toppe of the triangle to the base describe circumferences and agayne from the common section to the centres draw right lines and they shall contayne the inclination sought for In an Icosahedron vpon the side of one of the triangles thereof describe a pentagon and draw the line which subtendeth one of the angles of the sayd pentagon and making the centres the endes of that line and the space the perpendicular line of the triangle describe circumferences and draw from the common intersectioâ— of the circumferences vnto the centres right lines and they shall contayne likewise the inclination of the plaines of the icosahedron In a dodecahedron take one of the pentagons and draw likewise the line which subtendeth one of the angles of the pentagon and making the centres the endes of that line and the space the perpendicular line drawne from the section into two equall partes of that line to the side of the pentagon which is parallel vnto it describe circumferences and from the point of the intersection of the circumferences draw vnto the centres right lines and they shall also containe the inclination of the plaines of the dodecahedron Thus did this most singular learned man reason thinking the deâ—onstration in euery one of them to be plaine and cleare But to make the demonstration of them manifest I think it good to declare and make open his wordesâ— and first in a Tâ—trahedronâ— The ende of the fiuetenth Booke of Euclides Elementes after 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ¶ The 6. Proposition The 6. Probleme In an Octohedron geuen to inscribe a trilater equilater Pyramis SVppose thaâ— the Octohedron whereâ—â— the Tetrahedron is required to be insâ—riâ—â—â— be ABGDEI Take 〈…〉 bases of the Octoâ—â—dron that is 〈…〉 close in the loweâ—â— triangle BGD namely AEâ— HED IGD and let the fourth be AIB which is opposite to the lowest triangle before put namely to EGD And take the centres of those fower bases which let be the pointes H C N â— And vpon the triangle HCN erecte a pyramis HCNL Now â—orasmuch as these two bases of the Octohedron namely AGE and ABI are set vpon the right lines EG and BI which are opposite the one to the otherâ— in the square GEBI of the Octohedron from the poinâ— A draâ—e by the centres of the bases namely by the centres H L perpendicular lines AHF ALK cutting the lines EG and BI 〈◊〉 two equall partes in the pointes F K by the Corollary of the 1â— of the thirtenth Wherfore a right line drawen froÌ„ the point F to the point K shall be a parallel and equall to the sides of the Octohedron namely to â—â— and GI by the 33. of the first And the right line HL which cutteth the 〈…〉 AF AK proportionally for AH and AL are drawen from the centres of equall circles to the circumferences is a parallel to the right line FK by the 2. of the sixth and also to the sides of the Octohedron namely to Eâ— and IG by the 9. of the eleuenth Wherefore as the line AF is to the line AH so is the line FK to the line HL by the 4. of the sixth For the triangles AFK and AHL are like by thâ— Corollary of the 2. of the sixth But the line AF is in sesquialter proportion to the line AH for the side EG maketh HF the halfe of the right line AH by the Corollary of the 12. of the thirtenth Wherfore FK or GI the side of the Octohedron is sesquialter to the righâ—line HL. And by the same reason may we proue that the sides of the Octohedron are sesquialter to the rest of the right lines which make the pyramis HNCI namely to the right lineâ—â— N NC CI LN and CH wherefore those right lines are equall and therefore the triangleâ— which are described of them namely the triangles HCN HNL NCL and CHL. which make the pyramis HNCL are equall and equilater And forasmuch as the angles of the same pyramis namely the angles H C N L do end in the centres of the bases of the Octohedron therefore it is inscribed â—o the same Octohedron by the first definition of this booke Wherefore in an Octohedron â—euen is inscribed a trilâ—ter equilaâ—â—â—â—â—â—amisâ— which was required to â—e donâ— A Corollary The bases of a Pyramis inscribed in an Octohedron are parallels
to the bases of the Octohedron For forasmuch as the sides of the bases of the Pyramis touching the one the other are parallels to the sides of the Octohedron which also touch the one the other as for example HL was proued to be a parallel to GI and LC to DI therefore by the 15. of the eleuenth the plaine superficies which is drawen by the lines HL and LC is a parallel to the plaine superficies drawen by the lines GI and DI. And so likewise of the rest Second Corollary A right line ioyning together the centres of the opposite bases of the Octohedron is sesquialter to the perpendicular line drawen from the angle of the inscribed pyramis to the base thereof For forasmuch as the pyramis and the cube which containeth it do in the selfe same pointes end their angles by the 1. of this booke therefore they shall both be inclosed in one and the selfe same Octohedron by the 4. of this booke But the diameter of the cube ioyneth together the centres of the opposite bases of the Octohedron and therefore is the diameter of the Sphere which containeth the cube and the pyramis inscribed in the cube by the 13. and 14. of the thirtenth which diameter is sesquialter to the perpendicular which is drawen from the angle of the pyramis to the base thereof for the line which is drawen from the centre of the sphere to the base of the pyramis is the sixth part of the diameter by the 3. Corollary of the 13. of the thirtenth Wherefore of what partes the diâ—meter containeth sixe of the same partes the perpendicular containeth fower ¶ The 7. Proposition The 7. Probleme In a dodecahedron geuen to inscribe an Icosahedron SVppose that the dodecahedron geuen be ABCDE And let the centres of the circles which coÌ„tayne sixe bases of the same dodecahedron be the polnes L M N P Q O. And draw these right lines OL OM ON OP OQ and moreouer these right lines LM MN NP PQ QL And now forasmuch as equall and equilater pentagons are contayned in equall circles therefore perpendicular lines drawne from their centres to the sides shall be equall by the 14. of the third and shall diuide the sides of the dodecahedron into two equall partes by the 3. of the same Wherefore the foresayde perpendicular lines shall coâ—outre in the point of the section wherein the sides are diuided into two equall partes as LF and MF doo And they also containe equall angles namely the inclination of the bases of the dodecahedron by the 2. corollary of the 18. of the thirtenth Wherfore the right lines LM MN NP PQ QL and the rest of the right lines which ioyne together two centres of the bases and which subtende the equall angles â—ontayned vnder the sayd equall perpendicular lines are equall the one to the other by the 4. of the first Wherefore the triangles OLM OMN ONP OPQ OQL and the rest of the triangles which are set at the centres of the pentagons are equilater and equall Now forasmuch as the 12. pentagons of a dodecahedron containe 60. plaine superficiall angles of which 60. eueâ—y â—hre make one solide angle of the dodecahedron it followeth that a dodecahedron hath 20. solide angles but eche of those solide angles is subteÌ„ded of ech of the triangles of the Icosahedron namely of ech of those triangles which ioyne together the centres of the pentagoÌ„s which make the solide angle as we haue before proued Wherefore the 20 equall and equilater triangles which subtende the 20. solide angles of the dodecahedron and haue their sides which are drawne from the centres of the pentagons common doo make an Icosahedron by the 25. diffinition of the eleuenth and it is inscribed in the dodecahedron geuen by the first diffinition of this booke for that the angles thereof doo all at one time touch the bases of the dodecahedron Wherefore in a dodecahedron geuenâ— iâ— inscribed an Icosahedron which was required to be done ¶ The 8. Proposition The 8. Probleme In a dodecahedron geuen to include a cube DEscribe by the 17. of the thirtenth a dodecahedron And by the same take the 12. sides of the cube eche of which subtend one angle of eche of the 12. bases of the dodecahedron for the side of the cube subtendeth