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
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
with the altitudes of the sayd pyramids A and B shall be equall by the 6. of this booke Wherefore by the first part of this proposition the bases of the pyramids C to D are reciprokall with the altitudes of D to C. But in what proportion are the bases C to D in the same are the bases A to B forasmuch as they are equall And in what proportion are the altitudes of D to C in the same are the altitudes of B to A which altitudes are likewise equall Wherefore by the 11. of the fifth in what proportion the bases A to B are in the same reciprokally are the altitudes of the pyramids B to A. In like sort by the second part of this proposition may be proued the conuerse of this corollary The same thing followeth also in a Prisme and in a sided columne as hath before at large bene declared in the corollary of the 40. proposition of the 11. booke For those solides are in proportioÌ„ the one to the other as the pyramids or parallelipipedons for they are either partes of equemultiplices or equemultiplices to partes The 10. Theoreme The 10. Proposition Euery cone is the third part of a cilinder hauing one and the selfe same base and one and the selfe same altitude with it SVppose that there be a cone hauing to his base the circle ABCD and let there be a cilinder hauing the selfe same base and also the same altitude that the cone hath Then I say that the cone is the third part of the cilinder that is that the cilinder is in treble proportion to the cone For if the cilinder be not in treble proportion to the cone then the cilinder is either in greater proportions then triple to the cone or els in lesse First let it be in greater then triple And describe by the 6. of the fourth in the circle ABCD a square ABCD. Now the square ABCD is greater then the halfe of the circle ABCD For if about the circle ABCD we describe a square the square described in the circle ABCD is the halfe of the square described about the circle And let there be Parallelipipedon prismes described vpon those squares equall in altitude with the cilinder But prismes are in that proportion the one to the other that their bases are by the 32. of the eleuenth and 5. Corollary of the 7. of this booke Wherefore the prisme described vpon the square ABCD is the halfe of the prisme described vpon the square that is described about the circle Now the clinder is lesse then the prisme which is made of the square described abouâ— the circle ABCD being equal in altitude with it for it contayneth it Wherfore the prisme described vpon the square ABCD and being equall in altitude with the cylinder is greater then half the cylinder Deuide by the 30. of the third the circumferences AB BC CD and DA into two equall parts in the points E F G H And draw these right lines AE EB BF FC CG GD DH HA. Wherfore euery one of these triangles AEB BFC CGD and DHA is greater then halfe of that segment of the circle ABCD which is described about it as we haue before in the 2. proposition declared Describe vpon euery one of these triangles AEB BFC CGD and DHA a prisme of equall altitude with the cylinder Wherefore euery one of these prismes so described is greater then the halfe part of the segment of the cylinder that is set vpon the sayd segments of the circle For if by the pointes E F G H be drawen parallell lines to the lines AB BC CD and DA and then be made perfect the parallelogrammes made by those parallell lines and moreouer vpon those parallelograÌ„mes be erected parallelipipedons equall in altitude with the cylinder the prismes which are described vpon eche of the triangles AEB BFC CGD and DHA are the halfes of euery one of those parallelipipedons And the segments of the cylinder are lesse then those parallelipipedons so described Wherefore also euery one of the prismes which are described vpon the triangles AEB BFC CGD and DHA is greater then the halfe of the segment of the cylinder set vpon the sayd segment Now therefore deuiding euery one of the circumferences remaining into two equall partes and drawing right lines and raysing vp vpon euery one of these triangles prismes equall in altitude with the cylinder and doing this continually we shall at the length by the first of the tenth leaue certaine segments of the cylinder which shal be lesse then the excesse whereby the cylinder excedeth the cone more then thrise Let those segments be AE EB BF FC CG GD DH and HA. Wherfore the prisme remayning whose base is the poligonon â—igure AEBFCGDH and altitude the selfe same that the cylinder hath is greater then the cone taken three tymes But the prisme whose base is the poligonon figure AEBFCGDH and altitude the selfe same that the cylinder hath is treble to the pyramis whose base is the poligonon figure AEBFCGDA and altitude the selfe same that the cone hath by the corollary of the 3. of this booke Wherfore also the pyramis whose base is the poligonon figure AEBFCGDH and toppe the self same that the cone hath is greater then the cone which hath to his base the circle ABCD. But it is also lesse for it is contayned of it which is impossible Wherefore the cylinder is not in greater proportion then triple to the cone I say moreouer that the cylinder is not in lesse proportion then triple to the coneâ— For if it be possible let the cylinder be in lesse proportion then triple to the cone Wherefore by conuersion the cone is greater then the third part of the cylinder Describe now by the sixth of the fourth in the circle ABCD a square ABCD. Wherefore the square ABCD is greater then the halfe of the circle ABCD vpon the square ABCD describe a pyramis hauing one the selfe same altitude with the cone Wherfore the pyramis so described is greater theÌ„ halfe of the cone For if as we haue before declared we describe a square about the circle the square ABCD is the halfe of the square described about the circle and if vppon the squares be described parallelipipedons equall in altitude with the cone which solides are also called prismes the prisme or parallelipipedon described vpoÌ„ the square ABCD is the halfe of the prisme which is described vpoÌ„ the square described about the circle for they are the one to the other in that proportioÌ„ that their bases are by the 32. of the eleueÌ„th 5. corollary of the 7. of this booke Wherfore also their third parts are in the self same proportion by the 15. of the fift Wherfore the pyramis whose base is the square ABCD is the halfe of the pyramis set vpon the square described about the circle But the pyramis set vpon the square described about the circle is greater then the cone whome
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
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
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
Demonstration * The circles so made or so considered in the sphere are called the greatest circles All other not hauing the center of the sphere to be their center alsoâ— are called lesse circles Note these descriptions * An other Corollary * An other Corollary Construction * This is also proued in the Asâ—umpt before added out oâ— Flussas Note what a greater or greatest circle in a Spere is First part of the Construction Noteâ— * You know full well that in the superficies of the sphere â—â—â—ly the circumferences of the circles are but by thâ—se circumferences the limitatioÌ„ and assigning of circles is vsed and so the circumference of a circle vsually called a circle which in this place can not offend This figure is restored by M. Dee his diligence For in the greeke and Latine Euclides the line GL the line AG and the line KZ in which three lynes the chiefe pinch of both the demonstrations doth stand are vntruely drawen as by comparing the studious may perceaue Note You must imagine 〈◊〉 right line AX to be perpeÌ„dicular vpon the diameters BD and CE though here AC the semidiater seme to be part of AX. And so in other pointes in this figure and many other strengthen your imagination according to the tenor of constructions though in the delineatioÌ„ in plaine sense be not satisfied Note BO equall to BK in respect of M. Dee his demonstration following †Note â—his point Z that you may the better vnderstand M. Dee his demoÌ„stration Second part of the construction Second part of the demonstration ✚ Which of necessity shall fall vpon Z as M. Dee proueth it and his profe is set after at this marke ✚ following I. Dee * But AZ is greater theÌ„ AG as in the former propositioÌ„ KM was euident to be greater then KG so may it also be made manifest that KZ doth neyther touch nor cut the circle FGâ—H An other proue that the line AY is greater theÌ„ the line AG. * This as an assumpt is presently proued Two cases in this proposition The first case Demonstration leading to an impossibilitie Second case * As it is â—asiâ— to gather by the â—â—â—umpt put after the secoâ—â— of this booâ—â— Note a generall rule The second part of the Probleme two wayes executed An vpright Cone The second part of the Probleme The second â—aâ—â— oâ— the â—robleme ☜ * This may easely be demonstrated as in thâ— 17. proposition the section of a sphere was proued to be a circle * For taking away all doubt this aâ— a Lemma afterward is demâ—strated A Lemma as it were presently demonstrated Construction Demonstration The second part of the Probleme * Construction Demonstration An other way of executing this probleme The conuerse of the assuÌ„pt A great error commonly maintained Betwene straight and croked all maner of proportioÌ„ may be geuen Construction Demonstration The diffiniâ—ioÌ„ of a circâ—e â—â—apâ—â—d in a spâ—erâ— Construction Demonstration This is manifest if you consider the two triangles rectangles HOM and HON and then with all vse the 47. of the first of Euclide Construction DemonstratioÌ„ Construction DemonstratioÌ„ This in maner of a Lemmâ— is presently proued Note here of Axe base soliditie more then I nede to bring any farther proofe for Note * I say halfe a circâ—lar reuolution for that suâ—â—iseth in the whole diameter ST to describe a circle by iâ— it be moued â—â—out his center Q c. Lib 2 prop 2. de Spheâ—a Cylindrâ— Note * A rectangle parallelipipedon geuâ—n equall to a Sphere geuen To a Sphere or to any part of a Sphere assigned as a third fourth fifth c to geue a parallelipipedon equall Sided Columes Pyramids and prismes to be geuen equall to a Sphere or to any certayne part thereof To a Sphere or any segment or sector of the same to geue a cone or cylinder equall or in any proportion assigned Farther vse of Sphericall Geometrie The argument of the thirtenth booke Construction Demonstration * The AssuÌ„pt proued * Because AC is supposed greater then AD therefore his residue is lesse then the residue of AD by the common sentence Wherefore by the supposition DB is greater then â—C The chieâ—e line in all Euclides Geometrie What is ment here by A section in one onely poiâ—t Construction Demonstration * Note how CE and the gnonom XOP are proued equall for it serueth in the conuerse demonstrated by M. Dee here next after This proposition â—the conuerse of the former * As we haâ—e noted the place of the peculier proâ—e there â—in the demoÌ„stration of the 3. * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated Construction * Though I say perpeÌ„dicular yes you may perceue how infinite other pâ—sâ—â—ioÌ„s will serue so that DI and AD make an angle for a triangle to haue his sides proportionally cut c. Demonstration Demonstration I. Dee This is most euident of my second Theoreme added to the third propositioÌ„ For to adde to a whole line a line equall to the greater segmeÌ„t to adde to the greater segment a line equall to the whole line is all one thing in the line produced By the whole line I meane the line diuided by extreme and meane proportion This is before demonstrated most euidently and briefly by M. Dee after the 3. proposition Note Note 4. Proportional lines Note two middle proportionals Note 4. wayes of progresâ—ion in the proportion of a line deuided by extreme and middle proportion What resolution and composition is hath before bene taught in the beginning of the first booke * Proclus in the Greeke in the 58. page Construction Demonstration Two cases in this proposition Construction Thâ— first case Demonstration The second case Construction Demonstration Construction Demonstration This Corollary is the 3. proposition of the â—4 booke after Campane Demonstration of the first part Demonstration of the second part Construction Demâ—nstration Construstion Demonstration Constrâ—yction Demonstration This Corollary is the 11. propâ—sition of the 14. booke after Campane This Corollary is the 3. Corollary after the 17. proposition of the 14 booke after Campane * By the name oâ— a Pyramis both here iâ— this booke following vnderstand a Tetrahedron An other construction and demonstration of the second part after Fâ—ussas Third part of the demonstration This Corollary is the 15. proposition of the 14. booke after Campane This Corollary Campane putteth as a Corollary after
the 17. proposition of the 14. booke Construction â—rist part of the demonstration * For the 4. angles at the point K are equall to fower right angles by the Corollary of tâ—e 15. of the first and those 4. angle are equall the one to the other by the â— of the â—irst and theâ—efore ech is a right angle Second part of the demoÌ„stratioÌ„ * For the square of the line AB which is proued equall to the square of the line LM is double to the square of the line BD which is also equall to the square of the line LE. This Corollary is the 16. proposition of the 14. booke after Campane First part of the demonstration Second part of the ConstructioÌ„ Second part of the demonstraâ—ion * By the 2. Assumpt of the 13. of this booke Third part of the demonstration Second part of the demoÌ„stratioÌ„ For the line QW is equall to the line IZ the line ZW is commoÌ„ to them both This part is againe afterward demonstrated by Flussas The pentagon VBWCR proued to be in one and the selfe same playne superficies The pentagon VBWCZ it proued equiangle * Looke for a farther construction after Flussas at the ende of the demonstration That the side of the dodecahedron is a residuall line Draw in the former figure these lines ctA ctL câ—D The side of a pyramis The side of a cube The sides of a dodecahedron Comparison of the fiue sides of the foresayd bodies An other demoÌ„monstration to proue that the side of the IcosahedroÌ„ is greater then the side of the dodecahedron That 3. squares of the line FB are greater theÌ„ 6. squares of the line NB. That there can be no other solide besids these fiue contayned vnder equilater and equiangle bases That the angle of an equilater and equiangle Pentagon is one right angle and a 〈◊〉 part 〈◊〉 which thing was also before proued in the 〈◊〉 of the 32. of the â—irst The sides of the angle of the inclâ—â—â—tion of the 〈◊〉 of the 〈◊〉 are 〈◊〉 rationall The sides of the angle of the inclination of the 〈◊〉 â—f tâ—e 〈…〉 That the plaines of an octohedron are in liâ—e sort inclined That the plaines of an Icosahedron are in like sort inclined That the plaines of a Dâ—â—â—â—â—hedron are 〈◊〉 like sort inclined The sides of the angle of the inclination of the supeâ—ficieces of the TetrahedroÌ„ are proued rationall The sides of the angle of the inclination of the superficieces of the cube proued rationall The sides of thâ— angle c. of the octohedron proued rationall The sides of the angle c. of the Icosahedron proued irratioâ—all How to know whether the angle of the inclinatioÌ„ be a right angle an acute angle or an oblique angle The argument of the fourtenth booke First proposition after Flussas Construction Demonstration * This is manifest by the 12. propositioÌ„ of the thirtenh booke as Campane well gathereth in a Corollary of the same The 4. pâ—posâ—tioÌ„ after Flussas * This is afterward proued in the 4. proposition This Assumpt is the 3. proposition after Flussas Construction of the Assumpt Demonstration of the Assumpt Construction of the proposition Demonstration of the â—â—opâ—sition * Thâ— 5. proposition aâ—tâ—r 〈◊〉 Construction Demoâ—stration The 5. proposition aâ—ter Fâ—ussas Demonstration * This is the reason of the Corollary following A Corollary which also Flussas putteth as a Corollary after the 5. proposition in his order The 6. pâ—â—positioÌ„â—â—ter Flussas Coâ—struction Demonstration * This is not hard to proue by the 15. 16. and 19. of the â—â—â—eth †In the Corollary of the 17. of the tâ—irteÌ„th * 〈◊〉 againe is required the Assumpt which is afterward proued in this 4 proposition †But first the Assumpt following the construction whâ—reâ—f here beginneâ—h is to be proued The Assumpt which also Flussas putteth as an Assumpt aâ—ter the 6. propositioÌ„ Demonstration of the Assumpt Construction pertaining to the second dâ—monstration of the 4. propositioÌ„ Second demonstration oâ— the 4. proposition The 7â— proposition after Flussas Construction Demonstration * Here againe is required the Assumpt afterward proued in this 4. proposition †As may by the Assumpt afterward in this propositioÌ„ be plainely proued The 8. proâ—ition aâ—ter Flussas †By the Corollary added by Flussas after has Assumpt put after the 17. proposition of the 12. booke Corollary of the 8. after Flussas This Assumpt is the 3. propositioÌ„ aâ—ter â—lussas and is it which 〈◊〉 times hath bene taken aâ— gâ—aunted in this booke and oâ—ce also in the last proposition of the 13. booke as we haue beâ—ore noted Demonstration * In the 4. section â—f this proposition †In the 1. and 3 section of the same propositioÌ„ †In the 5. sectioÌ„ of the same proposition A Corollary The first proposition after Campane Construction Demonstration The 2. proposition after Campane Demonstration leading to an impossibilitie The 4. proposition after Campane Construâ—tâ—on Demonstration This Corollary Campane also â—utteth after the 4. proposition in his order The 5. proposition after Campane Construction Demonstration This is the 6. and 7. propositions after Campane Construction Demonsâ—ration This Corollary Campane also addeth after the 7. proposition iâ— his order The 5. proposition aâ—ter Campane Construction Demonstration This Assumpt Campane also hath after the 8. proposition in his order Construction Demonstration The 9. proposition after Campane Construction Demonstration This Campane putteth aâ— a Corollary in the 9. proposition after his order This Corollary is the 9. proposition after Campane The 12. proposition after Campane Construction Demonstration The 13. proposition after Campane The 14. proposition after Campane Demonstration of the first part Demonstration of the second part The 17. proposition after Campane Firâ—t part of the construction First part of the Demonstration Second parâ— of the câ—nstruction Second part of the Dâ—monstration The 18. proposition after Campane Demonstration of the first part Demonstration of the second part The Corollary of the 8. proposition after Campane Demâ—nstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration The argument of the 15. booke †In this proposition as also in all the other following by the name of a pyramis vnderstand a tetrahedron as I haue before admonished Construction Demonstration Construction Demonstration Construction Demonstrâ—tion Construction Demonstration Construction Demonstration That which here followeth concerning the inclination of the plaines of the fiue solides was before tought â—hough not altogether after the same maner out of Flussas in the latter â—nde of the 13 booke Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration First part of the construction First part of the demonsâ—ration Second part of the construction Second part of the Demonstration Third part of the construction Third part of the demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Produce in the figure the line TF to the point B. Construction Demonstration This proposition Campane hath is the last also in order of the 15. booke with him The argument of the 16. booke Construction Demonstration * By a Pyramis vnderstand a Tetrahedron throughout all this booke Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration †That is aâ— 18. to 1. Demonstration †That iâ— as 9. to 2. * That is as 18. to 2. or 9. to 1. Draw in the figure a line from B to H. * What the duple of an extreme and meane proportion is Construction Demonstration Demonstration Constrution Demonstration Construction Demonstration Demonstration * That is at 13. 1 â— is to â— Demonstration Demonstration Construction Demonstration Construction Demonstration Extend in the figure a line â—rom the point E to the point B. Extend in the figure a line from the point E to the point B. Demonstration Construction Demonstration Second part of the Demonstration Icosidodecahedron ExoctohedroÌ„ That the â—xoctohedron is contayned in a sphere That the exoctohedâ—on is contayned in the sphere geuen That the diaâ—â—â—ter of the sâ—here is doâ—ble to the side â—f the exoctohedron That the Icosidodecahedron is contayned in the sphere geuen * That is as 8. 103 â— Faultes escaped â—cl â—ag Line Faultesâ— Co 〈◊〉 〈◊〉    Errata Lib. 1.  1 2 41 point B. at Campane point C aâ— Campane 3 1 22 aâ—l lines drawne all righ◠〈…〉 3 1 28 lines drawen to the superficies right lines draweÌ„ to the circumference 9 1 42 liâ—es AB and AC lines AB and BC 15 1 35 are equall are proued equall 20 2 28 by the first by the fourth 21 1 39 tâ—e centre C. the centre E 2â— 2 â— Iâ—â—ower right If two right 25 2 3 fâ—â—â— petition fiueth petition 49 2 7 14. â—â— 32.64 c. 4.8.16.32.64 53 1 39 the triangle NG the triangle Kâ— 54 2 25 by the 44 by the 42 57 2 23 and Câ—G in the and CGB is thâ— Â Â Â In stede of â—lussates through out 〈◊〉 whole booke read â—lusâ—as    Errata Lib. 2.  60 2 29 Gnomon FGEH Gnomon AHKD   30 Gnomon EHFG Gnomon â—CKD 69 1 18 the whole line the whole â—igure 76 2 9 the diameter CD the diameter AHF    Errata Lib. 3.  82 2 36 angle equall to the angle 92 1 last the line AC is the line AF 〈◊〉    Errata Lib. 4.  110 2 10 CD toucheth the ED toucheth the   12 side of the other angle of the other 115 1 21 and HB and HE 117 2 44 the angle ACD the angle ACB 118 1 2 into ten equall into two equall 121 1 3â— CD and EA CD DE EA â—â—â— 1 29 the first the third    Errata Lib. 5.  126 1 43 it maketh 12. more then 17. by 5. it maketh 24. more then 17. by 7. 129 1  In stede of the figure of the 6. definition draw in the magâ—â— a figure like vnto thâ—s 134 2 4 As AB is to A so is CD to C As AB is to B so is CD to D 141 2 last But if K excede M But if H excede M LIEFE IS DEATHE AND DEATH IS LIEFE AETATIS SVAE XXXX AT LONDON Printed by Iohn Daye dwelling ouer Aldersgate beneath Saint Martins ¶ These Bookes are to be solde at his shop vnder the gate 1570.