the angle of the pentagon of the dodecahedron by the 2. corollary of the 17. of the thirtenth If therefore in the dodecahedron described by the selfe same 17. proposition we draw the 12. right lines subâ—ended vnder the foresayd 12. angles and ending in 8. angles of the dodecahedron and concurring together in such sort that they be in like sort situate as it was plainely proued in that proposition then shall it be manifest that the right lines drawne in this dodecahedron from the foresayd 8. angles thereof doo make the foresayd cube which therefore is included in the dodecahedron for that the sides of the cube are drawne in the sides of the dodecahedron and the angles of the same cube are set in the angles of the said dodecahedron As for example take 4. pentagons of a dodecahedron namely AGIBO BHCNO CKEDN and DFAON And draw these right lines AB BC CD DA. Which fower right lines make a square for that eche of those right lines doo subtend equall angles of equall pentagons the angles which those 4. right lines coÌ„taine are right angles as we proued in the construction of the dodecahedron in the 17. propositioÌ„ before alledged Wherfore the sixe bases being squares do make a cube by the 21. diffinition of the eleuenth and for that the 8. angles of the sayd cube are set in 8. angles of the dodecaheeron therefore is the sayd cube inscribed in the dodecahedron by the first diffinition of this booke Wherefore in a dodecahedron is inscribed a cube which was required to be doone ¶ The 9. Proposition The 9. Probleme In a Dodecahedron geuen to include an Octohedron SVppose that the dodecahedron geuen be ABGD Now by the 3. correllary of the 17. of the thirteÌ„th take the 6. sides which are opposite the one to the other those 6. sides I saye whose sections wherin they are deuided into two equal partes are coupled by three right lines which in the centre of the sphere wherin the Dodecahedron is contained doe cut the one the other perpendicularly And let the poyntes wherin the forsayde sides are cut into two equal partes be A B G D C I. And let the foresaid thre right lines ioyning together the saide sections be AB GD and CI. And let the centre of the sphere be E. Now forasmuch as by the foresaid correllary those thre right lines are equal it foloweth by the 4. of the first that the right lines subteÌ„ding the right angles which they make at the centre of the sphere whiche right angles are contained vnder the halues of the said three right lines are equal the one to the other
wherfore the line Ce is to the line eg as the greater segment to the lesse and therefore their proportion is as the whole line IC is to the greater segment Ce and as the greater segment Ce is to the lesse segment eg wherefore the whole line Ceg which maketh the greater segment and the lesse is equall to the whole line IC or IE And forasmuch as two parallel plaine superficieces namely that which is extended by IOB and that which is extended by the line ag are cutte by the playne of the triangle BCE which passeth by the lines ag and IB their common sections ag and IB shall be parallels by the 16. of the eleuenth But the angle BIE or BIC is a right angle wherefore the angle agC is also a right angle by the 29. of the first and those right angles are contayned vnder equall sides namely the line gC is equall to the line CI and the line ag to the line BI by the 33. of the first wherfore the bases Ca and CB are equall by the 4. of the first But of the line CB the line CE was proued to be the greater segment wherefore the same line CE is also the greater segment of the line Ca but cn was also the greater segment of the same line Ca. Wherefore vnto the line CE the line cn which is the side of the dodecahedron and is set at the diameter is equall And by the same reason the rest of the sides which are set at the diameter may be proued eâ—uall to lines equall to the line CE. Wherfore the pentagon inscribed in the circle where in is contained the triangle BCE is by the 11. of the fourth equiangle and equilater And forasmch as two pentagons set vpon euery one of the bases of the cube doo make a dodecahedron and sixe bases of the cube doo receaue twelue angles of the dodecahedron and the 8. semidiameters doo in the pointes where they are cutte by an extreame and meane proportion receaue the rest therefore the 12. pentagon bases contayning 20. solide angles doo inscribe the dodecahedron in the cube by the 1. diffinition of this booke Wherefore in a cube geuen is inscribed a dodecahedron which was required to be done First Corollary The diameter of the sphere which containeth the dodecahedron containeth in power these two sides namely the side of the Dodecahedron and the side of the cube wherein the Dodecahedron is inscribed For in the first figure a line drawne from the centre O to the poynt B the angle of the Dodecahedron namely the line OB containeth in power these two lines OV the halfe side of the cube and VB the halfe side of the dodecahedron by the 47. of the first Wherefore by the 15. of the fiueth the double of the line OB which is the diameter of the sphere containing the Dodecahedron containeth in power the double of the other lines OV and VB which are the sides of the cube and of the dodecahedron ¶ Second Corollary The side of a cube diuided by an extreme and meane proportion maketh the lesse segment the side of the dodecahedron inscribed in it and the greater segment the side of the cube inscribed in the same Dodecahedron For it was before proued that the side of the dodecahedron is the greater segment of BE the side of the triangle BECâ— but the side BE which is equall to the lineâ— GB and SF is the greater segmeÌ„t of GF the side of the cube which line â—E subtending thâ— angle of the pentagon was by the â— of this booke the side of the cube inscribed in the dodecahedron Third Corollary The side of a cube is equal to the sides of a Dodecahedron inscribed in it and circumscribed about it For it was manifest by this proposition that the side of a cube maketh the lesse segment the side of a Dodecahedron inscribed in it namely as in the first figure the line BS the side of the Dodecahedron inscribed is the lesse segmeÌ„t of the line GF the side of the cube And it was proued in the 17. of the thirtenth that the same side of the cube subteÌ„deth the angle of the pentagon of the Dodecahedron circumscribed and therefore it maketh the greater segment the side of the Dodecahedron or of the pentagon by the first corollary of the same Wherefore it is equal to bothe those segments The 14. Probleme The 14. Proposition In a cube geuen to inscribe an Icosahedron SVppose that the cube geuen be ABC the Centres of whose bases let be the points D E G H I K by whiche poyntes draw in the bases vnto the other sids parallels not touching the one the other And deuide the lines drawn from the centres as the line DT c. by an extreme and meane proportion in the poyntes A F L M N B P Q R S C O by the 30. of the sixth and let the greater segmentes be about the ceÌ„tres And draw these right lines AL AG AM and TG And forasmuch as the lines cut are parallels to the sides of the cube they shall make right angles the one with the other by the 29. of the first and forasmuche as they are equal their sections shall be equal for that the sections are like by the 2. of the fourtenth Wherfore the line TG is equal to the line DT for they are eche halfe sides of the cube Wherfore the square of the whole line TG and of the lesse segment TA is triple to the square of the line AD the greater segment by the 4. of the thirteÌ„th But the line AG containeth in power the lines AT TG for the angle ATG is a right angle Wherefore the square of the line AG is triple to the square of the line AD. And forasmuch as the line MGL is erected perpendicularly to the plain passing by the lines AT which is parallel to the bases of the cube by the corollary of the 14. of the eleueÌ„th therfore the angle AGL is a right angle But the line LG is equal to the line AD for they are the greater segments of equal lines Wherfore the line AG which is in power triple to the line AD is in power triple to the line LG Wherefore adding vnto the same square of the line AG the square of the line LG the square of the line AL which by the 47. of the first containeth in power the two lines AG and GL shal be quadruple to the line AD or LG Wherefore the line AL is double to the line AD by the 20. of the sixth and therfore is equal to the line AF or to the line LM And by the same reason may we proue that euery one of the other lines which couple the next sections of the lines cut as the lines AM PF PM MQ and the rest are equal Wherfore the triangles ALM APF AMP PMQ and the rest such like are equal equiangle and equilater
by the 4. and eigth of the first And forasmuch as vpon euery one of the lines cut of the cube are set two triangles as the triangles ALM and BLMâ— there shal be made 12. triaÌ„gles And forasmuch as vnder euery one of the â— angles of the cube are subtended the other 8. triangles as the triangle AMP. c. of 1â— and 8. triangles shall be produced 20. triangles equal and equilater coÌ„taining the solidâ— of an Icosahedron by the 25. diffinition of the eleuenth which shal be inscribed in the cube geuen ABC by the first diffinition of this booke The inuention of the demonstration of this dependeth of the ground of the former Wherfore in a cube geuen we haue described an Icosahedron which was required to be done First Corollary The diameter of a sphere which containeth an Icosahedron containeth two sides namely the side of the Icosahedron and the side of the cube which containeth the Icosahedron For if we drawe the line AB it shall make the angles at the poynt A right angles for that it is a parallel to the sides of the cube wherfore the linâ— which coupleth the opposite angleâ— of the Icosahedron at the poynts F and B coÌ„taineth in power the line AB the sidâ— of thâ— cube and the line AF the side of the Icosahedron by the 47. of the first Which line FB is equâ—l to the â—iameter of the sphere which contâ—ineth the Icosahedron by the demonstration of the â—â— of the thirteÌ„th Second Corollary The six opposite sides of the Icosahedron deuided into two equal parts their sections are coupled by three equal right lines cutting the one the other into two equal partes and perpendicularly in the centre of the sphere which containeth the Icosahedron For those three lines are the three lines which couple the centres of the bases of the cube which do in suche sort in the centre of the cube cut the one the other by the corollary of the third of this booke and therfore are equal to the sides of the cube But right lines drawne from the ceÌ„tre of the cube to the angles of the Icosahedron euery one of them shall subtend the halfe side of the cube and the halfe side of the Icosahedron which halfe sides containe a right angle wherefore those lines are equal Wherby it is manifest that the foresaid centre is the centre of the sphere which containeth the Icosahedron Third Corollary The side of a cube deuided by an extreme and meane proportion maketh the greater segment the side of an Icosahedron described in it For the half side of the cube maketh the halfe of the side of the Icosahâ—dron the greater segment wherefore also the whole side of the cube maketh the whole side of the Icosahedron the greater segment by the 15. of the fifthe for the sections are like by the â— of the fourtenth ¶ Fourth Corollary The sides and bases of the Icosahedron which are opposite the one to the other are parallels Forasmuch as euery one of the opposite sides of the Icosahedron may be in the parallel lines of the cube namely in those parallels which are opposite in the cube and the triangles which are made of parallel lines are parallels by the 15. of the eleuenth therfore the opposite triâ—ngleâ— of the Icosahedron as also the sides are pâ—rallels the one to the other ¶ The 15. Probleme The 15. Proposition In an Icosahedron geuen to inscribe an Octohedron SVppose that the Icosahedron geuen be ACDF and by the former second Corollary let there be takeÌ„ the three right lines which cut the one the other into two equall partes perpendicularly and which couple the sections into two equall partes of the sides of the Icosahedron which let be BE GH and KL cutting the one the other in the point I. And drawâ— these riâ—ht lines â—G GE EH and HB And forasmuch as the angles at the point I are by construction right angles â—nd are conâ—â—ined under equâ—ll linesâ— the 〈◊〉 Gâ— and â—â— shall 〈…〉 squâ—re by the â— of the â—irst Likewyse â—nto thoâ—e 〈◊〉 shall be â—quall the lines drâ—wâ—n from 〈◊〉 pointes K and â— to euery one of the poinâ—â—s â— G. â— H And therefore the triangles which 〈◊〉 the â—â—ramis DGENK shall be equall 〈…〉 And by 〈…〉 ¶ The 16. Probleme The 16. Proposition In an Octohedron geuen to inscribe an Icosahedron LEt there be taken an Octohedron whose 6. angles let be A B C F P L. And draw the lines AC BF PL cutting the one the other perpendicularly in the point R by the 2. Corollary of the 14. of the thirtenth And let euery one of the 12. sides of the Octohedron be diuided by an extreme and meane proportion in the pointes H X M K D S N G V E Q T. And let the greater segmentes be the lines BH BX FM FK AD AQ CS CT PN PG LV LE And drawe these lines HK XM GE NV DS QT Now forasmuch as in the triangle ABF the sides are cut proportionally namely as the line BH is to the line HA so is the line FK to the line KA by the 2. of the fouretenth therefore the line HK shall be a parallel to the line BF by the 2. of the sixth And forasmuch as the line AC cutteth the line HK in the point Z and the line ZK is a parallel vnto the line RF the line RA shall be cut by an extreme and meane proportion in the point Z by the 2. of the sixth namely shall be cut like vnto the line FA and the greater segmeÌ„t therof shall be the line ZR Vnto the line ZK put the line RO equall by the 3. of the first and drawe the line KO now then the line KO shall be equall to the line ZR by the 33. of the â—irst Draw the lines KG KE and KI And forasmuch as the triangles ARF and AZK are equiangle by the 6. of the sixth the sides AZ and ZK shall be equall the one to the other by the 4. of the sixth for the sides AR and RF are equall Wherfore the line ZK shall be the lesse segment of the line RA. But if the greater segment RZ be diuided by an extreme meane proportion the greater segment therof shall be the line ZK which was the lesse segment of the whole line RA by the 5. of the thirâ—enth And forasmuch as the two lines FE and FG are equall to the two lines AH and AK namely ech are lesse segmentes of equall sides of the Octohedron and the angles HAK and EFG are equall namely are right angles by the 14. of the thirtenth the bases HK and GF shall be equall by the 4. of the first And by the same reason vnto them may be proued equall the lines XM NV DS and QT And forasmuch as the lines AC BF and PL do cut the one the other into two equall parts and perpendicularly by construction the lines HK and GE which
subtend angles of triangles like vnto the triangles whose angles the lines AC BF and PL subtend are cut into two equall partes in the pointes Z and I by the 4. of the sixth so also are the other lines NV XM DS QT which are equall vnto the lines HK GE cut in like sort and they shall cut the lines AC BF and PL like Wherefore the line KO which is equall to RZ shall make the greater segment the line RO which is equall to the line ZK for the greater segment of the RZ was the line ZK and therefore the line OI shall be the lesse segment when as the whole line RI is equall to the whole line RZ Wherefore the squares of the whole line KO and of the lesse segment OI are triple to the square of the greater segment RO by the 4. of the thirtenth Wherfore the line KI which containeth in power the two lines KO and OI is in power triple to the line RO by the 47. of the first for the angle KOI is a right angle And forasmuch as the lines FE and FG which are the lesse segmentes of the sides of the Octohedron are equall and the line FK is coÌ„mon to them both and the angles KFG and KFE of the triangles of the Octohedron are equall the bases KG and KE shall by the 4 of the first be equall and therefore the angles KIE and KIG which they subtend are equall by the 8. of the first Wherefore they are right angles by the 13. of the first