the Lengthes of dayes and nightes the Houres and times both night and day are knowne with very many other pleasant and necessary vses Wherof some are knowne but better remaine for such to know and vse who of a sparke of true fire can make a wonderfull bonfire by applying of due matter duely Of Astrologie here I make an Arâ—e seuerall from Astronomie â— not by new deuise but by good reason and authoritie for Astrologie is an Arte Mathematicall which reasonably demonstrateth the operations and effectes of the naturall beames of light and secrete influence of the Sterres and Planets in euery element and elementall body at all times in any Horizon assigned This Arte is furnished with many other great Artes and experiences As with perfecte Perspectiue Astronomie Cosmographie Naturall Philosophie of the 4. Elementes the Arte of Graduation and some good vnderstaÌ„ding in Musike and yet moreouer with an other great Arte hereafter following though I here set this before for some considerations me mouing Sufficient you see is the stuffe to make this rare and secrete Arte of and hard enough to frame to the Conclusion Syllogisticall Yet both the manifolde and continuall trauailes of the most auncient and wise Philosophers for the atteyning of this Arte and by examples of effectes to confirme the same hath left vnto vs sufficient proufe and witnesse and we also daily may perceaue That mans body and all other Elementall bodies are altered disposed ordred pleasured and displeasured by the Influentiall working of the Sunne Mone and the other Starres and Planets And therfore sayth Aristotle in the first of his Meteorologicall bookes in the second Chapter Est autem necessariò Mundus iste supernis lationibus ferè continuus Vt inde vis eius vniuersa regatur Ea siquidem Causa prima putanda omnibus est vnde motus principium existit That is This Elementall World is of necessitie almost next adioyning to the heauenly motions That from thence all his vertue or force may be gouerned For that is to be thought the first Cause vnto all from which the beginning of motion is And againe in the tenth Chapter Opâ—rtet igitur horum principia sumamus causas omnium similiter Principium igitur vy mouens praecipuumque omnium primum Circulus ille est in quo manifeste Solis latio c. And so forth His Meteorologicall bookes are full of argumentes and effectuall demonstrations of the vertue operation and power of the heauenly bodies in and vpon the fower Elementes and other bodies of them either perfectly or vnperfectly composed And in his second booke De Generatione Corruptione in the tenth Chapter Quocirca prima latiâ— Orâ—us Interitus causa non est Sed obliqui Circuli latio ea namque continua est duobus motibus fit In Englishe thus Wherefore the vppermost motion is not the cause of Generation and Corruption but the motion of the Zodiake for that both is continuall and is caused of two mouinges And in his second booke and second Chapter of hys Physikes Homo namque generat hominem atque Sol. For Man sayth he and the Sonne are cause of mans generation Authorities may be brought very many both of 1000. 2000. yea and 3000. yeares Antiquitie of great Philosophers Expert Wise and godly men for that Conclusion which daily and hourely we men may discerne and perceaue by sense and reason All beastes do feele and simply shew by their actions and passions outward and inward All Plants Herbes Trees Flowers and Fruites And finally the Elementes and all thinges of the Elementes composed do geue Testimonie as Aristotle sayd that theyr Whole Dispositions vertues and naturall motions depend of the Actiuitie of the heauenly motions and Influences Whereby beside the specificall order and forme due to euery seede and beside the Nature propre to the Indiuiduall Matrix of the thing produced What shall be the heauenly Impression the perfect and circumspecte Astrologien hath to Conclude Not onely by Apotelesmes 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 but by Naturall and Mathematicall demonstration 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Whereunto what Sciences are requisâ—te without exception I partly haue here warned And in my Bropâ—deâ—â—â—â— besides other matter there disclosed I haue Mathematically furnished vp the whole Method To this our age not so carefully handled by any that euer I saw or heard of I was for â—1 yeares ago by certaine earnest disputations of the Learned Gâ—â—ardus Mâ—rcâ—tâ—â— and 〈◊〉 Gogaâ—a and other therto so prouoked and by my constant and inuincible zeale to the veritie in obseruations of Heauenly Influencies to the Minâ—te of time than so diligent And chiefly by the Supernaturall influence from the Starre of Iacob so directed That any Modest and Sober Student carefully and diligently selâ—ing for the Truth will both finde coÌ„fesse therin to be the Veritie of these my wordes And also become a Reasonable Reformer of three Sortes of people about these Influentiall Operations greatly erring from the truth Wherof the one is Light Beleuers the other Light Despisers and the third Light Practisers The first most coÌ„mon Sort thinke the Heauen and Sterres to be answerable to any their doutes or desires which is not so and in dede they to much ouer reache The Second sorte thinke no Influentiall vertue froÌ„ the heauenly bodies to beare any Sway in Generation and Corruption in this Elementall world And to the Sunne Mone and Sterres being so many so pure so bright so wonderfull bigge so farre in distance so manifold in their motions so constant in their periodes c. they assigne a sleight simple office or two and so allow vnto theÌ„ according to their capacities as much vertue and power Influentiall as to the Signe of the Sunne Mone and seuen Sterres hanged vp for Signes in London for distinction of houses such grosse helpes in our wordly affaires And they vnderstand not or will not vnderstand of the other workinges and vertues of the Heauenly Sunne Mone and Sterres not so much as the Mariner or Husband manâ— no not so much as the Elephant doth as the Cynocephalus as the Porâ—entine doâ—h nor will allow these perfect and incorruptible mighty bodies so much vertuall Radiation Force as they see in a litle peece of a Magnes stone which at great distance sheweth his operation And perchaunce they thinke the Sea Riuers as the Thames to be some quicke thing and sâ— to ebbe aâ—d slow run in and out of them selues at â—heiâ— owne fantasies God helpe God helpe Surely these men come to short and either are to dull or willfully blind or perhaps to malicious The third man is the common and vulgare Astrologien or Practiser who being not duely artificially and perfectly furnished yet either for vaine glory or gayne or like a simple dolt blinde Bayardâ— both in matter and maner erreth to the discredit of the Wary and modest Astrologien and to the robbing of those most noble corporall
be wood Copper Tinne Lead Siluer c. being as I sayd of like nature condition and like waight throughout And you may by Say Balance haue prepared a great number of the smallest waightes which by those Balance can be discerned or tryed and so haue proceded to make you a perfect Pyle company Number of waightes to the waight of six eight or twelue pound waight most diligently tryed all And of euery one the Content knowen in your least waight that is wayable They that can not haue these waightes of precisenes may by Sand Vniforme and well dusted make them a number of waightes somewhat nere precisenes by halfing euer the Sand they shall at length come to a least common waight Therein I leaue the farder matter to their discretion whom nede shall pinche The Venetians consideration of waight may seme precise enough by eight descentes progressionall halfing from a grayne Your Cube Sphaere apt Balance and conuenient waightes being ready fall to worke â— First way your Cube Note the Number of the waight Way after that your Sphaere Note likewise the NuÌ„ber of the waight If you now find the waight of your Cube to be to the waight of the Sphaere as 21. is to 11 Then you see how the Mechanicien and Experimenter without Geometrie and Demonstration are as nerely in effect tought the proportion of the Cube to the Sphere as I haue demonstrated it in the end of the twelfth boke of Euclide Often try with the same Cube and Sphaere Then chaunge your Sphaere and Cube to an other matter or to an other bignes till you haue made a perfect vniuersall Experience of it Possible it is that you shall wynne to nerer termes in the proportion When you haue found this one certaine Drop of Naturall veritie procede on to Inferre and duely to make assay of matter depending As bycause it is well demonstrated that a Cylinder whose heith and Diameter of his base is aequall to the Diameter of the Sphaere is Sesquialter to the same Sphaere that is as 3. to 2 To the number of the waight of the Sphaere adde halfe so much as it is and so haue you the number of the waight of that Cylinder Which is also Comprehended of our former Cube So that the base of that Cylinder is a Circle described in the Square which is the base of our Cube But the Cube and the Cylinder being both of one heith haue their Bases in the same proportion in the which they are one to an other in their Massines or Soliditie But before we haue two numbers expressing their Massines Solidities and Quantities by waight wherfore we haue the proportion of the Square to the Circle inscribed in the same Square And so are we fallen into the knowledge sensible and Experimentall of Archimedes great Secret of him by great trauaile of minde sought and found Wherfore to any Circle giuen you can giue a Square aequall as I haue taught in my Annotation vpon the first proposition of the twelfth boke And likewise to any Square giuen you may giue a Circle aequall If you describe a Circle which shall be in that proportion to your Circle inscribed as the Square is to the same Circle This you may do by my Annotations vpon the second proposition of the twelfth boke of Euclide in my third Probleme there Your diligence may come to a proportion of the Square to the Circle inscribed nerer the truth then is the proportion of 14. to 11. And consider that you may begyn at the Circle and Square and so come to conclude of the Sphaere the Cube what their proportion is as now you came from the Sphâ—ere to the Circle For of Siluer or Gold or Latton Lamyns or plates thorough one hole draweÌ„ as the maner is if you make a Square figureâ— way it and then describing theron the Circle inscribed cut of file away precisely to the Circle the ouerplus of the Square you shall then waying your Circle see whether the waight of the Square be to your Circle as 14. to 11. As I haue Noted in the beginning of Euclides twelfth boke c. after this resort to my last proposition vpon the last of the twelfth And there helpe your selfe to the end And here Note this by the way That we may Square the Circle without hauing knowledge of the proportion of the Circumference to the Diameter as you haue here perceiued And otherwayes also I can demonstrate it So that many haue cumberd them selues superfluously by trauailing in that point first which was not of necessitie first and also very intricate And easily you may and that diuersly come to the knowledge of the Circumference the Circles Quantitie being first knowen Which thing I leaue to your consideration making hast to despatch an other Magistrall Probleme and to bring it nerer to your knowledge and readier dealing with then the world before this day had it for you that I can tell of And that is A Mechanicall Dubblyng of the Cube c. Which may thus be done Make of Copper plates or Tyn plates a foursquare vpright Pyramis or a Cone perfectly fashioned in the holow within Wherin let great diligence be vsed to approche as nere as may be to the Mathematicall perfection of those figures At their bases let them be all open euery where els most close and iust to From the vertex to the Circumference of the base of the Cone to the sides of the base of the Pyramis Let 4. straight lines be drawen in the inside of the Cone and Pyramis makyng at their fall on the perimeters of the bases equall angles on both sides them selues with the sayd perimeters These 4. lines in the Pyramis and as many in the Cone diuide one in 12. aequall partes and an other in 24. an other in 60 and an other in 100. reckenyng vp from the vertex Or vse other numbers of diuision as experience shall reach youâ— Then set your Cone or Pyramis with the vertex downward perpendicularly in respect of the Base Though it be otherwayes it hindreth nothyng So let theÌ„ most stedily be stayed Now if there be a Cube which you wold haue Dubbled Make you a prety Cube of Copper Siluer Lead Tynne Wood Stone or Bone. Or els make a hollow Cube or Cubiâ— coffen of Copper Siluer Tynne or Wood c. These you may so proportioÌ„ in respect of your Pyramis or Cone that the Pyramis or Cone will be hable to conteine the waight of them in waâ—â— 3. or 4. times at the least what stuff so euer they be made ofâ— Let not your Solid angle at the vertex be to sharpe but that the water may come with ease to the very vertex of your hollow Cone or Pyramis Put one of your Solid Cubes in a Balance apt take the waight therof exactly in water Powre that water without losse into the hollow Pyramis or Cone quietly
coÌ„pareth them all with Triangles also together the one with the other In it also is taught how a figure of any forme may be chaunged into a Figure of an other forme And for that it entreateth of these most common and generall thynges thys booke is more vniuersall then is the seconde third or any other and therefore iustly occupieth the first place in order as that without which the other bookes of Eâ—clide which follow and also the workes of others which haue written in Geometry cannot be perceaued nor vnderstanded And forasmuch â—s all the demonstrations and proofes of all the propositions in this whole booke depende of these groundes and principles following which by reason of their playnnes neede no greate declaration yet to remoue all be it neuer so litle obscuritie there are here set certayne shorte and manifesâ— expositions of them Definitions 1. A signe or point is that which hath no part The better to vnderstand what manâ—r of thing a signe or point is ye must note that the nature and propertie of quantitie wherof Geometry entreateth is to be deuided so that whatsoeuer may be deuided into sundâ—y partes is called quantitie But a point although it pertayne to quantitie and hath his beyng in quantitie yet is it no quantitie for that it cannot be deuided Because as the definition saith it hath no partes into which it should be deuided So that a pointe is the least thing that by minde and vnderstanding can be imagined and conceyued then which there can be nothing lesse as the point A in the margent A signe or point is of Pithagoras Scholers after this manner defined A poynt is an vnitie which hath position NuÌ„bers are conceaued in mynde without any forme figure and therfore without matter wheron to 〈◊〉 figure consequently without place and position Wherfore vnitie beyng a part of number hath no position or determinate place Wherby it is manifest that â—umbâ—â— iâ— more simple and pure then is magnitude and also immateriall and so vnity which iâ— the bâ—ginning of number is lesse materiall then a â—igne or poyâ—â— which is the beginnyng of magnitude For a poynt is maâ—eriall and requireth position and place and â—â—â—rby differeth from vnitie â— A line is length â—ithout breadth There pertaine to quantiâ—â—e three dimensions length bredth thicknes or depth and by these thre are all quaÌ„titieâ— measured made known There are also according to these three dimensions three kyndes of continuall quantities a lyne a superficies or plaine and a body The first kynde namely a line is here defined in these wordes A lyne is length without breadth A point for that it is no quantitie nor hath any partes into which it may be deuided but remaineth indiuisible hath not nor can haue any of these three dimensions It neither hath length breadth nor thickenes But to a line which is the first kynde of quantitie is attributed the first dimension namely length and onely that for it hath neither breadth nor thicknes but is conceaued to be drawne in length onely and by it it may be deuided into partes as many as ye list equall or vnequall But as touching breadth it remaineth indiuisible As the lyne AB which is onely drawen in length may be deuided in the pointe C equally or in the point D vnequally and so into as many partes as ye list There are also of diuers other geuen other definitions of a lyne as these which follow A lyne is the mouyng of a poynte as the motion or draught of a pinne or a penne to your sence maketh a lyne Agayne A lyne is a magnitude hauing one onely space or dimension namely length wantyng breadth and thicâ—â—es 3 The endes or limites of a lyne are pointes For a line hath his beginning from a point and likewise endeth in a point so that by this also it is manifest that pointes for their simplicitie and lacke of composition are neither quantitie nor partes of quantitie but only the termes and endes of quantitie As the pointes A B are onely the endes of the line AB and no partes thereof And herein differeth a poynte in quantitie from vnitie in numberâ— for that although vnitie be the beginning of nombers and no number as a point is the beginning of quantitie and no quantitie yet is vnitie a part of number For number is nothyng els but a collection of vnities and therfore may be deuided into them as into his partes But a point is no part of quantitie or of a lyneâ— neither is a lyne composed of pointes as number is of vnities For things indiuisible being neuer so many added together can neuer make a thing diuisible as an instant in time is neither tyme nor part of tyme but only the beginning and end of time and coupleth ioyneth partes of tyme together 4 A right lyne is that which lieth equally betwene his pointes As the whole line AB lyeth straight and equally betwene the poyntes AB without any going vp or comming downe on eyther side A right line is the shortest extension or draught that is or may be from one poynt to an other Archimedes defineth it thus Plato defineth a right line after this maner A right line is that whose middle part shadoweth the exâ—remeâ— As if you put any thyng in the middle of a right lyne you shall not see from the one ende to the other which thyng happeneth not in a crooked lyne The Ecclipse of the Sunne say Astronomers then happeneth when the Sunne the Moone our eye are in one right line For the Moone then being in the midst betwene vs and the Sunne causeth it to be darkened Diuers other define a right line diuersly as followeth A right lyne is that which standeth firme betwene his extremes Agayne A right line is that which with an other line of lyke forme cannot make a figure Agayne A right lyne is that which hath not one part in a plaine superficies and an other erected on high Agayne A right lyne is that all whose partes agree together with all his other partes Agayne A right lyne is that whose extremes abiding cannot be altered Euclide doth not here define a crooked lyne for it neded not It may easely be vnderstand by the definition of a right lyne for euery contrary is well manifested set forth by hys contrary One crooked lyne may be more crooked then an other and from one poynt to an other may be drawen infinite crooked lynes but one right lyne cannot be righter then an other and therfore from one point to an other there may be drawen but one tight lyne As by figure aboue set you may see 5 A superficies is that which hath onely length and breadth A superficies is the second kinde of quantitie and to it are attributed two â—imensions namely length and breadth As in the
the line AB beyng longer then the line CD if ye take froÌ„ them two equall lines as EB and FD the partes remayning which are the lines AE and CF shall be vnequall the one to the other namely the lyne AE shall be greater then the line CF which is euer true in all quantities whatsoeuer 5 And if to vnequall thinges ye adde equall thinges the whole shall be vnequall As if ye haue two vnequal lines namely AE the greater and CF the lesse if ye adde vnto theÌ„ two equall lines EB FD then maye ye conclude that the whole lines composed are vnequall namely that the whole lyne AB is greater then the whole line CD and so of all other quantities 6 Thinges which are double to one and the selfe same thing are equall the one to the other As if the line AB be double to the line EF and if also the line CD be double to the same line EF â— theÌ„ may you by this common sentence conclude that the two lines AB CD are equall the one to the other And this is true in all quantities and that not only when they are double but also if they be triple or quadruple or in what proportion soeuer it be of the greater inequallitie Which is when the greater quantitie is compared to the lesse 7 Thinges which are the halfe of one and the selfe same thingâ— are equal the one to the other As if the line AB be the halfe of the line EF and if the lyne CD be the halfe also of the same line EF then may ye conclude by this common sentence that the two lines AB and CD are equall the one to the other This is also true in all kyndes of quantitie and that not onely when it is a halfe but also if it be a third a quarter or in what proportion soeuer it be of the lesse in equallitie Which is when the lesse quantitie is coÌ„pared to the greater 8 Thinges which agree together are equall the one to the other Such thinges are sayd to agree together whiche when they are applied the one to the other or set the one vpon the other the one â—â—â—â—deth not the other in any thyng As if the two triangles ABC and DEF were applied the one to the other and the triangle ABC were set vpon the triangle DEF if then the angle A do iustly agree with the angle D and the angle B with the angle E and also the angle C with the angle F and moreouer â—f the line AB do iustly fall vpon the line DE and the line AC vpon the line DF and also the line BC vppon the line EF so that on euery part of these two triangles there is iust agreement then may ye conclude that the two triangles are equall 9 Euery whole is greater then his part THe principles thus placed ended now follow the propositions which are sentences set forth to be proued by reasoning and demonstrations and therfore they are agayne repeated in the end of the demonstrationâ— For the proposition is euer the conclusion and that which ought to be pâ—oued Propositions are of two sortes the one is called a Probleme the other a Theoreme A Probleme is a propositâ—on which requireth some action or doing as the makyng of some â—igure or to deuide a figure or line to apply figure to â—igure to adde figures together or to subtrah one from an other to describe to inscribe to circumscribe one figure within or without another and suche like As of the first proposition of the first booke is a probleme which is thusâ— Vpon a right line geuen not being infinite to describe an â—quilater triangle or a triangle of three equall sides For in it besides the demonstration and contemplation of the mynde is required somewhat to be done namelyâ— to make an equilater triangle vpon a line geuen And in the ende of euery probleme after the demonstration is concluded aâ—ter this manâ—er Which is the thing which was required to be done A Theoreme is a proposition which requireth the searching ouâ— and demonstration of some propertie or passion of some figure Wherin is onely speculation and contemplation of minde without doing or working of any thingâ— As the fifth proposition of the first booke which is thus An Isosceles or triangle of two equall sides hath his angles at thâ— base equall the one to the other c. is a Theoreme For in it is required only to be proued and made plaine by reason and demonstratioÌ„ that these two angles be equall without further working or doing And in the endâ— of â—uâ—ry Theoreme after the demonstration is concluded after this maner Which thyng was required to be demonstrated or proued The first Probleme The first Proposition Vpon a right line geuen not beyng infinite to describe an equilater triangle or a triangle of three equall sides SVppose that the right line geuen be AB It is required vpon the line AB to describe an equilater triangle namely a triangle of three equall sides Now therfore making the centre the point A and the space AB describe by the third peticion a circle BCD and agayne by the same makyng the centre the point B and the space Bâ— describe an other circle ACE And by the first peticion from the point C wherin the circles cut the one the other draw one right line to the point A and an other right line to the point B. And forasmuche as the point A is the centre of the circle CBD therfore by the 15. definition the line AC is equall to the line AB Agayne forasmuch as the point B is the centre of the circle CAE therfore by the same definition the line BC is equall to the line BA And it is proued that the line AC is equall to the line AB wherfore either of these lines CA and CB is equall to the line AB but thinges which are equall to one and the same thing are also equall the one to the other by the first common sentence wherfore the line CA also is equall to the line CB. VVherfore these three right lines CA AB and BC are equal the one to the other VVherfore the triaÌ„gle ABC is equilater VVherfore vppon the line AB is described an equilater triangle ABC VVherfore vppon a line geuen not being infinite there is described an equilater triangle VVhich is the thing which was required to be done A triangle or any other rectilined figure is then said to be set or described vpon a line when the line is one of the sides of the figure This first proposition is a Probleme because it requireth acte or doyng namely to describe a triangle And this is to be noted that euery Proposition whether it be a Probleme or a Theoreme commonly containeth in it a thing geuen and a thing required to be searched out although it be not alwayes so And the thing geuen is euer
side the rectiline angle MAB either of which is equall to the rectiline angle geuen CDE which was required to be done An other construction also and demonstration after Pelitarâ—us And if the perpendiculer line chaunce to fall without the angle geuen namely if the angle geuen be an acute angle the selfe same manner of demonstration will serue but onely that in stede of the second common sentence must be vsed the 3. common sentence Appollonius putteth another construction demonstration of this propositioÌ„ which though the demonstration thereof depende of propositions put in the third booke yet for that the construction is very good for him that wil redely and mechanically without demonstration describe vpon a line geuen and to a point in it geuen a rectiline angle equall to a rectiline angle geuen I thought not amisse here to place it And it is thus Oenopides was the first inuenter of this proposition as witnesseth Eudemius The 15. Theoreme The 24. Proposition If two triangles haue two sides of the one equall to two sides of the other ech to his correspondent side and if the angle coÌ„tained vnder the equall sides of the one be greater then the angle contayned vnder the equall sides of the other the base also of the same shal be greater then the base of the other SVppose that there be two triangles ABC and DEF hauing two sides of the one that is AB and AC equall to two sides of the other that is to DE and DF ech to his correspondent side that is the side AB to the side DE and the side AC to the side DF and suppose that the angle BAC be greater then the angle EDF Then I saye that the base BC is greater then the base EF. For forasmuch as the angle BAC is greater then the angle EDF make by the 23. proposition vpon the right line DE and to the point in it geueÌ„ D an angle EDG equall to the angle geuen BAC And to one of these lines that is either to AC or DF put an equall line DG And by the first peticioÌ„ draw a right line from the point G to the point E and an other from the point F to the point G. And forasmuch as the line AB is equall to the line DE and the line AC to the line DG the one to the other and the angle BAC is by construction equall to the angle EDG therefore by the 4. proposition the base BC is equall to the base EG Agayne for as much as the line DG is equall to the line DF therâ— by the 5. proposition the angle DGF is equall to the angle DFG VVherefore the angle DFG is greater then the angle EGF VVherefore the angle EFG is much greater then the angle EGF And forasmuch as EFG is a triangle hauing the angle EFG greater then the angle EGF and by the 18. prâ—position vnder the greater angle is subtended the greater side therefore the side EG is greater then the side EF. But the side EG is equall to the side BC wherefore the side BC is greater then the side EF. If therefore two triangles haue two sides of the one equall to two sides of the other eche to his correspondent side and if the angle contayned vnder the equall sides of the one be greater then the angle contayned vnder the equall sides of the other the base also of the same shal be greater then the base of the other which was required to be proued In this Theoreme may be three cases For the angle EDG being put equall to the angle BAC and the line DG being put equall to the line AC and a line being drawen from E to G the line EG shall either fall aboue the line GF or vpon it or vnder it Euclides demonstration serueth when the line GE falleth aboue the line GF as we haue before manifestly seene But now let the line EG fall vnder the line E F as in the figure here put And forasmuch as these two lines AB and AC are equall to these two lines DE and DG the one to the other and they contayne equall angles therefore by the 4. proposition the base BC is equal to the base EG And forasmuch as within the triangle DEG the two linnes DF and FE are set vpon the side DE therfore by the 21. proposition the lines DF and Fâ— are lesse then the outward lines DG and GE but the line DG is equal to the line DF. Wherfore the line GE is greater then the line FE But GE is equall to BC. Wherefore the line BC is greater the the line EF. Which was required to be proued It may peraduenture semeâ— that Euclide should here in this proposition haue proued that not onely the bases of the triangles are vnequall but also that the areas of the same are vnequall for so in the fourth proposition after he had proued the base to be equall he proued also the areas to be equall But hereto may be answered that in equall angles and bases and vnequall angles and bases the consideration is not like For the angles and bases being equall the triangles also shall of necessitie be equall but the angles and bases being vnequall the areas shall not of necessitie be equall For the triangles may both be equall and vnequall and that may be the greater whiche hathe the greater angle and the greater base and it may also be the lesse And for that cause Euclide made no mencion of the comparison of the triangles VVhereof this also mought be a cause for that to the demonstration thereof are required certayne Propositions concerning parallel lines which we are not as yet come vnto Howbeit after the 37â— proposition of his booke you shal find the comparison of the areas of triangles which haue their sides equall and their bases and angles at the toppe vnequall The 16. Theoreme The 25. Proposition If two triangles haue two sides of the one equall to two sydes of the other eche to his correspondent syde and if the base of the one be greater then the base of the other the angle also of the same coÌ„tayned vnder the equall right linesâ— shall be greater then the angle of the other SVppose that there be two triangles A B C and DEF hauing two sides of tb'one that is AB and AC equall to two sides of the other that is to DE and DF ech to his correspondent side namely the side AB to the side DF and the side AC to the syde DF. But let the base BC be greater then the base EF. TheÌ„ I say thay the angle BAC is greater then the angle EDF For if not then is it either equall vnto it or lesse then it But the angle BAC is not equall to the angle EDF for if it were equall the base also BC should by the 4. proposition be equal to the base EF but by supposition it is not VVherfore
namely the first ten propositions as they follow in order VVhich is vndoubtedly great pleasure to coÌ„sider also great increase furniture of knowledge VVhose PropositioÌ„s are set orderly after the propositioÌ„s of Euclide euery one of Barlaam correspoÌ„dent to the same of Euclide And doubtles it is wonderful to see how these two coÌ„trary kynds of quantity quantity discrete or number and quantity continual or magnitude which are the subiectes or matterâ— of Arithmitique and Geometry shoulde haue in them one and the same proprieties common to them both in very many points and affections although not in all For a line may in such sort be deuided that what proportion the whole hath to the greater parte the same shall the greater part haue to the lesse But that can not be in number For a number can not so be deuided that the whole number to the greater part thereof shall haue that proportion which the greater part hath to the lesse as Iordane very playnely proueth in his booke of Arithmetike which thynge Campane also as we shall afterward in the 9. booke after the 15. proposition see proueth And as touching these tenne firste propositions of the seconde booke of Euclide demonstrated by Barlaam in numbers they are also demoÌ„strated of Campane after the 15. proposition of the 9. booke whose demonstrations I mynde by Gods helpe to set forth when I shal come to the place They are also demoÌ„strated of Iordane that excellet learned authour in the first booke of his Arithmetike In the meane tyme I thought it not amisse here to set forth the demonstrations of Barlaam for that they geue great light to the seconde booke of Euclide besides the inestimable pleasure which they bring to the studious considerer And now to declare the first Proposition by numbers I haue put this example following Take two numbers the one vndeuided as 74. the other deuided into what partes and how many you list as 37. deuided into 20. 10. 5. and 2â— which altogether make the whole 37. Then if you multiply the number vndeuided namely 74 into all the partes of the number deuided as into 20. 10. 5. and 2. you shall produce 1480. 740. 370. 148. which added together make 2738 which self number is also produced if you multiplye the two numbers first geuen the one into the other As you see in the example on the other side set So by the aide of this Proposition is gotten a compendious way of multiplication by breaking of one of the numbers into his partes which oftentimes serueth to great vse in workingâ— chiâ—â—ly in the rule of proportions The demonstration of which proposition followeth in Barlaam But â—irst are put of the author these principles following ¶ Principles 1. A number is sâ—yd to multiply an other number when the number multiplied is so oftentymes added to it selfe as there be vnities in the number which multiplieth wherby is produced a certaine number which the number multiplied measureth by the vnities which are in the number which multipliâ—th 2. And the number produced of that a multiplication is called a plaine or superficiall number 3. A square number is that which is produced of the multiplicatian of any number into it selfe 4. Euery lesse number compared to a greater is sayd to be a part of the greater whether the lesse measure the greater or measure it not 5. Numbers whome one and the selfe same number measureth equally that is by one and the selfe same number are equall the one to the otheâ— 6. Numbers that are equemultiplâ—ces to one and the selfe same number that is which contayne one and the same number equally and alike are equall the one to the other The first Proposition Two numbers beyng geuen if the one of them be deuided into any numbers how many soeuer the playne or superficiall number which is produced of the multiplication of the two numbers first geuen the one into the other shall be equall to the superficiall nuÌ„bers which are produced of the multiplication of the number not deuided into euery part of the number deuided Suppose that there be two numbers AB and C. And deuide the number AB into certayne other numbers how many soeuer as into AD DE and EB Then I say that the superficiall number which is produced of the multiplication of the number C into the number AB is equall to the superficiall numbers which are produced of the multiplication of the number C into the nuÌ„ber AD and of C into DE and of C into EB For let F be the superficiall number produced of the multiplication of the number C into the number AB and let GH be the superficiall number produced of the multiplication of C into AD And let HI be produced of the multiplication of C into DE aâ—d finally of the multiplication of C into EB let there be produced the number IK Now forasmuch as AB multiplying the number C produced the number F therefore the number C measureth the number F by the vnities which are in the number AB And by the same reason may be proued that the number C doth also measure the number GH by the vnities which are in the number AD and that it doth measure the number HI by the vnities which are in the nuÌ„ber DF and finally that it measureth the number IK by the vnities which are in the number EB Wherefore the nuÌ„ber C measureth the whole number GK by the vnities which are in the number AB But it before measured the number F by the vnities which are in the number AB wherfore either of these numbers F and GK is equemultiplex to the number C. But numbers which are equemultiplices to one the selfe same numbers are equall the one to the other by the 6. definition Wherfore the number F is equall to the number GK But the number F is the superficiall number produced of the multiplication of the nuÌ„ber C into the number AB and the number GK is composed of the superficiall numbers produced of the multiplication of the nuÌ„ber C not deuided into euery one of the numbers AD DE and EB If therefore there be two numbers geuen and the one of them be deuided c. Which was required to be proued The 2. Theoreme The 2. Proposition If a right line be deuided by chaunce the rectangles figures which are comprehended vnder the whole and euery one of the partes are equall to the square whiche is made of the whole SVppose that the right line AB be by chaunse denided in the point C. Then I say that the rectangle figure comprehended vnder AB and BC together with the rectangle comprehended vnder AB and AC is equall vnto the square made of AB Describe by the 46. of the first vpon AB a square ADEB and by the 31 of the first by the point C draw a line parallel vnto either of these lines AD aâ—d BE and let the same
be Câ— Now is the parallelogramme AE equall to the parallelogrammes AF and CE by the first of this booke But AE is the square made of AB And AF is the rectangle parallelogramme comprehended vnder the lines BA and AC for it is comprehended vnder the lines DA and AC but the line AD is equall vnto the line AB And likewise the parrallelogramme CE is equall to that which is contayned vnder the lynes AB and BC for the line BE is equal vnto the line AB VVherfore that which is contayned vnder BA and AC together with that which is contayned vnder the lines AB and BC is equall to the square made of the line AB If therefore a right line be deuided by chaunce the rectaÌ„gle figures which are comprehended vnder the whole and euery one of the partes are equall to the square which is made of the whole which was required to be demonstrated An other demonstration of Campane Suppose that the line AB be deuided into the lines AC CD and DB. Then I say that the square of the whole line AB which let be AEBF is equal to the rectangle figures which are contayned vnder the whole and euery one of the partes for take the line K which let be equal to the line A B. Now then by the first proposition the rectangle figure contained vnder the lines AB and K is equall to the rectangle figures contayned vnder the line K and al the partes of the line AB But that which is contayned vnder the lines K and AB is equall to the square of the line AB and the rectangle figures contayned vnder the line K and al the partes of AB are equall to the rectangle figures contayned vnder the line AB and all the partes of the line AB for the lines AB and K are equall wherefore that is manifest which was required to be proued An example of this Proposition in numbers Take a number as 11. and deuide it into two partes namely 7. and 4 and multiply 11. into 7 and then into 4. and there shal be produced 77. and 44 both which numbers added together make 121. which is equall to the square number produced of the multiplication of the number 11. into himselfe as you see in the example The demonstration whereof followeth in Barlaam The second Proposition If a number geuen be deuided into two other numbers the superficiall numbers which are produced of the multiplication of the whole into either part added together are equall to the square number of the whole number geuen Suppose that the number geuen be AB and let it be deuided into two other numbers AC and CB. Then I say that the two superficiall numbers which are produced of the multiplication of AB into AC and of AB into BC those two superficiall numbers I say beyng added together shal be equall to the square number produced of the multiplicatioÌ„ of the number AB into it selfe For let the number AB multiplying it selfe produce the number D. Let the number AC also multiplying the number AB produce the number EF agayne let the number CB multipliing the selfe same number AB produce the number FG. Now forasmuch as the number AC multiplying the number AB produced the number EF therefore the number AB measureth the number EF by the vnities which are in AC Againe forasmuch as the number CB multiplied the number AB and produced the number FG therfore the number AB measureth the number FG by the vnities which are in the number CB. But the same number AB before measured the number EF by the vnities which are in the number AC Wherefore the number AB measureth the whole number â—G by the vnities whcih are in AB Farther forasmuch as the number AB multiplying it selfe produced the number D therefore the number AB measureth the number D by the vnities which are in himselfe Wherfore it measureth either of these numbers namely the number D and the number EG by the vnities which are in himselfe Wherfore how multiplex the number D is to the number AB so multiplex is the number EG to the same number AB But numbers which are equemultiplices to one and the selfe same numberâ— are equal the one to the other Wherefore the number D is equall to the number EG And the number D is the square number made of the number AB and the number EG is composed of the two superficiall numbers produced of AB into BC and of BA into AC Wherefore the square number produced of the number AB is equall to the superficial numbers produced of the number AB into the number BC and of AB into AC added together If therefore a number be deuided into two other numbers c. which was required to be proued The 3. Theoreme The 3. Proposition If a right line be deuided by chaunce the rectangle figure coÌ„prehended vnder the whole and one of the partes is equall to the rectangle figuâ—e comprehended vnder the partes vnto the square which is made of the foresaid part SVppose that the right line geuen AB be deuided by chaunce in the point C. Then I say that the rectangle figure compreheÌ„ded vnder the lines AB and BC is equall vnto the rectangle figure comprehended vnder the lines AC and CB and also vnto the square which is made of the line BC. Describe by the 46. of the first vpon the line BCâ— square CDEB and by the second peticion extend ED vnto F. And by the point A draw by the 31. of the first a line parallel vnto either of these lines CD and BE and let the same be AF. Now the parallelograme AE is equall vnto the parallelogrammes AD and CE. And AE is the rectangle figure comprehended vnder the lines AB and BC. For it is comprehended vnder the lines AB and BE which line BE is equall vnto the line BC. And the parallelograme AD is equall to that which is contayned vnder the lines AC and CB for the line DC is equall vnto the line CB And DB is the square which is made of the lyne CB. VVherfore the rectangle figure comprehended vnder the lynes AB and BC is equall to the rectangle figure comprehended vnder the lines AC and CB also vnto the square which is made of the line BC. If therfore a right line be deuided by chaunce the rectangle figure comprehended vnder the whole and one of the partes is equall to the rectangle figure comprehended vnder the partes and vnto the square which is made of the foresayd part which was required to be proued An example of this Proposition in numbers Suppose a number namely 14. to be deuided into two partes 8. and 6. The whole number 14. multiplied into 8. one of his partes produceth 112 the partes 8. 6. multiplied the one into the other produce 48 which added to 64 which is the square of 8. the
nombers of the nombers AD and DB are double to the square nombers of AC and CD For forasmuch as the nomber AD is deuided into the nombers AB and BD therefore the square nombers of the nombers AD and DB are equall to the superficiall nomber produced of the multiplication of the nombers AD and DB the on into the other twise together with the square of the nomber AB by the 7 propositiō But the square of the nomber AB is equal to fower squares of either of the nombers AC or CB for AC is equall to the nomber CB wherfore also the squares of the nombers AD and DB are equall to the superficiall nomber produced of the multiplication of the nombers AD and DB the one into the other twise and to fower squares of the nomber BC or CA. And forasmuch as the superficiall nomber produced of the multiplication of the nombers AD and DB the one into the other together with the square of the nomber CB is equal to square of the nomber CD by the 6 propositiō therfore the nomber produced of the multiplication of the nomber◠AD and DB the one into the other twise together with two squares of the nomber CB is equall to two squares of the nomber CD Wherefore the squares of the nombers AD and DB are equall to two squares of the nomber CD and to two squares of the nomber AC Wherefore they are double to the squares of the numbers AC and CD And the square of the nomber AD is the square of the whole and of the nomber added And the square of DB is the square of the nomber added the square also of the nomber CD is the square of the nomber composed of the halfe and of the nomber added If therefore an euen nomber be deuided c. Which was required to be proued The 1. Probleme The 11. Proposition To deuide a right line geuen in such sort that the rectangle figure comprehended vnder the whole and one of the partes shall be equall vnto the square made of the other part SVppose that the right line geuen be AB Now it is required to deuide the line AB in such sort that the rectangle figure contayned vnder the whole and one of the partes shall be equall vnto the square which is made of the other part Describe by the 46. of the first vpon AB a square ABCD. And by the 10. of the first deuide the line AC into two equall partes in the point E and draw a line from B to E. And by the second petition extend CA vnto the point F. And by the 3. of the first put the line EF equall vnto the line BE. And by the 46. of the first vpon the line AF describe a square FGAH And by the 2. petition extend GH vnto the point K. Then I say that the line AB is deuided in the point H in such sort that the rectangle figure which is comprehēded vnder AB and BH is equall to the square which is made of AH For forasmuch as the right line AC is deuided into two equall partes in the poynt E and vnto it is added an other right line AF. Therefore by the 6. of the second the rectangle figure contayned vnder CF and FA together with the square which is made of AE is equall to the square which is made of EF. But EF is equall vnto EB VVherefore the rectangle figure contayned vnder CF and FA together with the square which is made of EA is equall to the square which is made of EB But by 47. of the first vnto the square which is made of EB are equall the squares which are made of BA and AE For the angle at the poynt A is a right angle VVherefore that which is contayned vnder CF and FA together with the square which is made of AE is equall to the squares which are made of BA and AE Take away the square which is made of AE which is common to them both VVherfore the rectangle figure remayning contayned vnder CF and FA is equall vnto the square which is made of AB And that which is contained vnder the lines CF and FA is the figure FK For the line FA is equall vnto the line FG. And the square which is made of AB is the figure AD. VVherefore the figure FK is equall vnto the figure AD. Take away the figure AK which is common to them both VVherefore the residue namely the figure FH is equall vnto the residue namely vnto the figure HD But the figure HD is that which is contayned vnder the lines AB and BH for AB is equall vnto BD. And the figure FH is the square which is made of AH VVherfore the rectangle figure comprehended vnder the lines AB and BH is equall to the square which is made of the line HA. VVherefore the right line geuen AB is deuided in the point H in such sort that the rectangle figure contayned vnder AB and BH is equall to the square which is made of AH which was required to be done Thys proposition hath many singular vses Vpon it dependeth the demonstration of that worthy Probleme the 10. Proposition of the 4. booke which teacheth to describe an Isosceles triangle in which eyther of the angles at the base shall be double to the angle at the toppe Many and diuers vses of a line so deuided shall you finde in the 13. booke of Euclide Thys is to be noted that thys Proposition can not as the former Propositions of thys second booke be reduced vnto numbers For the line EB hath vnto the line AE no proportion that can be named and therefore it can not be expressed by numbers For forasmuch as the square of EB is equall to the two squares of AB and AE by the 47. of the first and AE is the halfe of AB therefore the line BE is irrationall For euen as two equall square numbers ioyned together can not make a square number so also two square numbers of which the one is the square of the halfe roote of the other can not make a square number As by an example Take the square of 8. which is 64. which doubled that is 128. maketh not a square number So take the halfe of 8. which is 4. And the squares of 8. and 4. which are 64. and 16. added together likewyse make not a square number For they make 80. who hath no roote square Which thyng must of necessitie be if thys Probleme should haue place in numbers But in Irrational numbers it is true and may by thys example be declared Let 8. be so deuided that that which is produced of the whole into one of his partes shall be equall to the square number produced of the other part Multiply 8. into him selfe and there shall be produced 64. that is the square ABCD. Deuide 8. into two equall partes that is into 4 and 4. as the line
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.