Wherefore the right line KE which containeth in power the two lines KI and â—E by the 47. of the first is in power quadruple to the line RO or IE for the line RI is proued to be in power triple to the same line RO But the line GE is double to the line IE Wherfore the line GE is also in power 〈…〉 PF And by the same reason may be proued that the â—est of the eleuen solide angles of the 〈◊〉 are 〈…〉 the sections of euery one of the sides of the Octohedron namely in the pointes E N V H â— M â— D S Q T. Wherefore there are 12. angles of the Icosahedron Moreouer forasmuch as euery one of the bases of the Octohedron do eche containe triangles of the Icosahedron 〈…〉 pyramiâ—â— ABCâ—FP which is the halfe of the Octohedron the triangle FCP receaueth in thâ— section of his sides the â— triangle GMS and the triangle CPB containeth the triangle NXS and thâ— triangle â—AP contayneth the triangle HND and moreouer the triangle APF containeth the triangle â—DG and the same may be proued in the opposite pyramis ABCFL Wherefore there shall be eight triangleâ— And forasmuch as besides these triangles to euery one of the solide angles of the Octohedron 〈◊〉 subtended two triangles as the triangles KEG amd MEG to the angle F and the triangles HNV and XNV to the angle B also the triangles NDS and â—DS to the angle P likewise the triangleâ— DHK and QHK to the angle A Moreouer the triangles EQT and VQT to the angle L and finally the triangles SXM and TXM to the angle C these 12. triangles being added to thâ—â— for 〈◊〉 triangles shall produce â—0 triangles equall and equilâ—ter coupled together which shall male an Icosahedron by the 25. definition of the eleuenth and it shall be inscribed in the Octohedron geuen ABCâ—â—L by the first definition of this booke for the 1â— angles thereof are set in 1â— like sections of the sides of the Octohedron Wherefore in an Octohedron geuen is inscribed an Icosahedron ¶ First Corollary The side of an equilater triangle being diuided by an extreme and meane proportion a right line subtending within the triangle the angle which is contained vnder the greater segment and the lesse is in power duple to the lesse segment of the same side For the line KE which subtendeth the angle KFE of the triangle AFL which angle KFE is contained vnder the two segmentes KF FE was proued equall 〈◊〉 the line HK which containeth in power the two lesse segmentes HA and AK by the 47. of the â—â—rst fo◠〈◊〉 angle HAK is 〈…〉 Second Corollary The bases of the Icosahedron are concentricall that is haue one and the selfe same centre with the bases of the Octohedron which contayneth it For suppose that 〈…〉 Octohedron 〈◊〉 ECD the base of an Icosahedron and let the centre of the base ABG be the point F. And drawe these right lines FA FB FC and FE Now then the 〈…〉 to the two lines FB and BC for they are lines drawen from the centre and are also lesse segmentes and they contayne the 〈…〉 ¶ The 17. Probleme The 17. Proposition In an Octohedron geuen to inscribe a Dodecahedron SVppose that the Octohedron geuen be ABGDEC whose 12. â—ides let be cut by an extreme and meane proportion as in the former Proposition It was manifest that of the right lines which couple thâ—se sections are made 20. triangles of which 8. are concentricall with the bases of the Octohedron by the second Corollary of the former Proposition If therefore in euery one of the centres of the 20. triangles be inscribed by the 1. of this booke euery one of the â—â— â—â—gles of the Dodecahedron we shall finde that â— angles of the Dodecahedron are set in the 8. centres of the bases of the Octohedron namely these angles I u ct O M a P and X and of the other 12. solide angles there are two in the centres of the two triangles which haue one side common vnder euery one of the solide angles of the Octohedron namely vnder the solide angle A the two solide angles K Z vnder the solide angle B the two solide angles H T vnder the solide angle G the two solide angles Y V vnder the solide angle D the two solide angles F L vnder the solide angle E the two solide angles S N vnder the solide angle C the two solide angles Q R and forasmuch as in the Octohedron are sixe solide angles vnder them shall be subtended 12. solide angles of the Dodâ—cahedron and so are mâ—de 20. solide angles composed of 12. equall and â—quilâ—ter superficiall pentagons as it was 〈◊〉 by the 5. of this booke which therefore containe a Dodecahedron by the 24. definition of the eleuenth And it is inscribed in the Octohedron by the 1. definition of this booke for that euery one of the bases of the Octohedron do receaue angles therof Wherefore in an Octohedron geuen is inscribed a Dodecahedron ¶ The 18. Probleme The 18. Proposition In a trilater and equilater Pyramis to inscribe a Cube SVppose that there be a trilater equilater Pyramis whose base let be ABC and â—oppe the point D. And let it be comprehended in a Sphereâ— by the 13. of the 〈◊〉 And lâ—â— the centre of that Sphere be the point E. And from the solide angles A B C D draw right lines passing by the centre E vnto the opposite bases of
two lines HIF and TIO cutting the one the other are in one and the selfe same ' plaine by the 2. of the eleuenth And therefore the poyntes H T F O are in one the selfe same plaine Wherforeâ— the rectangle figure HOFT beâ—ng quadrilater and equilater and in one and the selfe same playne is a square by the diâ—â—inition of a square And by the same reason may the rest of the bases of the solide be proued to be squares equall and plaine or superficial Now then the solide is comprehended of 6. equal squares which are contained of 12. equal sides which squares make 8. solide angles of which foure are in the ceâ—tres of the bases oâ— the pyramis and the other 4. are in the midle sections of the foure perdendiculars Wherfore the solide HOFTPGRN is a cube by the 21. diffinition of the eleuenth and is inscribed in the pyramis by the first definition of this boke Wherfore in a trilater equilater pyramis geuen is inscribed a cube ¶ A Corrollary The line which cutteth into two equall partes the opposite sides of the Pyramis is triple to the side of the cube inscribed in the pyramis and passeth by the centre of the cube For the line SEV whose third part the line SI is cutteth the opposite sides CD and AB into two equll partes but the line EI which is drawne from the centre of the cube to the base is proued to be a third part of the line ES wherefore the side of the cube which is double to the line EI shall be a third part of the whole line VS which is as hath bene proued double to the line ES. The 19. Probleme The 19. Proposition In a trilater equilater Pyramis geuen to inscribe an Icosahedron SVppose that the pyramis is geuen 〈◊〉 ABâ—Dâ— euery one of whose sâ—des 〈◊〉 be diuidâ—d into two equall partes in the poyâ—â—â—â—â— M K L P N. And iâ— euery one of the bâ—ses of that pyramis descride the trianglâ—â— Lâ—â— PMN NKL and 〈…〉 which triangles shall be equilater by the 4. of the firâ—t â—or the sides subâ—end equall angles of the pyramis contayned vnder the halues of the sides of the same pyramisâ— wherfore the sides of the said triangles are equall Let those sides be â—iuided by an extreame and meane proporâ—â—on by the 30. of the sixth in the poyntes C E Q R S T H I O V Y X. Now then those sides are cutte into the selfe same proportions by the 2. of the fourteÌ„th and therfore they make the liâ—e sectioÌ„s equall by the â— part of the ninth of the fiueth Now I say that the foresayd poynâ—es doâ— recâ—aue the angles of the Icosahedron inscribed in the pyramis ABâ—D In the foresayd triangles let there agayne be made other triangles by coupling the sections and let those triangles be TRS IOH CEQ and VXY which shall be equilater for euery one of their sides doo subâ—â—â—d equall angles of equilater triangles and those sayd equall angles are contayned vnder equall sideâ— namely vnder the greater segmenâ— and the lesse â— and therefore the sides which subtend those angles are equall by the 4. of the first Now let vs proue that at eche of the foresayd poynts as for example at T is set the solide angle of an Icosahâ—dronâ— Forasmuch as the triangles TRS and TQO are equilater and equall the 4. right lines TR T S TQ and TO shall be equall And forasmuch as â—PNK is a square cutting the pyramis ABâ—D into two equall paâ—â—â—â— by the corollay of the second of this bookeâ— the line TH shall be in power duple to the line TN or NH by the 47. of the first For the lines TN or NH are equall for that by construction they are eche lesse segmentes and the line RT or T S is in power duple to the same line TN or NH by the corollary of the 16. of this booke for it subtendeth the angle of the triangle contayned vnder the two segmentes Wherfore the lines TH T S TR TQ and TO are equall and so also are the lines HS SR RQ QO and OH which subtend the angles at the poynt T equall For the line QR contayneth in power the two lines PQ and PR the lesse segmentes which two lines the line TH also contayned in power And the rest of the lines doo subtend angles of equilater triangles contayned vnder the greater segment and the lesse Wherefore the fiue triangles TRS TSH THO TOQ TQR are equilater and equall making the solide angle of an Icosahedron at the poynt T by the 16. of the thirtenth in the side PN of the triangle P NM And by the same reason in the other sides of the 4. triangles PNM NKL FMK LFP which are inscribed in the bases of the pyramis which sides are 12â— in nuÌ„ber shal be set 12. angles of the IcosahedroÌ„ coÌ„tained vnder 20. equal equilater triangles of which fowere are set in the 4. bases of the pyramis namely these fower triangles TRS HOI CEQ VXY 4. triangles are vnder 4. angles of the pyramis that is the fower triangles CIX YSH ERV TQO and vnder euery one of the sixe sides of the pyramis are set two triangles namely vnder the side of the triangles THS and THOâ— vnder the side DB the triangles RQE and RQT vnder the side DA the triangles COQ and COI vnder the side AB the triangles EXC and EXVâ— vnder the side BG the triangles SVR and SVY and vnder the side AG the triangles IYH and IYX. Wherefore the solide being contayned vnder 20. equilater and equall triangles shall be an Icosahedron by the 23. diffinition of the eleuenth and shall be inscribed in the pyramis ABâ—D by the first diâ—â—â—nition of this booke for all his angles doo at one time touch the bases of the pyramis Wherefore in a trilater equilater pyramis geuen we haue inscribed an Icosahedron ¶ The 20. Proposition The 20. Probleme In a trilater equilater Pyramis geuen to inscribe a dodecahedron SVppose that the pyramis geuen be ABGD â—che of whose sides let be cutte into two equall partes and draw the lines which couple the sections which being diuided by an extreame and meane proportion and right lines being drawne by the sections shall receaue 20. triangles making an Icosahedron as in the former proposition it was manifest Now then if we take the centres of those triangles we shall there finde the 20. angles of the dodecahedron inscribed in it by the 5. of this booke And forasmuch as 4. bases of the foresayd IcosahedroÌ„ are coÌ„centricall with the bases of the pyramis as it was proued in the 2. corollary of the 6. of this boke there shal be placed 4â— angles of the dodecahedroÌ„ namely the 4. angles E F H D in the 4. centres of the bases and of the other 16. angles vnder euery one of the 6. sides of the pyramis are subtended two namely vnder the side AD the angles CK vnder the side BD the angles LI vnder the
side GD the angles M N vnder the side AB the angles T S vnder the side BG the angles P O and vnder the side AG the angles R Q so there rest 4. angles whose true place we will now appoynt Forasmuch as a cube contayned in one and the selfe same sphere with a dodecahedron is inscribed in the same dodecahedron as it was manifest by the 17. of the thirtenth and 8. of this booke it followeth that a cube and a dodecahedron circumscribed about it are contayned in one and the selfe same bodies for that their angles concurre in one and the selfe same poyntes And it was proued in the 18. of this booke that 4. angles of the cube inscribed in the pyramis are set in the middle sections of the perpendicularâ— which are drawne from the solide angles of the pyramis to the opposite bases wherefore the other 4. angles of the dodecahedron are also as the angles of the cube set in those middle sections of the perpendiculars Namely the angle V is set in the middest of the perpendicular AHâ— the angle Y in the middest of the perpendicular BF the angle X in the middest of the perpendicular GE and lastly the angle D in the middest of the perpendicular D which is drawne from the toppe of the pyramis to the opposite base Wherefore those 4. angles of the dodecahedron may be sayd to be directly vnder the solide angles of the pyramis or they may be sayd to be set at the perpendiculars Wherefore the dodecahedron after this maner set is inscribed in the pyramis geuen by the first diffinition of this booke for that vpoÌ„ euery one of the bases of the pyramis are set an angle of the dodecahedroÌ„ inscribed Wherefore in a trilater equilater pyramis is inscribed a dodecahedron The 21. Probleme The 21. Proposition In euery one of the regular solides to inscribe a Sphere IN the 13. of thâ— thirtenth and thâ— other 4. propositioâ—â— following iâ— was declared that â—he â—â— regular solidesâ—â—re so contaâ—â—ed in a sphere that â—ight linâ—â— drawne from the cenâ—â—â— oâ— the 〈…〉 of 〈◊〉 solide inscribed are equall Which right lines therefore make pyramids whose â—oppes are the centre of the sphere or of the solide and the basâ—â—â—â—e cuâ—â—â— one of the bases of those solides And 〈…〉 solide â—quall and like the one to the other and described in equall circles those cirâ—les shall cutte the sphere for the angles which touch the circumference of the circle touch also the superficies of the sphere Wherefore perpeÌ„diculars drawne from the centre of the sphere to the bases or to the playne superficieces of the equall circles are equall by the corollary of the assumpt of the 1â— of the twelfth Wherefore making the centre the 〈◊〉 of the sphere which 〈◊〉 the solide and thâ— space some one of the equall perpendicularâ— dâ—scribâ— a sphere and it shall touch euery one of the bases of 〈◊〉 solide 〈…〉 perficies of the sphere passe beyond those bases when as those pâ—â—peâ—diculars 〈…〉 are drawne from the centre to the bases by the 3. corollary of the saâ—â—â—â—â—umpt Wherâ—fore â—e haue iâ— euery one of the regular bodies inscribed a sphere which regular boâ—â—â— are in number one i◠〈◊〉 by the corollary of the 1â— of the 〈◊〉 A Corollary The regular figures inscribed in spheres and also the spheres circumscribed about them or contayning them haue one and the selfe same centre Namely their pyramids the â—ngles of whose bâ—ses touch the superâ—â—â—â—â—â— of thâ—â—â—here doo from those angles cause equall right lines to be drawâ—â— to one and â—he selfe 〈◊〉 poynâ— making the topâ—â—â— of the pyramidâ— in the same poynt and therefore they 〈…〉 thâ— câ—â—tres of the spheres in the selfe same toppes when 〈◊〉 the right lines drawne from those angles to the croâ—â—ed superficies whâ—rein are 〈◊〉 the angles of the bases of the pyramidâ— are equallâ— An adueâ—â—â—sment of Flussas â— Of these solides onely the Octohedron receaueth the other solides inscribed one with 〈…〉 other For the Octohedron contayneth the Icosahedron inscribed in it and the same Icosahedron contayneth the Dodecahedron inscribed in the same Icosahedron and the same dodecahedron contayneth the cube inscribed in the same Octohedron and 〈…〉 â—â—râ—â—mscribeth the Pyramis inscribed in the sayd Octohedron But this happâ—neth not in the other solides The ende of the fiuetenth Booke of Euclides Elemenâ—â—â— after Caâ—paâ—â— and 〈◊〉 ¶ The sixtenth booke of the Elementes of Geometrie added by Flussas IN the former fiuetenth booke hath bene taught how to inscribe the fiue regular solides one with in an other Now semeth to rest to coÌ„pare those solidâ— so inscribed one to an other and to set forth their passionâ— and proprieties which thing Flussas considering in this sixteÌ„th booke added by him hath excellently well and most conningly performed For which vndoubtedly he hath of all them which haue a loue to the Mathematicals deserued much prayse and commendacion both for the great traâ—ailes