BC is to CA so is CA to CD The 1. Probleme The 9. Proposition A right line being geuen to cut of froÌ„ it any part appointed LEt the right line geuen be AB It is required that from the same line AB be cut of any part appoynted Suppose that a thyrd part be appoynted to be cut of From the point A draw a right line AC making with the line AB an angle and in the line AC take a poynt at all aduentures which let be D. And beginning at D put vnto AD two equall lines DE EC by the 2. of the first And draw a right line from B to C and by the point D by the â—1 of the first draw vnto BC a parallell line DF. Now forasmuch as vnto one of y sides of the triangle ABC namely vnto the side BC is drawen a parallell line FD it followeth by the 2. of this booke that as CD is in proportion vnto DA so is BF to FA. But by construction CD is double to DA. Wherefore y line BF is also double to the line FA. Wherfore the line BA is treble vnto the line AF. Wherfore from the right lineâ— geuen AB is cut of a third part appoynted namely AF which was required to be done The 2. Probleme The 10. Proposition To deuide a right line geueÌ„ not deuided like vnto a right line geuen beyng deuided SVppose that the right line geuen not deuided be AB and the right lyne geuen being deuided let be AC It is required to deuide the line AB which is not deuided like vnto the line AC which is deuided Suppose the lyne AC be deuided in the pointes D and E let y lines AB AC so be put that they make an angle at all aduentures and draw a line from B to C and by the pointes D and E draw vnto the line BC by the 31. of the first two parallel lines DF and EG and by the point D vnto the line AB by the same draw a parallel line DHK Wherfore either of these figureâ— FH and HB are parallelogrammes Wherfore the line DH is equall vnto the line FG and the line HK is equall vnto the line GB And because to one of the sides of the triangle DKC namely to the side KC is drawn a parallel line HE therefore the line CE by thâ— 2. of the sixt is in proportion vnto the line ED as the line KH is to the line HD but the line KH is equall vnto the line BG and the line HD is equal vnto the line GF Wherfore by the 11. of the fift as CE is vnto ED so is BG to GF Agayne because to one of the sides of the triangle AGE namely to GE is drawn a parallel lyne FD therfore the line ED by the 2. of the sixth is in proportion vnto the lyne DA as the line GF is to the line FA. And it is already proued that as CE is to ED so is BG to GF VVherfore as CE is to ED so is BG to GF and as ED is to DA so is GF to FA. VVherfore the right line geuen not deuided namely AB is deuided like vnto the right line geuen being deuided which is AC which was required to be done ¶ A Corollary out of Flussates By this Proposition we may deuide any right line geuen accordyng to the proportion of any right lynes geuen For let those right lynes hauyng proportion be ioyned together directly that they may make all one right lyne and then ioyne them to the lyne geuen anglewise And so proceede as in the proposition where you see that the right line geuen AB is deuided into the right lynes AF FG and GB which haue the selfe same proportion that the right lines AD DE and EC haue By this and the former proposition also may a right line geuen be easily deuided into what partes so euer you will name As if you will deuide the line AB into three equall partes let the lyne DE be made equall to the lyne AD and the lyne EC made equall to the same by the third of the first And then vsing the selfe same maner of construction that was before the lyne AB shall be deuided into three equall partes And so of any kynde of partes whatsoeuer The 3. Probleme The 11. Proposition Vnto two right lines geuen to finde a third in proportion with them SVppose that there be two right lines geuen BA and AC and let them be so put that they comprehend an angle howsoeuer it be It is required to finde vnto BA and vnto AC a third line in proportion Produce the lynes AB and AC vnto the pointes D and E. And vnto the line AC by the 2. of the first put an equall line BD and draw a lyne from B to C. And by the pointe D by the 31. of the first draw vnto the lyne BC a parallel lyne DE which let concurre with the line AC in the point E. Now forasmuch as vnto one of the sides of the triangle ADE namely to DE is drawne a parallel line BC therfore as AB is in proportion vnto BD so by the 2. of the sixt is AC vnto CE. But the lyne BD is equall vnto the line AC VVherfore as the lyne AB is to the line AC so is the line AC to the line CE. VVherfore vnto the two right lines geuen AB and AC is found a third line CE in proportioÌ„ with them which was required to be done ¶ An other way after Pelitarius Let the lines AB and BC be set directly in such sort that they both make one right line Then froÌ„ the point A erect the lyne AD makyng with the lyne AB an angle at all aduentures And put the lyne AD equall to the lyne BC. And draw a right line from D to B which produce beyond the poynt B vnto the point E. And by the point C draw vnto the lyne DA a parallel lyne CE concurring with the lyne DE in the point E. Then I say that the line CE is the third line proportionall with the lines AB and BC. For forâ—asmuch as by the 15. of the first the angle B of the triangle ABD is equall to the angle B of the triangle CBE and by the 29. of the same the angle A is equall to the angle C and the angle D to the angle E therefore by the 4. of this booke AB is to DA as BC is to CE. Wherfore by the 11. of the fifth AB is to BC as BC is to CE which was required to be done ¶ An other way also after Pelitarius Let the lines AB and BC be so ioyned together that they may make a right angle namely ABC And drawe a line from A to C and from the point C drawe vnto the line AC a perpendicular CD by the 11. of the first And
proportion that the first of the three lines put is to the 〈◊〉 â—or tâ—e 〈◊〉 line to the third namely the line AE to the line EB is in double propoâ—tion that it is to the second by the 10. deâ—inition of the fiâ—t ¶ The second Corollary Hereby may we learne how from a rectiline â—igure geuen to take away a part appointed leaâ—ing the rest of the rectiline â—igure like vnto the whole For if froÌ„ the right line AB be cut of a part appoynted namely EB by the 9. of this booke as the line AE is to the line EB so is the rectiline â—igure described of the line AF to the rectiline figure described of the line FB the sayd â—igures being supposed to be like both the one to the other and also to the rectiline â—igure described of the line AB and being also in like sort situate Wherfore taking away â—rom the rectiline â—igure described of the line AB the rectiline figure described of the line FB the residue namely the rectiline figure described of the line AF shall be both like vnto the whole rectiline â—igure geuen described of the line AB and in like sort situate ¶ The third Corollary To compose two like rectiline â—igures into one rectiline figure like and equall to the same figures Let their sides of like proportioÌ„ be set so that they make a right angle as the lines AF and FB are And vpoÌ„ the line subtending the said angle namely the line AB describe a rectiline â—igure like vnto the rectiline figures geuen and in like sort situate by the 18. of this booke and the same shall be equall to the two rectiline figures geuen by the 31. of this booke ¶ The second Proposition If two right lines cut the one the other obtuseangled wise and from the endes of the lines which â—ut the one the other be drawen perpendicular lines to either line the lines which are betwene the endes and the perpendicular lines are cut reciprokally Suppose that there be two right lines AB and GD cutting the one the other in the point E and making an obtuse angle in the section E. And from the endes of the lines namely A and G let there be drawen to either line perpendicular lines namely from the point A to the line GD which let be AD and from the point G to the right line AB which let be GB Then I say that the right lines AB and GD do betwene the end A and the perpendicular B and the end G and the perpendicular D cut the one the other reciprokally in the point E so that as the line AE is to the line ED so is the line GE to the line EB For forasmuch as the angles ADE and GBE are right angles therfore they are equall But the angles AED and GEB are also equall by the 15. of the first Wherefore the angles remayning namely EAD EGB are equall by the Corollary of the 32. of the first Wherefore the triangles AED and GEH are equiangle Wherfore the sides about the equall angles shall be proportionall by the 4. of the sixt Wherfore as the line AE is to the line ED so is the line GE to the line EB If therefore two right lines cut the one the other obtuseangled wife c which was required to be proued ¶ The third Proposition If two right lines make an acute angle and from their endes be drawen to ech line perpendicular lines cutting them the two right lines geuen shall be reciprokally proportionall as the segmentes which are about the angle Suppose that there be two right lines AB and GB making an acute angle ABG And from the poyntes A and G let there be drawen vnto the lines AB and GB perpendicular lines AC and GE cutting the lines AB and GB in the poyntes E and â— Then I say that the lines namely AB to GB are reciprokally proportionall as the segmentes namely CB to EB which are about the acute angle B. For forasmuch as thâ— right angles ACB and GER are equall and the angleâ— ABG is common to the triangles ABC and GBE â— therefore the angles remayning BAC and EGB are equall by the Corollary of the 32. of the first Wherfore the triangles ABC and GBE are equiangle Wherefore the sideâ— about the equall angles are proportionall by the 4. of the sixe â— so that as the line AB is to the line FC so is the line GB to the line BE. Wherefore alternately as the line AB is to the line GB so is the line CB to the line BE. If therefore two right lines makâ— aâ—â—câ—te angleâ— câ— which was required to be proued â— The fourth Propositionâ— If in a circle be drawen two right lines cutting the one the other the sections of the one to the sections of the other shall be reciprokally proportionall In the circle AGB let these two right lines 〈…〉 one the other in the poynt E. Thâ—â— I say that reciprokally 〈◊〉 â—hâ— line AE is to the line ED so is the line GE to the line EB For forasmuch as by the 35. of the third the rectangle figure contayned vnder the lines AE and EB is equall to the rectangle figure contayned vnder the lines GE and ED but in equall rectangle parallelogrammes the sides about the equall angles are reciprokall by the 14. of the sixt Therefore the line AE is to the line ED reciprokally as the line GE is to the line EB by the second definition of the sixt If therefore in a circle be drawen two right lines c which was required to be proued ¶ The fift Proposition If from a poynt geuen be drawen in a plaine superâ—icies two right lines to the concaue circumference of a circle they shall be reciprokally proportionall with their partes takeÌ„ without the circle And moreouer a right line drawen from the sayd poynt touching the circle shall be a meane proportionall betwene the whole line and the vtter segment Suppose that there be a circle ABD and without it take a certayne poynt namely G. And from the point G drawe vnto the concaue circumference two right lines GB and GD cutting the circle in the poyntes C and E. And let the line GA touch the circle in the point A. TheÌ„ I say that the lines namely GB to GD are reciprokally as their parts taken without the circle namely as GC to GE. For forasmuch as by the Corollary of the 36. of the third the rectangle figure contayned vnder the lines GB and GE is equall to the rectangle figure contayned vnder the lines GD and GC therefore by the 14. of the sixt reciprokally as the line GB is to the line GD so is the line GC to the line GE for they are sides contayning equall angles I say moreouer that betwene the lines GB and GE or betwene the lines GD and GC the touch line GA is a meane proportionall For forasmuch as the rectangle
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
wherefore A is a plaine number and the sides therof are D and F by the 17. definition of the seuenth Againe forasmuch as D and E are the lest numbers that haue one the same proportion with C B therefore by the 21. of the seuenth how many times D measureth C so many times doth E measure B. How often E measureth B so many vnities let there be in G. Wherefore E measureth B by those vnities which are in G wherefore G multiplying E produceth B wherefore B is a plaine number by the 17. definition of the seuenth And the sides thereof are E and G. Wherefore those two numbers A and B are two plaine numbers I say moreouer that they are like For forasmuch as F multiplying E produced C and G multiplying E produced B therefore by the 17. of the seuenth as F is to G so is C to B but as C is to B so is D to E wherefore as D is to E so is F to G. Wherefore A and B are like plaine numbers for their sides are proportionall which was required to be proued ¶ The 19. Theoreme The 21. Proposition If betwene two numbers there be two meane proportionall numbers those numbers are like solide numbers SVppose that betwene two numbers A and B there be two meane proportionall numbers C D. Then I say that A and B are like solide numbers Take by the 3â— of the seuenth or 2. of the eight three of the least numbers that haue one and the same proportion with A C D B and let the same be E F G. Wherefore by the 3. of the eight their extremes E G are prime the one to the other And forasmuch as betwene the numbers E and G there is one meane proportionall number therfore by the 20 of the eight they are like plaine numbers Suppose that the sides of E be H and K. And let the sides of G be L and M. Now it is manifest that these numbers E F G are in continuall proportion and in the same proportion that H is to L and that K is to M. And forasmuch aâ— E F G are the least numbers that haue one and the same proportion with A C D therefore of equalitie by the 14. of the seuenth as E is to G so is A to D. But E G are by the 3. of the eight prime numbers yea they are prime and the least but the least numbers by the 21. of the seuenth measure those numbers that haue one the same proportion with them equally the greater the greater and the lesse the lesse that is the antecedent the antecedent the consequent the consequent therfore how many timeâ— E measureth A so many times G measureth D. How many times E measureth A so many vnities let there be in N. Wherefore N multiplieng E produceth A. But E is produced of the numbers H K. Wherfore N multiplieng that which is produced of H K produceth A. Wherefore A is a solide number and the sides therof are H K N. Agayne forasmuch as E F G are the least numbers that haue one and the same proportion with C D B therefore how many times E measureth C so many times G measureth B. How ofteÌ„times G measureth B so many vnities let there be in X. Wherfore G measureth B by those vnities which are in X. Wherfore X multiplieng G produceth B. But G is produced of the numbers L M. Wherefore X multiplieng that number which is produced of L and M produceth B. Wherfore B is a solide number and the sides therof are L M X. Wherfore A B are solide numbers I say moreouer that they are like solide numbers For forasmuch as N and X multiplieng E produced A and C therfore by the 18. of the seuenth as N is to X so iâ— A to C that is E â—o F. But as E is to F so is H to L and K to M therefore as H is to L so is K to M and N to X. And H K N are the sides of A and likewise L M X aâ—â— thâ— sides of Bâ— wherfore A Bâ— are like solide numbers which was required to be proued ¶ The 20. Theoreme The 22. Proposition If three numbers be in continuall proportion and if the first be a square number the third also shall be a square number SVppose that there be three numbers in continuall proportion A B C and let the first be a square number Then I say that the third is also a square number For forasmuch as betwene A and C there is one meane proportionall numberâ— namely B therefore by the 20. of the eight A and C are like playne numbers But A is a square number Wherefore C also is a square number which was required to be proued ¶ The 21. Theoreme The 23. Proposition If foure numbers be in continuall proportion and if the first be a cube nuÌ„ber the fourth also shall be a cube number SVppose that there be foure numbers in continuall proportion A B C D. And let A be a cube number TheÌ„ I say that D also is a cube number For forasmuch as betwene A and D there are two meane proportionall numbers Bâ— C. Therfore A D are like solide numbers by the 21. of this booke But A is a cube number wherfore D also is a cube numberâ— which was required to be demonstrated ¶ The 22. Theoreme The 24. Proposition If two numbers be in the same proportioÌ„ that a square number is to a square number and if the first be a square number the second also shall be a square number SVppose that two numbers A and B be in the same proportion that the square number C is vnto the squâ—â—â— nuÌ„ber D. And let A be a square nuÌ„ber Then I say that B also is a square number For forasmuch as C and D are square numbers Therfore G and D are like plaine numbers Wherfore by the 18. of the eight betwene C and D there is one meane proportionall number But as C is to D so is A to B. Wherfore betwene A and B there is one meane proportionall number by the 8. of the eight But A is a square number Wherfore by the 22. of the eight B also is a square number which was â—equired to be proued ¶ The 23. Theoreme The 25. Proposition If two numbers be in the same proportion the one to the other that a cube number is to a cube number and if the first be a cube number the second also shall be a cube number SVppose that two numbers A and B be in the same proportioÌ„ the one to the other that the cube nuÌ„ber C is vnto the cube number D. And let A be a cube number Then I say that B also is a cube nuÌ„ber For forasmuch as C D are cube nuÌ„bers therfore C D are like solide numbers wherfore by the 19. of the eight betwene C and D
the double of B is prime vnto the double of B then the two numbers whereof the number is composed namely the number composed of A and C and the double of B shall be prime the one to the other by the 30 of the seuenth And therefore the number composed of A and C shall be prime to B taken once For if any number should measure the two numbers namely the number composed of A and C and the number B it should also measure the number composed of A and C and the double of B by the 5. common sentence of the seuenth which is not possible for that they are proued to be prime numbers Here haue I added an other demonstration of the former Proposition after Campane which proueth that in nuÌ„bers how many soeuer which is there proued onely touching three numbers and the demonstration seement somwhat more perspicous then Theons demonstration And thus he putteth the proposition If numbers how many soeuer being in continuall proportion be the least that haue one the same proportion with them euery one of them shal be to the number composed of the rest prime Secondly I say that this is so in euery one of them namely that C is a prime number to the number composed of A B D. For if not then as before let E measure C and the number composed of A B D which E shal be a number not prime either to F or to G by the former proposition added by Campane wherefore let H measure them And forasmuch as H measureth E it shall also measure the whole A B C D whom E measureth And forasmuch as H measureth one of these numbers F or G it shall measure one of the extreames A or D which are produced of F or G by the second of the eight if they be multiplâ—ed into the meanes L or K. And moreouer the same H shall measure the meames BC by the 5. common sentence of the seuenth when as by supposition it measureth either F or G. which measure B C by the second of the eight But the same H measureth the whole A B C D as we haue proued for that it measureth E. Wherefore it shall also measure the residue namely the number composed of the extreames A and D by the 4. common sentence of the seuenth And it measureth one of these A or D for it measureth one of these F or G which produce A and D wherefore the same H shall measure one of these A or D and also the other of them by the former common sentence which numbers A and D are by the 3. of the eight prime the one to the other Which were absurd This may also be proued in euery one of these numbers A B C D. Wherefore no number shall measure one of these numbers A B C D and the numbers composed of the rest Wherefore they are prime the one to the other If therefore numbers how many soeuer c which was required to be proued Here as I promised I haue added Campanes demonstrations of those Propositions in numbers which Euclâ—de in the second booke demonstrated in lines And that in thys place so much the rather for that Theon as we see in the demonstration of the 15. Proposition seemeth to alledge the 3. 4. Proposition of the second boke which although they concerne lines onely yet as we there declared and proued are they true also in numbers ¶ The first Proposition added by Campane That number which is produced of the multiplication of one number into numbers how many soeuer is equall to that number which is produced of the multiplication of the same number into the number composed of them This proueth that in numbers which the first of the second proued touching lines Suppoâ—â— that the number A being multiplyed into the number B and into the number C and into the number D doo produce the numbers E F and G. Then I say that the number produced of A multiplyed into the number composed of B C and D is equall to the number composed of E F and G. For by the conuerse of the definition of a number multiplyed what part vnitie is of A the selfe same part is B of E and C of F and also D of G. Wherefore by the 5. of the seuenth what part vnitie is of A the selfe same part is the number composed of B C and D of the number composed of E F and G. Wherfore by the definition that which is produced of A into the number composed of B C D is equall to the number composed of E F G which was required to be proued The second Proposition That number which is produced of the multiplication of numbers how many soeuer into one nuÌ„ber is equall to that number which is produced of the multiplication of the number composed of them into the same number This is the conuerse of the former As if the â—â—â—bers â— and G and D multiplyed into the number A doo produce the numbers E and F and G. Then the number composed of B C D. multiplyed into the number A shall produce the number composed of the numbers E F G. Which thing is easly proued by the 16. of the seuenth and by the former proposition ¶ The third Proposition That number which is produced of the multiplication of numbers how many soeuer into other numbers how many soeuer is equall to that number which is produced of the multiplication of the number composed of those first numbers into the number composed of these latter numbers As if the numbers A B C doo multiply the numbers D E F ech one eche other and if the numbers produced be added together Then I say that the number composed of the numbers produced is equall to the number produced of the number composed of the numbers A B C into the number composed of the numbers D E F. For by the former propositioÌ„ that which is produced of the number composed of A B C into D is equall to that which is produced of euery one of the sayd numbers into D and by the same reason that which is produced of the number composed of A B C into E is equal to that which is produced of euery one of the sayd numbers into E and so likewise that which is produced of the number composed of A B C into F is equall to that which is produced of euery one of the sayd numbers into F. But by the first of these propositions thâ—â— which is produced of the number composed of these numbers A B C into euery one of these numbers D E F is equall to that which is produced of the number composed into the number composed wherefore that is manifest which was required to be proued ¶ The fourth Proposition If a number be deuâ—â—â—d into partes how many soeuer that nuÌ„ber which is produced of the whole into him selfe is equall to that number which is produced of
E produced D wherfore A measureth D but it also measureth it not which is impossible Wherfore it is impossible to finde out a fourth number proportionall with these numbers A B C whensoeuer A measureth not D. ¶ The 20. Theoreme The 20. Proposition Prime numbers being geuen how many soeuer there may be geuen more prime numbers SVppose that the prime numbers geuen be A B C. Then I say that there are yet more prime numbers besides A B C. Take by the 38. of the seuenth the lest number whom these numbers A B C do measure and let the same be DE. And vnto DE adde vnitie DF. Now EF is either a prime number or not First let it be a prime number then are there found these prime numbers A B C and EF more in multitude then the prime numbers â—irst geuen A B C. But now suppose that EF be not prime Wherefore some prime number measureth it by the 24. of the seuenth Let a prime number measure it namely G. Then I say that G is none of these numbers A B C. For if G be one and the same with any of these A B C. But A B C measure the nuÌ„ber DE wherfore G also measureth DE and it also measureth the whole EF. Wherefore G being a number shall measure the residue DF being vnitieâ— which is impossible Wherefore G is not one and the same with any of these prime numbers A B C and it is also supposed to be a prime number Wherefore there are â—ound these prime numbers A B C G being more in multitude then the prime numbers geuen A B C which was required to be demonstrated A Corollary By thys Proposition it is manifest that the multitude of prime numbers is infinite ¶ The 21. Theoreme The 21. Proposition If euen nuÌ„bers how many soeuer be added together the whole shall be eueÌ„ SVppose that these euen numbers AB BC CD and DE be added together Then I say that the whole number namely AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an euen number therefore euery one of them hath an halfe Wherefore the whole AE also hath an halfe But an euen number by the definition is that which may be deuided into two equall partes Wherefore AE is an euen number which was required to be proued ¶ The 22. Theoreme The 22. Proposition If odde numbers how many soeuer be added together if their multitude be euen the whole also shall be euen SVppose that these odde numbers AB BC CD and DE being euen in multitude be added together Then I say that the whole AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an odde number is ye take away vnitie from euery one of them that which remayneth oâ— euery one of theÌ„ is an euen number Wherefore they all added together are by the 21. of the ninth an euen number and the multitude of the vnities taken away is euen Wherefore the whole AE is an euen number which was required to be proued ¶ The 23. Theoreme The 23. Proposition If odde numbers how many soeuer be added together and if the multitude of them be odde the whole also shall be odde SVppose that these odde numbers AB BC and CD being odde in multitude be added together Then I say that the whole AD is an odde number Take away from CD vnitie DE wherefore that which remaineth CE is an euen number But AC also by the 22. of the ninth is an euen number Wherfore the whole AE is an euen number But DE which is vnitie being added to the euen number AE maketh the whole AD aâ— odde number which was required to be proued◠¶ The 24. Theoreme The 24. Proposition If from an euen number be takeÌ„ away an euen number that which remaineth shall be an euen number SVppose that AB be an euen number and from it take away an euen number CB. Then I say that that which remayneth namely AC is an euen number For forasmuch as AB is an euen euen number it hath an halfe and by the same reason also BC hath an halfe Wherfore the residue CA hath an halfe Wherfore AC is an euen number which was required to be demonstrated ¶ The 25. Theoreme The 25. Proposition If from an euen number be taken away an odde number that which remaineth shall be an odde number SVppose that AB be an euen number and take away from it BC an odde number Then I say that the residue CA is an odde number Take away from BC vnitie CD Wherfore DB is an euen number And AB also is an euen number wherefore the residue AD is an euen number by the â—ormer proposition But CD which is vnitie being taken away from the euen nuÌ„ber AD maketh the residue AC an odde number which was required to be proued ¶ The 26. Theoreme The 26. Proposition If from an odde number be taken away an odde number that which remayneth shall be an euen number SVppose that AB be an odde number and from it take away an odde number BC. TheÌ„ I say that the residue CA is an euen number For forasmuch as AB is an odde number take away from it vnitie BD. Wherfore the residue AD is euen And by the same reason CD is an euen number wherfore the residue CA is an euen number by the 24. of this booke â— which was required to be proued ¶ The 27. Theoreme The 27. Proposition If from an odde number be taken a way an euen number the residue shall be an odde number SVppose that AB be an odde number and from it take away an euen number BC. Then I say that the residue CA is an odde number Take away froÌ„ AB vnitie AD. Wherfore the residue DB is an eueÌ„ number BC is by supposition euen Wherfore the residue CD is an euen number Wherefore DA which is vnitie beyng added vnto CD which is an euen number maketh the whole AC an â—dde number which was required to be proued ¶ The 28. Theoreme The 28. Proposition If an odde number multiplieng an euen number produce any number the number produced shall be an euen number SVppose that A being an odde number multiplieng B being an euen number do produce the number C. Then I say that C is an euen number For forasmuch as A multiplieng B produced C therfore C is composed of so many numbers equall vnto B as there be in vnities in A. But B is an euen nuÌ„ber wherfore C is composed of so many euen numbers as there are vnities in A. But if eueÌ„ numbers how many soeuâ—r be added together the whole by the 21. of the ninth is an euen number wherfore C is an euen number which was required to be demonstrated ¶ The 29. Theoreme The 29. Proposition Iâ— an odde number multiplying an
that the line EF is made equall to the line AD which is the diameter of the square ABCD of which square the line AB is a side it is certayne that the â—ide of a square is incoÌ„meÌ„surable in leÌ„gth to the diameter of the same square if there be yet founde any one superficies which measureth the two squares ABCD and EFGH as here doth the triangle ABD or the triangle ACD noted in the square ABCD or any of the foure triangles noted in the square EFGH as appeareth somwhat more manifestly in the second example in the declaration of the last definition going before the line EF is also a rational line Note that these lines which here are called rationall lines are not rational lines of purpose or by supposition as was the first rationall line but are rationall onely by reason of relation and comparison which they haue vnto it because they are commensurable vnto it either in length and power or in power onely Farther here is to be noted that these wordes length and power and power onely are ioyned onely with these wordeâ— commensurable or incommensurable and are neuer ioyned with these woordes rationall or irrationall So that no lines can be called rational in length or in power nor like wise can they be called irrationall in length or in power Wherin vndoubtedly Campanus was deceiued who vsing those wordes speaches indifferently caused brought in great obscuritie to the propositions and demonstrations of this boke which he shall easily see which marketh with diligence the demonstrations of Campanus in this booke 7 Lines which are incommensurable to the rationall line are called irrationall By lines incommensurable to the rationall line supposed in this place he vnderstandeth such as be incommensurable vnto it both in length and in power For there are no lines incommensurable in power onely for it cannot be that any lines should so be incommenâ—urable in power onely that they be not also incommensurable in length What so euer lines be incommeâ—surable in power the same be also incommensurable in length Neither can Euclide here in this place meane lines incommensurable in length onely for in the diffinition before he called them rationall lines nâ—ither may they be placed amongst irrationall lines Wherfore it remayneth that in this diffintion he speaketh onely of those lines which are incommensurable to the rationall line first geuen and supposed both in length and in power Which by all meanes are incommensurable to the rationall line therfore most aptly are they called irrationall lines This diffinition is easy to be vnderstanded by that which hath bene sayd before Yet for the more plainenes see this example Let the â—â—rst rationall line supposed be the line AB whose square or quadrate let be ABCD. And let there be geuen an other line EF which lâ—t be to the rationall line incommensurable in length and power so that let no one line measure the length of the two lines AB and EF and let the square of the line EF be EFGH Now if also there be no one superficies which measureth the two squares ABCD and EFGH as is supposed to be in this example theÌ„ is the line EF an irrationall line which word irrational As before did this word rational misliketh many learned in this knowledge of Geometry Flussates as he left the word rationall and in steade thereof vsed this word certaine so here he leaueth the word irrationall and vseth in place thereof this word vncertaine and euer nameth these lines vncertaine lines Petrus Montaureus also misliking the word irrationall would rather haue them to be called surd lines yet because this word irrationall hath euer by custome and long vse so generally bene receiuedâ— he vseth continually the same In Greeke such lines are called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 alogoi which signifieth nameles vnspeakeable vncertayne in determinate and with out proportion not that these irrationall lines haue no proportion at all either to the first rationall line or betwene them selues but are so named for that theyr proportions to the rationall line cannot be expressed in number That is vndoubtedly very vntrue which many write that their proportions are vnknowne both to vs and to nature Is it not thinke you a thing very absurd to say that there is any thing in nature and produced by nature to be hidde from nature and not to be knowne of nature it can not be sayd that their proportions are vtterly hidde and vnknowne to vs much lesse vnto nature although we cannot geue them their names and distinctly expresse them by numbers otherwise should Euclide haue taken all this trauell and wonderfull diligence bestowed in this bookeâ— in vaine and to no vseâ— in which he doth nothing ellâ— but teach the proprieties and passions of these irrationall linesâ— and sheweth the proportions which they haue the one to the other Here is also to be noted which thing also Tartalea hath before diligently notedâ— that Campanus and many other writers of Geometryâ— ouer much â—â—â—ed and were deceiued in that they wrote and taught that all these lines whose squares were not sâ—gnified and mought be expressed by a square number although they mighâ— by any other number as by 11. 12. 14. and such others not square numbers are irrationall lines Which is manifestly repugnant to the groundes and principles of Euclide who wil that all lines which are commensurable to the rationall line whether it be in length and power or in power onely should be rationall Vndoubtedly this hath bene one of the chiefest and greatest causes of the wonderfull confusion and darkenes of this booke which so hath tossed and tormoyled the wittes of all both writers and readers masters and scholers and so ouerwhelmed them that they could not with out infinite trauell and sweate attayne to the truth and perfect vnderstanding thereof 8 The square which is described of the rationall right line supposed is rationall Vntill this diffinition hath Euclide set forth the nature and proprietie of the first kinde of magnitude namely of lines how they are rationall or irrationall now he bâ—ginneth to â—hew how the second kinde of magnitudes namely superficies are one to the other rationall or irrationall This diffinition is very playne Suppose the line AB to be the rationall line hauing his parts and diuisions certaynely knowne the square of which line let be the square ABCD. Now because it is the square of the rationall line AB it is also called rationall and as the line AB is the first rationall line vnto which other lines compared are coumpted rationall or irrationall so is the quadrat or square thereof the â—irst rationall superficies vnto which all other squares or figures compared are coumpted and named rationall or irrationall 9 Such which are commensurable vnto it are rationall In this diâ—â—inition where it is sayd such as are commensurable to the square of the rationall line are not vnderstand onely other squares or
the line AB is rationall by the definition Wherfore by the definition also of rationall figures the parallelogramme CD shall be rationall Now resteth an other caâ—e of the thirde kinde of rationall lines commensurable in length the one to the other which are to the rationall line AB first set commensurable in power onely and yet are therfore rationall lines And let the lines CE and ED be coÌ„mensurable in length the one to the other Now then let the selfe same construction remaine that was in the former so that let the lines CE and ED be rationall commensurable in power onely vnto the line AB But let them be commensurable in length the one to the other Then I say that in this case also the parallelogramme CD is rationall First it may be proued as before that the parallelogramme CD is commensurable to the square DF. Wherfore by the 12. of this booke the parallelogramme CD shall be commensurable to the square of the line ABâ— But the square of the line AB is rationall Wherefore by the definition the parallelograÌ„me CD shall be also rationall This case is well to be noted For it serueth to the demonstration and vnderstanding of the 25. Proposition of this booke ¶ The 17. Theoreme The 20. Proposition If vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth thereof shall be a rationall line and commensurable in length vnto that line wherupon the rationall parallelogramme is applied SVppose that this rationall rectangle parallelogramme AC be applied vpon the line AB which let be rationall according to any one of the foresaid wayes whether it be the first rationall line set or any other line commensurable to the rationall line first set and that in length and in power or in power onely for one of these three wayes as was declared in the Assumpt put before the 19. Proposition of this booke is a line called rationall and making in breadth the line BC. Then I say that the line BC is rationall and commensurable in length vnto the line BA Desrcribe by the 46. of the first vpon the line BA a square AD. Wherfore by the 9. definitioÌ„ of the tenth the square AD is rationall But the parallelogramme AC also is rationall by supposition Wherefore by the conuersion of the definition of rationall figures or by the 12. of this booke the square DA is commensurable vnto the parallelogramme AC But as the square DA is to the parallelogramme AC so is the line DB to the line BC by the first of the sixt Wherfore by the 10. of the tenth the line DB is commensurable vnto the line BC. But the line DB is equall vnto the line BA Wherefore the line AB is coÌ„mensurable vnto the line BC. But the line AB is rationall Wherefore the line BC also is rationall and commensurable in length vnto the line BA If therefore vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth therof shall be a rationall line commensurable in length vnto that line whereupon the rationall parallelogramme is applied which was required to be demonstrated ¶ An Assumpt A line contayning in power an irrationall superficies is irrationall Suppose that the line AB coÌ„taine in power an irrationall superficies that is let the square described vpon the line AB be equall vnto an irrationall superficies Then I say that the line AB is irrationall For if the line AB be rationall theÌ„ shall the square of the line AB be also rationall For so was it put in the definitions But by supposition it is not Wherefore the line AB is irrationall A line therefore contayning in power an irrationall superficies is irrationall ¶ The 18. Theoreme The 21. Proposition A rectangle figure comprehended vnder two rationall right lines commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall is called a mediall line SVppose that this rectangle figure AC be comprehended vnder these rationall right lines AB and BC commensurable in power onely Then I say that the superficies AC is irrationall and the line which contayneth it in power is irrationall and is called a mediall line Describe by the 46. of the first vpon the line AB a square AD. Wherefore the square AD is rationall And forasmuch as the line AB is vnto the line BC incommensurable in length for they are supposed to be commensurable in power onely and the line AB is equall vnto the line BD therefore also the lineâ— BD is vnto the line BC incommensurable in length And 〈◊〉 â—hâ— lin◠〈…〉 is to the line â— C so 〈◊〉 the square AD to the parallelogramme AC by the first of the fiuâ— Wherefore by the 10. of the tenth the square DA is vnto the parallelogramme AC incommensurable But the square DA is rationall Wherefore the parallelogramme AC is irrationall Wherefore also the line that contayneth the superficies AC in power that is whose square is equall vnto the parallelogramme AC is by the Assumpt going before irrationall And it is called a mediall line for that the square which is made of it is equall to that which is contayned vnder the lines AB and BC and therefore it is by the second part of the 17. of the sixt a meane proportionall line betwene the lines AB and BC. A rectangle figure therefore comprehended vnder rationall right lines which are commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall and is called a mediall line At this Proposition doth Euclide first entreate of the generation and production of irrationall lines And here he searcheth out the first kinde of them which he calleth a mediall line And the definition therof is fully gathered and taken out of this 21. Proposition which is this A mediall line is an irrationall line whose square is equall to a rectangled figure contayned of two rationall lines commensurable in power onely It is called a mediall line as Theon rightly sayth for two causes first for that the power or square which it producethâ— is equall to a mediall superficies or parallelogramme For as that line which produceth a rationall square is called a rationall line and that line which produceth an irrationall square or a square equall to an irrationall figure generally is called an irrationall line so iâ— thaâ— line which produceth a mediall square or a square equall to a mediall superficies called by speciall name a mediall line Secondly it is called a mediall line because it is a meane proportionall betwene the two lines coÌ„mensurable in power onely which comprehend the mediall superficies ¶ A Corollary added by Flussates A rectangle parallelogramme contayned vnder a rationall line and an â—rrationall line is irrationall For if the line AB be rationall and
which is contained vnder the lines AD and DB twise by a rationall superâ—icies for either of them is rationall Wherfore also the squares of the lines AC and CB added together exceede the squares of the lines AD and DB added together by a rationall superficies when yet ech of them is a mediall superficies which is impossible Wherefore a line containing in power a rationall and a mediall is in one point onely deuided into his names which was required to be demonstrated ¶ The 35. Theoreme The 47. Proposition A line contayning in power two medials is in one point onely deuided into his names SVppose that AB being a line containing in power two medialls be deuided into his names in the point C so that let the lines AC and CB be incommensurable in power hauing that which is composed of the squares of the lines AC CB mediall and that also which is contained vnder the lines AC and CB mediall and moreouer incommensurable â—o that which is composed of the squares of the lines AC and CB. Then I say that the line AB can in no other point be deuided into his names but onely in the point C. For if it be possible let it be deuided into his names in the point D so that let not the line AC be one and the same that is equall with the line DB but by supposition let the line AC be the greater And take a rationall line EF. And by the 43. of the first vpon the line EF apply a rectangle parallelograÌ„me EG equall to that which is coÌ„posed of the squares of the lines AC and CB and likewise vpon the line HC which is equall to the line EF apply the parallelogramme HK equall to that which is contained vnder the lines AC and CB twise Wherefore the whole parallelogramme EK is equall to the square of the line AB Againe vpon the same line EF describe the parallelogramme EL equall to the squares of the lines AD and DB. Wherefore the residue namely that which is contayned vnder the lines AD and DB twise is equall to the parallelogramme remaining namely to MK And forasmuch as that which is coÌ„posed of the squares of the lines AC and CB is by supposition mediall therefore the parallelograÌ„me EG which is equall vnto it is also mediall and it is applied vpon the rationall line EF. Wherefore by the 22. of the tenth the line HE is rationall and incommensurable in length vnto the line EF. And by the same reason also the line HN is rationall and incommensurable in length to the same line EF. And forasmuch as that which is composed of the squares of the lines AC and CB is incommensurable to that which is contained vnder the lines AC and CB twise for it is supposed to be incommensurable to that which is coÌ„tained vnder the lines AC and CB once therefore the parallelogramme EG is incommensurable to the parallelogramme H â— Wherefore the line EH also is incommensurable in length to the line HN and they are rationall lines wherfore the lines EH and HN are rationall commensurable in power onely Wherefore the whole line EN is a binomiall line and is deuided into his names in the point H. And in like sort may we proue that the same binomiall line EN is deuided into his names in the point M and that the line EH is not one and the same that is equall with the line MN as it was proued in the end of the demonstration of the 44. of this booke Wherefore a binomiall line is deuided into his names in two sundry pointes which is impossible by the 42. of the tenth Wherefore a line containing in power two medials is not in sundry pointes deuided into his names Wherefore it is deuided in one point onely which was required to be demonstrated ¶ Second Definitions IT was shewed before that of binomiall lines there were sixe kindes the definitions of all which are here now set and are called second definitioÌ„s All binomiall lines as all other kindes of irrationall lines are coÌ„ceaued coÌ„sidered and perfectly vnderstanded onely in respecte of a rationall line whose partes as before is taught are certayne and knowen and may be distinctly expressed by number vnto which line they are compared Thys rationalâ— line must ye euer haue before your eyes in all these definitions so shall they all be â—asie inough A binomiall line ye know is made of two partes or names wherof the one is greater then the other Wherfore the power or square also of the one is greater then the power or square oâ— the other The three first kindes of binomiall lines namely the first the seconâ— the third are produced when the square of the greater name or part of a binomâ—all eâ—cedeth the square of the lesse name or part by the square of a line which is commâ—nsurable in length to it namely to the greater The three last kindes namely the fourth the â—iâ—t and the sixt are produced when the square of the greater name or part â—â—â—â—edeth the square of the lesse name or part by the square of a line incommensurable in length vnto it that is to the greater part A first binomiall line is whose square of the greater part exceedeth the square of tâ—e lesse part â—y the square of a line commensurable in length to the greater part and the greater part is also commensurable in length to tâ—e rationall line first set As lâ—t the raâ—ionâ—ll line first set be AB whose partes are distinctly knowen suppose also that the line CE be a binomiall line whose names or partes let be CD and DE. And let the square of the line CD the greater part excede the square of the line DE the lesse part by the square of the line FG which line FG let bâ—e commensuâ—able in length to the line CD which is the greater part of the binomiall line And moreouer let the line CD the greater paâ—t be commensurble in length to the rationall line first set namely to AB So by this dâ—â—inition the binomiall line CE is a first binomiall line A second binomiall line is when the square of the greater part exceedeth the square of the lesse part by the square of a line commensurable in length vnto it and the lesse part is commensurable in length to the rationall line first set As supposing euer the rationall line let CE be a binomiall line deuided in the poynt D. The square of whose greater part CD let exceede the square of the lesse part DE by the square of the line FG which line â—G let be coÌ„mensurable in length vnto the line CD tâ—e grâ—ater pâ—â—â— oâ— the binomiall line And let also the line DE the lesse part of the binomiall line be commensuâ—able in lâ—ngth to the rationall line first set AB So by this definition the binomiall line CE is a second binomiall line A third binomiall
shall in like sort by the 14. of the tenth be in power more then the line DF by the square of a line incommensurable in length to the line CF and then if the line AE be commensurable in length to the rationall line the line CF shall also in like sort be commensurable in length to the same rationall line and so either of the lines AB and CD is a fourth residuall line And if the line BE be coÌ„meÌ„surable in leÌ„gth to the rationall line the line DF shall also be coÌ„mensurable in leÌ„gth to the same line and so either of the lines AB CD is a â—iâ—t residuall line And if neither of the lines AE nor BE be commensurable in length to the rationall line in like sort neither of the lines CF nor DF shall be coÌ„mensurable in leÌ„gth to the same rational line And so either of the lines AB CD is a sixt residual line Whereâ—ore the line CD is a residuall line of the selfe same order that the line AB is A line therfore commensurable in length to a residuall line is it selfe also a residuall line of the selfe same order which was required to be proued As before touching binomiall lines so also touching residuall lines this is to be noted that a line commensurable in length to a residuall line is alwayes a residuall line of the selfe same order that the residuall line is vnto whom it is coÌ„mensurable as hath before in this 103. propositioÌ„ bene proued But if a line be coÌ„mensurable in power only to a residuall lineâ— then followeth it not yea it is impossible that that line should be a residuall of the self same order that the residual line is vnto whom it is commensurable in power onely Howbeit those two lines shall of necessitie be both either of the three first orders of residâ—â—ll lines or of the three last orders which is not hard to proue if ye marke diligently the former demonstration and that which was spoken of binomiall lines as touching this matter ¶ The 80. Theoreme The 104. Proposition A line commensurable to a mediall residuall line is it selfe also a medial residuall line and of the selfe same order SVppose that AB be a mediall residuall line vnto whome let the line CD be commensurable in length and in power or in power onely Then I say that CD is also a mediall residuall line and of the selfe same order For forasmuch as the line AB is a mediall residuall line let the line conueniently ioyned vnto i◠〈◊〉 BE wherefore the lines AE and BE are mediall commensurable in power onely As AB is to CD so by the 22. of the sixth let BE be to DF. And in like sort as in the former so also in this may we proue that the line AE is commensurable in length and in power or in power onely vnto the line CF the line BE 〈◊〉 the line DF. Wherefore by the 23. of the tenth 〈◊〉 line CF is a mediall line and the line DF is also a mediall line for that it is commensurable to the mediall line BE. And in like sort the lines CF and DF are commensurable in power onely for that they haue the selfe same proportioÌ„ the one to the other that the lines AE and EB haue which are commensurable in power onely Wherefore the line CD is a mediall residuall line I say moreouer that it is of the selfe same order that the line AB is For for that as the line AE is to the line BE so is the line CF to the line DF. But as the line AE is to the line BE so is the square of the line AE to the parallelogramme contayned vnder the lines AE and BE by the first of the sixth and as the line CF is to the line DF so is the square of the line CF to the parallelogramme contayned vnder the lines CF and DF. Wherefore as the square of the line AE is to the parallelogramme contayned vnder the lines AE and BE so is the square of the line CF to the parallelogramme contayned vnder the lines CF and DF. Wherefore alternately as the square of the line AE is to the square of the line CF so is the parallelogramme contayned vnder the lines AE and BE to the parallelogramme contained vnder the â—ines CF and DF. But the square of the line AE is commensurable to the square of the line CF for the line AE is commensurable to the line CF Wherefore also the parallelogramme contayned vnder the lines AE and BE is commensurable to the parallelogramme contayned vnder the lines CF and DF. Wherefore if the parallelogramme contayned vnder the lines AE and EB be rationall the parallelogramme also contayned vnder the lines CF and FD shall be rationall And then either of the lines AB and CD is a first mediall residuall line But if the parallelogramme contayned vnder the lines AE and BE be mediall the parallelogramme also contayned vnder the lines CF and FD shall be also mediall by the corollary of the 23. of the teÌ„th and so either of the lines AB and CD is a second mediall residuall line Wherefore the line CD is a mediall residuall line of the selfe same order that the line AB is A line therefore commensurable to a mediall residuall line is it selfe also a mediall residuall line of the selfe same order which was required to be demonstrated This Theoreme is vnderstanded generally that whether a line be commensurable in length in power or in power onely to a mediall residuall line it is it selfe also a mediall residuall line and of the selfe same order which thing also is to be vnderstanded of the three Theoremes which follow An other demonstration after Campane Suppose that A be a mediall residuall line vnto whome let the line B be commensurable in length or in power onely And take a rationall line CD vnto which apply the parallelogramme CE equall to the square of the line A and vnto the line FE which is equall to the line CD apply the parallelogramme Fâ— equall to the square of the line B. Now then the parallelogrammes CE and FG shall be commensurable for that the lines A B are commensurable in power wherefore by the 1. of the sixth and 10. of this booke thâ— lines DE and FG are commensurable in length Now then if A be a first mediall residuall line then is the line DE a second residuall line by the 98. of this booke and if the line A be a sâ—cond mediall residuall line then is the line â— â— a third residuall line by the 99. of this booke But if DE be a second residuall line Gâ— also shall be a second residuall line by the â—03 of this boke And if DE be a third residuall line GE also shall by the same be also a third residuall line Wherefore it followeth by the 9â— and 93. of this booke that B is either a first
measureth the square number produced of EF. Wherefore also by the 14. of the eight the number G measureth the number EF and the number G also measureth it selfe Wherefore the number G measureth these numbers EF and G when yet they are prime the one to the other which is impossible Wherefore the diameter A is not commensurable in length to the side B. Wherefore it is incommensurable which was required to be demonstrated An other demonstration after Flussas Suppose that vppon the line AB be described a square whose diameter let be the line AC Then I say that the side AB is incommensurable in length vnto the diameter AC Forasmuch as the lines AB and BC are equall therefore the square of the line AC is double to the square of the line AB by the 47. of the first Take by the 2. of the eight nuÌ„bers how many soeuer in continuall proportion froÌ„ vnitie and in the proportion of the squares of the lines AB and AC Which let be the numbers D E F G. And forasmuch as the first from vnitie namely E is no square number for that it is a prime number neither is also any other of the sayd numbers a square number except the third from vnitie and so all the rest leuing one betwene by the 10. of the ninth Wherefore D is to E or E to F or F to G in that proportion that a square number is to a number not square Wherefore by the corrollary of the 25. of the eight they are not in that proportion the one to the other that a square number is to a square number Wherefore neither also haue the squares of the lines AB and AC which are in the same proportion that porportion that a square number hath to a square number Wherefore by the 9. of this booke their sides namely the side AB and the diameter AC are incommensurable in length the one to the other which was required to be proued This demonstration I thought good to adde for that the former demonstrations seme not so full and they are thought of some to be none of Theons as also the proposition to be none of Euclides Here followeth an instruction by some studious and skilfull Grecian perchance Theon which teacheth vs of farther vse and fruite of these irrationall lines Seing that there are founde out right lines incommensurable in length the one to the â—ther as the lines A and B there may also be founde out many other magnitudes hauing leÌ„gth and breadth such as are playne superficieces which shal be incommeÌ„surable the one to the other For if by the 13. of the sixth betwene the lines A and B there be taken the meane proportionall line namely C then by the second corrollary of the 20. of the sixth as the line A is to the line B so is the figure described vpon the line A to the figure described vpon the line C being both like and in like sort described that is whether they be squares which are alwayes like the one to the other or whether they be any other like rectiline figures or whether they be circles aboute the diameters A and C. For circles haue that proportion the one to the other that the squares of their diameters haue by the 2. of the twelfth Wherfore by the second part of the 10. of the tenth the â—igures described vpon the lines A and C being like and in like sort described are incommensurable the one to the other Wherfore by this meanes there are founde out superficieces incommensurable the one to the other In like sort there may be founde out figures coÌ„meÌ„surable the one to the other if ye put the lines A and B to be coÌ„mensurable in leÌ„gth the one to the other And seing that it is so now let vs also proue that euen in soliâ—es also or bodyes there are some commensurable the one to the other and other some incommensurable the one to the other For if from eche of the squares of the lines A and B or from any other rectiline figures equal to these squares be â—rected solides of equall altiâ—ude whether those solides be compâ—sed of equidistant supersicieces or whether they be pâ—ramids or prismes thosâ— solides sâ— erâ—câ—ed shal be in that proportioÌ„ the one to the other that theyr bases are by the 32. oâ— the eleuenth and 5. and 6. of the twelfth Howbeit there is no such proposition concerning prismes And so if the bases of the solides bâ— commensurable the one to the other the solides also shall be commensurable the one to the other and if the bases be incommensurable the one to the other the solides also shall be incommensurable the one to the other by the 10. of the tenth And if there be two circles A and B and vpon ech of the circles be erected Cones or Cilinders of equal altitude those Cones Cilinders sâ—all be in that proportion the one to the other that the circles are which are their bases by the 11. of the twelfth and so if the circles be commensurable the one to the other the Cones and Cilinders also shall be commensurable the one to the other But if the circles be incoÌ„mensurable the one to the other the Cones also and Cilinders shal be incoÌ„mensurable the one to the other by the 10. of the tenth Wherefore it is manifest that not onely in lines and superâ—icieces but also in solides or bodyes is found commensurabilitie or incommensurability An aduertisement by Iohn Dee Although this proposition were by Euclide to this booke alotted as by the auncient grecian published vnder the name of Aristoteles 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 it would seme to be and also the property of the same agreable to the matter of this booke and the proposition it selfe so famous in Philosophy and Logicke as it was would in maner craue his elemeÌ„tal place in this teÌ„th boke yet the dignitie perfectionâ— of Mathematicall Method can not allow it here as in due order following But most aptly after the 9. propositioÌ„ of this booke as a Corrollary of the last part thereof And vndoubtedly the propoâ—itioÌ„ hath for this 2000. yeares bene notably regarded among the greke Philosophers and before Aristotles time was concluded with the very same inconuenience to the gaynesayer that the first demonstration here induceth namely Odde number to be equall to euen as may appearâ— in Aristotles worke named Analitica prima the first booke and 40. chapter But els in very many places of his workes he maketh mention of the proposition Euident also it is that Euclide was about Aristotles time and in that age the most excellent Geometrician among the Grekes Wherefore seing it was so publike in his time so famous and so appertayning to the property of this booke it is most likely both to be knowne to Euclide and also to haue bene by him in apt order placed But of the disordring of it can remayne no doubt if ye consider in Zamberts translation