and paynâ—s which it is most likely he hath taâ—â—n in iâ—uenting such straunge and wonderfull propositions with their demonstrations in this booke contayned as also for participating and communicating abrode the same to others Which booke also that the reader should want nothing conducing to the perfection of Euclides Elements I haue with some trauaile translated for the worthines â—hereof haue added it aâ— a sixtenth booke to the 15. bookes of Euclide Vouchsafe therefore gentle reader diligently to read and peyse it for in it shall you finde noâ— onely matter strange and delectable but also occasion of inuention of greater things pertayning to the natures of the fiue regular solidâ—s◠¶ The 1. Proposition A Dodecahedron and a cube inscribed in it and a Pyramis inscribed in the same cube are contained in one and the selfe same sphere FOr the angles of the pyramiâ— are seâ— in the angâ—es of the cube wherein it is inscribed by the first of the fiuetenthâ— and all the angles of the cube are set in the angles of the dodecahedâ—â—â— circumscribed 〈…〉 〈◊〉 the 8. of the fiuetenth And all the angles of the Dodecahedron are set in the superficies of the sphere by the 17. of the thirtenth Wherefore those three solides inscribed one within an other are contained in one and the selfe same sphere by the first diffinition of the fiuetenth A dodecahedron therfore and a cube inscribed in it and a pyramis inscribed in the same cube are contained 〈…〉 â—â—lfe same sphere 〈…〉 These three solides li 〈…〉 elfe same Icosahedron or Octohedron or Pyramis 〈…〉 me Icosahedron by the 5.11 12. of the fiuetenth and they ar 〈…〉 ctohedron by the 4. 6. and 16. of the same lastly they are inscribed in 〈…〉 the first 18. and 19. of the same For the angles of all these solide 〈…〉 the circumscribed Icosahedron or octohedron or pyramis ¶ The 〈…〉 The proportion of a Dodecahedron circumscribed about a cube to a DodecahedroÌ„ inscribed in the same cube is
Construction Two cases in this Proposition First caseâ— Second case Demonstration Construction Demonstration Construction Demonstration An other way after Peliâ—arius Construction Demonstration Construction Demonstration Demonstration leading to an impossibilitie Three cases in this Propositiâ—n The third case Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration A Proposition added by Petarilius Note Construction Demonstration An other way also after Pelitarius Construction Demonstration An other way to do the samâ— after Pelitarius Demonstration Demonstrâ—tion Demonstration leading to an absurditie A Corollary Construction Demonstration Demonstration An â—ther way to do the same after OroÌ„tius An other way after Pelitarius Construction Demonstration An addition of Flussates * A Poligonon figure is a figure consisting of many sides The argument of this fift booke The first aâ—thor of this booke Eudoxus The first definition A part taken two maner of wayes The fiâ—st way The second way How a lesse quantity is sayd to measure a greater In what significatioÌ„ Euclide here taketh a part Parâ— metienâ— or mensuranâ— Pars multiplicatiâ—a Pars aliquota This kinde of part coÌ„monly vsed in Arithmetique The other kinde of part Pars constitâ—ens or componens Pars aliquanta The second definition Numbers very necessary for the vnderstanding of this booke and the other bookes following The tâ—ird definition Rational proportion deuided â—â—to two kindes Proportion of equality Proportion of inequality Proportioâ— of the greater to the lesse Multiplex Duple proportion Triple quadruple Quintuple Superperticular Sesquialtera Sesquitertia Sesquiquarta Superpartiens Superbipartiens Supertripartiens Superquadripartiens Superquintipartiens Multiplex superperticular Dupla Sesquialtera Dupla sesquitertia Tripla sesquialtera Multiplex superpartiens Dupla superbipartiens Dupla supertripartiens Tripla superbipartiens Tripla superquadâ—ipartiens How to knoâ— the denomination of any proportion Proportion of the lesse in the greater Submultiplex Subsuperparticular Subsuperpertient c. The fourth definition Example of this definition in magnitudes Example thereof in numbers Note The fifth definition An example of this â—efinition in magnitudes Why Euclide in defining of Proportion vsed multiplication The sixth definition An example of this definitiin in magnitudes An example in numbers An other example in numbers An other example in numbers Note this particle according to any multiplication An example where the equimultiplices of the first and third exceedâ— the equimultiplices of the second and fourth and yet the quantities geuen are not in one and the selfe same proportion A rule to produce equimultiplices of the first and third equall to the equimultiplices of the secondâ— and fâ—urth Example thereof The seuenth definition 9 12 3 4 Proportionality of two sortes contiâ—uall and discontinuall An example of continuall proportionalitie in numbers 16.8.4.2.1 In coutinnall proportionalitie the quantities cannot be of one kinde Discontinuall propâ—rtionalitie Example of discontinual proportionality in numbers In discoÌ„tinual proportionalitie the proportions may be of diuers kindes The eight definition An example of this definition in magnitudes An example in numbers Note The ninth definition An example of this definition in magnitudes Example â—n numbers The tenth definition A rule to adde proportions to proportions 8. 4. 2. 1. 2 2 2 1 1 1 The eleuenth definition Example of this definition in magnitudâ— Example in numbers The twelfâ—h definition Example of this deâ—inition in magnituds Example in numbers The thirtenth definition Example of this definition in magnituds Example in numbers The fourtenth definition Example of this definition in magnituds Example in numbers The fiâ—tâ—ne definition This is the coÌ„uerse of the former definition Example in magnitudes Example in numbers The sixtene definition An example of this definition in magnitudes An example in numbers The seueÌ„tenth definition An example of this definition in magnitudes An example in numbers Note The eighttenth definition An example of this definition in magnitudes Example in numbers The nintenth definition An example of this definition in magnitudes Example in numbers The 20. definition The 2â— defiâ—ition These two last definitions not found in the greeke examplers Construction Demonstration Demonstrationâ— Construction Demonstration Construction Demonstration ALemmae or an assumpt A Corollary Conuerse proportion Construction Demonstration Two cases in this Propotion The second The second part demonstrated The first part of this Proposition demonstrated The second part of the proposition demonstrated First differâ—câ— of the first part Demonstratiâ— of tâ—e same first differeÌ„ce Second diffeâ—ence Third diâ—â—erence The second part â—f this proposition The first parâ— of this Proposition demonstrated The second part proued The first part of this proposition proued The second part demonstrated Construction Demonstrationâ— Constrâ—ction Demonstrationâ— Construction Demonstration An addition of Campane Demonstration Construction Demonstration DemoÌ„stration of alternate proportion Construction Demonstration DemonstratioÌ„ of proportion by diuision Constrâ—ctionâ— Demonsâ—ration Demonstration of proportion by composition This proposition is the conuerse of the former DemonstratioÌ„â—eâ—aing to an â—mpossibility That which the fift of this booke proued only touching multiplices this proueth generally of all magnitudes ALemma A Corollary Conuersion of proportion This proposition pertaineth to Proportion of equality inordinate proportionality The second difference The third difference Thâ—r proposition pertaineth to Proportion of equality in perturbate proportionality The third difference Proportion of equality in ordinate proportionality Construction Demonstration When there are more then three magnitudes in either order Aâ—CDEâ—GH Proportion of equality in perturbate proprotionalitie Construction Demonstration Note That which the second propositioÌ„ of this booke proued only touching multiplices is here proued generally touching magnitudes An other demonstration of the same affirmatiuely An other demonstration