by his motion described the round Conical superficies about the Cone And as the circuÌ„fereÌ„ce of the semicircle described the round sphericall superficies about the Sphere In this example it is the superficies described of the line DC By this definition it is playne that the two circles or bases of a cilinder are euer equall and parallels for that the lines moued which produced them remayned alwayes equall and parallels Also the axe of a cilinder is euer an erected line vnto either of the bases For with all the lines described in the bases and touching it it maketh right angles Campane Vitellâ—o with other later writers call this solide or body a round Columnâ— or piller And Campane addeth vnto this definition this as a corrollary That of a round Columne of a Sphere and of a circle the ceÌ„tre is one and the selfe same That is as he him selfe declareth it proueth the same where the Columne the Sphere and the circle haue one diameter 20 Like cones and cilinders are those whose axes and diameters of their bases are proportionall The similitude of cones and cilinders standeth in the proportion of those right lines of which they haue their originall and spring For by the diameters of their bases is had their length and breadth and by their axe is had their heigth or deepenes Wherefore to see whether they be like or vnlike ye must compare their axes together which is their depth and also their diameters together which is thier length breadth As if the axe â—G of the cone ABC be to to the axe EI of the cone DEF as the diameter AC of the cone ABC is to the diameter DF of the cone DEF then aâ—e the cones ABC and DEF like cones Likewise in the cilinders If the axe LN of the cilinder LHMN haue that proportion to the axe OQ of the cilinder ROPQ which the diameter HM hath to the diameter RP then are the cilinders HLMN and ROPQ like cilinders and so of all others 21 A Cube is a solide or bodely figure contayned vnder sixe equall squares As is a dye which hath sixe sides and eche of them is a full and perfect square as limites or borders vnder which it is contayned And as ye may conceiue in a piece of timber contayning a foote square euery way or in any such like So that a Cube is such a solide whose three dimensions are equall the length is equall to the breadth thereof and eche of them equall to the depth Here is as it may be in a playne superficies set an image therof in these two figures wherof the first is as it is commonly described in a playne the second which is in the beginning of the other side of this leafe is drawn as it is described by arte vpoÌ„ a playne superficies to shew somwhat bodilike And in deede the latter descriptioÌ„ is for the sight better theÌ„ the first But the first for the demoÌ„strations of Euclides propositions in the fiue bookes following is of more vse for that in it may be considered and sene all the fixe sides of the Cube And so any lines or sections drawen in any one of the sixe sides Which can not be so wel sene in the other figure described vpon a playnd And as touching the first figure which is set at the ende of the other side of this leafe ye see that there are sixe parallelogrammes which ye must conceyue to be both equilater and rectangle although in dede there can be in this description onely two of them rectangle they may in dede be described al equilater Now if ye imagine one of the sixe parallelogrammes as in this example the parallelogramme ABCD to be the base lieng vpon a ground playne superfices And so conceiue the parallelogramme EFGH to be in the toppe ouer it in such sort that the lines AE CG DH BF may be erected perpendicularly from the pointes A C B D to the ground playne superficies or square ABCD. For by this imagination this figure wil shew vnto you bodilike And this imagination perfectly had wil make many of the propositions in these fiue bookes following in which are required to be described such like solides although not all cubes to be more plainly and easily conceiued In many examples of the Greeke and also of the Latin there is in this place set the diffinition of a Tetrahedron which is thus 22 A Tetrahedron is a solide which is contained vnder fower triangles equall and equilater A forme of this solide ye may see in these two examples here set whereof one is as it is commonly described in a playne Neither is it hard to conceaue For as we before taught in a Pyramis if ye imagine the triangle BCD to lie vpon a ground plaine superficies and the point A to be pulled vp together with the lines AB AC and AD ye shall perceaue the forme of the Tetrahedron to be contayned vnder 4. triangles which ye must imagine to be al fower equilater and equiangle though they can not so be drawen in a plaine And a Tetrahedron thus described is of more vse in these fiue bookes following then is the other although the other appeare in forme to the eye more bodilike Why this definition is here left out both of Campane and of Flussas I can not but maruell considering that a Tetrahedron is of all Philosophers counted one of the fiue chiefe solides which are here defined of Euclide which are called coÌ„monly regular bodies without mencion of which the entreatie of these should seeme much maimed vnlesse they thought it sufficiently defined vnder the definition of a Pyramis which plainly and generally taken includeth in deede a Tetrahedron although a Tetrahedron properly much differeâ—h from a Pyramis as a thing speciall or a particular from a more generall For so taking it euery Tetrahedron is a Pyramis but not euery Pyramis is a Tetrahedron By the generall definition of a Pyramis the superficieces of the sides may be as many in number as ye list as 3.4 5.6 or moe according to the forme of the base whereon it is set whereof before in the definition of a Pyramis were examples geuen But in a Tetrahedron the superficieces erected can be but three in number according to the base therof which is euer a triangle Againe by the generall definition of a Pyramiâ— the superficieces erected may ascend as high as ye list but in a Tetrahedron they must all be equall to the base Wherefore a Pyramis may seeme to be more generall then a Tetrahedron as before a Prisme seemed to be more generall then a Parallelipipedon or a sided Columne so that euery Parallelipipedon is a Prisme but not euery Prisme is a Parallelipipedon And euery axe in a Sphere is a diameter but not euery diameter of a Sphere is the axe therof So also noting well the definition of a Pyramis euery Tetrahedron may be called a Pyramis
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
fourth The second and third are to be found which may betwene A B be two meanes in continuall propoâ—tion as now suppose such two lines found and let them be C and D. Wherefore by Euclides Corollary as A is to B if A were taken as first so shall the Parallelipipedon described of A be to the like Parallelipipedon and in like sort described of C being the second of the fower lines in continuall proportion it is to weâ—e A C D and B. Or if B shall be taken as first and that thus they are orderly in continuall proportion B D C A then by the sayd Corollary as B is to A so shall the Parallelipipedon described of B be vnto the like Parallelipipedon and in like sort described oâ— D. And vnto a Parallelipipedon of A or B at pleasure described may an other of C or D be made like and in like sort situated or described by the 27. of this eleuenth booke Wherefore any two right lines being geueÌ„ c which was required to be done Thus haue I most briefly brought to your vnderstanding if first B were double to A then what Parallelipipedon soeuer were described of A the like Parallelipipedon and in like sort described of C shall be double to the Parallelipipedon described of A. And so likewise secondly if A were double to B the Parallelipipedon of D shoulde be double to the like of B described both being like situated Wherefore if of A or B were Cubes made the Cubes of C and D are proued double to them as that of C to the Cube of A and the Cube of D to the Cube of B in the second case And so of any proportion els betwene A and B. Now also do you most clerely perceaue the Mathematicall occasion whereby first of all men Hippocrates to double any Cube geuen was led to the former Lemma Betwene any two right lines geuen to finde two other right lines which shall be with the two first lines in continuall proportion After whose time many yeares diuine Plato Heron Philo Appollonius Diâ—â—lâ—â— Pappus Sporus Menechâ—us Archytas Tarentinus who made the wodden doue to slye Eratoâ—â—hene Nicomedes with many other to their immortall fame and renowme published diuers their witty deuises methods and engines which yet are extant whereby to execute thys Problematicall Lemma But not withstanding all the trauailes of the â—oresayd Philosophers and Mathematiciens yea and all others doinges and contriuinges vnto this day about the sayd Lemma yet there remaineth sufficient matter Mathematically so to demonstrate the same that most exactly readily it may also be Mechanically practisâ—d that who soeuer shall achieue that feate shall not be counted a second Archimedes but rather a perâ—les Mathematicien and Mathematicorum Princeps I will sundry wayes in my briefe additions and annotations vpon Euclide excite you thereto yea and bring before your eyes sundry new wayes by meinuented and in this booke so placed as matter thereof to my inuentions appertayning may geue occasion Leauing the farther full absolute my concluding of the Lemma to an other place and time which will now more coÌ„pendiously be done so great a part therof being before hand in thys booke published ¶ A Corollary added by Flussas Parallelipipedons consisting vpon equall bases are in proportion the one to the other as their altitudes are For if those altitudes be cut by a plaine superficies parallel to the bases the sections shall be in proportion the one to the other as the sections of the bases cut by the 25. of this booke Which sections of the bases are the one to the other in that proportion that their sides or the altitudes of the solides are by the â—irst of the sixt Wherefore the solides are the one to the other as their altitudes are But if the bases be vnlike the selfe same thing may be proued by the Corollary of the 25. of this booke which by the 25. Proposition was proued in like bases ¶ The 29. Theoreme The 34. Proposition In equall Parallelipipedons the bases are reciprokall to their altitudes And Parallelipipedons whose bases are reciprokall to their altitudes are equall the one to the other But now againe suppose that the bases of the Parallelipipedons AB and CD be reciprokall to their altitudes that is as the base EH is to the base NP so let the altitude of the solide CD be to the altitude of the solide AB Then I say that the solide AB is equall to the solide CD For againe let the standing lines be erected perpendicularly to their bases And now if the base EH be equall to the base NP but as the base EH is to the base NP so is the altitude of the solide CD to the altitude of the solide AB Wherefore the altitude oâ— the solide CD is equall to the altitude of the solide AB But Parallelipipedons consisting vpon equall bases and vnder one and the selfe same altitude are by the 31. of the eleuenth equall the one to the other Wherefore the solide AB is equall to the solide CD But now suppose that the base EH be not equall to the base NP but let the base EH be the greater Wherefore also the altitude of the solide CD that is the line CM is greater then the altitude of the solide AB that is then the line AG. Put againe by the 3. of the first the line CT equall to the line AG and make perfecte the solide CZ Now for that as the base EH is to the base NP so is the line MC to the line AG. But the line AG is equall to the line CT Wherefore as the base EH is to the base NP so is the line CM to the line CT But as the base EH is to the base NP so by the 32. of the eleuenth is the solide AB to the solide CZ â—or the solides AB and CZ are vnder equall altitudes And as the line CM is to the line CT so by the 1. of the sixt is the base MP to the base Pâ—T and by the 32. of the eleuenth the solide CD to the solide CZ Wherefore also by the 11. and 9. of the fift as the solide AB is to the solide CZ so is the solide CD to the solide CZ Wherfore either of these solides AB and CD haue to the solide CZ one and the same proportion Wherefore by the 7. of the fift the solide AB is equall to the solide CD which was required to be demonstrated Againe suppose that the bases of the Parallelipipedons AB and CD be reciprokall to their altitudes that is as the base EH is to the base NP so let the altitude of the solide CD be to the altitude of the solide AB Then I say that the solide AB is equall to the solide CD For the same order of construction remayning for that as the base EH is to the base NP so is the altitude of the solide CD to the altitude
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
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
triangles ABC ACD CDE and likewise the base FGHKL into these triangles FGH FHL and HKL And imagine that vpon euery one of those triangles be set a pyramis of equall altitude with the two pyramids put at the beginning And for that as the triangle ABC is to the triangle ADC so is the pyramis ABCM to the pyramis ADCM by the 5. of this boke Wherfore by composition by the 18. of the fift as the fower sided figure ABCD is to the triangle ACD so is the pyramis ABCDM to the pyramis ACDM But as the triangle ACD is to the triangle CDE so is the pyramis ACDM to the pyramis CDEM Wherefore of equalitie by the 22. of the fift as the base ABCD is to thâ— base CDE so is the pyramis ABCDM to the pyramis CDEM Wherfore againe by composition by the 18. of the fift as the base ABCDE is to the base CDE so is the pyramis ABCEDM to the pyramis CDEM And by the same reason also as the base FGHKL is to the base HKL so is the pyramis FGHKLN to the pyramis HKLN And forasmuch as there are two pyramids CDEM and HKLN hauing triangles to their bases and being vnder one and the selfe same altitude therefore by the 5. of the twelfth as the base CDE is to the base HKL so is the pyramis CDEM to the pyramis HKLN Now for that as the base ABCED is to the base CDE so is the pyramis ABCEDM to the pyramis CDEM But as the base CDE is to the base HKL so is the pyramis CDEM to the pyramis HKLN Wherefore of equalitie by the 22. of the fift as the base ABCED is to the base HKL so is the pyramis ABCEDM to the pyramis HKLN But also as the base HKL is the base FGHKL so is the Pyramis HKLN to to the pyramis FGHKLN Wherefore againe of equalitie by the 22. of the fift as the base ABCED is to the base FGHKL so is the pyramis ABCEDM to the pyramis FGHKLN Wherefore pyramids consisting vnder one and the selfe same altitude and hauing Polygonon figures to their bases are in that proportion the one to the other that their bases are which was required to be proued The 7. Theoreme The 7. Proposition Euery prisme hauing a triangle to his base may be deuided into three pyramids equall the one to the other hauing also triangles to their bases SVppose that ABCDEF be a prisme hauing to his base the triangle ABC and the opposite side vnto it the triangle DEF Then I say that the prisme ABCDEF may be deuided into three piramids equall the one to the other and hauing triangles to their bases Draw these right lines BD EC and CD And forasmuch as ABED is a parallelogramme and his diameter is the line BD therefore the triangle ABD is equall to the triangle EDB Wherefore also the pyramis whose base is the triangle ABD and toppe the poynt C is equall to the pyramis whose base is the triangle EDB toppe the point C by the 5. of this booke But the pyramis whose base is the triangle EDB and toppe the poynt C is one and the same which the pyramis whose base is the triangle EBC and toppe the poynt D for they are comprehended of the selfe same playne superficieces namely of the triangles BDEDEC DBC and EBC Wherefore also the pyramis whose base is the triangle ABD and toppe the poynt C is equall to the pyramis whose base is the triangle EBC and toppe the point D. Againe forasmuch as BCFE is a parallelogramme and the diameter thereof is EC therefore the triangle ECF is equall to the triangle CBE Wherefore also the pyramis whose base is the triangle EBC and toppe the poynt D is equall to the pyramis whose base is the triangle ECF and toppe the poynt D by the 5. of this booke But the pyramis whose base is the triangle BEC and toppe the poynt D is proued to be equall to the pyramis whose base iâ— the triangle ABD and toppe the poynt C. Wherfore also the pyramis whose base is the triangle CEF and toppe the poynt D is equall to the pyramis whose base is the triangle ABD toppe the poynt C. Wherefore the prisme ABDEF is deuided into three equall pyramids hauing triangles to their bases And forasmuch as the pyramis whose base is the triangle ABD and toppe the poynt C is one the selfe same with the pyramis whose base is the triangle CAB toppe the poynt D for they are contayned vnder the selfe same playne superficieces but it hath bene proued that the pyramis whose base is the triangle ABD and toppe the poynt C is the third pyramis of the prisme whose base is the triangle ABC aâ—d the opposite side vnto it the triangle DEF Wherefore the pyramis whose base is the triangle ABC and toppe the poynt D is the third pyramis of the prisme whose base is the triangle ABC and opposite side the triangle DEF Wherefore euery prisme hauing a triangle to his base may be deuided into three pyramids equall the one to the other hauing also triangles to their bases which was required to be proued ¶ Corollary Hereby it is manifest that euery pyramis is the third part of a prisme hauing one and the same base with it and also being vnder the selfe same altitude with it For if the base of the prisme be any other rectiline figure theÌ„ a triangle that also may be deuided into prismes which shal haue triangles to their bases Here Campane and Flussas adde certayne Corollaryes First Corollary Euery Prisme is treble to the Piramis which hath the selfe same triangle to his base that the Prisme hath and the selfe same altitude As it is manifest by this propoâ—â—tion where the Prisme is deuided into three equall Pyramids of which two are vpoÌ„ one and the selfe same base and vnder one and the selfe same altitude But if the Prisme haue to his â—ase a parallelogramme and if the Pyramis haue to his base the halfe of the same parallelogramme and their altitudes be equall then agayne the Pyramis shal be the third part of the Prisme For it was manifest by the 40. of the â—leuâ—th that Prismes being vnder equall altitudes and the one hauing to his base a triangle and the other a parallelogramme double to the same triangle are equall the one to the other Wherof followeth the former conclusion Second Corollary If there be many Prismes vnder one and the same altitude and hauing triangles to their bases and if the triangular bases be so ioyned together vpon one and the same playne that they compose a Poligonon figure A pyramis set vpon that base being a Poligonon figure and vnder the same altitude is the third part of that solide which is composâ—d of all the Prismes added together For forasmuch as euâ—ry one of the Prismes which hath to his base a triangle to euery one of the Pyramids set vpon the same base the altitude being alwayes one and the
same is treble it is manifest by the 12. of the fiueth that all the Prismes are to all the Pyramids treble Wherefore Parallelipipedons are treble to Pyramids set vpon the selfe same base with them and vnder the same altitude for that they contayne two Prismes Third Corollary If two Prismes being vnder one and the selfe same altitude haue to their bases either both triangles or both parallelogrammes the Prismes are the one to the other as their bases are For forasmuch as those Prismes are equemultiqlices vnto the Pyramâ—ds vpon the selfe same bases and vnder the same altitude which Pyramids are in proportion as their bases it is manifest by the 15. of the fift that the Prismes are in the proportion of the bases For by the former Corollary the Prismes are treble to the Pyramids sâ—t vpon the triangular bases Fourth Corollary Prismes are in sesquealtera proportion to Pyramids which haue the selfe same quadrangled base that the Prismes haue and are vnder the selfe same altitude For that Pyramis contayneth two Pyramids set vpon a triangular base of the same Prisme for it is proued that that Prisme is treble to the Pyramis which is set vpon the halfe of his quadrangled base vnto which the other which is set vpon the whole base is double by the sixth of this booke Fiueth Corollary Wherefore we may in like sort conclude that solides mencioned in the second Corollary which solids Campane calleth sided Columnes being vnder one and the selfe same altitude are in proportion the one to the other as their bases which are poligonon figures For they are in the proportion of the Pyramids or Prismes set vpon the selfe same bases and vnder the selfe same altitude that is they are in the proportioÌ„ of the bases of the sayde Pyramids or Prismes For those solids may be deuided into Prismes hauing the selfe same altitude when as their opposite bases may be deuided into triangles by the 20 of the sixth Vpon which triangles Prismes beyng set are in proportion as their bases By this 7. Proposition it playnely appeareth that â—uâ—lide as it was before noted in the diffinitionâ—â— vnder the diffinition of a Prisme comprehended also those kinds of solids which Campane calleth sided Columnes For in that he sayth Euery Prisme hauing a triangle to his base may be deuidedâ— c. he neded not taking a Prisme in that sense which Campane and most men take it to haue added that particle hauing to his base a triangle For by their sense there is no Prisme but it may haue to his base a triangleâ— and so it may seeme that Euclide ought without exceptioâ— haue sayd that euery prisme whatsoeuer may be deuided into three pyramids equall the one to the other hauing also triangles to â—heir bases For so do Campane and Flussas put the proposition leauing out the former particle hauing to his base a triangle which yet is red in the Greeke copye not leâ—t out by any other interpreters knowne abroade except by Campane and Flussas Yea and the Corollary following of this proposition added by Theon or Euclide and ameÌ„ded by M. Dee semeth to confirme this sence Of this â—s 〈◊〉 made manifest that euery pyramis is the third part of the prisme hauing the same base with it and equall altitude For and if the base of the prisme haue any other right lined figure then a triangle and also the superficies opposite to the base the same figure that prisme may be deuided into prismes hauing triangled bases and the superficieces to those bases opposite also triangled a â—â—ike and equally For there as we see are put these wordes â—or and if the base of the prisme be any other right lined figureâ— c. whereof a man may well inferre that the base may be any other rectiline figure whatsoeuer not only a triangle or a parallelogramme and it is true also in that sence as it is plaine to see by the second corollary added out of Flussas which corollary as also the first of his corollaries is in a maner all one with the Corollary added by Theon or Euclide Farther Theon in the demonstration of the 10. proposition of this booke as we shall aâ—terward see most playnely calleth not onely sided columnes prismes but also parallelipipedons And although the 40. proposition of the eleuenth booke may seme hereunto to be a lâ—t For that it can be vnderstanded of those prismes onely which haue triangles to their like equall opposite and parallel sides or but of some sided columnes and not of all yet may that let be thus remoued away to say that Euclide in that propositioÌ„ vsed genus pro specie that is the generall word for some special kinde therof which thing also is not rare not only with him but also with other learned philosophers Thus much I thought good by the way to note in farther defence of Euclide definition of a Prisme The 8. Theoreme The 8. Proposition Pyramids being like hauing triangles to their bases are in treble proportion the one to the other of that in which their sides of like proportion are SVppose that these pyramids whose bases are the triangles GBC and HEF and toppes the poyntes A and D be like and in like sort described and let AB and DE be sides of like proportion Then I say that the pyramis ABCG is to the pyramis DEFH in treble proportioÌ„ of that in which the side AB is to the side DE. Make perfect the parallelipipedons namely the solides BCKL EFXO And forasmuch as the pyramis ABCG is like to the pyramis DEFH therfore the angle ABC is equall to the angle DEF the angle GBC to the angle HEF and moreouer the angle ABG to the angle DEH and as the line AB is to the line DE so is the line BC to the line EF and the line BG to the line EH And for that as the line AB is to the line DE so is the line BC to the line EF and the sides about the equall angles are proportionall therefore the parallelogramme BM is like to the parallelograÌ„me EP and by the same reason the parallelogramme BN is like to the parallelogramme ER and the parellelogramme BK is like vnto the parallelogramme EX Wherefore the three parallelogrammes BM KB and BN are like to the three parallelogrammes EP EX and ER. But the three parallelogrammes BM KB and BN are equall and like to the three opposite parallelogrammes and the three parallelogrammes EP EX and ER are equall and like to the three opposite parallelogrammes Wherefore the parallelipipedons BCKL and EFXO are comprehended vnder playne superficieces like and equall in multitude Wherefore the solide BCKL is like to the solide EFXO But like parallelipipedons are by the 33. of the eleuenth in treble proportion the one to the other of that in which side of like proportion is to side of like proportion Wherefore the solide BCKL is to the solide EFXO in treble
proportion of that in which the side of like proportion AB is to the side of like proportion DE. But as the solide BCKL is to the solide EFXO so is the pyramis ABCG to the pyramis DEFH by the 15. of the fifth for that the pyramis is the sixth part of this solide for the prisme being the halfe of the parallelipipedon is treble to the pyramis Wherefore the pyramis ABCG is to the pyramis DEFH in treble proportion of that in which the side AB is to the side DE. Which was required to be proued Corollary Hereby it is manifest that like pyramids hauing to their bases poligonon figures are in treble proportion the one to the other of that in which side of like proportion is to side of like proportion For if they be deuided into pyramids hauing triangles to their bases for like poligonoÌ„ figures are deuided into like triangles and equal in multitude and the sides are of like proportion as one of the pyramids of the one hauing a triangle to his base is to one of the pyramids of the other hauing also a triangle to his base so also are all the pyramids of the one pyramis hauing triangles to their bases to all the pyramids of the other pyramis hauing also triangles to their bases That is the pyramis hauing to his base a poligonoÌ„â—igure to the pyramis hauing also to his base a poligonoÌ„â—igure But a pyramis hauing a triangle to his base is to a pyramis hauing also a triangle to his base being like vnto it in treble proportioÌ„ of that in which side of like proportioÌ„ is to side of like proportioÌ„ Wherfore a pyramis hauing to his base a poligonoÌ„ figure is to a pyramis hauing also a poligonon figure to his base the sayd pyramids being like the one to the other in treble proportion of that in which side of like proportion is to side of like proportion Likewise Prismes and sided columnes being set vpon the bases of those pyramids and vnder the same altitude forasmuch as they are equemultiplices vnto the pyramids namely triples by the corollary of the 7. of this booke shal haue the â—ormer porportion that the pyramids haue by the 15 of the fifth and therefore they shall be in treble proportion of that in which the sides of like proportion are ¶ The 9. Theoreme The 9. Proposition In equall pyramids hauing triangles to their bases the bases are reciprokall to their altitudes And pyramids hauing triangles to their bases whose bases are reciprokall to their altitudes are equall the one to the other SVppose that BCGA and EFHD be equall pyramids hauing to their bases the triangles BCG and EFH and the tops the pointes A and D. Then I say that the bases of the two pyramids BCGA and EFHD are reciprokall to their altitudes that is as the base BCG is to the base EFH so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA Make perfect the parallelipipedons namely BGML and EHPO And forasmuch as the pyramis BCGA is equall to the pyramis EFHD the solide BGML is sextuple to the pyramis BCGA For the parallelipipedon is duple to the Prisme set vpon the base of the Pyramis the Prisme is triple to the pyramis and likewise the solide EHPO is sextuple to the pyramis EFHD Wherefore the solide BGML is equal to the solide EHPO But in equall parallelipipedons the bases are by the 34. of the eleueÌ„th reciprokall to their altitudes Wherfore as the base BN is to the base EQ so is the altiâ—â—de of the solide EHPâ— to the altitude of the solide BGML But as the base BN is to the base EQ so is the triangle GBC to the triangle HEF by the 15. of the â—ifth for the triangles GBC HEF are the halues of the parallelogrammes BN and EQ â— Wherfore by the 11. of the fifth as the triangle GBC is to the triangle HEF so is the altitude of the solide EHPO to the altitude of the solide BGML But the altitude of the solide EHPO is one and the same with the altitude of the pyramis EFHD and the altitude of the solide BGML is one and the same with the altitude of the pyramis BCGA Wherefore as the base GBC is to the base HEF so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA Wherefore the bases of the two pyramids BCGA and EFHD are reciprokall to their altitudes But now suppose that the bases of the pyramids BCGA and EFHD be reciprokall to their altitudes that is as the base GBC is to the base HEF so let the altitude of the pyramis EFHD be to the altitude of the pyramis BCGA Then I say that the pyramis BCGA is equall to the pyramis EFHD For the selfe same order of construction remaining for that as the base GBC is to the base â—EF so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA But as the base GBC is to the base HEF so is the parallelogramme GC to the parallelogramme HF. Wherefore by the 11. of the fifth as the parallelogramme GC is to the parallegoramme HF so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA But the altitude of the pyramis EFND and of the solide EHPO is one and the selfe same and the altitude of the pyramis BCGA and of the solide BGML is also one and the same Wherefore as the base GC is to the base HF so is the altitude of the solide EHPO to the altitude of the solide BGML But parallelipipedons whose bases are reciprokall to their altitudes are by the 34. of the eleuenth equall the one to the other Wherefore the parallelipipedon BGML is equall to the parallelipipedon EHPO But the pyramis BCGA is the sixth part of the solide BGML and likewise the pyramis EFHD is the sixth part of the solide EHPO Wherefore the pyramis BCGA is equall to the pyramis EFHD Wherefore in equall pyramids hauing triangles to their bases the bases are reciprokall to their altitudes And pyramids hauing triangles to their bases whose bases are reciprocall to their altitudes are equall the one to the other which was required to be demonstrated A Corrollary added by Campane and Flussas Hereby it is manifest that equall pyramids hauing to their bases Poligonon figures haue their bases reciprokall with their altitudes And Pyramids whose bases being poligonon figures are reciprokall with their altitudes are equall the one to the other Suppose that vpon the poligonon figures A and B be set equall pyramids Then I say that their bases A and B are reciprokall with their altitudes Describe by the 25. of the sixth triangles equall to the bases A and B. Which let be C and D. Vpon which let there be set pyramids equall in altitude with the pyramids A and B. Wherfore the pyramids C and D being set vpoÌ„ bases equall with the bases of the pyramids A and B and hauing also their altitudes equall
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
FD. Wherfore cones cylinders consisting vpon equal bases are in proportion the one to the other as their altitudes which was required to be demonstrated ¶ The 15. Theoreme The 15. Proposition In equall Cones and Cylinders the bases are reciprokall to their altitudes And cones and Cylinders whose bases are reciprokall to their altitudes are equall the one to the other SVppose that these cones ACL EGN or these cylinders AX EO whose bases are the circles ABCD EFGH and axes KL and MN which axes are also the altitudes of the cones cylinders be equall the one to the other TheÌ„ I say that the bases of the cylinders XA EO are reciprokal to their altitudes that is that as the base ABCD is to the base EFGH so the altitude MN to the altitude KL For the altitude KL is either equall to the altitude MN or not First let it be equall But the cylinder AX is equal to the cylinder EQ But cones and cylinders consisting vnder one and the selfe same altitude are in proportion the one to the other as their bases are by the 11. of the twelueth Wherfore the base ABCD is equall to the base EFGH Wherefore also they are reciprokal as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL But now suppose that the altitude LK be not equall to the altitude M. N but let the altitude MN be greater And by the 3. of the first from the altitude MN take away PM equall to the altitude KL so that let the line PM be put equal to the line KL And by the point P let there be extended a playne superâ—icies TVS which let cut the cylinder EO and be a parallell to the two opposite playne superâ—icieces that is to the circles EFGH and RO. And making the base the circle EFGH the altitude MP imagine a cylinder ES. And for that the cylinder AX is equall to the cylinder EO and there is an other cylinder ES therfore by the 7. of the fift as the cylinder AX is to the cylinder ES so is the cylinder EO to the cylinder ES. But as the cylinder AX is to the cylinder ES so is the base ABCD to the base EFGH For the cylinders AX and ES are vnder one and the selfe same altitude And as the cylinder EO is to the cylinder ES so is the altitude MN to the altitude MP For cylinders coÌ„sisting vpoÌ„ equall bases are in proportion the one to the other as their altitudes Wherfore as the base ABCD is â—o the base EFGH so is the altitude MN to the altitude MP But the altitude PM is equall to the altitude KL Wherefore as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL Wherefore in the equall cylinders AX and EO the bases are reciprokall to their altitudes But now suppose that the bases of the cylinders AX and EO be reciprokal to their altitudes that is as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL Then I say that the cylinder AX is equall to the cylinder EO For the selfe same order of constructioÌ„ remayning for that as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL but the altitude KL is equall to the altitude PM Wherefore as the base ABCD is to the base EFGH so is the altitude MN to the altitude PM But as the base ABCD is to the base EFGH so is the cylinder AX to the cylinder ES for they are vnder equall altitudes and as the altitude MN is to the altitude PM so is the cylinder EO to the cylinder ES by the 14. of the twelueth Wherefore also as the cylinder AX is to the cylinder ES so is the cylinder EO to the cylinder ES. Wherefore the cylinder AX is equall to the cylinder EO by the 9. of the fift And so also is it in the cones which haâ—â— the selfe same bases and altitudes with the cylinders Wherefore in equall cones and cylinders the bases are reciprokall to their altitudes c. which was required to be demonstrated A Corrollary added by Campane and Flussas Hitherto hath bene shewed the passions and proprieties of cones and cylinders whose altitudes fall perpendicularly vpon the bases Now will we declare that cones and cilinders whose altitudes fall obliquely vpon their bases haue also the selfe same passions and proprieties which the foresayd cones and cilinders haue Forasmuch as in the tenth of this booke it was sayd that euery Cone is the third part of a cilinder hauing one and the selfe same base one the selfe same altitude with it which thing was demoÌ„strated by a cilinder geuen whose base is cut by a square inscribed in it and vpon the sides of the square are described Isosceles triangles making a poligonon figure and againe vpon the sides of this poligonon figure are infinitely after the same maner described other Isosceles triangles taking away more theÌ„ the halfe as hath ofteÌ„times bene declared therfore it is manifest that the solides set vpon these bases being vnder the same altitude that the cilinder inclined is and being also included in the same cilinder do take away more then the halfe of the cilinder and also more theÌ„ the halfe of the residue as it hath bene proued in erected cylinders For these inclined solides being vnder equall altitudes and vpon equall bases with the erected solides are equall to the erected solides by the corollary of the â—0 of the eleuenth Wherfore they also in like sort as the erected take away more then the halfe If therfore we coÌ„pare the inclined cilinder to a cone set vpon the selfe same base and hauing his altitude erected and reason by an argument leading to an impossibilitie by the demonstration of the tenth of this booke we may proue that the sided solide included in the inclined cylinder is greater then the triple of his pyramis and it is also equall to the same which is impossible And this is the first case wherein it was proued that the cilinder not being equall to the triple of the cone is not greater then the triple of the same And as touching the second case we may after the same maner conclude that that â—ided solide contayned in the cylinâ—er is greater then the cylinder which is very absurdâ— Wherefore if the cylinder be neither greater then the triple of the cone nor lesse it must nedes be equall to the same The demonstration of these inclined cylinders most playnely followeth the demonstration of the erected cylinders for it hath already bene proued that pyramids and sided solides which are also called generally Prismes being set vpon equall bases and vnder one and the selfe same altitude whether the altitude be erected or inclined are equall the one to the other namely are in proportion as their bases are by
the â— of this booke Wherefore a cylinder inclined shall be triple to euery cone although also the cone be erected set vpon one and the same base with it and being vnder the same altitude But the cilinder erected was the triple of the same cone by the tenth of this booke Wherefore the cilinder inclined is equall to the cilinder erected being both set vpon one and the selfe same base and hauing one and the selfe same altitude The same also cometh to passe in cones which are the third partes of equall cilinders therefore are equall the one to the other Wherefore according to the eleuenth of this booke it followeth that cylinders and cones inclined or erected being vnder one and the selfe same altitude are in proportion the one to the other as their bases are For forasmuch as the erected are in proportion as their bases are and to the erected cilinders the inclined are equall therefore they also shall be in proportion as their bases are And therefore by the 12. of this booke like cones and cylinders being inclined are in triple proportion of that in which the diameters of the bases are For forasmuch as they are equall to the erected which haue the proportion by the 12. of this booke and their bases also are equall with the bases of the erected therefore they also shall haue the same proportion Wherefore it followeth by the 13. of this booke thaâ— cylinder inclined being cut by a playne superficies parallel to the opposite playne superficieces therof shall be cut according to the proportions of the axes For suppose that vpon one and the selfe same base â—e set an erected cylinder and an inclined cylinder being both vnder one and the selfe same altitude which 〈…〉 a playne superficies parallel to the opposite bases Now it is manifest that the sections of the one cylinder are equall to the section of the other cylinder for they are set vpon equall bases and vnder one and the selfe same altitude namely betwene the parallel playne superficieces And their axes also are by those parallel playne superfici◠〈◊〉 proportionally by the 1â—â— of â—he â—leuenâ—h Wherefore the inclined cylinders being equall to the erected cylinders shall haue the proportion of theiâ— axes aâ— also haue the erected For in echâ— the proportion of the axes is one and the same Wherefore inclined Cones and Cylinders being set vpon equall bases shall by the 14. of this booke be in 〈◊〉 as their altitude◠〈…〉 forasmuch aâ— the iâ—clined are equall to the erected which haue the selfe same bases and altitude and the erected are iâ— proportion as their altitudes therfore the inclined shall be in proportion the one to the other as â—he selfe same altiâ—udes which are common to ech namely to the inclined and to the erected And therefore in equall cones and cylinders whether they be inclined or erected the bases shall be reciprokally proportionall with the altitudes and contrariwise by the 15. of this booke For forasmuch as we haue oftentimes shewed that the inclined cones and cylinders are equall to the erected hauing the selfâ— same bases and altitudes with them and the erected vnto whome the inclined are equall haâ—e their bases râ—â—iproâ—all proportionally with their altitudes therefore it followeth that the inclined being equall to the erected haue also their bases and altitudes which are common to eche reciprokally proportionall Likewise if theiâ— altitudes bases be reciprokally proportionall they theÌ„selues also shall be equall for that they are equall to the erected cylinders and cones set vpon the same bases and being vnder the same altitudeâ— which erected cylinders are equall the one to the other by the same 15. of this booke Wherefore we may conclude that those passions proprieties which in this twelfth booke haue bene proued to be in cones and cylinders whose altitudes are erected perpendicularly to the 〈…〉 set obliquely vpoâ— their bases Howbeit this is to be noted that such inclined cones or cylinders are not perfect rouâ—d as are the erected so that if they be cut by a playne superficies passing at right angles with their altitude this section is a Conicall section which is called Ellipsis and shall not describe in their superficies a circle as it doth in erected cylinders cones but a certaine figure whose lesse diameter is in cylinders equall to the dimetient of the base that is is one and the same with it And the same thing happeneth also in cones inclined being cut after the same maner The 1. Probleme The 16. Proposition Two circles hauing both one and the selfe same centre being geuen to inscribe in the greater circle a poligonon figure which shall consist of equall and euen sides and shall not touch the superficies of the lesse circle SVppose that there be two circles ABCD and EFGH hauing one the selfe same centre namely K. It is required in the greater circle which let be ABCD to inscribe a poligonon figure which shal be of equal and euen sides and not touch the circle EFGH Drawe by the centre K a right line BD. And by the 11. of the first from the point G rayse vp vnto the right line BD a perpendicular line AG and extend it to the point C. Wherefore the line AC toucheth the circle EFGH by the 15. of the third Now therfore if by the 30. of the third we diuide the circumference BAD into two equall partes and againe the halfe of that into two equal partes and thus do coÌ„tinually we shall by the corollary of the 1. of the tenth at the length leaue a certayne circumference lesse then the circumference AD. Let the circumference left be LD And from the point L. Drawe by the 12. of the first vnto the line BD a perpendiculare line LM and extende it to the point N. And draw these right lines LD and DN And forasmuch as the angles DML and DMN are right angles therfore by the 3. of the third the right line BD diuideth the right line LN into two equall parts in the pointe M. Wherfore by the 4. of the first the rest of the sides of the triangles DML and DMN namely the lines DL and DN shal be equall And forasmuch as the line AC is a parallell to the LN by the 28. of the first But AC toucheth the circle EFGH wherfore the line LM toucheth not the circle EFGH and much lesse do the lines LD and DN touch the circle EFGH If therefore there be applied right lines equall to the line LD continually into the circle ABCD by the 1. of the fourth there shal be described in the circle ABCD a poligonon figure which shal be of equall and euen sides and shall not touch the lesse circle namely EFGH by the 14. of the third or by the 29. which was required to be done ¶ Corollary Hereby it is manifest that a perpendicular line drawen from the poynt L to the line BD toucheth not
line which subtendeth the angle ZOB to the third line which subtendeth the angle ZKB But by construction BO is equall to BK therefore OZ is equall to KZ And the third alâ—o is equall to the third Wherefore the point Z in respecte of the two triangles rectangles OZB and KZB determineth one and the same magnitude iâ— the line BZ Which can not be if any other point in the line BZ were assigned nearer or farther of from the point B. One onely poynt therefore is that at which the two perpendiculars KZ and OZ fall But by construction OZ falleth at Z the point and therefore at the same Z doth the perpendicular drawen from K fall likewyse Which was required to be demonstrated Although a briefe monition mought herein haue serued for the pregnant or the humble learner yet for them that are well pleased to haue thinges made plaine with many wordes and for the stiffenecked busie body it was necessary with my controlment of other to annexe the cause reason therof both inuincible and also euident A Corollary 1. Hereby it is manifest that two equall circles cutting one the other by the whole diameter if from one and the same end of their common diameter equall portions of their circumferences be taken and from the pointes ending those equall portions two perpendiculars be let downe to their common diameter those perpendiculars shall fall vpon one and the same point of their common diameter 2. Secondly it followeth that those perpendiculars are equall ¶ Note From circles in our first supposition eche to other perpendicularly erected we procede and inferre now these Corollaries whether they be perpendicularly erected or no by reasou the demonstration hath a like force vpon our suppositions here vsed ¶ The 16. Theoreme The 18. Proposition Spheres are in treble proportion the one to the other of that in which their diameters are SVppose that there be two spheres ABC and DEF and let their diameters be BC and EF. Then I say that the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. For if not then the sphere ABC is in treble proportion of that in which BC is to EF either to some sphere lesse then the sphere DEF or to some sphere greater First let it be vnto a lesse namely to GHK And imagine that the spheres DEF and GHK be both about one and the selfe same centre And by the proposition next going before describe in the greater sphere DEF a polihedron or a solide of many sides not touching the superficies of the lesse sphere GHK And suppose also that in the sphere ABC be inscribed a polihedron like to the polihedron which is in the sphere DEF Wherefore by the corollary of the same the polihedron which is in the sphere ABC is to the polihedron which is in the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. But by supposition the sphere ABC is to the sphere GHK in treble proportion of that in which the diameter BC is to the diameter EF. Wherefore as the sphere ABC is to the sphere GHK so is the polihedroÌ„ which is described in the sphere ABC to the polihedroÌ„ which is described in the sphere DEF by the 11. of the fift Wherfore alternately by the 16. of the fift as the sphere ABC is to the polihedron which is described in it so is the sphere GHK to the polihedron which is in the sphere DEF But the sphere ABC is greater then the polihedroÌ„ which is described in it Wherfore also the sphere GHK is greater then the polihedroÌ„ which is in the sphere DEF by the 14. of the fift But it is also lesse for it is contayned in it which impossible Wherefore the sphere ABC is not in treble proportioÌ„ of that in which the diameter BC is to the diameter EF to any sphere lesse then the sphere DEF In like sort also may we proue that the sphere DEF is not in treble proportion of that in which the diameter EF is to the diameter BC to any sphere lesse then the sphere ABC Now I say that the sphere ABC is not in treble proportioÌ„ of that in which the diameter BC is to the diameter EF to any sphere greater theÌ„ the sphere DEF For if it be possible let it be to a greater namely to LMN Wherfore by conuersion the sphere LMN is to the sphere ABC in treble proportion of that in which the diameter EF is to the diameter BC. But as the sphere LMN is to the sphere ABC so is the sphere DEF to some sphere lesse theÌ„ the sphere ABC as it hath before bene proued for the sphere LMN is greater then the sphere DEF Wherfore the sphere DEF is in treble proportioÌ„ of that in which the diameter EF is to the diameter BC to some sphere lesse theÌ„ the sphere ABC which is proued to be impossible Wherefore the sphere ABC is not in treble proportion of that in which BE is to EF to any sphere greater theÌ„ the sphere DEF And it is also proued that it is not to any lesse Wherefore the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF which was required to be demonstrated A Corrollary added by Flussas Hereby it is manifest that spheres are the one to the other as like Polihedrons and in like sort described in them are namely eche are in triple proportion of that in which the diameters A Corollary added by Mâ— Dee It is then euident how to geue two right lines hauing that proportion betwene them which any two spheres geuen haue the one to the other For if to their diameters as to the first and second lines of fower in continuall proportion you adioyne a third and a fourth line in continuâ—ll proportion as I haue taught before The first and fourth lines shall aunswere the Pâ—obleme How generall this rule is in any two like solides with their correspondent or Omologall lines I neede not with more wordes declare ¶ Certaine Theoremes and Problemes whose vse is manifolde in Spheres Cones Cylinders and other solides added by Ioh. Dee A Theoreme 1. The whole superficies of any Sphere is quadrupla to the greatest circle in the same sphere contayned It is needeles to bring Archimedes demonstration hereof into this place seing his boke of the Sphere and Cylinder with other his woâ—kes are euery where to be had and the demoÌ„stration therof easie A Theoreme 2. Euery sphere is quadruplâ— to that Cone whose base is the greatest circle height the semidiameter of the same sphere This is the 32. Proposition of Archimedes fiâ—st booke of the Sphere and Cylinder A Probleme 1. A Sphere being geuen to make an vpright Cone equall to the same or in any other proportioâ— geuen betwene two right lines And as concerning the other part of
the 7. of the fift our conclusion is inferred the superficies Sphericall of the segment CAE to be to the superficies Sphericall of the segment FGH as AD is to GI A Theoreme 6. To any solide sector of a Sphere that vpright Câ—â—e is equall whose base is equall to the câ—nnex Sphericall superficies of that sector and heith equall to the semidiameter of the same Sphere Hereof the demonstration in respect of the premises and the common argument of inscriptioâ— and circumscription of figures is easy and neuerthelesse if your owne write will not helpe you sufficiently you may take helpe at Archimedes hand in his first booke last proposition of the sphere and cylinder Whether if ye haue recourse you shalâ— perceaue how your Theoreme here amendeth the common translation there and also our delinâ—ation geueth more sâ—uâ—y shew of the chiefe circumstances necessary to the construction then there you shall finde Of the sphere here imagined to be A we note a solide sector by the letterâ— PQRO. So that PQR doth signifie the sphericall superficies to that solide sector belonging which is also common to the segment of the same sphere PRQ and therefore a line drawne from the toppe of that segmentâ— which toppe suppose to be Q is the semidiameter of the circle which is equall to the sphericall superficies of the sayd solide sector or segmentâ— as before is taught Let that line be QP By Q draw a line contingent which let be SQT. At the poynt Q from the line QS cut a line equall to PQ which let be SQ And vnto SQ make QT equall then draw the right lines OSOâ— and OQ About which OQ as an axe faâ—â—ened if you imagine the triangle OST to make an halfe circular reuolution you shall haue the vpright cone OST whose heith is OQ the semidiameter of the sphere and base the circle whose diameter is ST equall to the solide sector PQRO. A Theoreme 7. To any segment or portion of a Sphere that cone iâ— equall which hath that circle to his base which is the bâ—se of the segmeÌ„t and heith a right line which vnto the heith of the segmeÌ„t hath that proportioÌ„ which the semidiameter of the Sphere together with the heith of the other segment remayâ—â—â—g hath to the heith of the same other segment remaynâ—ng This is well demonstrated by Archiâ—â—des therefore nedeth no inuention of myne to confirme the same and for that the sayd demonstration is ouer long here to be added I will refere you thether for the demonstration and here supply that which to Archimedes demonstration shall geue light and to your farther speculation and practise shal be a great ayde and direction Suppose K to be a sphere the greatest circle K in coÌ„teyned let be ABCE and his diameter BE ceÌ„ter D. Let the sphere K be cutte by a playne superficies perpâ—ndicularly erected vpon the sayd greatest circle ABCE let the section be the circle about AC And let the segmentes of the sphere be the one that wherein is ABC whose â—oppe is Bâ— and the other let be that wherein is AEC and his toppe let be E I say that a cone which hath his base the circle about AC held a line which to BF the heith of the segment whose toppe is B hath that proportion that a line compoâ—ed of DE the semidiameter of the sphere and EF the heith of the other remayning segment whose toppe is E hath to EF the heith of that other segment remayning is equall to the segment of the sphere K whose toppe is B. To make this cone take my easy order thus Frame your worke for the findâ—ng of the fourth proportionall lineâ— by making EF the first and a line composed of DE and EF the secondâ— and the third let be BF then by the 12. of the sixâ—h let the fourth proportionall line be found which let be FGâ— vpon F the center of the base of the segment whose toppe is B erect a line perpendicular equall to FG found and drawe the lines GA and GC and so make perfect the cone GAC I say that the cone GAC is equall to the segment of the sphere K whose toppe is B. In like maner for the other segmeÌ„t whose toppe is E to finde the heith due for a cone equal to it by the order of the Theoreme you must thus frame your lines let the first be BF the second DB and BF composed in one right line and the third must be EF where by the 12. of the sixth finding the fourth it shall be the heith to rere vpon the base the circle about AC to make an vpright cone equall to the segment whose toppe is E. ¶ Logistically ¶ The Logisticall finding hereof is most easy the diameter of the sphere being geuen and the portions of the diameter in the segmentes conteyned or axes of the segmentes being knowne Then order your numbers in the rule of proportion as I here haue made most playne in ordring of the lines for the â—ought heith will be the producte A Corollary 1. Hereby and other the premises it is euident that to any segment of a Sphere whose whole diameter is knowne and the Axe of the segment geuen An vpright cone may be made equall or in any proportion betwene two right lines assignedâ— and therefore also a cylinder may to the sayd segment of the Sphere be made equall â—r in any proportion geuen betwene two right lines A Corollary 2. Manifestly also of the former theoreme it may be inferred that a Sphere and his diameter being deuided by one and the same playne superficies to which the sayd diameter is perpendicularâ— the two segmentes of the Sphere are one to the other in that proportion in which a rectangle parallelipipedon hauing for his base the square of the greater part of the diameter and his heith a line composed of the lesse portion of the diameter and the semidiameter to the rectangle parallelipâ—pedon hauing for his base the square of the lesse portion of the diameter his heith a line composed of the semidiameter the greater part of the diameter A Theoreme 8. Euery Sphere to the cube made of his diameter is in maner as 11. to 21. As vpon the first and second propositioÌ„s of this booke I began my additions with the circle being the chiefe among playne figures and therein brought manifold considerations about circles as of the proportion betwene their circumferences and their diameters of the content or Area of circles of the proportion of circles to the squares described of their diameters of circles to be geuen in al proâ—portions to other circles with diuerse other most necessary problemes whose vse is partly there specified So haue I in the end of this booke added some such Problemes Theoremesâ— about the sphere being among solides the chiefe as of the same either in it selfe considered or to cone and cylinder compared by reason of