of the same affirmatiuely An other demonstration of the same DemonstratioÌ„ leading to an impossibilitie An other demonstration of the same affirmatiuely Demonstration Demonstrationâ— Demonstration The argument of this sixth booke This booke necessary for the vse of instrumentes of Geometry The first definition The second deâ—inition Reciprocall figures called mutuall figures The third definition The fourth definition The fifth definition An other example Of substraction of proportion The sixth definition Demonstration of the first part Demonstâ—ation of the second part A Corollary added by Flussates The first part of this Theoreme Demonstration of the second part A Corollary added by Flussates Construction Demonstration of the first part Demonstratiâ—n of the second part which is the conuerse of the first Construction Demonstration This is the conuerse of the former proposition Construction Demonstration Constructioâ— The first part of this proposition Demonstration leading to an impossibilitie The second part of this proposition Construction Demonstration Construction Demonstration Construction Demonstration A Corollary out of Flussates By this and the former propoâ—ition may a right line be deuided into what partes soeuer you will. Construction Demonstration An other way after Pelitarius An otâ—eâ— way after Pelitarius Construction Demonstration An other way after Campane Construction Demonstratioâ— A proposition added by Pelitarius The
first part of this proposition Demonstration of the of the same The second part which is the conuerse of the first The first parâ— of this proposition Demonstration of the same The second part which is the conuerse of the first Demonstration of the first part The second part which is the conuerse of the first The first part of thâ—â— Theoreme The second part which is the conuerse of the first A Coâ—ollary Description of the rectiline figure râ—quired Demonstration Demonstration A Corollary The first parâ— of this Theoreme The second part demonstrated The third part The first Corollary The second Corollary Demonstration The first part of this proposition The second part which is the conuerse of the first * Note that this is proued in the assumpt following An Assumpt An other demoÌ„stration of the second part after Flussates An other demonstration after flussates Demonstration of this propositioÌ„ wherein is first proued that the parallegramme EG is like to the whole parallelograÌ„me ABCD. That the parallelograÌ„me KH is like to the whole parallelogramme ABCD That the parallelogrammes EG and KH are like the one to the other An other Demonstration after Flussates An addition of Pelitarius Another addition of Pelitarius Construction Demonstration Demonstration * By the dimetieÌ„t is vnderstand here the dimetient which is â—rawen from the angle which is common to them both to the opposite angle Demonstration leading to an absurditie An other way after Flussates In this propositioÌ„ are two cases in the first the parallelogramme compared to the parallelograÌ„me described of the halfe line is described vpon a line greater theÌ„ the halfe line In the second vpoÌ„ a line lesse The first case where the parellelogramme compared namely AF is described vpon the line AK which is greater then the halfe line AC Demonstration of this case The second case where the parallelogramme compared namely AE is described vpon the line AD which is lesse then the line AC Demonstration of the second case Construction Two cases in this Proposition The first case The second case A Corollary added by Flussates and is put of Theon as an assumpt beâ—ore the 17. proposition of the teÌ„th booke which â—or that it followeth of this proposition I thought it not amisse here to place Construction Demonstration Construction Demoâ—straâ—ion An other way Construction Demonstration The conuerse of the former proposition Demonstration That the angles at thâ— ceÌ„ter are in proportioÌ„ the one to the other as the circumferences wheron they are That the angles at the circumferences are so also That the sectors are so also Construction of the Probleme Demonstartion of the same The first Corollary The second Corollary The third Corollary Demonstration of this proposition Demonstration of this propositions Demonstration of this proposition Demonstration of the first part of this proposition Demonstration of the second part Why Euclide in the middest of his workes was compelled to adde these three bookes of numbers Arithmetike of more excellency then Geometry Things intellectuall of more worthines theâ— things sensible Arithmetike ministreth prinâ—ciples and groundes in a maner to all sciences Boetius Cap. 2. Lib. prim Arithmeti Timaus The argument of the seuenth booke The first definition Without vnity should be confusion of thinges â—oetius in his booke dâ— vnitate vno An other desinition of vnity The second definition Differenâ—e betwene a point and vnity Boetius An other desinition of number Iordane An other definition of numbers Vnity hath in it the vertue and power of all numbers Number considered three maner of wayâ— The third definition The fourth definition The fifth definition The sixth definition Boetius An other definition of an eueÌ„ number Note Pithagoraâ— An other definition An other definition An other definition The seuenth definition An other definition of an od number An other definition The eight definition Campane An other deâ—inition of an eueÌ„ly euen number Flussates An other definition Boetius An other definition The ninth definition Campane An other definition Flussates An other definition An other definition The tenth definition This definition not found in the Greeks An other definition Boeâ—ius defâ—nition of a number euenly eueÌ„ and euenly â—d The eleuenth definition Flusâ—ates An other definition The twelfth definition Prime numbers called incomposed numbers The thirtenth definition The fourtenth definiâ—ion The fiftenth definition The sixtenth definition Two numbers required in multiplication The seuententh definition Why they are called superficiall numbers The eightenth definition Why they are called solid numbers The ninetenth definition Why it is called a square number The twenteth definition Why it is called a cube number The twenty one definition Why the definition of proportionall magnitudes is vnlike to the definitio of proportionall numbers The twenty two definition The twenty three definitioÌ„ Perfect numbers rare of great vse in magike in secret philosophy In what respect a number is perfect Two kinds of imperfect numbers A â—â—mber wanâ—â—ngâ— Common sentences â—irst common seâ—tence â—â—cond â—ommon sentence Third common sentence Fâ—urth common sentence â—iâ—th common sentence Sixth common sentence Seuenth comâ—mon sentence Constrâ—ctioâ— Demonstratiâ—â— leading to an absurditie The conuerse of â—his proposition How to â—now whether two numbers geuen be prime the one to the other Two cases in this probleme The first case The second case DemonstratioÌ„ of the second case That CF is a common measure to the numbers AB and CD That CF is the greatest common measure to AB and CD The second case Two cases in this Proposition The first case The second case This propositioÌ„ and the 6. proposition in discrete quantitie answer to the first of the fifth in continual quantitie Demonstration Construction Demonstration Thiâ— proposition and the next following in discret quaÌ„titie answereth to the fifth propositioÌ„ of the fifth boke in continuall quaÌ„tity Construction Demonstration Constuâ—ction Demonstration An other demonstration after Flussates Construction Demonstration Construction Demonstration This proposition iâ— discret quaÌ„titie answereth to the ninth propâ—sitioÌ„ of the fifth boke in continual quaÌ„titie Demonstration This in discret quaÌ„tity answereth to the twelfe proposition of the fifth in continual quaÌ„tity Demonstration This in discrete quantiâ—y answereth to the sixtenth proposition of the fifth booke in continuall quantitie Note This in discrete quantity anâ—wereth to tâ—â— tweÌ„ty one proposition oâ— the fifth booke in continuall quantitie Demonstration Certaine additions of â—aâ—pane The second case Propâ—rtionality deuided Prâ—portionaliâ—y composed Euerse proportionality The conuersâ— of the same prâ—position Demonstration A Corollary followiâ—g thâ—se propositions adâ—ed by Campaâ—e Coâ—strâ—ctioâ— Demonstration Demonsâ—raâ—ion â—emonstraâ—ion A Corollary added by Flussâ—tes Demonstration This proposition and the former may be extended to numbers how many soeuer The second part of this proposition which is the conuerse of the first Demonstration An assumpt added by Campane This proposition in numbers demonstrateth that which the 17. of the sixth demonstrateth in lines Demonstration The second part which is the conuerse of the