the square of ED to the square of DK but by construction ED is subquintuple to DF. Wherefore the square of ED is subquintuple to the square of DK And therefore the square of DK is quintuple to the square of ED. And ED is equall to ED by construction therefore the square of DK is quintuple to the square of E D. Wherefore the double of BD is deuided by an extreme and meane proportion whose greater segment is BK â— by the second of this thirteâ—th But by construction AB is the double of â—D â— Wherefore AB is diuided by extreme and meane proportion and his greater segment is BK and thereby K â— the point of the diuision We haue therefore deuided by extreme and meane proportion any right line geuen in length and position Which was to be done Noteâ— Ech of these wayes may well be executed But in the first you haue this auantage that the diameter is taken at pleasure Which â—n the second way is euer iust thrise so long as the line geuen to be deuided Iohn Dee ¶ The 4. Theoreme The 4. Proposition If a right line be deuided by an extreame and meane proportion the squares made of the whole line and of the lesse segmeÌ„t are treble to the square made of the greater segment SVppose that the right line AB be deuided by an extreame meane proportioÌ„ in the point C. And let the greater segment thereof be AC Then I say that the squares made of the lines AB and BC are treble to the square of the line AC Describe by the 46. of the first vpon the line AB a square ADEB And make perfect the figure Now forasmuch as the line AB is deuided by an extreame and meane proportion in the point C and the greater segmeÌ„t thereof is the line AC 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 CB is the parallelogramme AK and the square of the line AC is the square FD. Wherefore the parallelogramme AK is equall to the square FD. And the parallelogramme AF is equall to the parallelogramme FE put the square CK common to them both wherfore the whole parallelograÌ„me AK is equall to the whole parallelogramme CE Wherefore the parallelogrammes CE and AK are double to the parallelogramme AK Bâ—t the parallelogrammes AK and CE are the gnomon LMN and the square CK Wherefore the gnomon LMN and the square CK are double to the parallelogramme AK But it is proued that the parallelogramme AK is equal to the square DF. Wherefore the gnomon LMN and the square CK are double to the square DF. Wherefore the gnomon LMN and the squares CK and DF are treble to the square DF. But the gnomon LMN and the squares CK and DF are the whole square AE together with the square CK which are the squares of the lines AB and BC. And DF is the square of the line AC Wherefore the squares of the lines AB and BC are treble to the square of the line AC If therefore a right line be deuided by an extreame and meane proportion the squares made of the whole line and of the lesse segment are treble to the square made of the greater segment which was required to be proued Looke for an other demonstration of this proposition after the fifth proposition of this booke ¶ The 5. Theoreme The 5. Proposition If a right line be deuided by an extreame and meane proportion and vnto it be added a right â—ine equall to the greater segment the whole right line is deuided by an extreame and meane proportion and the greater segment thereof is the right line geuen at the beginning SVppose that the right line AB be deuided by an extreame and meane proportion in the point C and let the greater segment thereof be AC And vnto the line AB adde the line AD equall to the line AC Then I say that the line Dâ— is deuided by an extreame and meane proportion in the point A and the greater segment thereof is the right line pâ—t at the beginning namely AB Describe by the 46. of the first vpon on the line AB a square AE and make perfect the figure And forasmuch as the line AB is deuided by an extreame 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 CEâ— and the square of the â—â—ne AC is the square CH. Wherefore the parallelogramme CE is equall to the square CH. But vnto the square CH is equall the square DH by the first of the sixth and vnto the parallelogramme CE is equall the parallelogramme HE. Wherefore the parallelogramme DH is equall to the parallelogramme HE. Adde the parallelogramme HB common to them both Wherefore the whole parallelogramme DK is equall to the whole square AE And the parallelogramme DK is that which is contayned vnder the lines BD and DA for the line AD is equall to the line DL the square AE is the square of the line AB Wherfore that which is contayned vnder the lines AD and DB is equall to the square of the line AB Wherefore as the line DB is to the line BA so is the line BA to the line AD by the 17. of the sixth But the line DB is greater then the line BA Wherefore the line BA is greater then the line AD. Wherefore the line BD is deuided by an extreame and meane proportion in the point A and his greater segment is the line AB If therefore a right line be deuided by an extreame and meane proportion and vnto â—t be added a right line equall to the greater segment the whole right line is deuided by an extreame and meane proportion and the greater segment therof is the right line geuen at the beginning which was required to be demonstrated This proposition is agayne afterward demonstrated A Corollary added by Campane Hereby it is 〈…〉 from the greaâ—â—◠〈◊〉 of a line deuided by an extreame meane proportion be 〈◊〉 away 〈◊〉 segment the sayd great a segment shall be deuided by an extreame and meane proportion and the greater segment thereof shall be the line taken away As let the line â—â— be deuided by an extreame and meanâ— proportion in the point C. And leâ— the 〈…〉 line 〈…〉 A. D. I say that AC is also deuided by an extreame and meanâ— proportion in the point D and that his greater portion is DC For by the definitino of a line so deuided AB is to AC as AC is to CB. But as AC is to CB so is AC to DC by the 7. of the 〈…〉 is equall to CB wherefore by the 11. of the fifth as AB is to AC so is AC to CD
double to the side of the Octohedron the side is in power sequitertia to the perpeÌ„diclar line by the 12. of this booke wherfore the diameter thereof is in power duple superbipartiens tertias to the perpendicular line Wherfore also the diameter and the perpeÌ„dicular line are rationall and commensuâ—able by the 6. of the tenth As touching an Icosahedron it was proued in the 16. of this booke that the side thereof is a lesse line when the diameter of the sphere is rationall And forasmuch as the angle of the inclination of the bases thereof is contayned of the perpendicular lines of the triangles and subtended of the right line which subtendeth the angle of the Pentagon which contayneth fiue sides of the Icosahedron and vnto the perpendicular lines the side is commensurable namely is in power sesquitertia vnto them by the Corollary of the 12. of this booke therefore the perpendicular lines which contayne the angles are irrationall lines namely lesse lines by the 105. of the tenth booke And forasmuch as the diameter contayneth in power both the side of the Icosahedron and the line which subtendeth the foresayd angle if from the power of the diameter which is rationall be taken away the power of the side of the Icosahedron which is irrationall it is manifest that the residue which is the power of the subtending line shal be irrationall For if it shoulde be rationall the number which measureth the whole power of the diameter and the part taken away of the subtending line should also by the 4. common sentence of the seuenth measure the residue namely the power of the side which is irrationall for that it is a lesse line which were absurd Wherefore it is manifest that the right lines which compose the angle of the inclination of the bases of the Icosahedron are Irrationall lines For the subtending line hath to the line contayninge a greater proportion then the whole hath to the greater segment The angle of the inclination of the bases of a dodecahedron is contayned vnder two perpendiculars of the bases of the dodecahedron and is subtended of that right line whose greater segment is the side of a Cube inscribed in the dodecahedron which right line is equall to the line which coupleth the sections into two equal parts of the opposite sides of the dodecahedron And this coupling line we say is an irrationall line for that the diameter of the sphere contayneth in power both the coupling line and the side of the dodecahedron but the side of the dodecahedron is an irrationall line namely a residuall line by the 17. of this booke Wherefore the residue namely the coupling line is an irrationall line as it is â—asy to proue by the 4. coÌ„mon sentence of the seueÌ„th And that the perpeÌ„dicular lines which contayne the angle of the inclination are irrationall is thus proued By the proportion of the subtending line of the foresayd angles of inclination to the lines which containe the angle is found out the obliquitie of the angle For if the subtending line be in power double to the line which contayneth the angle then is the angle a right angle by the 48. of the first But if it be in power lesse then the double it is an acute angle by the 23. of the second But if it be in power more then the double or haue a greater proportion then the whole hath to the greater segmeÌ„tâ— the angle shal be an obtuse angle by the 12. of the second and 4. of the thirtenth By which may be proued that the square of the whole is greater then the double of the square of the greater segment This is to be noted that that which Flussas hath here taught touching the inclinations of the bases of the â—iue regular bodies Hypsicles teacheth after the 5 proposition of the 15. booke Where he confesseth that he receiued it of one Isidorus and seking to make the mater more cleare he endeuored himselfe to declare that the angles of the inclination of the solides are geuen and that they are either acute or obtuse according to the nature of the solide although â—uclidâ— in all his 15. bookes hath not yet shewed what a thing geuen is Wherefore Flussas framing his demoÌ„stration vpon an other ground procedeth after an other maner which semeth more playne and more aptly hereto be placed then there Albeit the reader in that place shal not be frustrate of his also The ende of the thirtenth Booke of Euclides Elementes ¶ The fourtenth booke of Euclides Elementes IN this booke which is commonly accompted the 14. booke of Euclide is more at large intreated of our principal purpose namely of the comparison and proportion of the fiue regular bodies customably called the 5. figures or formes of Pythagoras the one to the other and also of their sides together eche to other which thinges are of most secret vse and inestimable pleasure and commoditie to such as diligently search for them and attayne vnto them Which thinges also vndoubtedly for the woorthines and hardnes thereof for thinges of most price are most hardest were first searched and found out of Philosophers not of the inferior or meane sort but of the depest and most grounded Philosophers and best exercised in Geometry And albeit this booke with the booke following namely the 15. booke hath bene hetherto of all men for the most part and is also at this day numbred and accompted amoÌ„gst Euclides bookes and supposed to be two of his namely the 14. and 15. in order as all exemplars not onely new and lately set abroade but also old monumentes written by hand doo manifestly witnes yet it is thought by the best learned in these dayes that these two bookes are none of Euclides but of some other author no lesse worthy nor of lesse estimation and authoritie notwithstanding then Euclide Apollonius a man of deepe knowledge a great Philosopher and in Geometrie maruelous whose woÌ„derful bookes writteÌ„ of the sections of cones which exercise occupy thewittes of the wisest and best learned are yet remayning is thought and that not without iust cause to be the author of them or as some thinke Hypsicles him selfe For what can be more playnely then that which he him selfe witnesseth in the preface of this booke Basilides of Tire sayth Hypsicles and my father together scanning and peysing a writing or books of Apollonius which was of the comparison of a dodecahedron to an Icosahedron inscribed in one and the selfe same sphere and what proportion these figures had the one to the other found that Apollonius had fayled in this matter But afterward sayth he I found an other copy or booke of Apollonius wherein the demonstration of that matter was full and perfect and shewed it vnto them whereat they much reioysed By which wordes it semeth to be manifest that Apollonius was the first author of this booke which was afterward set forth by Hypsicles For so his owne wordes after in
spherâ— containeth the Dodecahedron of this pentagon and the Icosahedron of this triangle by the 4. of this booke â— and the line CL falleth perpendiculaâ—ly vpon the side of the Icosahedron and the line CI vpon the side of the Dodecahedron that which is 30. times contained vnder the side and the perpendicular line falling vpon it is equal to the â—uperficies of that solide vpon whose side the perpendiculâ—r falleth If therefore in a circle c. as in the proposition which was required to be demonstrated A Corollary The superficieces of a Dodecahedron and of an Icosahedron described in one and the selfe same sphere are the one to the other as that which is contained vnder the side of the one and the perpendicular line drawne vnto it from the centre of his base to that which is contained vnder the side of the other and the perpendicular line drawne to it from the centre of his base For aâ— thirtyâ— timâ—s is to thirty times so is once to once by the 15. of thâ— fifth The 6. Proposition The superficies of a Dodecahedron is to the superficies of an Icosahedron described in one and the selfe same sphere in that proportion that the side of the Cube is to the side of the Icosahedron contained in the self same sphere SVppose that there be a circle ABG in it by the 4. of this boke let there be inscribed the sideâ— of a Dodecahedron and of an Icosahedron contained in onâ— and the selfe same sphere And let the side oâ— the Dodecahedron be AG and the side of the Icosahedron be DG And let the centre be the poynt E from which draw vnto those sâ—des perpendicular lines EI and EZ And produce the line EI to the poynt B and draw the linâ— BG And let the side of the cube contained in the self same sphere be GC Then I say that the superficies of the Dodecahedron iâ— to the superficies of the Icosahedron as the line â—G iâ— to the liâ—â— GD For forasmuche as the line EI beinâ— diuided by an extreme and meane proportion the greater segment thâ—rof shall be the linâ— EZ by the corollary of the first of this booke and the line CG being diuided by an extreme and meane proportion his greater segment is the line AG by the corollary of the 17. of the thirtenth Wherefore the right lines EI and CG â—râ— cut proportionally by the second of this bâ—oke Whâ—râ—fore as the line CG is to the line AG so is the line EI to the line EZ Wherâ—fore that which it contained vnder the extreames CG and EZ is â—quall to that which iâ— contaynâ—d vnder the meanes AG and EI. by the 16. of the sixth But as that which iâ— contained vnder the linâ—â— CG and â—Z is to that which is contained vnder the lines DG and EZ so by the first of the sixth iâ— the linâ— CG to the line DG for both those parallelogrames haue oâ—â— and the selfe same altitude namely the line EZ Wherfore as that which is contained vnder the lines EI and AG which iâ— proued equal to that which is contained vnder the lineâ— CG and EZ is to that which is contained vnder the lines DG and EZ so is the line CG to the liâ—â— DG But as that which is contained vnder the lines EI and AG is to that which is contained vnder the lines DG and EZ so by the corollary of the former proposition is the superficies of the Dodecahedron to the superficies of the Icosahedron Wherfore as the superficies â—â— the Dodecahedron is to the superficies of the Icosahedron so is CG the side of the cube to GD the side of the Icosahedron The superficies therefore of a Dodecahedron is to the superficiesâ— c. as in the proposition which was required to be proued An Assumpt The Pentagon of a Dodecahedron is equall to that which is contained vnder the perpendicular line which falleth vpon the base of the triangle of the Icosahedron and fiue sixth partes of the side of the cube the sayd three solides being described in one and the selfe same sphere Suppose that in the circle ABEG the pentagon of a Dodecahedron be Aâ—CIG and let two sides thereof AB and AG be subtended of the right line BG And let the triangle of the Icosahedron inscribed in the selfe same sphere by the 4. of this booke be AFH And let the centre of the circle be the poynt D and let the diameter be ADE cutting FH the side of the triangle in the poynt Z and cutting the line BG in the poynt K. And draw the right line BD. And from the right line KG cut of a third part TG by the 9. of the sixth Now then the line BG subtending two sides of the Dodecahedron shal be the side of the cube inscribed in the same sphere by the 17. of the thirtenth and the triangle of the Icosahedron of the same sphere shal be Aâ—H by the 4. of this booke And the line AZ which passeth by the centre D shall fall perpendicularly vpon the side of the triangle For forasmuch as the angles GAE BAE are equall by the 27. of the thirdâ— for they are see vpon equall circumferences therefore the â—ases BK and KG are by the â— of the first equall Wherefore the line BT contayneth 5. sixth partes of the line BG Then I say that that which is contayned vnder the lines AZ and BT is equall to the pentagon Aâ—Câ—G For forasmuch as the line â—Z is sesqâ—ialter to the line AD for the line Dâ— is diuided into two equall partes in the poynt Z by the corollary of the â—2â— of the thirtenth Likewise by construction the line KG is sesquialter to the line KT therefore as the line AZ is to the line AD so is the line KG to the 〈◊〉 â—T Wherefore that which is contayned vndeâ— the 〈◊〉 AZ and KT is equall to that which is contayned vnder the meanes AD and KG by the 16. of the sixth But vnto the line KG is the line â—K â—roued equall Wherefore that which is contayned vnder the lines AZ and KT is equall to that which is contayned vnder the lines AD and BK But that which is contayned vnder the lines AD and BK is by the 41. of the first double to the triangle ABD Wherefore that which is contayned vnder the lines AZ and KT is double to the same triangle ABD And forasmuch as the pentagon ABCIG contayneth◠〈…〉 equall â—o the triangle ABD and that which is contayned vnder the lines AZ and KT contayneth two such triangles therefore the pentagon ABCIG is duple sesquialter to the rectangle parallelogramme contayned vnder the lines AZ and KT And 〈…〉 1. of the sixth that which is coÌ„teyned vnder the lines AZ and BT is to that which is contayned vnder the lines AZ and KT as the base BT is to the base â—â—Tâ— therefore that which is contayned vnder the lines AZ
DEF whose side let be DE and let the right line subtending the angle of the pentagon made of the sides of the Icosahedron be the line EF. Then I say that the side ED is in power double to the line H the lesse of those segmentes Forasmuch as by that which was demonstrated in the 15. of this booke it was manifest that ED the side of the Icosahedron is the greatâ—r segment of the line EFâ— and that the diameter DF containeth in power the two lines ED and EF namely the whole and the greater segment but by suppoâ—ition the side AB coÌ„taineth in power the two lines C H ioined together in the self same proportioÌ„ Wherefore the line EF is to the line ED as the line C is to the line H by the â— oâ— this bokeâ— And altâ—rnaâ—â—y by the 16. of the fiueth the line EF is to the line C as the line ED is to the line H. And forasmuche as the line DF containeth in power the two lines ED and EF and the line AB containeth in power the two lines C and H therefore the squares of the lines EF and ED are to the square of the line DF as the squares of the lines C and H to the square AB And alternately the squares of the lines EF and â—D are to the squares of the lines C and H as the square of the line DF is to the square of the line ABâ— But DF the diameter is by the 14. of the thirtenâ—h iâ— power double to AB the side of the octohedron inscribed by supposition in the same sphere Wherefore the squares of the lines EF and ED are double to the squares of the lines C and H. And therfore one square of the line ED is double to one square of the line H by the 12. of the fifth Wherfore ED the side of the Icosahedron is in power duple to the line H which is the lesse segment If therfore the poweâ— of the side of an octohedron be expressed by two right lines ioyned together by an extreme and meane proportion the side of the Icosahedron contained in the same sphere shal be duple to the lesse segment The 17. Proposition If the side of a dodecahedron and the right line of whome the said side is the lesse segment be so set that they make a right angle the right line which containeth in power halfe the line subtending the angle is the side of an Octohedron contained in the selfe same sphere SVppose that AB be the side of a Dodecahedron and let the right line of which that side is the lesse segment be AG namely which coupleth the opposite sides of the Dodecahedron by the 4. corollary of the 17. of the thirtenth and let those lines be so set that they make a right angle at the point A. And draw the right line BG And let the line D containe in power halfe the line BG by the first proposition added by Flussas after the laste of the sixth Then I say that the line D is the side of an Octohedron contayned in the same sphere Forasmuche as the line AG maketh the greater segment GC the side of the cube contained in the same sphere by the same 4. corollary of the 17. of the thirtenth and the squares of the whole line AG. and of the lesse segment AB are triple to the square of the greater segment GC by the 4. of the thirtenth Moreouer the diameter of the sphere is in power triple to the same line GC the side of the cube by the 15. of the thirtenth Wherfore the line BG is equal to the 〈◊〉 For it conâ—â—ineth in power the two lines AB and AG by the 47. of the first and therefore it containeth in power the triple of the line GC But the side of the Octohedron contained in the same sphere is in power triple to halfe the diameter of the sphere by the 14. of the thirtenth And by suppoâ—â—tion the line D contaiâ—â—â—â— in powâ—â— the halfe of the line BG Wherefore the line D containing in power the halfe of the same diameter is the side of an octohedron If therfore the side of a Dodecahâ—dron and the right line of whome the said side is the lesse segment be so set that they make a right angle the right line which containeth in power halfe the line subtending the angle is the side of an Ocâ—â—â—edron contained in the selfe same sphere Which was required to be proued A Corollary Vnto what right line the side of the Octoâ—edron is in power sesquialter vnto the same line the side of the Dodecahedron inscribed in the same sphere is the greater segment For the side of the Dodecahedron is the greater segment of the segment CG vnto which D the side of the Octohedron is in power sesquiâ—lter that is is halfe of the power of the line BG which was triple vnto the line CG ¶ The 18. Proposition If the side of a Tetrahedron containe in power two right lines ioyned together by an extreme and meane proportion the side of an Icosahedron described in the selfe same Sphere is in power sesquialter to the lesse right line SVppose that ABC be a Tetrahedron and let his side be AB whose power let be diuided into the lines AG and GB ioyned together by an extreme and meane proportion namely let it be diuided into AG the whole line and GB the greater seâ—ment by the Corollary of the first Proposition added by Flussas after the last of the sixth And let ED be the side of the Icosahedron EDF contained in the selfe same Sphere And let the line which subtendeth the angle of the Pentagon described of the sides of the Icosahedron be EF. Then I say that ED the side of the Icosahedron is in power sesquialter to the lesse line GB Forasmuch as by that which was demonstrated in the 15. of this booke the side ED is the greâ—ter segment of the line EF which subtendeth the angle of the Pentagon But as the whole line EF is to the greater segment ED so is the same grâ—â—ter segment to the lesse by the 30. of the sixth and by supposition AG was the whole line and Gâ— the greater segment Wherefore as EF is to ED so is AG to Gâ— by the second of the fouretenth And alternately the line EF is to the line AG as the line ED is to the line GB And forasmuch as by supposition the line AB containeth in power the two lines AG and GB therefore by the 4â— of the first the angle AGB is a right angle But the angle DEF is a right angle by that which was demonstrated in the 15. of this booke Wherefore the triangles AGâ— and FED are equiangle by the â— of the sixth Wherefore their sides are proportionall namely as the line ED is to the line GB so is the line FD to the line AB by the 4. of the sixth But by that which hath before
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
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
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
a right line coupling their centres being diuided by an extreame and meane proportion maketh the greater segment the right line which coupleth the centres of the next bases If by the centres of fiue bases set vppon one base be drawne a playne superficies and by the centres of the bases which are set vpon the opposite base be drawne also a playne superficies and then be drawne a right line coupling the centres of the opposite bases that right line is so cut that eche of his partes set without the playne superficies is the greater segment of that part which is contayned betwene the playnes The side of the dodecahedron is the greater segment of the line which subtendeth the angle of the pentagon A perpendicular line drawne from the centre of the dodecahedron to one of the bases is in power quintuple to half the line which is betwene the playnes And therfore the whole line which coupleth the centres of the opposite bases is in power quintuple to the whole line which is betwene the sayd playnes The line which subtâ—deth the angle of the base of the dodecahedroÌ„ together with the side of the base are in power quintuple to the line which is drawne from the ceÌ„tre of the circle which contayneth the base to the circumference A section of a sphere contayning three bases of the dodecahedron taketh a third part of the diameter of the sayd sphere The side of the dodecahedron and the line which subtendeth the angle of the pentagon are equall to the right line which coupleth the middle sections of the opposite sides of the dodecahedron ¶ The ende of the Elementes of Geometrie of the most auncient Philosopher 〈◊〉 of Megara The intent of this Preface Number Note the worde Vnit to expresse the Greke Monaâ— not Vnitie as we hauâ— all commonly till now vsed Magnitude A point A Line Magnitude Ano. 1488. ☞ Arithmetike Note * Anno. 1550. R. B. Note 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ☞ This noble Earle dyed Anno. 1554. skarse of 24. yeares of ageâ— hauing no issue by his wife Daughter to the Duke of Somerset Iustice. ☞ * Plato 7. de Rep. ☞ * Note The difference betwene Strataruhmetrie and Tacticie I.D. Fâ—endâ—â— you will finde it hard to performâ— my descriptiâ—n of â—his Fâ—ate But by Châ—râ—graphieâ— you may helpe your selfe some â—hat wherâ— thâ— Figures knowne in Sidâ—â—â—nd Angles are not Regular And whereâ— Resolution into Triangles can sâ—â—uâ— c. And yet you will finde it strange to deale thus generâ—lly with Arithmeticall figures and that for Battayle â—ay Their coâ—tentâ—â—â— differ so much from like Geometrâ—call Figurâ—s A marueilous Glasse ☞ S.W.P. ☞ Note 1. 2. 3. 5. 6. 7. 8. I.D. Read in Aristotle his 8. booke of Politikes the 5 6 and 7. chapters Where you shall haue some occasion farder to thinke of Musike than commonly is thought ☜ * Anno. 1548 and 1549. in Louayn Note I.D. The Cutting of a Sphare according to any proportion assigned may by this proposition be done Mechanically by tempering Lâ—quor to a certayne waight in respect of the waight of the Sphare 〈◊〉 Swyâ—â—â—ng A common errorâ— noted A paradox N. T. The wonderfull vse of these Propositions The practise Staticall to know the proportion betwene the Cube and the Sphare I. D. * For so haue you 256. partes of a Graine * The proportion of the Square to the Circle insâ—ribed * The Squaâ—ing of the Cirâ—le Mâ—â—haniâ—ally * To any Squirâ— gâ—uenâ— to 〈…〉 Note Squaring of the Circle without knowledge of the proportion betwene Circumference and Diameter To Dubble the Cube redily by Art Mechanicall depending vppon Demonstration Mathematicall I. D. The 4. sides of this Pyramiâ— must be 4. Isosceles Triangles â— likâ— and â—quall I. D. * In all workâ—ngeâ— with this Pyramis or Cone Let their Situationâ— be in all Pointeâ— and Conditions a like oâ— all one while you are about â—ne worke Els you will 〈◊〉 I. D. * Consider well whan you must put your wateâ—â— togyther and whan you must empty youâ— first waterâ— out of your Pyramiâ— or Cone Elâ— you will 〈◊〉 * Vitruuius Lib. 9. Cap. 3. ☞ God bâ— thanked â—or this Inuentionâ— the fruiâ—e â—nsuing * Note Note as concerning the Sphaericall Superâ—icies of the water ☞ * Note Note this Abridgeâ—ent of Dubbling 〈◊〉 Cube â—â— * Note * ☜ To giue Cubes one to the other in any proportion Rationall or Irrationall * Emptying the first The demonstrations of this Dubbling of the Cube and oâ— the rest I.D. * Here 〈…〉 of the water * By the 33. of the eleuenth books of Euclide I.D. * And your diligence in praâ—â—ise can â—o in waight of wateâ—â— peâ—forme it Therefore now you arâ— able to â—eue good reason of your whole doing * Note this Corollary * The great Commodities following of these new Inuentions * ☞ Such is the Fruite of the Mathematicall Sciences and Artes. MAN is the Lesse World. * ☜ Microcosmus * Lib. 3. Cap. 1. ☞ Saw Milles. * Atheneus Lib. 5. cap. 8. Proclus Pag. 18. To go to the bottom of the Sea without daunger Plutâ—â—â—bus in Marco Mâ—rcello Syâ—asius in Epistolis Polybius Plinius Quintâ—lianus T. Liuius * Athenaâ—s * Galeâ—us Anthemius Burning Glasses Gunnes 4. Reg. 20. A perpetuall Motion An obiection The Answer ☜ A Mathematicien Vitrunius VVho is an Architect * The Immaterialitie of perfect Architecture What Lineament is Note Anno. 1559. * Anno. 1551 De his quae Mundo mirabiliter eueniunt cap. 8. Tusc. â— * ☞ A Digression Apologeticall * A prouerb Fayre fisht and caught a Frog ☞ Psal. 140. Act. 7. C. Lib. 30. Cap. 1. ☜ R. B. ☞ Vniuersities ☜ ☞ The Ground platt of this Praeface in a Table The argumâ—â—â— of the first Booke Definition of a poynt Definition of a poynt after Pithagoras Definition of a liâ—â— An other definition of a line An other The endes of a line Difference of a point frâ—â—nity Vnitie is a part of number A poynt is no part of quantitie Definition of a right line DefinitioÌ„ therof after Plato An other definition An other An other An other An other VVhy Euclide here defineth not a crooked lyne Definition of a superficies A superficies may be deuided two wayes An other definition of a superficies The extremes of a superficies Another definition of a superficies Definition of a plaine superficies Another definition of a playne superficies NOTE Another definition of a playne superficies An other definition An other definition An other definition Definition of a playne angle Definition of a â—ecâ—ilined angle 〈◊〉 of angles VVhat a right angle VVhat also a perpendicular lyne iâ— VVhat an obtuse angle â—â— VVhat an acute angle is The limite of any thing No science of thinges infinite Definition of a figure Definition of a circle A circle the most perfect of all figures The centre of a circle Definition of a diameter Definition of a semicircle Definition of a section of a circle Definition of râ—â—â—â—lined figures
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
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