Second part of the first case The second case First part of the secoÌ„d case Second part of the secoÌ„d case Construction Two cases in this Proposition The first case The first part of the first case 〈◊〉 second 〈◊〉 of the 〈◊〉 case The second case A Corollary The first Senary by substraction Demonstration An other demonstration after Campane Diffinition of the eight irrationall line Diffinition of 〈◊〉 â—inth irrationall line An other demonstratioÌ„ after Campane Construction Demonstration Diffinition of the tenth irâ—ationall line Diffinition of the eleueÌ„th irrationall line â—â—â—â—iâ—ition of the twelueth irraâ—ionall line Diffinition of the thirtenth and last irrationall line An Assumpt of Campane I. Dee Though Campanes lemma be true yeâ— the maner of demonstrating it narrowly considered is not artificiall Second Senary Demonstration leading to an impossibilitie Demonstration leading to an absurditie Construction Demonstration leading to an absurditie Demonstration leading to an absurditie DemonstratioÌ„ leading to an impossibilitie Construction Demonstration 〈◊〉 an abjurdâ—tâ—â—â— Sixe kindes of reâ—iduall lines First diffinition Second diffinition Third diffinition Fourth diffinition Fifth diffinition Sixth diffinition Third Senary Construction Demonstratioâ— Construction Demoâ—stratiâ—â— Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstratioâ— An other more redie way to finde out the sixe residuall lines Fourth Senary The â—irst parâ— of the Construction The first part of the demonstration Note AI and FK concluded rational parallelogramme Note DH and FK parallelogrammes mediall Second part of the construction Second part of the demonstration LN is the onely liâ—e â—hat we sought consider First part of the construction The first part of thâ— demonstration AI and FK concluded parallelograÌ„mes mediall DH EK rationall The second part of the construction The second part of the demonstration * Analytically the proâ—e hereof followeth amoÌ„gâ— other thinges The line LN found which is the principall drift of all the former discourse The first part of the Construction The fiâ—st part of the demonstration Note AI and FK mediall Note DH and EK mediall Note AI incommensurable to EK Second part of the ConstructioÌ„ The principall line LN fouÌ„de * Because the lines AF and â—G are proued commensurable in length * By the first oâ— the sixth and tenth of the tenth The first part of the construction The first part of the demonstration Note AK rational Note DK mediall AI and FK incommensurable The second part of the construction The second part of the demonstration LN the chiefe line of this theoreme founde Demonstration The line LN Demonstration The fiueth Senary These sixe propositions following are the conuâ—rses of the sixe former propositions Construction Demonstration * By the 20. of the tenth ** By the 21. of the tenth * By the 22. of the tenth â—F coÌ„cluded a residual line Construction Demonstration CF conclâ—ded a residuall line Construction Demonstration CF concluded a residual line Construction Demonstration CF proued a residuall line CF proued a residuall line Construction Demonstration CF â—roued â— residuall The sixt Senary Construction Demonstration CD coÌ„cluded a residuall line Note Construction Demonstration CD proued a mediall Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Seuenth Senary Constraction Demonstration Construction Demonstration Demoâ—stratioâ— Construction Demonstration on leading to an impossibilitie A Corollary The determination hath sundry partes orderly to be proued Construction Demonstration This is an Assumpt problematicall artificially vsed and demonstrated * Therfore those three lines are in continuall proportion FE concluded a residuall liâ—â— which is sâ—â—what prepâ—â—icroâ—sly in respect oâ— the â—â—der propounded both in the propositioÌ„ and also in the determinatioÌ„ Construction Demonstration Construction Demonsâ—ration Here are the â—ower partes of the propositiâ— more orderly hâ—dled theâ— in the former demöstration Construction Demonsâ—ration An Assumpt An other demonstratioÌ„ after Flussas Construction Demonsâ—ration This is in a maner the conuerse of both the former propositions ioyntly Construction Demonstration Construction Demonstration Demonstration An other demonstration DemonstratioÌ„ leading to an impossibiliâ—ie An other demonstration leading to an impossibiliâ—ie The argument of the eleuenth booke A point the beginning of all quantitie continuall The methode vsed by Euclide in the ten â—â—â—mer booâ—es â—irst boâ—â—e Second â—â—oâ—e Third booâ—e â—ourth bâ—oâ—e â—iueth boâ—â—e Sixth booâ—e Seuenth bookâ— â—ight booâ—â— Ninth booke Tenth booâ—e What is entreaâ—ea of in the fiâ—e booâ—es follâ—wiâ—◠〈◊〉 â—â—â—ular bodiesâ— the â—â—all ende 〈…〉 oâ— I uâ—â—â—â—es â—eomeâ—â—â—all â—â—ementes Coâ—â—aâ—isâ—n â—â— the 〈◊〉 â—â—oâ—e and 〈◊〉 booke 〈◊〉 First difâ—inition A solide the most perfectest quantitie No science of thinges infinite Second diffinition Third diffinition Two difâ—initions included in this diâ—â—inition DeclaratioÌ„ of the first part Declaration of the second part Fourth diffinition Fifth diffinition Sixth diffinition Seuenth defâ—inition Eighth diâ—finition Ninth diâ—â—iâ—ition Tenth diffinition Eleuenth diffinition An other diffinition of a prisme which is a speciall diffinition of a prisme as it is commonly called and vsed This bodie called Figura Serratilis Psellus Twelueth diffinition What is to be taâ—â—n heede of in the diffinition of a sphere geuen by Iohannes de Sacro Busco Theodosiuâ— diâ—â—inition of a sphere The circumference of a sphere Galens diffinition 〈◊〉 a sphâ—râ— The digâ—itie of a sâ—here A sphere called a Globe Thirtenth diffinition Theodosius diffinition of the axe of a sphere Fourtenth diffinition Theodosius diffinition of the center of a sphere Flussas diffinition of the center of a sphere Fiuetenth diffinition Difference betwene the diameter axe of a sphere Seuententh diffinition First kinde of Cones A Cone called of Campane a roâ—â—de Piramis Seuententh diffinition A conicall superficies Eightenth diffinition Ninetenth diffinition A cillindricall superficies Corollary A roundâ— Columne or sphere A Corollary added by Campane Twenty diffinition Twenty one diffinitioâ— Twenty two diffinition A Tetrahedron one of the fiue regular bodyes Diâ—â—erence betwene a Tetrahedron and a Piramis Psellus calleth a Tetrahedron a Piramis Twenty three definition TweÌ„ty â—oâ—er definition Twenty fiue diffinition Fiue regular bodies The dignity of these bodies A Tetrahedron ascribed vnto the fire An octohedron ascribed vnto the ayre An Ikosahedron assigned vnto the water A cube assigned vnto the earth A dodecahedron assigned to heauen Diffinition of a parallelipipedon A Dâ—dâ—â—â—â—edron An Icosaâ—edron Demonstration leading to an impossibilitie An other demonstration after Flussas Construction Demonstration leading to an impossibilitie Demonstration leading to an impossibilitie Construction Demonstration Demonstration leading to an impossibilitie Construction * An Assumpt as M. Dee prâ—ueth it Demonstration Demonstration leading to an impossibilitie This proposition is as it were the conuerse of the sixth Construction Demonstration Construction Demonstration Construction Demonstration Construction Two cases in this proposition The first case Iohn Dee * This requireth the imagination of a plaine superficies passing by the pointe A and the straight line BC. And so helpe your selfe in the lyke cases either Mathematically imagining or Mechanically practising Second