casâ— Demonstâ—atâ—on Construction Demonstration Demonstration leading to an impossibilitie Note this maner of imagination Mathematicall Demonstration leading to an impossibilitie Construction Demonstration Demonstration leading to an absurditieâ— In tâ—is â—ronoâ—â—â—oâ—â—â— must vndâ—rsâ—and the propâ—rtioâ—â—ll â—artes or sâ—â—â—ions to be thâ—se which are câ—ntaiâ—ed 〈◊〉 the parallel superâ—â—cies Construction Demâ—nstration Construction Demonstration Demonstration leading to an impossibilitie Demonstration Construction Demonstraâ—ion Two cases in this proposition The first case Second case Constructiâ—â—â— Demonstration An other demonstration Construction Demonstraâ—â—â—n Construction Three cases in this proposition The first case A necessary thing to be proued before he pâ—oceede any â—arther in the construction of the Problemâ— * Which how to finde out is taught at the end of this demonstration and also was taught in the asâ—umpt put before the 14. proposition of the teÌ„th boke Demonstration of the first case Second case Third case An other demoÌ„stratioÌ„ to proue that thâ— line AB is not lesse theÌ„ the line LX. This was before taâ—ght in the tenth booke in the assumpt put before the 14. proposition * M. Dee to auoide cauillation addeth to Euclides proposition this worde sixe whome I haue followed accordingly and not Zamberts in this This kinde of body mencioned in the proposition is called a ParallelipipedoÌ„ according to the diâ—finition before geuen thereof Demonstration that the opposite sides are parallelogrammes DemonstratioÌ„ that the opposite superficies are equall * AB is equall to DC because the superficies AC is proued a parallelograÌ„me and by the same reason is BH equall to CF because the superficies FB is proued a parallelogramme therefore the 34. of the first is our proofe Fâ—â—â—â— Corollary Second Corolry These solides which he speaketh of in this Corollary are of some called sidâ—d columnâ—s Third Corollary Constrâ—ction Demonstrâ—tion * Looke at the end of the demonstratioâ— what is vnderstanded by staÌ„ding lines Iohn d ee his figure By this figure it appâ—arâ—th why â—uch Prismes were called â—â—edges of 〈◊〉 vâ—ry shape of a wedge as is the solide DEFGAC c. StaÌ„ding lines Construction Demonstration I. Dees figure Two cases in this proposition Thâ— first case Construction We are beholding to M. d ee for inuenting this figure with other which till his reforming were as much mishappen as this was and so both in the Greeke and Latine copies remaine Demonstration * Note now how the base respectiuely is takenâ— for so may alteratioÌ„ of respects alter the name of the bowndes eyther of solides or playnes Second case * There you perceaue how the base is diuersly considered chosen as before we aduertised you Construction Demonstration Construction Demonstration * * Note this famous Lâ—mma The doubling oâ—â—he Cube * Note * 〈…〉 Lemma Note what iâ— yet lacking requisite to the doubling of the Cube The conuerse of both the partes of the first case The conuerse of the second case The generall conclusion Construction DemonstratioÌ„ of the first part DemonstratioÌ„ of the second part To finde two middâ—e proportionals betwene two numbers geueÌ„ Note the practise of appâ—oching to precieâ—es in Cubik rootes * This is the way to apply any square geuen to a line also geueÌ„ sufficienty extended A probleme worth the searching for Construction Demonstration Construction Demonstration * It is euideÌ„t that those perpendiculars are all one with the staâ—ding lines of the solides if their solide angles be made of superâ—icâ—â—il right angles onely Dâ—uâ—ling of the 〈◊〉 c. * Dâ—mâ—â—â—â—ratioâ—â—f pâ—sâ—ibilitie in the â—â—oblem An other argumeâ—t to comâ—oâ—t the studious Demonstration of the first part Demonstration of the second part which is the conuerse of the first part Demonstration leading to an impossibilitie Construction Demonstration Construction Demonstration * Which of some are called sided Columnes Sided Columnes Construction Demonstration The squaring of the circle Demonstratiâ—n â—eading to an impossibiâ—itie Two cases in this proposition The first case That a square within any circle described is bigger than halfe the circle That the Isosceles triangles without the square are greater then halfe the segments wherin they are Second case * This As 〈…〉 afteâ—ward at the end of the demâ—straâ—ion proued Construction Demonstration Construction Demonstration Diâ—â—reence betwene the first probleme and the second Construction Demonstration Consâ—ruction Demonstration Consâ—ruction Demonstration Consâ—â—uction Note this well for it iâ— of great vse An other way of demonstâ—atâ—oÌ„ of the fâ—â—st â—robleme of thâ—â— addition Note this properâ—ie of a triangle rectangle Construction Demonstration * Though I say without the square yet you must thinke that it may be also within the square that diuersly Wherfore this Probleme may haue diuerse cases so but briefly to aâ—â—yde all may thus be said cut any side of that square into 3. partsâ— in the proportion of X to Y. Note the maner of the drift in this demonstration and construction mixtly and with no determination to the constructioÌ„â— as commoÌ„ly iâ— in problemeâ—â— which is here of me so vsedâ— for an example to young studeÌ„tes of variety in art Construction Demonstration * Note and remember one â—eâ—th in these solids The conclusion of the first part DemonstratioÌ„ of the second part namely that it is deuided moreouer into two equal Prismes Conclusion of the second part DemonstratioÌ„ of the last part that the two Prismes are greater then the halfe of the whole Pyramis Conclusion of the last part Conclusion of the whole proposition * An AssuÌ„pt An Assumpt Conclusion of the whole Demonstration leading to an impossibilitie * In the Assuâ—pâ—â—â—llowinâ— the second â—ropâ—â—ition of this bâ—â—ke Construction Demonstration Demonstration Note Sided Columnes sometime called prismes are triple to pyramids hauing one base and equall heâ—th with them Note â—arallelipipedons treble to pyramids of one base and heith with them Construction Demonstration An addition by Campane and Flussas Demonstration of the first part Demonstration â—f the second part which iâ— the conueâ—se of the first Constr 〈◊〉 〈◊〉 Parallelipipedons called Prismes * By this it is manifest that Euclide comprehended sided Columnes also vnder the name of a Prisme * A prisme hauing for his base a poligonon figure as we haue often before noted vnto you Note M. d ee his chiefe purpose in his additions Demonstration touching cylinders Second case Second parâ— which concerneth Cillinders Construction Demonstraâ—ion Construction Demonstration touching Cylinders Demonstration touching Cones First part of the propositioÌ„ demonstrated touching Cones Two cases in this proposition The first case Second case Construction Demonstration touching cylinders Second part demonstraâ—ed Construction * Note this LM because of KZ in the next proposition and here the point M for the point Z in the next demonstration I. Dee * For that the sections were made by the number two that is by taking halues and of the residue the halâ—eâ— and so to LD being an halfe and a residue which shall be a coÌ„mon measure backe againe to make sides of the Poligonon figure Construction
the square of the line A haue not vnto the square of the line B the same proportion that a square number hath to a square number Then I say that the lines A and B are incommensurable in length For if the lines A and B be commensurable in length then the square of the line A should haue vnto the square of the line B the same proportion that a square number hath to a square number by the first part of this proposition but by supposition it hath not wherfore the lines A and B are not commensurable in length Wherfore they are incomensurable in length Wherfore squares made of right lines commensura in length haue that proportion the one to the other that a square number hath to a square number And squares which haue that proportion the one to the other that a square number hath to a squaâ—e number shall also haue the sides commensurable in length But squares described of right lines incommensurable in length haue not that proportion the one to the other that a square number hath to a square number And squares which haue not that propoâ—tion the one to the other that a sâ—uare number hath to a square number haue not also their sides commâ—nsurable in length which was all that was required to be proued ¶ Corrollary Hereby it is manifest that right lines coÌ„mensurable in length are also euer commensurable in power But right lines commensurable in power are not alwayes commensurable in length And right lines incoÌ„mensurable in leÌ„gth are not alwayes incommensurable in power But right lines incommensurable in power are euer also incommensurable in length For forasmuâ—h as squares made of right lines commensurable in length haue that proportion the one to the other that a square number hath to a square number by the first part of this proposition but magnitudes which haue that proportion the one to the other that number simply hath to number are by the sixt of the tenth commensurable Wherfore right lines commensurable in length are commensurable not onely in length but also in power Againe forasmuch as there are certaine squares which haue not that proportion the one to the other that a square number hath to a square number but yet haue that proportion the one to the other which number simply hath to number their sides in dede are in power commensurable for that they describe squares which haue that proportion which number simply hath to number which squares are therfore commensurable by the 6. of this booke but the said sides are incommensurable in length by the latter part of this proposition Wherâ—fore it is tâ—ue that lines commensurable in power are not straight way commensurable in length also And by the selâ—e same reason is proued also that third part of the corollary that lines incommensurable in length are not alwayes incommensurable in power For they may be incommâ—nsurable in length but yet commensurable in power As in those squares which are in proportion the one to the other as number is to number but not as a square number is to a square number But right lines incommensurable in power are alwayes also incommensurable in length For iâ— they be commensurable in length they shal also be commensurable in power by the first part of this Corollary But they are supposed to be incommensurable in length which is absurde Wherâ—ore right lines incommensurable in power are euer incommensurable in lengthâ— For the better vnderstanding of this proposition and the other following I haue here added certayne annotacions taken out of Montaureus And first as touching the signiâ—ication oâ— wordes and termes herein vsed whâ—ch arâ— such that vnlesse they be well marked and peysed the matter will be obscure and hard and in a maner inexplicable First this ye must note that lines to be commensurable in length and lines to be in proportion the one to the other as number is to number is all one So that whatsoeuer lines are commensurable in length are also in proportion the one to the other as number is to number And conuersedly what so euer lynes are in proportion the one to the other as number is to number are also commeÌ„surable in length as it is manifest by the 5 and 6 of this booke Likewise lines to be incommensurable in length and not to be in proportion the one to the other as number is to number is all one as it is manifest by the 7. and 8. of this booke Wherfore that which is sayd in this Theoreme ought to be vnderstand of lines commensurable in length and incommensurable in length This moreouer is to be noted that it is not all one numbers to be square numbers and to be in proportioÌ„ the one to the other as a square number is to a square number For although square numbers be in proportion the one to the other as a square number is to a square number yet are not all those numbers which are in proportion the one to the other as a square number is to a square number square numbers For they may be like superficiall numbers and yet not square numbers which yet are in proportioÌ„ the one to the other as a square number is to a square number Although all square numbers are like superficiall numbers For betwene two square numbers there â—alleth one meane proportionall number by the 11. of the eight But if betwene two numbers there fall one meane proportionall number those two numbers are like superficiall numbers by the 20. of the eight So also if two numbers be in proportion the one to the other as a square number is to a square number they shall be like superficiall nuÌ„bers by the first corollary added after the last proposition of the eight booke And now to know whether two superficiall numbers geuen be like superficiall numbers or no it is thus found out First if betwene the two numbâ—rs geuen there fall no meane proportionall then are not these two numbers like superficiall numbers by the 18. of the eight But if there do fall betwene them a meane proportionall then are they like superâ—iciall numbers by the 20. of the eight Moreouer two like superficiall numbers multiplied the one into the other do produce a square number by the firsâ— of the ninth Wherfore if they do not produce a square number then are they not like superficiall numbers And if the one being multiplied into the other they produce a square number then are they like superficiall by the 2. of the ninth Moreouer if the said two superficial numbers be in superperticular or superbipartient proportion then are they not like superficiall numbers For if they should be like then should there be a meane proportionall betwene them by the 20. of the eight But that is contrary to the Corollary of the 20. of the eight And the easilier to conceiue the demonstrations following take this example of that which we haue sayd ¶
An Assumpt Forasmuch as in the eight booke in the 26. proposition it was proued that like playne numbers haue that proportion the one to the other that a square number hath to a square number and likewise in the 24. of the same booke it was proued that if two numbers haue that proportion the one to the other that a square number hath to a square number those numbers are like plaine numbers Hereby it is manifest that vnlike plaine numbers that is whose sides are not proportionall haue not that proportion the one to the other that a square number hath to a square number For if they haue then should they be like plaine numbers which is contrary to the supposition Wherfore vnlike plaine numbers haue not that proportion the one to the other that a square number hath to a square nuÌ„ber And therfore squares which haue that proportion the one to the other that vnlike plaine numbers haue shall haue their sides incommensurable in length by the last part of the former proposition for that those squares haue not that proportion the one to the other that a square number hath to a square number ¶ The 8. Theoreme The 10. Proposition If foure magnitudes be proportionall and if the first be commensurable vnto the second the third also shal be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth SVppose that these foure magnitudes A B C D be proportionall As A is to B so let C be to D and let A be commensurable vnto B. Then I say that C is also commensurable vnto D. For forasmuch as A is commensurable vnto B it hath by the fift of the tenth that proportion that number hath to number But as A is to B so is C to D. Wherfore C also hath vnto D that proportion that number hath to number Wherfore C is commensurable vnto D by the 6. of the tenth But now suppose that the magnitude A be incommensurable vnto the magnitude B. Then I say that the magnitude C also is incommensurable vnto the magnitude D. For forasmuch as A is incommensurable vnto B therfore by the 7. of this booke A hath not vnto B such proportion as number hath to number But as A is to B so is C to D. Wherefore C hath not vnto D such proportion as number hath to number Wherfore by the 8. of the tenth C is incommensurable vnto D. If therefore there be foure magnitudes proportionall and if the first be commensurable vnto the second the third also shall be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth which was required to be proued ¶ A Corollary added by Montaureus If there be foure lines proportionall and if the two first or the two last be commensurable in power onely the other two also shall be commensurable in power onely This is proued by the 22. of the sixt and by this tenth proposition And this Corollary Euclide vseth in the 27. and 28. propositions of this booke and in other propositions also ¶ The 3. Probleme The 11. Proposition Vnto a right line first set and geuen which is called a rationall line to finde out two right lines incommensurable the one in length onely and the other in length and also in power SVppose that the right line first set and geuen which is called a rationall line of purpose be A. It is required vnto the said line A to finde out two right lines incommensurable the one in length onely the other both in length and in power Take by that which was added after the 9. proposition of this booke two numbers B and C not hauing that proportion the one to the other that a square number hath to a square number that is let them not be like plaine numbers for like plaine numbers by the 26. of the eight haue that proportion the one to the other that a square number hath to a square number And as the number B is to the number C so let the square of the line A be vnto the square of an other line namely of D how to do this was taught in the assumpt put before the 6. proposition of this booke Wherfore the square of the line A is vnto the square of the line D commensurable by the sixt of the tenth And forasmuch as the number B hath not vnto the number C that proportion that a square number hath to a square nuÌ„ber therfore the square of the line A hath not vnto the square of the line D that proportioÌ„ that a square number hath to a nuÌ„ber Wherfore by the 9. of the tenth the line A is vnto the line D incommensurable in length onely And so is found out the first line namely D incommensurable in length onely to the line geuen A. Agayne take by the 13. of the sixt the meane proportionall betwene the lines A and D and let the same be E. Wherfore as the line A is to the line D so is the square of the line A to the square of the line E by the Corollary of the 20. of the sixt But the line A is vnto the line D incommensurable in length Wherfore also the square of the line A is vnto the square of the line E incommensurable by the second part of the former proposition Now forasmuch as the square of the line A is incoÌ„meÌ„surable to the square of the line E it followeth by the definition of incommensurable lynes that the line A is incommensurable in power to the line E. Wherfore vnto the right line geuen and first set A which is a rationall line and which is supposed to haue such diuisions and so many partes as ye list to conceyue in minde as in this example 11 whereunto as was declared in the 5. definition of this booke may be compared infinite other lines either commensurable or incommensurable is found out the line D incommensurable in length onely Wherfore the line D is rationall by the sixt definitioÌ„ of this booke for that it is incommensurable in length onely to the line A which is the first line set and is by suppositioÌ„ rational There is also found out the line E which is vnto the same line A incommensurable not onely in length but also in power which line E compared to the rationall line A is by the definition irrationall For Euclide alwayes calleth those lines irrationall which are incommensurable both in length and in power to the line first set and by supposition rationall ¶ The 9. Theoreme The 12. Proposition Magnitudes commensurable to one and the selfe same magnitude are also commensurable the one to the other SVppose that either of these magnitudes A and B be commensurable vnto the magnitude C Then I say that the magnitude A is commensurable vnto the magnitude B. For â—orasmuch as the
the circle A hath to the circle B. As the diameter of the circle A is to the diameter of the circle B so let the line C be to a fourth line by the 12. of the 〈…〉 line be D. And by the 11. of the sixth let a thirde line proportionall be found to the lines C D which let be Eâ— I say now that the line C hath to the line E that proportion which the circle A hath to the circle B. For by construction the lineâ— C D and E are proportionall therefore the square of Câ— is to the square of D as C is to E by the Corollary of the 10. of the sixth But by construction as the diameter of the circle A is to the diameter of the circle B so is C to D wherefore as the square of the diameter of the circle A is to the square of the diameter of the circle B so is the square of the line C to the square of the line D by the 22. of the sixth But as the square of the diameter of A the circle is to the square of the diameter of the circle D so is the circle A to the circle B by the second of the twelfth wherefore by 11. of the fiueth as the circle A is to the circle B so is the square of the line C to the square of the liâ—e D. But it is proued that â—s the square of the line C is to the square of the line D so is the line C to the line E. Wherefore by the 11. of the fiueth as the circle A is to the circle B so is the line C to the line E. Two circles being geuen thereâ—ore and a right line we haue found a right line to which the right line geuâ—n hath that proportion which the one circle hath to the other Which ought to be done Note The difference betwene this Probleme and that next before is this there although we had two circles geuen and two lines were found in that proportion the one to the other in which the geuen circles were and here likewise are two circles geueÌ„ and two lineâ— also are had in the same proportioÌ„ that the geueÌ„ circles are yet there we tooke at pleasure the first of the two lines wherunto we framed the second proportionally to the circles geuen But here the first of the two lines is assigned poynted aâ—d determined to vs and not our choyse to be had therein as was in the former Probleme A Probleme 3. A circle being geuen to finde an other circle to which the geuâ—n câ—rcle is in any proportion geueÌ„ in two right lines Suppose the circle Aâ—C geuen and therefore his semidiameter is geuen whereby his diameter also is geuen which diameter let be AC Let the proportion geuen be that which is betwene â— F two right lines I say a circle is to be found vnto which Aâ—C hath that proportion that â— hath to â— As â— is to â— so let AC the diameter be to an other right line by the 12. of the sixth Which line suppose to be 11. Betwene AC and â—â— finde a middle proportionall line by the 13. of the sixth which let be LN By the 10. of the first deuide â—N into two equall paâ—tes and let that be done in the point â— Now vpon â—â— o being made the center describe a circle which let be â—Nâ— I say that ABC is to LMN as â— is to F. For seing that AC LN and H are three right lines in continuall proportion by construction therefore by the Corollary of the 20. of the sixth as AC is to H so is the square of Aâ— to the square of LN But AC is to H as â— is to â— by construction Wherefore the square of AC is to the square of LN as E is F but as the square of the diameter AC is to the square of the diameter LN so is the circle ABC to the circle â—MN by this 2. of the twelueth wherefore by the 11. of the fiueth the circle ABC is to the circle LMN as â— is to F. A circle being geuen therefore an other circle is founde to which the geuen circle is in any proportion geuen betwene two right lines which ought to be done A Probleme 4. Two circles being geuen to finde one circle equall to them both Suppose the two circles geueÌ„ haue their diameters Aâ— CD I say that a circle must be â—ound equall to the two circles whose diameters are Aâ— and CD vnto the line Aâ— at the point A erect a perpendicular line Aâ— from which sufficiently produced cut a line equall to CD which let be AF. By the first peticion draw from F to â— a right line so is FAâ— made a triangle rectangle I say now that a circle whose diameter is Fâ— is equal to the two circles whose diameters are Aâ— and â—D For by the 47. of the first the square of Fâ— is equall to the squares of Aâ— AF. Which AF is by construction equall to CDâ— wherefore the square of Fâ— is equall to the squares of AB and CD But circles are one to an other as the squares of their diameters are one to the other by this second of the twelueth Therefore the circle whose diameter is â—â— is equall to the circles whose diameters are Aâ— and CD Therefore two circles being geuen we haue found a circle equall to them both Which was required to be done A Corollary 1. Hereby it is made euident that in all triangles rectangle the circles semicircles quadrants oâ— any other portions of circles described vpon the subtendent line is equall to the two circles semicircles quadrants or any two other like portions of circles described on the two lines comprehending the right angle like to like being compared For like partes haue that proportion betwene them selues that their whole magnitudes haue of which they are like partes by the 15. of the fifth But of the whole circles in the former probleme it is euident and therefore in the fornamed like portions of circlâ—s it is a true consequent A Corollary 2. By the former probleme it is also manifest vnto circles three fower fiue or to how many so euer one will geue one circle may be geuen equall For if first to any two by the former probleme you finde one equall and then vnto your found circle and the third of the geuen circles as two geuen circles finde one other circle equall and then to that second found circle and to the fourth of the first geuen circlesâ— as two circles one new circle be found equall and so proceede till you haue once cuppled orderly euery one of your propouÌ„ded circles except the first and second already doone with the new circle thus found for so the last found circle is equall to all the first geuen circles If ye doubt or sufficiently vnderstand me not helpe your selfe by the discourse and demonstration of the
last proposition in the second booke and also of the 31. in the sixth booke ¶ A Probleme 5. Two vnequall circles being geuen to finde a circle equall to the excesse of the greater to the lesse Suppose the two vnequal circles geueÌ„ to be ABC DEF let ABC be the greater whose diameter suppose to be AC the diameter of DEF suppose to be DF. I say a circle must be found equal to that excesse in magnitude by which ABC is greater thâ— DEF By the first of the fourth in the circle ABC Apply a right line equall to DF whose one end let be at C and the other let be at B. FroÌ„ B to A draw a right line By the 30. of the third it may appeare that ABC is a right angle and thereby ABC the triangle is rectangled wherfore by the first of the two corollaries here before the circle ABC is equall to the circle DEF For BC by construction is equall to DF and more ouer to the circle whose diameter is AB That circle therefore whose diameter is AB is the circle conteyning the magnitude by which ABC is greater then DEF Wherefore two vnequal circles being geuen we haue found a circle equall to the excesse of the greater to the lesse which ought to be doone A Probleme 6. A Circle being geuen to finde two Circles equall to the same which found Circles shall haue the one to the other any proportion geuen in two right lines Suppose ABC a circle geuen and the proportion geuen let it be that which is betwene the two right lines D and E. I say that two circles are to be found equall to ABC and with al one to the other in the proportioÌ„ of D to E. Let the diameter of ABC be AC As D is to E so let AC be deuided by the 10. of the sixth in the poynt F. At F to the line AC let a perpeÌ„dicular be drawne FB and let it mete the circuÌ„fereÌ„ce at the poynt B. From the poynt B to the points A and C let right lines be drawne BA and BC. I say that the circles whose diameteâ— are the lines BA and BC are equall to the circle ABC and that those circles hauing to their diameters the lines BA and BC are one to the other in the proportion of the line D to the line E. For first that they are equal it is euident by reason that ABC is a triangle rectangle wherfore by the 47. of the first the squares of BA and BC are equall to the square of AC And so by this second it is maniâ—est the two circles to be equall to the circle ABC Secondly as D is to â— so is AF to FC by construction And as the line AF is to the line FC so is the square of the line â—A to the square of the line BC. Which thing we will briefely pâ—oue thus The parallelogramme contayned vnder AC and AF is equall to the square of BA by the Lemma after the 32. of the tenth booke and by the same Lemma or Assumpt the parallelogramme contayned vnder AC and â—C is equall to the square of the line BC. Wherfore as the first parallelogramme hath it selfe to the secondâ— so hath the square of BA equall to the first parallelogramme it selfe to the square of BC equall to the second parallelogramme But both the parallelograÌ„mes haue one heigth namely the line AC and bases the lines AF and FC wherefore as AF is to FC so is the parallelogâ—amme contayned vnder AC AF to the parallelogramme contayned vnder AC FC by the fiâ—st of the sixth And therefore as AF is to FC so is the square of BA to the square of BC. And as the square of BA is to the square of BC so is the circle whose diameter is BA to the circle whose diameter is BC by this second of the twelfth Wherefore the circle whose diameter is BA is to the circle whose diameter is BC as D is to E. And before we proued them equall to the circle ABC Wherâ—fore a circle being geuen we haue found two circles equall to the same which haue the one to the other any proportion geuen in two right lines Which ought to be done Note Heâ—e may you perâ—eiue an other way how to execute my first probleme for if you make a right angle conteyned of the diameters geueÌ„ as in this figure suppose them BA and BC and then subtend the right angle with the line AC and from the right angle let fall a line perpendicular to the base AC that perpâ—ndicular at the point of his fall deuideth AC into AF and FC of the proportion required A Corollary It followeth of thinges manifestly proued in the demonstration of this probleme that in a triangle rectangle if from the right angle to the base a perpendicular be let fall the same perpendicular cutteth the base into two partes in that proportion one to the other that the squares of the righâ— lines conteyning the right angle are in one to the other those on the one side the perpendicular being compared to those on the other both square and segment A Probleme 7. Betwene two circles geuen to finde a circle middell proportionall Let the two circles geuen be ACD and BEF I say that a circle is to be fouÌ„d which betwene ACD and BEF is middell proportionall Let the diameter of ACD be AD and of BEF let Bâ— be the diameter betwene AD and BF finde a line middell proportionall by the 13. of the sixth which let be HK I say that a circle whose diameter is HK is middell proportionall betwene ACD and BEF To AD HK and BF three right lines in continuall proportion by construction let a fourth line be found to which BF shal haue that proportion that AD hath to HK by the 12. of the sixth let that line be â— It is manifest that the â—ower lines AD HK BF and L are in continuall proportion For by coÌ„struction as AD is to HK so is Bâ— to L. And by construction on as AD is to HK so is HK to BF wherefore HK is to BF as BF is to L by the 11. of the fifth wherfore the 4. lines are in continuall proportion Wherefore as the first is to the third that is AD to BF so is the square of the first to the square of the second that is the square of AD to the square of HK by the corollary of the 20. of the sixth And by the same corollary as HK is to L so is the square of HK to the square of BF But by alternate proportion the line AD is to BF as HK is to L wherefore the square of AD is to the square of HK as the square of HK is to the square of BF Wherefore the square of HK is middell proportionall betwene the square of AD and the square of BF But as the squares are